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Special and General Relativity
Understanding 4-Momentum in General Relativity
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[QUOTE="PeterDonis, post: 5451938, member: 197831"] As long as you have a metric, you can always raise and lower indexes, so this isn't really a meaningful distinction. It isn't unless two conditions are met: first, the spacetime must have a timelike Killing vector field, and second, you must be using coordinates in which the "0" basis vector is tangent to the orbits of the Killing vector field. The first condition is necessary for there to be a meaningful notion of "energy" as a conserved quantity; the second is necessary for the "0" component of ##p## to be the only relevant one for the energy. No. It doesn't have anything to do with the EP. The more precise name for it is "energy at infinity", and it is only a constant of motion for geodesic motion (i.e., motion solely under gravity) in a stationary spacetime (i.e., one with a timelike Killing vector field, as above). Heuristically, it can be thought of as telling you how tightly an object is bound: an object in a bound orbit around a massive object will have an energy at infinity less than its rest mass, and the smaller the ratio of energy at infinity to rest mass, the more tightly it is bound. (Conversely, energy at infinity larger than rest mass indicates an object that is not bound at all--it is coming in from infinity and will escape to infinity.) Why do you say "aren't exactly"? [/QUOTE]
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Understanding 4-Momentum in General Relativity
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