Garth said:
Actually Pete Tuv;u = 0, the conservation of energy-momentum does not generally imply the conservation of energy and the conservation of momentum;
I was speaking about conservation of energy, not momentum. One would never assume that a quantity such as momentum is a constant of motion when it is moving in a gravitational field. The law of conservation of momentum applies
only to those particles on which no force is acting. I.e. in Newtonian gravity a particle falling in the Earth's gravitational field (as observed by someone sitting on its surface) will measure the energy to be constant and yet the momentum will constantly changing when in free-fall. However since this is a static field then the energy of such a falling particle will be zero.
..those are frame dependent concepts and in a freely falling frame the separate total energy and momentum of another object, freely falling in a different part of the gravitational field will not themselves individually appear conserved.
Actually its a geometric property when external 4-forces are zero. The equation div
T = 0 is a geometric statement, i.e. independant of spacetime coordinates. It is easily found by calculating in a locally Lorentzian frame. So while its true that energy is frame dependant it is not true that the
Law of conservation is frame dependant. Its sort of like measuring proper mass. It is measured when the particle is at rest in a locally Lorentzian field. If one is evaluating the energy from a freely-falling frame in a curved spacetime then the field will be time independant and such a field is non-conservative.
Its not as if the GR community calls this "local" energy-momentum conservation for nothing Garth. They state it that way for a reason.
The (-+++) nature of the metric means that the 3-momentum component is vector subtracted from the total energy to obtain the 4-momentum or energy-momentum of the object, and it is this resultant, which is the rest mass, that is conserved in GR.
3-momentum is not vector subtracted from energy since one is a tensor of rank zero while the other is a tensor of rank one. The meaning of the equation you speak of (i.e. the T^0u_u = 0 equation) is a conservation equation which states that energy entering a small enclosed surface will equal the rate at which energy passes through the surface which is the rate at which energy decreases from external to the surface. Plus you can't add 4-vectors which are located at different events in spacetime. That is a violation of the rule for adding vectors. In any case, when the energy for a single particle in free-fall in a static g-field is calculated then the energy (E ~ P
0) it will be a constant of motion, i.e. conserved.
When one speaks of conservation laws one
must take into account the specific example and see if it matches the condition postulated in the law. I.e. in that case "momentum" is rarely conserved since the law states that "The momentum of a free particle is conserved" and therefore you must take into account forces acting on a particle. Likewise the energy of a particle in a field is not constant unless the potential is time-independant, i.e. a conservative field.
Pete