From a7da57c0736bec58d1fc4ec99d211099c31bb45f Mon Sep 17 00:00:00 2001 From: Preston Pan Date: Wed, 24 Jan 2024 19:26:59 -0800 Subject: new content --- mindmap/special relativity.org | 68 ++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 68 insertions(+) create mode 100644 mindmap/special relativity.org (limited to 'mindmap/special relativity.org') diff --git a/mindmap/special relativity.org b/mindmap/special relativity.org new file mode 100644 index 0000000..ffb4cb7 --- /dev/null +++ b/mindmap/special relativity.org @@ -0,0 +1,68 @@ +:PROPERTIES: +:ID: e38d94f2-8332-4811-b7bd-060f80fcfa9b +:END: +#+title: special relativity +#+author: Preston Pan +#+html_head: +#+html_head: +#+html_head: +#+options: broken-links:t + +* Motivation +[[id:6e2a9d7b-7010-41da-bd41-f5b2dba576d3][Newtonian mechanics]] is proven to be experimentally effective for macroscopic phenomena. However, it fails in the attempt +to unify it with [[id:fde2f257-fa2e-469a-bc20-4d11714a515e][Maxwell's Equations]]; the speed of light and electromagnetic radiation in general confirmed by Maxwell's equations +are invariant to relative speed; this is in contradiction to the Galilean velocity addition that Newtonian mechanics postulates: +\begin{align*} +\vec{v} = \vec{v}_{1} + \vec{v}_{2} +\end{align*} +Where $\vec{v}_{1}$ and $\vec{v}_{2}$ are the speeds of the two objects, and $\vec{v}$ is the relative velocity between the two. The contradiction lies in the +attempt to do this type of velocity addition with the speed of light. The speed of light must remain invariant including the relative +speed (all experiments thus far confirm this and Maxwell's equations have an absolute velocity of the speed of light in a vacuum), +but when another object is moving relative to light, it results in something lesser or greater than the speed of light, even though +it must remain the same. To solve this, we must add another two postulates to Newtonian mechanics: +1. The speed of light must remain constant under all reference frames. +2. The laws of physics remain consistent within all inertial reference frames. +The result of these two additions is known as the /special theory of relativity/. + +* Derivation of Gamma Factor +Suppose Bob is on a train, where the train is moving at a constant speed to the right $\vec{v}$. Alice is outside of the train observing +Bob. Now Alice and Bob decide to use a clock to keep track of time; they do this by calculating the amount of times light +bounces from the floor to the roof of the train on mirrors. Note that the method by which they keep track of time doesn't matter, and +who keeps track of time doesn't matter, as we will see. All that matters is that nothing can move faster than the speed of light, so +no information can either; light in this case can be replaced with something else that can keep track of time in the same way. In any case, +once the light reaches the roof from the floor, where this distance is $d$ meters, $\frac{d}{c}$ seconds will have passed for Bob. + +#+caption: A very scientifically accurate drawing of the situation +#+attr_html: :width 300px +[[../img/relativity1.jpg]] + +Now this image is from Bob's perspective; when we switch to Alice's perspective, we gain a new insight; that /light has to travel the same speed for her/, but +it has a larger distance to travel because of the train's velocity. + +#+caption: Light ray from Alice's perspective +#+attr_html: :width 300px +[[../img/relativity2.jpg]] + +from this diagram, we can gather that the amount of time it takes for light to reach the roof will be longer. +Also, we can see that if Alice believes in Bob's clock, her time would be: +\begin{align*} +t' = \frac{d}{\sqrt{c^{2} - v^{2}}} +\end{align*} +which is considerably different from the $\frac{d}{c}$ we got for Bob. This means that using the same clock can cause /different time measurements/ between the two +people. The gamma factor is defined as: +\begin{align*} +\gamma = \frac{t'}{t} \\ +\gamma = \frac{d}{\sqrt{c^{2} - v^{2}}}\frac{c}{d} \\ += \frac{c}{\sqrt{c^{2} - v^{2}}} \\ += \frac{c}{\frac{c}{c}\sqrt{c^{2} - v^{2}}} \\ +\gamma = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} +\end{align*} +Note that the gamma factor is independent of distance traveled, which lets us calculate our relativistic time: +\begin{align*} +\gamma t = t' +\end{align*} +There is a profound implication from this realization, and that is that it is not sufficient to view time in an objective way; time measurements +are inertial reference frame dependent. This process where time slows down for the moving reference frame is called /time contraction/. +* Space Contraction +Now we introduce the idea of /space contraction/; just like how time can slow down for moving inertial reference frames with respect to other +reference frames, space can also contract in the same way. Imagine a light ray that goes from one side of the train to the other. -- cgit