andatoshiki/toshiki-notebook
1
0
Fork 0
mirror of https://github.com/andatoshiki/toshiki-notebook.git synced 2026-06-06 09:16:45 +00:00
Code Issues Actions Packages Projects Releases Wiki Activity
aad831afe4
toshiki-notebook/academic/physics/ipho-formulas-jpn/13.html

100 lines
132 KiB
HTML
Raw Blame History

This file contains invisible Unicode characters

This file contains invisible Unicode characters that are indistinguishable to humans but may be processed differently by a computer. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

<!DOCTYPE html>
<html lang="en-US" dir="ltr">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<title>Formulas for IPhO 日本語版: Section 13 | Toshiki's Note</title>
<meta name="description" content="Toshiki's web notebook served via Vitepress!">
<link rel="preload stylesheet" href="/assets/style.1276ae12.css" as="style">
<script type="module" src="/assets/app.da2b9cf1.js"></script>
<link rel="preload" href="/assets/inter-roman-latin.2ed14f66.woff2" as="font" type="font/woff2" crossorigin="">
<link rel="modulepreload" href="/assets/chunks/framework.ade46834.js">
<link rel="modulepreload" href="/assets/chunks/theme.2274efad.js">
<link rel="modulepreload" href="/assets/academic_physics_ipho-formulas-jpn_13.md.7cf2775d.lean.js">
<link rel="stylesheet" href="https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css">
<link rel="stylesheet" href="https://cdnjs.toshiki.dev/ajax/libs/font-awesome/6.3.0/css/all.min.css">
<link rel="icon" href="/favicon.ico">
<meta name="author" content="Anda Toshiki">
<meta name="keywords" content="Toshiki, Anda Toshiki, andatoshiki, GitHub, GitHub action, Vitepress, Vite, Notebook, Knowledge base, Programming, Programming Notes, Academic, Personal, Notebook, Productivity, Journal, Note-taking, Markdown, Notepad, Organization, Tutorial">
<meta name="google-site-verification" content="lm7PNJiYSPEx1dMast1Xptc0Vk0cU06o-daZSsIgr2I">
<meta name="HandheldFriendly" content="True">
<meta name="MobileOptimized" content="320">
<meta name="theme-color" content="#3c8772">
<meta property="og:type" content="website">
<meta property="og:locale" content="en-US">
<meta property="og:title" content="Toshiki&#39;s Note">
<meta property="og:description" content="Toshiki&#39;s web notebook served via Vitepress!">
<meta property="og:site" content="https://note.toshiki.dev">
<meta property="og:site_name" content="Toshiki&#39;s Note">
<meta property="og:image" content="https://note.toshiki.dev/og-cover.png">
<script>function siteruntime(){window.setTimeout("siteruntime()",1e3),X=new Date("8/24/2021 10:28:00"),Y=new Date,T=Y.getTime()-X.getTime(),M=24*60*60*1e3,a=T/M,A=Math.floor(a),b=(a-A)*24,B=Math.floor(b),c=(b-B)*60,C=Math.floor((b-B)*60),D=Math.floor((c-C)*60),siteruntime_span.innerHTML="This site has been running for: "+A+" day(s) "+B+" hour(s) "+C+" minute(s) "+D+" second(s)"}siteruntime();</script>
<script async="true" defer="true" data-website-id="86de8554-d4c9-4f2b-b62a-068b71241048" src="https://umami.toshiki.dev/umami.js"></script>
<script id="check-dark-light">(()=>{const e=localStorage.getItem("vitepress-theme-appearance")||"",a=window.matchMedia("(prefers-color-scheme: dark)").matches;(!e||e==="auto"?a:e==="dark")&&document.documentElement.classList.add("dark")})();</script>
</head>
<body>
<div id="app"><div class="Layout" data-v-83f63849><!--[--><!--]--><!--[--><span tabindex="-1" data-v-0eca8f1e></span><a href="#VPContent" class="VPSkipLink visually-hidden" data-v-0eca8f1e> Skip to content </a><!--]--><!----><header class="VPNav" data-v-83f63849 data-v-999a1a39><div class="VPNavBar has-sidebar" data-v-999a1a39 data-v-e99cf6bd><div class="container" data-v-e99cf6bd><div class="title" data-v-e99cf6bd><div class="VPNavBarTitle has-sidebar" data-v-e99cf6bd data-v-a8886b70><a class="title" href="/" data-v-a8886b70><!--[--><!--]--><!--[--><img class="VPImage logo" src="/logos/logo.png" alt data-v-164d1caf><!--]--><!--[-->Toshiki&#39;s Note<!--]--><!--[--><!--]--></a></div></div><div class="content" data-v-e99cf6bd><div class="curtain" data-v-e99cf6bd></div><div class="content-body" data-v-e99cf6bd><!--[--><!--]--><div class="VPNavBarSearch search" style="--vp-meta-key:&#39;Meta&#39;;" data-v-e99cf6bd><!--[--><div id="docsearch"><button type="button" class="DocSearch DocSearch-Button" aria-label="Search"><span class="DocSearch-Button-Container"><svg class="DocSearch-Search-Icon" width="20" height="20" viewBox="0 0 20 20" aria-label="search icon"><path d="M14.386 14.386l4.0877 4.0877-4.0877-4.0877c-2.9418 2.9419-7.7115 2.9419-10.6533 0-2.9419-2.9418-2.9419-7.7115 0-10.6533 2.9418-2.9419 7.7115-2.9419 10.6533 0 2.9419 2.9418 2.9419 7.7115 0 10.6533z" stroke="currentColor" fill="none" fill-rule="evenodd" stroke-linecap="round" stroke-linejoin="round"></path></svg><span class="DocSearch-Button-Placeholder">Search</span></span><span class="DocSearch-Button-Keys"><kbd class="DocSearch-Button-Key"></kbd><kbd class="DocSearch-Button-Key">K</kbd></span></button></div><!--]--></div><nav aria-labelledby="main-nav-aria-label" class="VPNavBarMenu menu" data-v-e99cf6bd data-v-1817056a><span id="main-nav-aria-label" class="visually-hidden" data-v-1817056a>Main Navigation</span><!--[--><!--[--><a class="VPLink link VPNavBarMenuLink" href="/development/" tabindex="0" data-v-1817056a data-v-f28b94cc data-v-075865b7><!--[-->Development<!--]--><!----></a><!--]--><!--[--><div class="VPFlyout VPNavBarMenuGroup active" data-v-1817056a data-v-4c03a652><button type="button" class="button" aria-haspopup="true" aria-expanded="false" data-v-4c03a652><span class="text" data-v-4c03a652><!----> Academic <svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="text-icon" data-v-4c03a652><path d="M12,16c-0.3,0-0.5-0.1-0.7-0.3l-6-6c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l5.3,5.3l5.3-5.3c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-6,6C12.5,15.9,12.3,16,12,16z"></path></svg></span></button><div class="menu" data-v-4c03a652><div class="VPMenu" data-v-4c03a652 data-v-da053c30><div class="items" data-v-da053c30><!--[--><!--[--><div class="VPMenuGroup" data-v-da053c30 data-v-2e982fbb><p class="title" data-v-2e982fbb>K-12</p><!--[--><!--[--><div class="VPMenuLink" data-v-2e982fbb data-v-6f715184><a class="VPLink link" href="/academic/chemistry/index" data-v-6f715184 data-v-075865b7><!--[-->Chemistry<!--]--><!----></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-2e982fbb data-v-6f715184><a class="VPLink link" href="/discrete-math/index" data-v-6f715184 data-v-075865b7><!--[-->Discrete Math.<!--]--><!----></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-2e982fbb data-v-6f715184><a class="VPLink link" href="/academic/literature/index" data-v-6f715184 data-v-075865b7><!--[-->Literature<!--]--><!----></a></div><!--]--><!--]--></div><!--]--><!--[--><div class="VPMenuGroup" data-v-da053c30 data-v-2e982fbb><p class="title" data-v-2e982fbb>Tools</p><!--[--><!--[--><div class="VPMenuLink" data-v-2e982fbb data-v-6f715184><a class="VPLink link active" href="/academic/physics/ipho-formulas-jpn/1" data-v-6f715184 data-v-075865b7><!--[-->Formulas for IPhO JPN.<!--]--><!----></a></div><!--]--><!--]--></div><!--]--><!--[--><div class="VPMenuLink" data-v-da053c30 data-v-6f715184><span class="VPLink" data-v-6f715184 data-v-075865b7><!--[--><!--]--><!----></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-da053c30 data-v-6f715184><span class="VPLink" data-v-6f715184 data-v-075865b7><!--[--><!--]--><!----></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-da053c30 data-v-6f715184><span class="VPLink" data-v-6f715184 data-v-075865b7><!--[--><!--]--><!----></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-da053c30 data-v-6f715184><span class="VPLink" data-v-6f715184 data-v-075865b7><!--[--><!--]--><!----></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-da053c30 data-v-6f715184><span class="VPLink" data-v-6f715184 data-v-075865b7><!--[--><!--]--><!----></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-da053c30 data-v-6f715184><span class="VPLink" data-v-6f715184 data-v-075865b7><!--[--><!--]--><!----></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-da053c30 data-v-6f715184><span class="VPLink" data-v-6f715184 data-v-075865b7><!--[--><!--]--><!----></span></div><!--]--><!--]--></div><!--[--><!--]--></div></div></div><!--]--><!--[--><div class="VPFlyout VPNavBarMenuGroup" data-v-1817056a data-v-4c03a652><button type="button" class="button" aria-haspopup="true" aria-expanded="false" data-v-4c03a652><span class="text" data-v-4c03a652><!----> Application <svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="text-icon" data-v-4c03a652><path d="M12,16c-0.3,0-0.5-0.1-0.7-0.3l-6-6c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l5.3,5.3l5.3-5.3c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-6,6C12.5,15.9,12.3,16,12,16z"></path></svg></span></button><div class="menu" data-v-4c03a652><div class="VPMenu" data-v-4c03a652 data-v-da053c30><div class="items" data-v-da053c30><!--[--><!--[--><div class="VPMenuGroup" data-v-da053c30 data-v-2e982fbb><p class="title" data-v-2e982fbb>Personal projects</p><!--[--><!--[--><div class="VPMenuLink" data-v-2e982fbb data-v-6f715184><a class="VPLink link" href="/application/markdown-it-katex/how-to-use" data-v-6f715184 data-v-075865b7><!--[-->markdown-it-katex<!--]--><!----></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-2e982fbb data-v-6f715184><a class="VPLink link" href="/application/vitepress-plugin-shiki-twoslash/index" data-v-6f715184 data-v-075865b7><!--[-->vitepress-plugin-shiki-twoslash<!--]--><!----></a></div><!--]--><!--]--></div><!--]--><!--]--></div><!--[--><!--]--></div></div></div><!--]--><!--[--><div class="VPFlyout VPNavBarMenuGroup" data-v-1817056a data-v-4c03a652><button type="button" class="button" aria-haspopup="true" aria-expanded="false" data-v-4c03a652><span class="text" data-v-4c03a652><!----> Save <svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="text-icon" data-v-4c03a652><path d="M12,16c-0.3,0-0.5-0.1-0.7-0.3l-6-6c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l5.3,5.3l5.3-5.3c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-6,6C12.5,15.9,12.3,16,12,16z"></path></svg></span></button><div class="menu" data-v-4c03a652><div class="VPMenu" data-v-4c03a652 data-v-da053c30><div class="items" data-v-da053c30><!--[--><!--[--><div class="VPMenuLink" data-v-da053c30 data-v-6f715184><a class="VPLink link" href="/save/reading/index" data-v-6f715184 data-v-075865b7><!--[-->Reading<!--]--><!----></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-da053c30 data-v-6f715184><a class="VPLink link" href="/academic/vocabulary/index" data-v-6f715184 data-v-075865b7><!--[-->Vocabulary<!--]--><!----></a></div><!--]--><!--]--></div><!--[--><!--]--></div></div></div><!--]--><!--]--></nav><!----><div class="VPNavBarAppearance appearance" data-v-e99cf6bd data-v-72c0c02a><label title="toggle dark mode" data-v-72c0c02a data-v-cb74fac6><button class="VPSwitch VPSwitchAppearance" type="button" role="switch" aria-checked="false" data-v-cb74fac6 data-v-9f7dbbcf><span class="check" data-v-9f7dbbcf><span class="icon" data-v-9f7dbbcf><!--[--><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="sun" data-v-cb74fac6><path d="M12,18c-3.3,0-6-2.7-6-6s2.7-6,6-6s6,2.7,6,6S15.3,18,12,18zM12,8c-2.2,0-4,1.8-4,4c0,2.2,1.8,4,4,4c2.2,0,4-1.8,4-4C16,9.8,14.2,8,12,8z"></path><path d="M12,4c-0.6,0-1-0.4-1-1V1c0-0.6,0.4-1,1-1s1,0.4,1,1v2C13,3.6,12.6,4,12,4z"></path><path d="M12,24c-0.6,0-1-0.4-1-1v-2c0-0.6,0.4-1,1-1s1,0.4,1,1v2C13,23.6,12.6,24,12,24z"></path><path d="M5.6,6.6c-0.3,0-0.5-0.1-0.7-0.3L3.5,4.9c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l1.4,1.4c0.4,0.4,0.4,1,0,1.4C6.2,6.5,5.9,6.6,5.6,6.6z"></path><path d="M19.8,20.8c-0.3,0-0.5-0.1-0.7-0.3l-1.4-1.4c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l1.4,1.4c0.4,0.4,0.4,1,0,1.4C20.3,20.7,20,20.8,19.8,20.8z"></path><path d="M3,13H1c-0.6,0-1-0.4-1-1s0.4-1,1-1h2c0.6,0,1,0.4,1,1S3.6,13,3,13z"></path><path d="M23,13h-2c-0.6,0-1-0.4-1-1s0.4-1,1-1h2c0.6,0,1,0.4,1,1S23.6,13,23,13z"></path><path d="M4.2,20.8c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l1.4-1.4c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-1.4,1.4C4.7,20.7,4.5,20.8,4.2,20.8z"></path><path d="M18.4,6.6c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l1.4-1.4c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-1.4,1.4C18.9,6.5,18.6,6.6,18.4,6.6z"></path></svg><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="moon" data-v-cb74fac6><path d="M12.1,22c-0.3,0-0.6,0-0.9,0c-5.5-0.5-9.5-5.4-9-10.9c0.4-4.8,4.2-8.6,9-9c0.4,0,0.8,0.2,1,0.5c0.2,0.3,0.2,0.8-0.1,1.1c-2,2.7-1.4,6.4,1.3,8.4c2.1,1.6,5,1.6,7.1,0c0.3-0.2,0.7-0.3,1.1-0.1c0.3,0.2,0.5,0.6,0.5,1c-0.2,2.7-1.5,5.1-3.6,6.8C16.6,21.2,14.4,22,12.1,22zM9.3,4.4c-2.9,1-5,3.6-5.2,6.8c-0.4,4.4,2.8,8.3,7.2,8.7c2.1,0.2,4.2-0.4,5.8-1.8c1.1-0.9,1.9-2.1,2.4-3.4c-2.5,0.9-5.3,0.5-7.5-1.1C9.2,11.4,8.1,7.7,9.3,4.4z"></path></svg><!--]--></span></span></button></label></div><div class="VPSocialLinks VPNavBarSocialLinks social-links" data-v-e99cf6bd data-v-268ff66d data-v-65dbf981><!--[--><a class="VPSocialLink" href="https://github.com/andatoshiki" aria-label="github" target="_blank" rel="noopener" data-v-65dbf981 data-v-2d45784b><svg role="img" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><title>GitHub</title><path d="M12 .297c-6.63 0-12 5.373-12 12 0 5.303 3.438 9.8 8.205 11.385.6.113.82-.258.82-.577 0-.285-.01-1.04-.015-2.04-3.338.724-4.042-1.61-4.042-1.61C4.422 18.07 3.633 17.7 3.633 17.7c-1.087-.744.084-.729.084-.729 1.205.084 1.838 1.236 1.838 1.236 1.07 1.835 2.809 1.305 3.495.998.108-.776.417-1.305.76-1.605-2.665-.3-5.466-1.332-5.466-5.93 0-1.31.465-2.38 1.235-3.22-.135-.303-.54-1.523.105-3.176 0 0 1.005-.322 3.3 1.23.96-.267 1.98-.399 3-.405 1.02.006 2.04.138 3 .405 2.28-1.552 3.285-1.23 3.285-1.23.645 1.653.24 2.873.12 3.176.765.84 1.23 1.91 1.23 3.22 0 4.61-2.805 5.625-5.475 5.92.42.36.81 1.096.81 2.22 0 1.606-.015 2.896-.015 3.286 0 .315.21.69.825.57C20.565 22.092 24 17.592 24 12.297c0-6.627-5.373-12-12-12"/></svg></a><a class="VPSocialLink" href="https://twitter.com/andatoshiki" aria-label="twitter" target="_blank" rel="noopener" data-v-65dbf981 data-v-2d45784b><svg role="img" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><title>Twitter</title><path d="M23.953 4.57a10 10 0 01-2.825.775 4.958 4.958 0 002.163-2.723c-.951.555-2.005.959-3.127 1.184a4.92 4.92 0 00-8.384 4.482C7.69 8.095 4.067 6.13 1.64 3.162a4.822 4.822 0 00-.666 2.475c0 1.71.87 3.213 2.188 4.096a4.904 4.904 0 01-2.228-.616v.06a4.923 4.923 0 003.946 4.827 4.996 4.996 0 01-2.212.085 4.936 4.936 0 004.604 3.417 9.867 9.867 0 01-6.102 2.105c-.39 0-.779-.023-1.17-.067a13.995 13.995 0 007.557 2.209c9.053 0 13.998-7.496 13.998-13.985 0-.21 0-.42-.015-.63A9.935 9.935 0 0024 4.59z"/></svg></a><!--]--></div><div class="VPFlyout VPNavBarExtra extra" data-v-e99cf6bd data-v-67546bb2 data-v-4c03a652><button type="button" class="button" aria-haspopup="true" aria-expanded="false" aria-label="extra navigation" data-v-4c03a652><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="icon" data-v-4c03a652><circle cx="12" cy="12" r="2"></circle><circle cx="19" cy="12" r="2"></circle><circle cx="5" cy="12" r="2"></circle></svg></button><div class="menu" data-v-4c03a652><div class="VPMenu" data-v-4c03a652 data-v-da053c30><!----><!--[--><!--[--><!----><div class="group" data-v-67546bb2><div class="item appearance" data-v-67546bb2><p class="label" data-v-67546bb2>Appearance</p><div class="appearance-action" data-v-67546bb2><label title="toggle dark mode" data-v-67546bb2 data-v-cb74fac6><button class="VPSwitch VPSwitchAppearance" type="button" role="switch" aria-checked="false" data-v-cb74fac6 data-v-9f7dbbcf><span class="check" data-v-9f7dbbcf><span class="icon" data-v-9f7dbbcf><!--[--><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="sun" data-v-cb74fac6><path d="M12,18c-3.3,0-6-2.7-6-6s2.7-6,6-6s6,2.7,6,6S15.3,18,12,18zM12,8c-2.2,0-4,1.8-4,4c0,2.2,1.8,4,4,4c2.2,0,4-1.8,4-4C16,9.8,14.2,8,12,8z"></path><path d="M12,4c-0.6,0-1-0.4-1-1V1c0-0.6,0.4-1,1-1s1,0.4,1,1v2C13,3.6,12.6,4,12,4z"></path><path d="M12,24c-0.6,0-1-0.4-1-1v-2c0-0.6,0.4-1,1-1s1,0.4,1,1v2C13,23.6,12.6,24,12,24z"></path><path d="M5.6,6.6c-0.3,0-0.5-0.1-0.7-0.3L3.5,4.9c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l1.4,1.4c0.4,0.4,0.4,1,0,1.4C6.2,6.5,5.9,6.6,5.6,6.6z"></path><path d="M19.8,20.8c-0.3,0-0.5-0.1-0.7-0.3l-1.4-1.4c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l1.4,1.4c0.4,0.4,0.4,1,0,1.4C20.3,20.7,20,20.8,19.8,20.8z"></path><path d="M3,13H1c-0.6,0-1-0.4-1-1s0.4-1,1-1h2c0.6,0,1,0.4,1,1S3.6,13,3,13z"></path><path d="M23,13h-2c-0.6,0-1-0.4-1-1s0.4-1,1-1h2c0.6,0,1,0.4,1,1S23.6,13,23,13z"></path><path d="M4.2,20.8c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l1.4-1.4c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-1.4,1.4C4.7,20.7,4.5,20.8,4.2,20.8z"></path><path d="M18.4,6.6c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l1.4-1.4c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-1.4,1.4C18.9,6.5,18.6,6.6,18.4,6.6z"></path></svg><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="moon" data-v-cb74fac6><path d="M12.1,22c-0.3,0-0.6,0-0.9,0c-5.5-0.5-9.5-5.4-9-10.9c0.4-4.8,4.2-8.6,9-9c0.4,0,0.8,0.2,1,0.5c0.2,0.3,0.2,0.8-0.1,1.1c-2,2.7-1.4,6.4,1.3,8.4c2.1,1.6,5,1.6,7.1,0c0.3-0.2,0.7-0.3,1.1-0.1c0.3,0.2,0.5,0.6,0.5,1c-0.2,2.7-1.5,5.1-3.6,6.8C16.6,21.2,14.4,22,12.1,22zM9.3,4.4c-2.9,1-5,3.6-5.2,6.8c-0.4,4.4,2.8,8.3,7.2,8.7c2.1,0.2,4.2-0.4,5.8-1.8c1.1-0.9,1.9-2.1,2.4-3.4c-2.5,0.9-5.3,0.5-7.5-1.1C9.2,11.4,8.1,7.7,9.3,4.4z"></path></svg><!--]--></span></span></button></label></div></div></div><div class="group" data-v-67546bb2><div class="item social-links" data-v-67546bb2><div class="VPSocialLinks social-links-list" data-v-67546bb2 data-v-65dbf981><!--[--><a class="VPSocialLink" href="https://github.com/andatoshiki" aria-label="github" target="_blank" rel="noopener" data-v-65dbf981 data-v-2d45784b><svg role="img" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><title>GitHub</title><path d="M12 .297c-6.63 0-12 5.373-12 12 0 5.303 3.438 9.8 8.205 11.385.6.113.82-.258.82-.577 0-.285-.01-1.04-.015-2.04-3.338.724-4.042-1.61-4.042-1.61C4.422 18.07 3.633 17.7 3.633 17.7c-1.087-.744.084-.729.084-.729 1.205.084 1.838 1.236 1.838 1.236 1.07 1.835 2.809 1.305 3.495.998.108-.776.417-1.305.76-1.605-2.665-.3-5.466-1.332-5.466-5.93 0-1.31.465-2.38 1.235-3.22-.135-.303-.54-1.523.105-3.176 0 0 1.005-.322 3.3 1.23.96-.267 1.98-.399 3-.405 1.02.006 2.04.138 3 .405 2.28-1.552 3.285-1.23 3.285-1.23.645 1.653.24 2.873.12 3.176.765.84 1.23 1.91 1.23 3.22 0 4.61-2.805 5.625-5.475 5.92.42.36.81 1.096.81 2.22 0 1.606-.015 2.896-.015 3.286 0 .315.21.69.825.57C20.565 22.092 24 17.592 24 12.297c0-6.627-5.373-12-12-12"/></svg></a><a class="VPSocialLink" href="https://twitter.com/andatoshiki" aria-label="twitter" target="_blank" rel="noopener" data-v-65dbf981 data-v-2d45784b><svg role="img" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><title>Twitter</title><path d="M23.953 4.57a10 10 0 01-2.825.775 4.958 4.958 0 002.163-2.723c-.951.555-2.005.959-3.127 1.184a4.92 4.92 0 00-8.384 4.482C7.69 8.095 4.067 6.13 1.64 3.162a4.822 4.822 0 00-.666 2.475c0 1.71.87 3.213 2.188 4.096a4.904 4.904 0 01-2.228-.616v.06a4.923 4.923 0 003.946 4.827 4.996 4.996 0 01-2.212.085 4.936 4.936 0 004.604 3.417 9.867 9.867 0 01-6.102 2.105c-.39 0-.779-.023-1.17-.067a13.995 13.995 0 007.557 2.209c9.053 0 13.998-7.496 13.998-13.985 0-.21 0-.42-.015-.63A9.935 9.935 0 0024 4.59z"/></svg></a><!--]--></div></div></div><!--]--><!--]--></div></div></div><!--[--><!--]--><button type="button" class="VPNavBarHamburger hamburger" aria-label="mobile navigation" aria-expanded="false" aria-controls="VPNavScreen" data-v-e99cf6bd data-v-2314de5a><span class="container" data-v-2314de5a><span class="top" data-v-2314de5a></span><span class="middle" data-v-2314de5a></span><span class="bottom" data-v-2314de5a></span></span></button></div></div></div></div><!----></header><div class="VPLocalNav" data-v-83f63849 data-v-fa4746c0><button class="menu" aria-expanded="false" aria-controls="VPSidebarNav" data-v-fa4746c0><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="menu-icon" data-v-fa4746c0><path d="M17,11H3c-0.6,0-1-0.4-1-1s0.4-1,1-1h14c0.6,0,1,0.4,1,1S17.6,11,17,11z"></path><path d="M21,7H3C2.4,7,2,6.6,2,6s0.4-1,1-1h18c0.6,0,1,0.4,1,1S21.6,7,21,7z"></path><path d="M21,15H3c-0.6,0-1-0.4-1-1s0.4-1,1-1h18c0.6,0,1,0.4,1,1S21.6,15,21,15z"></path><path d="M17,19H3c-0.6,0-1-0.4-1-1s0.4-1,1-1h14c0.6,0,1,0.4,1,1S17.6,19,17,19z"></path></svg><span class="menu-text" data-v-fa4746c0>Menu</span></button><div class="VPLocalNavOutlineDropdown" style="--vp-vh:0px;" data-v-fa4746c0 data-v-f8dbb575><button data-v-f8dbb575>Return to top</button><!----></div></div><aside class="VPSidebar" data-v-83f63849 data-v-0e47c5d5><div class="curtain" data-v-0e47c5d5></div><nav class="nav" id="VPSidebarNav" aria-labelledby="sidebar-aria-label" tabindex="-1" data-v-0e47c5d5><span class="visually-hidden" id="sidebar-aria-label" data-v-0e47c5d5> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="group" data-v-0e47c5d5><section class="VPSidebarItem level-0 collapsible has-active" data-v-0e47c5d5 data-v-0cc45b6b><div class="item" role="button" tabindex="0" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><h2 class="text" data-v-0cc45b6b>IPhO Formulas: JP Ver.</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-0cc45b6b><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-0cc45b6b><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-0cc45b6b><!--[--><div class="VPSidebarItem level-1 is-link" data-v-0cc45b6b data-v-0cc45b6b><div class="item" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-0cc45b6b data-v-075865b7><!--[--><p class="text" data-v-0cc45b6b>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-0cc45b6b data-v-0cc45b6b><div class="item" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-0cc45b6b data-v-075865b7><!--[--><p class="text" data-v-0cc45b6b>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-0cc45b6b data-v-0cc45b6b><div class="item" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-0cc45b6b data-v-075865b7><!--[--><p class="text" data-v-0cc45b6b>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-0cc45b6b data-v-0cc45b6b><div class="item" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-0cc45b6b data-v-075865b7><!--[--><p class="text" data-v-0cc45b6b>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-0cc45b6b data-v-0cc45b6b><div class="item" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-0cc45b6b data-v-075865b7><!--[--><p class="text" data-v-0cc45b6b>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-0cc45b6b data-v-0cc45b6b><div class="item" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-0cc45b6b data-v-075865b7><!--[--><p class="text" data-v-0cc45b6b>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-0cc45b6b data-v-0cc45b6b><div class="item" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-0cc45b6b data-v-075865b7><!--[--><p class="text" data-v-0cc45b6b>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-0cc45b6b data-v-0cc45b6b><div class="item" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-0cc45b6b data-v-075865b7><!--[--><p class="text" data-v-0cc45b6b>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-0cc45b6b data-v-0cc45b6b><div class="item" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-0cc45b6b data-v-075865b7><!--[--><p class="text" data-v-0cc45b6b>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-0cc45b6b data-v-0cc45b6b><div class="item" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-0cc45b6b data-v-075865b7><!--[--><p class="text" data-v-0cc45b6b>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-0cc45b6b data-v-0cc45b6b><div class="item" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-0cc45b6b data-v-075865b7><!--[--><p class="text" data-v-0cc45b6b>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-0cc45b6b data-v-0cc45b6b><div class="item" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-0cc45b6b data-v-075865b7><!--[--><p class="text" data-v-0cc45b6b>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-0cc45b6b data-v-0cc45b6b><div class="item" data-v-0cc45b6b><div class="indicator" data-v-0cc45b6b></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-0cc45b6b data-v-075865b7><!--[--><p class="text" data-v-0cc45b6b>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-83f63849 data-v-bb292142><div class="VPDoc has-sidebar has-aside" data-v-bb292142 data-v-6c4a7022><!--[--><!--]--><div class="container" data-v-6c4a7022><div class="aside" data-v-6c4a7022><div class="aside-curtain" data-v-6c4a7022></div><div class="aside-container" data-v-6c4a7022><div class="aside-content" data-v-6c4a7022><div class="VPDocAside" data-v-6c4a7022 data-v-f77a9b1a><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-f77a9b1a data-v-ea95c6a2><div class="content" data-v-ea95c6a2><div class="outline-marker" data-v-ea95c6a2></div><div class="outline-title" data-v-ea95c6a2>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-ea95c6a2><span class="visually-hidden" id="doc-outline-aria-label" data-v-ea95c6a2> Table of Contents for current page </span><ul class="root" data-v-ea95c6a2 data-v-74f66e6c><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-f77a9b1a></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://jsd.toshiki.dev/gh/andatoshiki/toshiki-notebook@master/assets/logo/sponsor/telegram.png"></article></a></div><!--]--></div></section><!--]--></div></div><!--]--><!--]--><!--]--><!--]--></div></div></div></div><div class="content" data-v-6c4a7022><div class="content-container" data-v-6c4a7022><!--[--><!--]--><!----><main class="main" data-v-6c4a7022><div style="position:relative;" class="vp-doc _academic_physics_ipho-formulas-jpn_13" data-v-6c4a7022><div><h1 id="formulas-for-ipho-日本語版-section-13" tabindex="-1">Formulas for IPhO 日本語版: Section 13 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-13" aria-label="Permalink to &quot;Formulas for IPhO 日本語版: Section 13&quot;">​</a></h1><h2 id="_13-相対性理論" tabindex="-1">13: 相対性理論 <a class="header-anchor" href="#_13-相対性理論" aria-label="Permalink to &quot;13: 相対性理論&quot;">​</a></h2><h3 id="_13-1-lorentz-変換" tabindex="-1">13.1:Lorentz 変換 <a class="header-anchor" href="#_13-1-lorentz-変換" aria-label="Permalink to &quot;13.1:Lorentz 変換&quot;">​</a></h3><ol><li><p>Lorentz 変換 (Minkowski 幾何学の 4 次元時空の回 転）(慣性系間の速度が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold-italic">V</mi><mo>=</mo><mi>V</mi><msub><mi mathvariant="bold-italic">e</mi><mi>x</mi></msub><mo fence="true">)</mo></mrow><mo>:</mo><mi>β</mi><mo>=</mo><mi>V</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo separator="true">,</mo><mi>γ</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\left.\boldsymbol{V}=V \boldsymbol{e}_x\right): \beta=V / c, \gamma=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.25555em;">V</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mn>1</mn><mi mathvariant="normal">/</mi><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>β</mi><mn>2</mn></msup></mrow></msqrt><mtext> として, </mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mspace width="2em"></mspace><mi>c</mi><msup><mi>t</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>c</mi><mi>t</mi><mo>−</mo><mi>β</mi><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>β</mi><mi>c</mi><mi>t</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msup><mi>E</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi mathvariant="normal">/</mi><mi>c</mi><mo>=</mo><mi>γ</mi><mrow><mo fence="true">(</mo><mi>E</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo>−</mo><mi>β</mi><msub><mi>p</mi><mi>x</mi></msub><mo fence="true">)</mo></mrow><mo separator="true">,</mo><msubsup><mi>p</mi><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><mi>γ</mi><mrow><mo fence="true">(</mo><msub><mi>p</mi><mi>x</mi></msub><mo>−</mo><mi>β</mi><mi>E</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo fence="true">)</mo></mrow><mo separator="true">,</mo><msubsup><mi>p</mi><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi>p</mi><mi>y</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext> ここで, </mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>E</mi><mo>=</mo><mfrac><mrow><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></mrow><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></msqrt></mfrac><mo>=</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>p</mi><mi>x</mi></msub><mo>=</mo><mfrac><mrow><mi>m</mi><msub><mi>v</mi><mi>x</mi></msub></mrow><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></msqrt></mfrac><mo separator="true">,</mo><mtext> etc. </mtext></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; 1 / \sqrt{1-\beta^2} \text { として, } \\ &amp; \qquad c t^{\prime}=\gamma(c t-\beta x), x^{\prime}=\gamma(x-\beta c t), y^{\prime}=y \\ &amp; E^{\prime} / c=\gamma\left(E / c-\beta p_x\right), p_x^{\prime}=\gamma\left(p_x-\beta E / c\right), p_y^{\prime}=p_y \\ &amp; \text { ここで, } \\ &amp; E=\frac{m c^2}{\sqrt{1-v^2 / c^2}}=m c^2+\frac{1}{2} m v^2+\cdots \\ &amp; p_x=\frac{m v_x}{\sqrt{1-v^2 / c^2}}, \text { etc. } \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.6485em;vertical-align:-5.5742em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.0742em;"><span style="top:-8.5586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:-7.0586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:-5.5586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:-4.0355em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:-1.8844em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:0.6531em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.5742em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.0742em;"><span style="top:-8.5586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord"></span><span class="mord">1/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0067em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.9667em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120
c340,-704.7,510.7,-1060.3,512,-1067
l0 -0
c4.7,-7.3,11,-11,19,-11
H40000v40H1012.3
s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232
c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1
s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26
c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z
M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2333em;"><span></span></span></span></span></span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">として</span><span class="mord">, </span></span></span></span><span style="top:-7.0586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord mathnormal">c</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mord mathnormal">c</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-5.5586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">βE</span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.0355em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">ここで</span><span class="mord">, </span></span></span></span><span style="top:-1.8844em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.175em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120
c340,-704.7,510.7,-1060.3,512,-1067
l0 -0
c4.7,-7.3,11,-11,19,-11
H40000v40H1012.3
s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232
c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1
s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26
c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z
M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.13em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span></span></span><span style="top:0.6531em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.175em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120
c340,-704.7,510.7,-1060.3,512,-1067
l0 -0
c4.7,-7.3,11,-11,19,-11
H40000v40H1012.3
s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232
c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1
s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26
c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z
M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.13em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord"> etc. </span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.5742em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_13-2-4-元ベクトルの長" tabindex="-1">13.2: 4 元ベクトルの長 <a class="header-anchor" href="#_13-2-4-元ベクトルの長" aria-label="Permalink to &quot;13.2: 4 元ベクトルの長&quot;">​</a></h3><ol start="2"><li>4 元ベクトルの長さ (スカラー量であり Lorentz 変換 で不変）:<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msup><mi>s</mi><mn>2</mn></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mi>c</mi><mi>t</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><msup><mi>z</mi><mn>2</mn></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mi>m</mi><mi>c</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mi>E</mi><mi mathvariant="normal">/</mi><mi>c</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><msubsup><mi>p</mi><mi>x</mi><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>p</mi><mi>y</mi><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>p</mi><mi>z</mi><mn>2</mn></msubsup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} s^2 &amp; =(c t)^2-x^2-y^2-z^2 \\ (m c)^2 &amp; =(E / c)^2-p_x^2-p_y^2-p_z^2 \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0713em;vertical-align:-1.2857em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7857em;"><span style="top:-3.9216em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.3974em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mord mathnormal">c</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2857em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7857em;"><span style="top:-3.9216em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mord mathnormal">t</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.3974em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2857em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_13-3-速度の加算" tabindex="-1">13.3: 速度の加算 <a class="header-anchor" href="#_13-3-速度の加算" aria-label="Permalink to &quot;13.3: 速度の加算&quot;">​</a></h3><ol start="3"><li>速度の加算 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>v</mi><mi>x</mi></msub><mo>=</mo><mfrac><mrow><msubsup><mi>v</mi><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mi>V</mi></mrow><mrow><mn>1</mn><mo>+</mo><msubsup><mi>v</mi><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>V</mi><mi mathvariant="normal">/</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac><mo separator="true">,</mo><msub><mi>v</mi><mi>y</mi></msub><mo>=</mo><mfrac><msubsup><mi>v</mi><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mrow><mi>γ</mi><mrow><mo fence="true">(</mo><mn>1</mn><mo>+</mo><msubsup><mi>v</mi><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>V</mi><mi mathvariant="normal">/</mi><msup><mi>c</mi><mn>2</mn></msup><mo fence="true">)</mo></mrow></mrow></mfrac></mrow><annotation encoding="application/x-tex">v_x=\frac{v_x^{\prime}+V}{1+v_x^{\prime} V / c^2}, v_y=\frac{v_y^{\prime}}{\gamma\left(1+v_x^{\prime} V / c^2\right)} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3649em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4289em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6779em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.461em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6779em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.7731em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p></li></ol><h3 id="_13-4-doppler-効果" tabindex="-1">13.4: Doppler 効果 <a class="header-anchor" href="#_13-4-doppler-効果" aria-label="Permalink to &quot;13.4: Doppler 効果&quot;">​</a></h3><ol start="4"><li>Doppler 効果 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>ν</mi><mo>=</mo><mi>γ</mi><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>β</mi><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo stretchy="false">)</mo><msub><mi>ν</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\nu=\gamma(1+\beta \cos \theta) \nu_0 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_13-5-minkowski-空間" tabindex="-1">13.5: Minkowski 空間 <a class="header-anchor" href="#_13-5-minkowski-空間" aria-label="Permalink to &quot;13.5: Minkowski 空間&quot;">​</a></h3><ol start="5"><li>Minkowski 空間は, 時間が虚数（ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>t</mi><mi>i</mi><mi>c</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(t i c t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mord mathnormal">i</span><span class="mord mathnormal">c</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> であれば Euclid 空間にすることができる. 回転角 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex">\varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">φ</span></span></span></span> に対して, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tan</mi><mo>⁡</mo><mi>φ</mi><mo>=</mo><mi>v</mi><mi mathvariant="normal">/</mi><mi>i</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">\tan \varphi=v / i c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord">/</span><span class="mord mathnormal">i</span><span class="mord mathnormal">c</span></span></span></span> となり, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo separator="true">,</mo><mi>cos</mi><mo>⁡</mo><mi>φ</mi></mrow><annotation encoding="application/x-tex">\sin \varphi, \cos \varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8623em;vertical-align:-0.1944em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span></span></span></span> を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tan</mi><mo>⁡</mo><mi>φ</mi></mrow><annotation encoding="application/x-tex">\tan \varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span></span></span></span> で表して Euclid 幾何学の公式を適用する (Lorentz 変換).</li></ol><h3 id="_13-6-長さの縮み" tabindex="-1">13.6: 長さの縮み <a class="header-anchor" href="#_13-6-長さの縮み" aria-label="Permalink to &quot;13.6: 長さの縮み&quot;">​</a></h3><ol start="6"><li>長さの縮み : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>l</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>l</mi><mn>0</mn></msub><mi mathvariant="normal">/</mi><mi>γ</mi></mrow><annotation encoding="application/x-tex">l^{\prime}=l_0 / \gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span></span></span></span>.</li></ol><h3 id="_13-7-時間の遅れ" tabindex="-1">13.7: 時間の遅れ <a class="header-anchor" href="#_13-7-時間の遅れ" aria-label="Permalink to &quot;13.7: 時間の遅れ&quot;">​</a></h3><ol start="7"><li>時間の遅れ: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>t</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>t</mi><mn>0</mn></msub><mi>γ</mi></mrow><annotation encoding="application/x-tex">t^{\prime}=t_0 \gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span></span></span></span>.</li></ol><h3 id="_13-8-同時刻の相対性" tabindex="-1">13.8: 同時刻の相対性 <a class="header-anchor" href="#_13-8-同時刻の相対性" aria-label="Permalink to &quot;13.8: 同時刻の相対性&quot;">​</a></h3><ol start="8"><li>同時刻の相対性 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>=</mo><mo>−</mo><mi>γ</mi><mi>v</mi><mi mathvariant="normal">Δ</mi><mi>x</mi><mi mathvariant="normal">/</mi><msup><mi>c</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Delta t=-\gamma v \Delta x / c^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_13-9-f-dp-dt" tabindex="-1">13.9: F=dp/dt <a class="header-anchor" href="#_13-9-f-dp-dt" aria-label="Permalink to &quot;13.9: F=dp/dt&quot;">​</a></h3><ol start="9"><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">F</mi><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">p</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>t</mi><mrow><mo fence="true">(</mo><mo>=</mo><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>γ</mi><mi>m</mi><mi mathvariant="bold-italic">v</mi><mo stretchy="false">)</mo><mo fence="true">)</mo></mrow><mo stretchy="false">(</mo></mrow><annotation encoding="application/x-tex">\boldsymbol{F}=\mathrm{d} \boldsymbol{p} / \mathrm{d} t\left(=\frac{\mathrm{d}}{\mathrm{d} t}(\gamma m \boldsymbol{v})\right)(</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2301em;vertical-align:-0.35em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span><span class="mord mathnormal mtight">t</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord mathnormal">γm</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span></span></span></span> ここでの <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\gamma=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></msqrt><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left.1 / \sqrt{1-v^2 / c^2}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8em;vertical-align:-0.65em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord">1/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120
c340,-704.7,510.7,-1060.3,512,-1067
l0 -0
c4.7,-7.3,11,-11,19,-11
H40000v40H1012.3
s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232
c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1
s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26
c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z
M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span></span></span></span></li></ol><h3 id="_13-10-超相対論的極限" tabindex="-1">13.10: 超相対論的極限 <a class="header-anchor" href="#_13-10-超相対論的極限" aria-label="Permalink to &quot;13.10: 超相対論的極限&quot;">​</a></h3><ol start="10"><li>超相対論的極限 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><mi>v</mi><mo>≈</mo><mi>c</mi><mo separator="true">,</mo><mi>p</mi><mo>≈</mo><mi>m</mi><mi>c</mi><mo separator="true">,</mo><msqrt><mrow><mn>1</mn><mo>−</mo><mo stretchy="false">(</mo><mi>v</mi><mi mathvariant="normal">/</mi><mi>c</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msqrt><mo>≈</mo></mrow><annotation encoding="application/x-tex">: v \approx c, p \approx m c, \sqrt{1-(v / c)^2} \approx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6776em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.305em;"></span><span class="mord mathnormal">m</span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120
c340,-704.7,510.7,-1060.3,512,-1067
l0 -0
c4.7,-7.3,11,-11,19,-11
H40000v40H1012.3
s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232
c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1
s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26
c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z
M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mrow><mn>2</mn><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>v</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo stretchy="false">)</mo></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{2(1-v / c)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.305em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">2</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mclose">)</span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120
c340,-704.7,510.7,-1060.3,512,-1067
l0 -0
c4.7,-7.3,11,-11,19,-11
H40000v40H1012.3
s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232
c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1
s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26
c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z
M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span></span></li></ol><h3 id="_13-11-電場と磁場の-lorentz-変換" tabindex="-1">13.11: 電場と磁場の Lorentz 変換 <!----> <a class="header-anchor" href="#_13-11-電場と磁場の-lorentz-変換" aria-label="Permalink to &quot;13.11: 電場と磁場の Lorentz 変換 &lt;Badge type=&quot;tip&quot; text=&quot;supplemental&quot; /&gt;&quot;">​</a></h3><ol start="11"><li>電場と磁場の Lorentz 変換 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold-italic">E</mi><mi mathvariant="normal">∥</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi mathvariant="bold-italic">E</mi><mi mathvariant="normal">∥</mi></msub><mo separator="true">,</mo><msubsup><mi mathvariant="bold-italic">B</mi><mi mathvariant="normal">∥</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi mathvariant="bold-italic">B</mi><mi mathvariant="normal">∥</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{E}_{\|}^{\prime}=\boldsymbol{E}_{\|}, \boldsymbol{B}_{\|}^{\prime}=\boldsymbol{B}_{\|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2499em;vertical-align:-0.422em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.422em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2499em;vertical-align:-0.422em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.422em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0413em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span>,<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi mathvariant="bold-italic">E</mi><mo lspace="0em" rspace="0em">⊥</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi mathvariant="normal">/</mi><mi>c</mi><mo>=</mo><mi>γ</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold-italic">E</mi><mo lspace="0em" rspace="0em">⊥</mo></msub><mi mathvariant="normal">/</mi><mi>c</mi><mo>+</mo><mi mathvariant="bold-italic">v</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo>×</mo><msub><mi mathvariant="bold-italic">B</mi><mo lspace="0em" rspace="0em">⊥</mo></msub><mo fence="true">)</mo></mrow><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi mathvariant="bold-italic">B</mi><mo lspace="0em" rspace="0em">⊥</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><mi>γ</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold-italic">B</mi><mo lspace="0em" rspace="0em">⊥</mo></msub><mo>−</mo><mi mathvariant="bold-italic">v</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo>×</mo><msub><mi mathvariant="bold-italic">E</mi><mo lspace="0em" rspace="0em">⊥</mo></msub><mi mathvariant="normal">/</mi><mi>c</mi><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gathered} \boldsymbol{E}_{\perp}^{\prime} / c=\gamma\left(\boldsymbol{E}_{\perp} / c+\boldsymbol{v} / c \times \boldsymbol{B}_{\perp}\right), \\ \boldsymbol{B}_{\perp}^{\prime}=\gamma\left(\boldsymbol{B}_{\perp}-\boldsymbol{v} / c \times \boldsymbol{E}_{\perp} / c\right) \end{gathered} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol></div></div></main><footer class="VPDocFooter" data-v-6c4a7022 data-v-b5edbda4><!--[--><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><!--]--><div class="edit-info" data-v-b5edbda4><div class="edit-link" data-v-b5edbda4><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/13.md" target="_blank" rel="noreferrer" data-v-b5edbda4 data-v-075865b7><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" aria-label="edit icon" data-v-b5edbda4><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-b5edbda4><p class="VPLastUpdated" data-v-b5edbda4 data-v-00fbdbd8>Last updated: <time datetime="2023-06-11T16:37:22.000Z" data-v-00fbdbd8></time></p></div></div><div class="prev-next" data-v-b5edbda4><div class="pager" data-v-b5edbda4><a class="pager-link prev" href="/academic/physics/ipho-formulas-jpn/12" data-v-b5edbda4><span class="desc" data-v-b5edbda4>Previous page</span><span class="title" data-v-b5edbda4>12: Keplerの法則</span></a></div><div class="has-prev pager" data-v-b5edbda4><!----></div></div></footer><!--[--><!--[--><!--[--><div id="comment-container"></div><!--]--><!--]--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-83f63849 data-v-02da8605><div class="container" data-v-02da8605><p class="message" data-v-02da8605>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and <i class="coffee fa fa-coffee fa-xs" aria-hidden="true"></i> by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code></p><p class="copyright" data-v-02da8605>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a>, <a href="https://github.com/lolilab">LoliLab</a> and <a href="https://github.com/toshikidev">Toshiki Dev</a> present <br /><span id="siteruntime_span"></span></p></div></footer><!--[--><!--]--></div></div>
<script>__VP_HASH_MAP__ = JSON.parse("{\"academic_chemistry_index.md\":\"cd37962f\",\"academic_physics_ipho-formulas-jpn_12.md\":\"faba7cc6\",\"academic_physics_ipho-formulas-jpn_13.md\":\"7cf2775d\",\"academic_physics_ipho-formulas-jpn_2.md\":\"556d84a9\",\"academic_literature_index.md\":\"6e443b78\",\"academic_literature_writing_methods-of-development.md\":\"70e598bc\",\"academic_physics_index.md\":\"56d78bca\",\"academic_physics_ipho-formulas-jpn_11.md\":\"f48c6f4f\",\"academic_physics_ipho-formulas-jpn_6.md\":\"0356f234\",\"academic_vocabulary_index.md\":\"eae026af\",\"academic_physics_ipho-formulas-jpn_7.md\":\"eb876e83\",\"academic_vocabulary_2023_02_2023-02-27.md\":\"d668eacd\",\"application_markdown-it-katex_how-to-use.md\":\"13137c85\",\"development_rclone-for-r2.md\":\"7a02294a\",\"academic_chemistry_notes_12-5.md\":\"b0856e76\",\"application_markdown-it-katex_tips.md\":\"8df87232\",\"academic_physics_ipho-formulas-jpn_1.md\":\"f74714c1\",\"academic_physics_ipho-formulas-jpn_10.md\":\"43ccaee3\",\"development_aws_aws-get-started.md\":\"f1dabd31\",\"academic_physics_ipho-formulas-jpn_5.md\":\"0ce3855c\",\"academic_physics_ipho-formulas-jpn_8.md\":\"92d7dca3\",\"development_aws_webserver.md\":\"11736134\",\"application_vitepress-plugin-shiki-twoslash_config_reference.md\":\"9c2d4c18\",\"application_vitepress-plugin-shiki-twoslash_guide_custom-theme.md\":\"976c27b6\",\"application_vitepress-plugin-shiki-twoslash_config_flags.md\":\"e811dcba\",\"development_file-naming-convention.md\":\"ccf744fb\",\"application_vitepress-plugin-shiki-twoslash_index.md\":\"9df1f1b2\",\"development_aws_readme.md\":\"e7b441d6\",\"application_vitepress-plugin-shiki-twoslash_guide_markdown-extensions.md\":\"930fa04a\",\"development_aws_main.md\":\"83af6fcd\",\"development_aws_appendix.md\":\"0422a763\",\"development_aws_acknowledgement.md\":\"9e811428\",\"development_aws_author.md\":\"a5aeb29d\",\"development_aws_aws-batch.md\":\"9ac15c51\",\"development_aws_assignments.md\":\"be296800\",\"development_aws_cloud.md\":\"49b16db4\",\"development_aws_closing.md\":\"ef7690ef\",\"development_aws_handson-bashoutter.md\":\"b0b7cff8\",\"development_aws_docker-system.md\":\"15d14809\",\"development_aws_handson-jupyter.md\":\"6aa4a83d\",\"development_aws_handson-ec2.md\":\"6144649e\",\"development_aws_handson-qabot.md\":\"98acc725\",\"academic_physics_ipho-formulas-jpn_3.md\":\"24d5d675\",\"development_aws_handson-serverless.md\":\"3b6d595f\",\"development_aws_license.md\":\"e88f3615\",\"development_aws_scientific-computing.md\":\"5ca38245\",\"development_aws_serverless.md\":\"aca213ae\",\"academic_physics_ipho-formulas-jpn_4.md\":\"3da0ed20\",\"getting-started.md\":\"382ac296\",\"academic_physics_ipho-formulas-jpn_9.md\":\"c5d72921\",\"index.md\":\"83b33afa\",\"javascript_notes_1_1-1.md\":\"24b32387\",\"javascript_notes_1_1-2.md\":\"1705709d\",\"jp_index.md\":\"f7d85b8b\",\"roadmap.md\":\"25f8c686\",\"save_reading_index.md\":\"be9215b4\",\"save_reading_outliers_1.md\":\"3ed630fb\",\"save_reading_outliers_2.md\":\"c7bb74df\",\"application_vitepress-plugin-shiki-twoslash_api_annotations.md\":\"d87b92c8\",\"save_reading_outliers_3.md\":\"f6bf0590\",\"save_reading_outliers_4.md\":\"990bef93\",\"academic_chemistry_problems_02-20.md\":\"c447cfd4\",\"application_vitepress-plugin-shiki-twoslash_api_errors.md\":\"f3721c1d\",\"academic_chemistry_problems_03-02-1.md\":\"d0604f3c\",\"academic_chemistry_problems_03-02-2.md\":\"498840e1\",\"application_vitepress-plugin-shiki-twoslash_api_cutting.md\":\"6f39f424\",\"application_vitepress-plugin-shiki-twoslash_api_emit.md\":\"75099646\",\"academic_chemistry_problems_03-02-3.md\":\"4f565ce6\",\"application_vitepress-plugin-shiki-twoslash_api_multi-file.md\":\"13e65419\",\"application_vitepress-plugin-shiki-twoslash_api_includes.md\":\"8287166d\",\"application_vitepress-plugin-shiki-twoslash_api_logging.md\":\"faafdfd2\",\"application_vitepress-plugin-shiki-twoslash_api_queries.md\":\"a750c749\",\"application_vitepress-plugin-shiki-twoslash_api_types.md\":\"1c82fa3c\",\"development_aws_introduction.md\":\"feada6ef\",\"application_markdown-it-katex_support-function.md\":\"20802fa1\",\"application_markdown-it-katex_support-table.md\":\"100b11f3\"}")
__VP_SITE_DATA__ = JSON.parse("{\"lang\":\"en-US\",\"dir\":\"ltr\",\"title\":\"Toshiki's Note\",\"description\":\"Toshiki's web notebook served via Vitepress!\",\"base\":\"/\",\"head\":[],\"appearance\":true,\"themeConfig\":{\"nav\":[{\"text\":\"Development\",\"link\":\"/development/\"},{\"text\":\"Academic\",\"items\":[{\"text\":\"K-12\",\"items\":[{\"text\":\"Chemistry\",\"link\":\"/academic/chemistry/index\",\"activeMatch\":\"/academic/chemistry/\"},{\"text\":\"Discrete Math.\",\"link\":\"/discrete-math/index\",\"activeMatch\":\"/categories/fragments/\"},{\"text\":\"Literature\",\"link\":\"/academic/literature/index\",\"activeMatch\":\"/academic/literature/\"}]},{\"text\":\"Tools\",\"items\":[{\"text\":\"Formulas for IPhO JPN.\",\"link\":\"/academic/physics/ipho-formulas-jpn/1\",\"activeMatch\":\"/academic/physics/ipho-formulas-jpn/\"}]},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"}],\"activeMatch\":\"/academic/\"},{\"text\":\"Application\",\"items\":[{\"text\":\"Personal projects\",\"items\":[{\"text\":\"markdown-it-katex\",\"link\":\"/application/markdown-it-katex/how-to-use\",\"activeMatch\":\"/application/markdown-it-katex/\"},{\"text\":\"vitepress-plugin-shiki-twoslash\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/index\",\"activeMatch\":\"/application/vitepress-plugin-shiki-twoslash/index\"}]}],\"activeMatch\":\"/save/\"},{\"text\":\"Save\",\"items\":[{\"text\":\"Reading\",\"link\":\"/save/reading/index\",\"activeMatch\":\"/save/reading/\"},{\"text\":\"Vocabulary\",\"link\":\"/academic/vocabulary/index\",\"activeMatch\":\"/academic/vocabulary/\"}],\"activeMatch\":\"/save/\"}],\"sidebar\":{\"/development/\":[{\"text\":\"Wiki Database\",\"collapsed\":false,\"items\":[{\"text\":\"File Naming Convention\",\"link\":\"/development/file-naming-convention\"}]}],\"/academic/chemistry/\":[{\"text\":\"Textbook\",\"collapsed\":true,\"items\":[{\"text\":\"12-5: Reaction Mechanism\",\"link\":\"/academic/chemistry/notes/12-5\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"}]},{\"text\":\"Kinetics\",\"collapsed\":false,\"items\":[{\"text\":\"Rate determining steps\",\"link\":\"/academic/chemistry/notes/kinetics/rate-determining-step\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"}]},{\"text\":\"Problems & Solutions\",\"collapsed\":true,\"items\":[{\"text\":\"Problem: 02-20\",\"link\":\"/academic/chemistry/problems/02-20\"},{\"text\":\"Problem: 03-02-1\",\"link\":\"/academic/chemistry/problems/03-02-1\"},{\"text\":\"Problem: 03-02-2\",\"link\":\"/academic/chemistry/problems/03-02-2\"},{\"text\":\"Problem: 03-02-3\",\"link\":\"/academic/chemistry/problems/03-02-3\"}]}],\"/academic/physics\":[{\"text\":\"IPhO Formulas: JP Ver.\",\"collapsed\":false,\"items\":[{\"text\":\"1: 数学\",\"link\":\"/academic/physics/ipho-formulas-jpn/1\"},{\"text\":\"2: 一般的な推奨事\",\"link\":\"/academic/physics/ipho-formulas-jpn/2\"},{\"text\":\"3: 運動学\",\"link\":\"/academic/physics/ipho-formulas-jpn/3\"},{\"text\":\"4: 力学\",\"link\":\"/academic/physics/ipho-formulas-jpn/4\"},{\"text\":\"5: 振動と波\",\"link\":\"/academic/physics/ipho-formulas-jpn/5\"},{\"text\":\"6: 幾何光学，測光\",\"link\":\"/academic/physics/ipho-formulas-jpn/6\"},{\"text\":\"7: 波動光学\",\"link\":\"/academic/physics/ipho-formulas-jpn/7\"},{\"text\":\"8: 電気回路\",\"link\":\"/academic/physics/ipho-formulas-jpn/8\"},{\"text\":\"9: 電磁気学\",\"link\":\"/academic/physics/ipho-formulas-jpn/9\"},{\"text\":\"10: 熱力\",\"link\":\"/academic/physics/ipho-formulas-jpn/10\"},{\"text\":\"11: 量子力学\",\"link\":\"/academic/physics/ipho-formulas-jpn/11\"},{\"text\":\"12: Keplerの法則\",\"link\":\"/academic/physics/ipho-formulas-jpn/12\"},{\"text\":\"13: 相対性理論\",\"link\":\"/academic/physics/ipho-formulas-jpn/13\"}]}],\"/academic/vocabulary/\":[{\"text\":\"Vocabulary\",\"collapsed\":true,\"items\":[{\"text\":\"2023-02-27\",\"link\":\"/academic/vocabulary/2023/02/2023-02-27\"}]}],\"/academic/literature/\":[{\"text\":\"Writing Resources\",\"collapsed\":true,\"items\":[{\"text\":\"Patterns of Organization and Methods of Development\",\"link\":\"/academic/literature/writing/methods-of-development\"}]}],\"/javascript/\":[{\"text\":\"1: Basic JavaScript-Value, Variables, and Control Flow\",\"collapsed\":true,\"items\":[{\"text\":\"1-1: Numbers\",\"link\":\"/javascript/notes/1/1-1\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"}]}],\"/save/reading/\":[{\"text\":\"Outliers\",\"collapsed\":true,\"items\":[{\"text\":\"Introduction & Chapter 1: The Roseto Mystery\",\"link\":\"/save/reading/outliers/1\"},{\"text\":\"Chapter 2: The 10,000-Hour Rule\",\"link\":\"/save/reading/outliers/2\"},{\"text\":\"Chapter 3: The Trouble with Geniuses, Part 1\",\"link\":\"/save/reading/outliers/3\"},{\"text\":\"Chapter 4: The Trouble with Geniuses, Part 2\",\"link\":\"/save/reading/outliers/4\"}]}],\"/application/markdown-it-katex/\":[{\"text\":\"markdown-it-katex\",\"collapsed\":false,\"items\":[{\"text\":\"1: How to use?\",\"link\":\"/application/markdown-it-katex/how-to-use\"},{\"text\":\"2: KaTeX supported functions\",\"link\":\"/application/markdown-it-katex/support-function\"},{\"text\":\"3: KaTeX support tables\",\"link\":\"/application/markdown-it-katex/support-table\"},{\"text\":\"4: Tips\",\"link\":\"/application/markdown-it-katex/tips\"}]}],\"/application/vitepress-plugin-shiki-twoslash/\":[{\"text\":\"Guide\",\"collapsed\":false,\"items\":[{\"text\":\"Getting Started\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/\"},{\"text\":\"Markdown Extensions\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/guide/markdown-extensions\"},{\"text\":\"Using a Custom Theme\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/guide/custom-theme\"}]},{\"text\":\"Features\",\"collapsed\":false,\"items\":[{\"text\":\"Queries\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/queries\"},{\"text\":\"Errors\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/errors\"},{\"text\":\"Emit\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/emit\"},{\"text\":\"Cutting\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/cutting\"},{\"text\":\"Multi-file\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/multi-file\"},{\"text\":\"@types\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/types\"},{\"text\":\"Meta Annotations\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/annotations\"},{\"text\":\"Logging\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/logging\"},{\"text\":\"Includes\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/includes\"}]},{\"text\":\"Config\",\"collapsed\":false,\"items\":[{\"text\":\"Reference\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/config/reference\"},{\"text\":\"Compiler Flags\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/config/flags\"}]}]},\"footer\":{\"copyright\":\"Copyright © 2023-2023 <a href=\\\"https://github.com/andatoshiki\\\">Anda Toshiki</a>, <a href=\\\"https://github.com/lolilab\\\">LoliLab</a> and <a href=\\\"https://github.com/toshikidev\\\">Toshiki Dev</a> present <br /><span id=\\\"siteruntime_span\\\"></span>\",\"message\":\"Wrote with <i class=\\\"heart fa fa-heart fa-xs fa-beat\\\"></i> and <i class=\\\"coffee fa fa-coffee fa-xs\\\" aria-hidden=\\\"true\\\"></i> by <a href=\\\"https://toshiki.dev\\\">Anda Toshiki</a> at <code>root@andatoshiki:/~</code>\"},\"logo\":\"/logos/logo.png\",\"outline\":\"deep\",\"outlineTitle\":\"TOC\",\"outlineBadges\":false,\"lastUpdatedText\":\"Last updated\",\"algolia\":{\"appId\":\"G9IUR45K98\",\"apiKey\":\"8528cc91281d8112b28f508317a96dd3\",\"indexName\":\"toshiki-notebook\"},\"editLink\":{\"pattern\":\"https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/:path\",\"text\":\"Edit this page on GitHub\"},\"socialLinks\":[{\"icon\":\"github\",\"link\":\"https://github.com/andatoshiki\"},{\"icon\":\"twitter\",\"link\":\"https://twitter.com/andatoshiki\"}]},\"locales\":{\"/\":{\"label\":\"English\",\"lang\":\"en-US\"},\"/jp/\":{\"label\":\"Japanese\",\"title\":\"Vue Test Utils\",\"lang\":\"jp-JP\",\"description\":\"La documentation officielle de Vue Test Utils\"}},\"scrollOffset\":90,\"cleanUrls\":true}")</script>
</body>
</html>
Reference in New Issue View Git Blame Copy Permalink