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my question is basic and elementary. I can't seem to understand why the formula for the energy of a particle is $E= -pU$ where $p$ is the 4-momentum of the particle and $U$ is the 4-velocity of the observer.

Actually what bothers me also is that when we say an observer has, say, a 4-velocity $U$, we mean it has this velocity w.r.t us (i.e. the person who's doing the calculation) right? the same thing for the 4-momentum of the particle!

In other words, for the formula of the energy above why shouldn't it be proportional to the velocity as measured by the observer (not to those making the calculation as we do)?

saad
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    Where are you getting your formula from? It's not a standard expression for the energy. – mike stone Dec 06 '20 at 16:28
  • Possible duplicate: https://physics.stackexchange.com/q/506106/2451 – Qmechanic Dec 06 '20 at 17:20
  • E2=(PC)2+(M0c2)2. In Einstein’s original formula he has (Pc) squared. Which is momentum times speed?? I too find it strange that speed is redundant. – Bill Alsept Dec 06 '20 at 17:30
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    $E=-pU$ is likely a shorthand for (or an incomplete transcription of) $E=-\tilde p\cdot \hat U=-p^a g_{ab} U^b$, the time-component [according to the observer with 4-velocity $\tilde U$] of the particle's 4-momentum $\tilde p$, using the $(-+++)$ signature convention. [...as seen in the comment posted by Qmechanic] – robphy Dec 06 '20 at 21:28
  • yes this the full expression =−̃ ⋅ .... i just used a shorthand i thought it would be understandable! – saad Dec 07 '20 at 13:02

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