

(Quick aside; it may concern you that a squared quantity can be negative.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).So first, here’s an old physics trick (used by old physicists) to guess answers without doing any thinking or work, and perhaps while drinking.Take everything that could have anything to do with the question (any speeds, densities, sizes, etc.) and put them together so that the units line up correctly.For a more formal derivation, you’d have to stir the answer gravy: of regular vectors, (which could be distance, momentum, whatever) remains unchanged by rotations.If you take a stick and just turn it, then of course it stays the same length.It may even be possible to mix together a bunch of other universal constants until you get velocity squared, or there may just be a new, previously unknown physical constant involved.


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 . In relativity you rotate how you see things in spacetime, by running past them (changing your speed with respect to what you’re looking at). “Spacetime rotations” (changing your own speed) are often called “Lorentz boosts“, by people who don’t feel like being clearly understood. Remember that the spacetime interval for a spacetime vector with spacial component , and temporal component , is . (This used a slight breach of notation: “” is a velocity of the velocity, or “speed”) The amazing thing about “spacetime speed” is that, no matter what v is, .
