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If your position vector is , then your velocity vector is (velocity is the time derivative of position).
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.
Notice that when you ignore time (t=0), then this reduces to the usual definition.
This fancy new “spacetime interval” conserves the length of things under ordinary rotations (which just move around the x, y, z part), but also conserves length under “rotations involving time”.
You can derive the gamma function by thinking about light clocks, or a couple other things, but I don’t want to side track on that just now. That is, is the ratio of how fast “outside time” passes from the point of view of the object’s “on-board time”. For succinctness (and tradition) I’ll bundle the first three terms together: Now check this out!
Remember that the spacetime interval for a spacetime vector with spacial component , and temporal component , is .
And the true “conserved distance” was revealed to be: .New theories should always include the old theories as a special case (or disprove them). If you allow the speed of the object to be zero (v=0), you find that everything other than the first term in that long equation for E vanishes, and you’re left with (drumroll): E=mc of energy.