The equation that allows us to calculate stuff in special relativity!

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: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:where:

Note that if $\frac{v}{c}$ tends towards zero, then this reduces to the usual Galilean transformations which our intuition expects:

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.

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 twin paradox).

Question: what is the length $L^{\prime }$ than an observer in frame $S^{\prime }$ 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: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 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 $x_2^{\prime }-x_1^{\prime }$ for any given time $t^{\prime }2=t^{\prime }1$.

In summary, we have:

By plugging those values into the Lorentz transformation, we can eliminate $t_{2}and{t}_1$, and conclude that for any $t_2^{\prime }=t_1^{\prime }$, the length contraction relation holds:

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 spacelike-separated events by looking at the light cone, and therefore can be measured even in different orders by different observers.

The following aspects of Maxwell's equations make no sense without special relativity:

When charged particle though experiment are seen from the point of view of special relativity, it becomes clear that magnetism is just a direct side effect of charges being viewed in special relativity. One is philosophically reminded of how spin is the consequence of quantum mechanics + special relativity.

Bibliography:

A subset of Spacetime diagram.

The key insights that it gives are:

The opposite of spacelike-separated events.

Unifies both special relativity and gravity.

Not compatible with the Standard Model, and the 2020 unification attempts are called theory of everything.

One of the main motivations for it was likely having forces not be instantaneous, but rather mediated by field to maintain the principle of locality, just like electromagnetism did earlier.

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Bibliography:

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