Sorry JW, but let’s see whether we can clear up the maths first. What I recommend you do is to do the multiplication in detail instead of trying to take the answers on trust from memory.
(P+C) squared certainly does equal (PP+2PC+CC) as you can check by doing the long multiplication, or if you prefer to check by example, suppose you let P=3 and C=5 (So P+C=8, right?) then (P+C)X(P+C)=(3X3+2X3X5+5X5)=(9+30+25)=64=8X8, right?
Which of course was not what we wanted. Right? We wanted PC squared=(3X5)X(3X5)=15X15=225.
Right? Try it with any numbers you like. Those are some assumptions I am making, though I am open to persuasion on my typos. Now, let’s see what else there is...
You say: “In fact your expansions of essentially the same calculation are different. PC squared = PPCC, but MC squared = MMCCCC. "
You had me shook there for a moment, but I checked and actually what I really had said was: “More properly it should be E squared = (PC) squared plus (MC squared) squared, or if you prefer: EE=PPCC + MMCCCC” Check it out carefully. The difference is small, but crucial. The problems with notation here do make it confusing, but by (PC) squared I meant PCPC (which because multiplication is commutative, means the same as PCCP or PPCC. (OK? Try it with 3 and 5 again if you like). However, by (MC squared) I meant MCC, as in cricket. And by (MC squared) squared I meant MCCMCC, which still means MMCCCC.
Do you feel better about that now?
Now JW, by rest mass I mean what the object appears to have when it is standing still relative to me and my scale (or from your point of view, relative to you and your scale, OK?) And the rest mass in the formula is M. (Usually written “m”, but in my note it helps to capitalise the values for clarity.)
Light has precisely zero rest mass, right? It certainly has momentum, but it cannot rest relative to any observer; but must move at speed C in whatever direction it is seen to go. So its MC term is zero, and so is its MCC term.
OK?
Now, the lightest particles that we physically have measured as far as I know at the time of writing this, are electron neutrinos with a rest mass of about 2 eV. They practically always travel at near the speed of light. I am sorry , but to find their value for P (their momentum) you will have to do your own looking up. However, it is not likely to be negligible compared to MCC for a typical neutrino. Since a photon has exactly zero rest mass, its PC is large in comparison. For massive particles like electrons (about 250000 time that of the neutrino) or protons (roughly 500000000 times the mass of the neutrino) the MCC is out of sight greater than PC.
OK again?
>The upshot is this, I really don't understand your point.<
Well I hope that my point is now clearer. Light only has its momentum at the speed of light and it has no other speed. The only way to change its momentum is to change its energy by changing its frequency, but let’s not discuss that just now! Change the speed of an electron or an electric train, and you change MCC quite significantly in terms of what we call the Lorentz factor: 1/sqr(1-VV/CC), which approaches infinity as V approaches C. This is because M is very different from zero. Notice that no such factor applies to PC.
OK?
