= Relativity
{wiki=Theory_of_relativity}
= Relativistic
{synonym}
= Non-relativistic
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= Theory of relativity
{synonym}
= Special relativity
{parent=Relativity}
{wiki}
Explains how it is possible [that everyone observes the same speed of light, even if they are moving towards or opposite to the light]!!!
This was first best observed by the , which uses the movement of the Earth at different times of the year to try and detect differences in the speed of light.
This leads leads to the following conclusions:
* to and time dilation
* the speed of light is the maximum speed anything can reach
All of this goes of course completely against our daily Physics intuition.
The "special" in the name refers to the fact that it is a superset of , which also explains gravity in a single framework.
Since time and space get all messed up together, you have to be very careful to understand what it means to say "I observed this to happen over there at that time", otherwise you will go crazy. A good way to think about is this:
* use to setup a bunch of clocks for every position in your
* on every point of space, you put a little detector which records events and the time of the event
* each detector can only detect events locally, i.e. events that happen very close to the detector
* then, after the event, the detectors can send a signal to you, who is sitting at the origin, telling you what they detected
= Invariance of the speed of light
{parent=Special relativity}
{wiki}
This single [experimental observation]/idea is the basis for all of .
Special relativity is the direct result of people bending their backs to accommodate for this really weird fact.
= History of special relativity
{parent=Special relativity}
{wiki}
Bibliography:
* chapter III "Relativity, the special theory" has a good sketch as you may imagine.
= Luminiferous aether
{parent=History of special relativity}
{wiki}
Can you just imagine what if was one single fixed rigid body? This is apparently what believed, page 111 quoting his entry to Encyclopedia Britannica:
\Q[There can be no doubt that the interplanetary and interstellar spaces are not empty but are occupied by a material substance or body, which is certainly the largest, and probably the most uniform, body of which we have any knowledge.]
Then it would provide a natural space coordinate for the entire universe!
Apparently was the first to completely say: let's just screw this aether thing completely then, it's getting too complicated, and we don't really need it. As Wikipedia puts it well, in very unencyclopedic tonehttps://en.wikipedia.org/w/index.php?title=Luminiferous_aether&oldid=1063821642{ref}: Aether fell to .
= Aether theory
{parent=Luminiferous aether}
{wiki=Aether_theories}
= Aether drag hypothesis
{parent=History of special relativity}
{wiki}
Given experiments such as the and the that couldn't really detect the Earth's movement across aether, people started to wonder if the wasn't dragging the .
= Lorentz ether theory
{c}
{parent=Aether drag hypothesis}
{wiki}
= Special relativity experiment
{c}
{parent=Special relativity}
{wiki}
* : the more experiments confirm , the more special relativity has to be correct
* TODO more precisely how it is evidence.
= Aberration
{disambiguate=astronomy}
{parent=Special relativity experiment}
{wiki}
= Fizeau experiment
{c}
{parent=Special relativity experiment}
{wiki}
= Michelson-Morley experiment
{c}
{parent=Special relativity experiment}
{tag=The most important physics experiments}
{wiki=Michelson–Morley_experiment}
\Video[http://youtube.com/watch?v=lzBKlY4f1XA]
{title=Michelson by Amrita Vlab (2013)}
{description=Shows the optical controls and alignment in more detail.}
\Video[http://youtube.com/watch?v=j-u3IEgcTiQ]
{title=Michelson by TSG Physics (2012)}
{description=TSG PHysiQuantum electrodynamics bibliographycs is the channel from the MIT Department of Physics Technical Services Group. In the video they produce a very clear round interference pattern.}
= Hafele-Keating experiment
{c}
{parent=Special relativity experiment}
{title2=Special relativity atomic clock on planes experiment}
{wiki=Hafele–Keating experiment}
= Einstein synchronization
{c}
{parent=Special relativity}
{wiki}
= Frame of reference
{parent=Einstein synchronization}
{wiki}
= Inertial frame of reference
{parent=Frame of reference}
{wiki}
= Inertial frames of reference
{synonym}
= Inertial frame
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= Lorentz transformation
{c}
{parent=Special relativity}
{wiki}
= Lorentz transform
{c}
{synonym}
The equation that allows us to calculate stuff in !
Take two observers with identical rules and stopwatch, and aligned axes, but one is on a car moving at towards the $+x$ direction at speed $v$.
TODO image.
When both observe an event, if we denote:
* $(t, x, y, z)$ the observation of the standing observer
* $(t', x', y', z')$ the observation of the ending observer on a car
It is of course arbitrary who is standing and who is moving, we will just use the term "standing" for the one without primes.
Then the coordinates of the event observed by the observer on the car are:
$$
\begin{aligned}
t' & = \gamma \left( t - \frac{v x}{c^2} \right) \\
x' & = \gamma \left( x - v t \right) \\
y' & = y \\
z' & = z
\end{aligned}
$$
where:
$$
\gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}
$$
Note that if $\frac{v}{c}$ tends towards zero, then this reduces to the usual which our intuition expects:
$$
\begin{aligned}
t' & = t
x' & = x - v t \\
y' & = y \\
z' & = z
\end{aligned}
$$
This explains why we don't observe special relativity in our daily lives: macroscopic objects move too slowly compared to light, and $\frac{v}{c}$ is almost zero.
= Lorentz covariance
{c}
{parent=Lorentz transformation}
{wiki}
= Lorentz covariant
{c}
{synonym}
Same motivation as , but version of that: we want the laws of physics to have the same form on all inertial frames, so we really want to write them in a way that is Lorentz covariant.
This is just the relativistic version of that which takes the into account instead of just the old .
= Lorentz invariant
{c}
{parent=Lorentz covariance}
= Lorentz invariance
{c}
{synonym}
Basically a synonym of ?
= Lorentz transform consequence: everyone sees the same speed of light
{c}
{parent=Lorentz transformation}
OK, so let's verify the main desired consequence of the : that everyone observes the same .
Observers will measure the speed of light by calculating how long it takes the light going towards $+x$ cross a rod of length $L = x_2 - x_1$ laid in the x axis at position $X1$.
TODO image.
Each observer will observe two events:
* $(t_1, x_1, y_1, z_1)$: the light touches the left side of the rod
* $(t_2, x_2, y_2, z_2)$: the light touches the right side of the rod
Supposing that the standing observer measures the speed of light as $c$ and that light hits the left side of the rod at time $T1$, then he observes the coordinates:
$$
\begin{aligned}
t_1 & = T1 \\
x_1 & = X1 \\
t_2 & = \frac{L}{c} \\
x_2 & = X1 + L \\
\end{aligned}
$$
Now, if we transform for the moving observer:
$$
\begin{aligned}
t_1' & = \gamma \left( t_1 - \frac{v x_1}{c^2} \right) \\
x_1' & = \gamma \left( x_1 - v t_1 \right) \\
t_2' & = \gamma \left( t_2 - \frac{v x_2}{c^2} \right) \\
x_2' & = \gamma \left( x_2 - v t_2 \right) \\
\end{aligned}
$$
and so the moving observer measures the speed of light as:
$$
\begin{aligned}
c' & = \frac{x_2' - x_1'}{t_2' - t_1'} \\
& = \frac{(x_2 - v t_2) - (x_1 - v t_1)}{(t_2 - \frac{v x_2}{c^2}) - (t_1 - \frac{v x_1}{c^2})} \\
& = \frac{(x_2 - x_1) - v (t_2 - t_1)}{(t_2 - t_1) - \frac{v}{c^2} (x_2 - x_1)} \\
& = \frac{\frac{x_2 - x_1}{t_2 - t_1} - v}{1 - \frac{v}{c^2} \frac{x_2 -x_1}{t_2 - t_1}} \\
& = \frac{c - v}{1 - \frac{v}{c^2} c} \\
& = \frac{c - v}{\frac{c - v}{c}} \\
& = c \\
\end{aligned}
$$
= Length contraction
{parent=Lorentz transformation}
{wiki}
Suppose that a rod has is length $L$ measured on a rest frame $S$ (or maybe even better: two identical rulers were manufactured, and one is taken on a spaceship, a bit like the ).
Question: what is the length $L'$ than an observer in frame $S'$ moving relative to $S$ as speed $v$ observe the rod to be?
The key idea is that there are two events to consider in each frame, which we call 1 and 2:
* the left end of the rod is an observation event at a given position at a given time: $x_1$ and $t_1$ for $S$ or $x'_1$ and $t'_1$ for $S'$
* the right end of the rod is an observation event at a given position at a given time : $x_2$ and $t_2$ for $S$ or $x'_2$ and $t'_2$ for $S'$
Note that what you visually observe on a photograph is a different measurement to the more precise/easy to calculate two event measurement. On a photograph, it seems you might not even see the contraction in some cases as mentioned at https://en.wikipedia.org/wiki/Terrell_rotation
Measuring a length means to measure the $x_2 - x_1$ difference for a single point in time in your frame ($t2 = t1$).
So what we want to obtain is $x'_2 - x'_1$ for any given time $t'2 = t'1$.
In summary, we have:
$$
\begin{aligned}
L &= x_2 &- x_1 \\
L' &= x'_2 &- x'_1
t'_2 = t'_1
\end{aligned}
$$
By plugging those values into the , we can eliminate $t_2 and t_1$, and conclude that for any $t'_2 = t'_1$, the length contraction relation holds:
$$
L' = \frac{L}{\gamma}
$$
The key question that needs intuitive clarification then is: but how can this be symmetric? How can both observers see each other's rulers shrink?
And the key answer is: because to the second observer, the measurements made by the first observer are not simultaneous. Notably, the two measurement events are obviously by looking at the , and therefore can be measured even in different orders by different observers.
= Terrell rotation
{c}
{parent=Length contraction}
{wiki}
What you would see the moving rod look like on a photo of a experiment, as opposed as using two locally measured separate spacetime events to measure its length.
= Time dilation
{parent=Lorentz transformation}
{wiki}
One of the best ways to think about it is the thought experiment.
= Transversal time dilation
{parent=Time dilation}
{wiki=Time_dilation#Simple_inference_of_velocity_time_dilation}
Light watch transverse to direction of motion. This case is interesting because it separates from