Can someone explain why the “c” is squared in Einstein’s relativity equation?
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c is the conversion factor between length and time.
From a modern perspective, intuitively, length of a vector is the “wrong” thing to measure: a length of a vector is not even a differentiable function. Every physical laws either make use of length projection (which is differentiable), or length squared. That’s why dot product is useful.
Special relativity say that different inertia frame of reference measure the same proper metric. The metric is just a dot product in spacetime, so length and time interval are both squared, so the conversion factor must be squared. The opposite is also true: every possible change of coordinate that preserve this metric must produce an inertia frame of reference.
Therefore, any quantities that can be computed independent of frame of reference has to be dependent on the metric. Which is why c^2 is always there.
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