# General relativity and the conservation of momentum

I'm trying to understand the conservation of momentum in general relativity.

Due to the curvature of space-time by matters and energy, the path of a linear motion appears to be distorted.

Therefore from the frame of reference of an observer who is at a not-so-curved region of space-time, it appears that the velocity vector of an object that is moving along a straight line in a very-curved space-time is changing?

Does that imply that from the observer's frame of reference (in a not-so-curved space-time) momentum of that object (in the very-curved space-time) is not conserved?

In this sense is momentum not always conversed in general relativity or did I misinterpret something?

• keep in mind that whatever is curving that region of spacetime is probably more than capable of affecting the object's momentum
– Jim
May 1, 2013 at 17:06
• @Jim Hmmm. So you are saying that time should be taken into account :D? May 1, 2013 at 17:15
• In general relativity, momentum/energy is the source of the Einstein tensor. Or said differently, non gravitational momentum/energy creates gravitational momentum/energy. So it you want to have a conserved quantity, it is not so simple, you have to add the 2 components (non-gravitational and gravitational). This is called the energy-momentum pseudotensor. But you have to be careful : this is not a covariant object (in fact, you have to choose between be a conserved quantity and being covariant...). So it is very difficult to speak about global conserved quantities in general relativity. May 1, 2013 at 17:48
• "from the frame of reference of an observer who is at a not-so-curved region of space-time..." GR doesn't have global frames of reference. You can't have a frame that covers both a curved region and a distant flat region.
– user4552
May 1, 2013 at 21:05
• In relativity, momentum is part of the energy-momentum four-vector, so that's what we expect to be conserved in SR. In GR you don't typically expect to have a globally conserved vector quantity, because parallel transport is path-dependent. Thus there is no unambiguous prescription for how to add vectors originating from different regions of spacetime. For this reason, it is not generically possible in GR (in an arbitrary spacetime) to define conservation of momentum or conservation of mass-energy. However, this can be done in certain special cases such as asymptotically flat spacetimes.
– user4552
May 1, 2013 at 21:07

When a particle is deflected by gravity the gravitational field will also be modified by the particle. To form a conservation law for momentum you need to take into account the momentum in the gravitational field as well as the particle. This can be done e.g. using pseudo-tensor methods.

This works but remember that momentum is a relative concept. Even in Newtonian dynamics it depends on the velocity of your reference frame. In general relativity it also depends on the frame but a much wider class of frames is valid. This means that momentum conservation depends on the choice of co-ordinates. Locally you can pick an inertial reference frame but over extended regions there is no inertial frame. A momentum in one location cannot be simply added to a momentum vector in another location.

Nevertheless, momentum conservation laws over extended regions do work correctly in general relativity. For an extended description of the formalism and why it works see my article at http://vixra.org/abs/1305.0034

Edit: In the comments below MWT p457 has been cited to support the idea that energy and momentum are only conserved in specific cases. I am adding this to directly refute what has been said there.

MWT begin by saying that there is no such thing as energy or momentum for a closed universe because "To weigh something one needs a platform on which to stand to do the weighing" This is pure Wheeler rhetoric of the type for which he is greatly admired, but in this case it is simply misleading. Weight is a Newtonian term with no useful counterpart in general relativity except in the specific case of an isolated system in an asymptotically flat spacetime. For other situations such as the closed cosmology energy and momentum conservation take a different but equally valid form.

They go on to say that in a closed universe total energy or momentum or charge is "trivially zero" They justify that it is zero because you can use Gauss's divergence theorem to write the charge, energy or momenta as a boundary integral. For a closed universe the boundary disappears making the result zero. This is of course correct, but they give no justification for calling this answer trivial. Energy and momenta are initially defined as a sum of volume integral contributions from each physical field including electromagnetic fields, fermionic fields, gravitational field etc. It is in this sense that we understand that conserved quantities can move and can transform from one form to another but the total remains constant. It is a property of gauge fields that when the dynamical field equations are used the conserved quantity is the divergence of a flux from just the gauge field so that it can be integrated over a volume and be calculated as a boundary surface integral. This gives charge/energy/momenta a holographic nature where they can be considered either as a volume integral over contributions from different fields of a surface integral over the gauge field flux. The important thing to understand is that to go from the volume to the surface form the field equations must be used. This means that the total charge/energy/momenta in a closed universe is zero but that this is not in any sense a trivial result. If you calculate total energy as a volume integral for a configuration of fields that do not satisfy the equations of motion the answer will not necessarily be zero. Stating that it is zero is therefore making a non-trivial assertion about the dynamics. This is what conservation laws are all about.

MWT go on to explain why it would make no sense to have an energy-momentum 4-vector globally. The invalid assumption they are making is that energy and momenta need to form a 4-vector. A 4-vector is a representation of the Poincare group and is the natural form that energy-momentum takes in special relativity where Poincare invariance is the global spacetime symmetry. In general relativity the global spacetime symmetry is diffeomorphism invariance so the correct expectation is that all quantities should take the form of a representation of the diffeomorphism group for the manifold. This is what happens. If you demand an energy-momentum 4-vector then of course you will only get an answer locally, and also for an asymptotically flat spacetime where Poincare symmetry is valid at spatial infinity, but demanding such a 4-vector is simply the wrong thing to do in general relativity.

In the fully general case of any spacetime we can apply Noether's theorem using invariance under diffeomorphism generated by any contravariant transport vector field (Observe that it is invariance of the equations that is required, not invariance of the solutions. Some people like to confuse the two) The result is a conserved current with a linear dependence on the transport vector field. This is the correct form for a representation of the diffeomorphism group. I refer to my cited paper for the mathematical details.

This current gives conservation laws for energy, and momenta including generalizations of angular momenta as well as linear momenta depending on the transport vector field chosen. If it transports space-like hypersurfaces in a timelike direction it will give an energy conservation law and if it transports in spacelike directions it gives momenta conservation laws. These energy and momenta do not normally form 4-vectors but they can be integrated to give non-trivially conserved quantities. The global form a conservation law must take is that the total energy and momenta in a volume must change at a rate which is the negative of the flux of the quantity over the boundary, and this is what you get with the currents derived from Noether's theorem.

It may be that other people will want to add comments here that dispute the validity of energy conservation in other ways. I refer once again to my new article at http://vixra.org/abs/1305.0034 where I refute all the objections that I have heard. Triviality is dealt with in item (6) and 4-momentum is dealt with in item (8). Unless someone comes up with a novel objection I will just refer to the numbered objections in this paper in future.

Remember, there are no authorities in science and any expert may be shown to be wrong either by reasoning or by experiment.

• No, this is incorrect. General relativity does not have global conservation of a tensorial momentum (or of a scalar mass-energy) in any given spacetime. This is explained in Misner, Thorne, and Wheeler on p. 457. GR only has two things: (1) local conservation of energy-momentum (because the field equations guarantee that the stress-energy tensor is divergenceless), and (2) things like the ADM mass and Komar mass, which are defined on special types of spacetimes (asymptotically flat ones).
– user4552
May 6, 2013 at 21:31
• @Ben, I have edited my answer to explain in detail why Misner, Thorne, and Wheeler are wrong. You could also have found this in general terms in my cited article. May 7, 2013 at 9:37