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555</style>
556 </head>
557 <body class='markdown-preview'><h1>Special Relativity</h1>
558<h2>Postulates of special relativity</h2>
559<ol>
560<li>
561<p>Laws of physics are constant in all inertial reference frames</p>
562</li>
563<li>
564<p>Speed of light <span class="math"><script type="math/tex">c</script></span> is the same to all observers (proved by Michelson-Morley)</p>
565</li>
566</ol>
567<p><span class="math"><script type="math/tex">\therefore</script></span>, <span class="math"><script type="math/tex">t</script></span> must dilate as speed changes.</p>
568<p>Time and space - four dimensional relationship</p>
569<p><strong>Inertial reference frame</strong> - not accelerating</p>
570<p><strong>Proper time / length</strong> - measured by observer in same frame as events</p>
571<h2>Lorentz factor</h2>
572<span class="math"><script type="math/tex; mode=display">\gamma = {1 \over \sqrt{1-{v^2 \over c^2}}}
573</script></span>
574<p>Proper (time|length) denoted by (<span class="math"><script type="math/tex">t|l_0</script></span>) is time observed from moving frame.</p>
575<p>Applied:</p>
576<p><span class="math"><script type="math/tex">t = t_0 \gamma</script></span> (time - longer in moving frame)<br>
577<span class="math"><script type="math/tex">l={l_0 \over \gamma}</script></span> (length - only contracts in direction of motion - shorter in moving frame)<br>
578<span class="math"><script type="math/tex">m=m_0 \gamma</script></span> (mass dilation)</p>
579<span class="math"><script type="math/tex; mode=display">v=c \sqrt{1-{1 \over \gamma ^2}}
580</script></span>
581<p>(<span class="math"><script type="math/tex">c=3 \times 10^8</script></span> m / s)</p>
582<h2>Mass-energy equivalence</h2>
583<p><span class="math"><script type="math/tex">E_0=mc^2</script></span> (rest energy)</p>
584<p><span class="math"><script type="math/tex">E=\gamma mc^2</script></span> (total energy)</p>
585<ul>
586<li>mass is a dense form of energy</li>
587</ul>
588<h3>Relativistic momentum</h3>
589<span class="math"><script type="math/tex; mode=display">\rho = {mv \over {\sqrt{1-{v^2 \over c^2}}}} = \gamma mv=\gamma \rho_0
590</script></span>
591<p>Low speeds - little difference between relativistic & non-relativistic momentum<br>
592<span class="math"><script type="math/tex">\rho</script></span> approaches <span class="math"><script type="math/tex">\infty</script></span> as speed approaches <span class="math"><script type="math/tex">c</script></span><br>
593Impossible to reach speed of light (speed <span class="math"><script type="math/tex">c</script></span> requires <span class="math"><script type="math/tex">\infty</script></span> energy)<br>
594Photon has no mass - zero space or time</p>
595<h3>Total energy of an object</h3>
596<span class="math"><script type="math/tex; mode=display">E_{total}=E_k+E_{rest}=\rho mc^2
597</script></span>
598<h3>Kinetic energy</h3>
599<span class="math"><script type="math/tex; mode=display">E_k=(\gamma - 1)mc^2
600</script></span>
601<p>(takes relativity into account - use for fast objects)</p>
602<h3>Rest energy</h3>
603<span class="math"><script type="math/tex; mode=display">E_{rest}=mc^2
604</script></span>
605<p>(where <span class="math"><script type="math/tex">v<0.1c</script></span>) - this is the innate mass energy (proper energy)</p>
606<h3>Work</h3>
607<span class="math"><script type="math/tex; mode=display">W = \Delta E = \Delta mc^2
608</script></span>
609<h3>Velocity</h3>
610<span class="math"><script type="math/tex; mode=display">v={\rho \over {m \sqrt{1+{p^2 \over {m^2c^2}}}}}
611</script></span>
612<h3>Fusion / Fission equations</h3>
613<p>eV to J: multiply by <span class="math"><script type="math/tex">1.6\times 10^{-19}</script></span> (charge of an electron)</p>
614<h2>High altitude muons</h2>
615<ul>
616<li>Time dilation - more muons reach Earth than expected</li>
617<li>Normal half-life is <span class="math"><script type="math/tex">2.2 \mu s</script></span> in stationary frame of reference</li>
618<li>At close to <span class="math"><script type="math/tex">c</script></span>, muons observed from Earth have lifetime <span class="math"><script type="math/tex">> 2.2 \mu s</script></span></li>
619<li>Slower time - more time to travel, so more muons reach surface</li>
620</ul></body>
621</html>