Can someone explain why the “c” is squared in Einstein’s relativity equation?
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When something moves x faster, it doesn’t have x more energy. It has x squared more energy. If I move twice as fast as you, I have four times as much energy.
Without getting into a proof which would be beyond the scope of ELI5, it’s to make sure that the units match up
Energy is measured in Joules, which is kg m^2 /s^2 and mass in measured in Kilograms (kg).
Mass at rest and energy are related, but they don’t have the same units of measurement. However if you multiply the mass by a velocity (m/s) squared you “force” the units to match up. It just turns out that the needed velocity is the speed of light.
Addendum:
in a slightly more general version of the equation (which accounts for movement) you have
E^2 = m^2 c^4 + p^2 c^2
Where p is momentum, measured in kg m/s. So in this case you would multiply momentum by only a single c rather then two because one copy of “m/s” is already accounted for.
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.
I think when I was 5 it was ELI5 to me as the following:
For fusion to occur there needs to be two particles both traveling at the speed of light that collide.
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