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“Spacetime rotations” (changing your own speed) are often called “Lorentz boosts“, by people who don’t feel like being clearly understood.
You can prove that the spacetime interval is invariant based only on the speed of light being the same to everyone.
So whatever that last term is () it’s also conserved (as long as you don’t change your own speed). Could this whole thing (thought Einstein) be the energy of the object in question, divided by c? And, since c is a constant, energy divided by c is also conserved. Notice that the energy and momentum here are not the energy and momentum: and .
This only has noticeable effects at extremely high speeds, and at lower speeds they look like: and , which is what you’d hope for.
But since relativity messes with distance and time, it’s important to come up with a better definition of time. This way you can talk about how fast an object is moving through time, as well as how fast it’s moving through space.
If you were to use the “” equation for kinetic energy you would be exactly right up to one part in 20,000,000,000,000,000.
All of the higher terms are divided by some power of c (a big number), so until the speed gets ridiculously high they just don’t matter. If somebody flings a battery at you, it really doesn’t matter if the battery is charged up or not.
Don’t worry about it.) Now, Einstein (having a passing familiarity with physics) knew that momentum () is conserved, and that the magnitude of momentum is conserved by rotation (in other words, the of momentum is conserved (top picture of this post).
He also knew that to get from velocity to momentum, just multiply by mass (momentum is ). So if ordinary momentum is given by the first term (the “spacial term”): , then what’s that other stuff ()?