Intuition for stress-energy tensor conservation

Question

I've been trying to go through the math to get an intuitive picture of the equation $\nabla\cdot T^{\mu\nu}=0 $ where $T^{\mu\nu}$ is the stress-energy-momentum tensor.

This is the story I've come up with:

In inertial reference frames, the total energy and total momentum of a system are conserved and don't change over time. But if spacetime is curved, there can be no truly inertaial reference frames, so we should try to say something about the case where our frame of reference begins accelerating. In this case, we see a bunch of momentum in the opposite direction pop into existence out of seemingly nowhere, bringing with it corresponding amounts of kinetic energy. So we can't talk about conservation of energy and momentum anymore.

But we can still save a weaker principle, namely that if the energy density of a point increases, that energy must flow to that point from somewhere, so if energy density at a point increases over time, we need to have a commensurate momentum directed towards that point. Since we see both energy and momentum increase as we accelerate, this principle still holds. But the momentum still comes out of nowhere, right?

Well, not necessarily. The analogous principle for momentum is that to add more momentum to a system, we need stress/pressure, which describes the flow of momentum the same way momentum describes the flow of mass-energy. But when we accelerate, we see distances contracting ever so slightly in the direction of our acceleration, which looks like an inward force in the direction of motion at every point, the pressure we need to balance out the apparent change in momentum density. So in general everything balances and the amount of stress-energy-momentum flowing into a point in space time is the same as the amount flowing out even when observed from a non-inertial frame.

Is this at all correct?

Answer 1

The covariant energy-momentum conservation law is not what you wrote. It is $$ \nabla_\mu T^{\mu\nu}=0. $$ Be careful though: "convariant conservation" equations do not imply that any component of the energy or momentum is actually conserved.

To get actual conserved quantities you need a symmetry. In particular an isometry associated with a Killing vector field $\xi^\mu$ will give you a conserved energy or momentum depending on whether the Killing vector field is timelike or spacelike. The Killing equation $$ \nabla_\mu \xi_\nu+\nabla_\nu\xi_\mu=0 $$ gives $$ \nabla_\mu (T^{\mu\nu}\xi_\nu)=0 $$ or $$ 0= \frac{1}{\sqrt g} \partial_\mu (\sqrt{g}T^{\mu\nu}\xi_\nu) $$

so the quantities $$ P_{\xi} =\int_{t=const} d^3 x \sqrt{g} T^{0\mu}\xi_\mu $$ are independent of the time slice.

I find discussion of these issues in words to be largely usless. The words used are seldom precisely defined.

Answer 2

It's not an unreasonable picture, but the use of words is somewhat wrong.

In this case, we see a bunch of momentum in the opposite direction pop into existence out of seemingly nowhere, bringing with it corresponding amounts of kinetic energy. So we can't talk about conservation of energy and momentum anymore.

This is correct.

But we can still save a weaker principle, namely that if the energy density of a point increases, that energy must flow to that point from somewhere, so if energy density at a point increases over time, we need to have a commensurate momentum directed towards that point.

This is almost correct, except we need to have energy flow directed towards that point. Not momentum flow. Momentum flow may be there as well but is not directly related to energy density. Energy flow is.

The analogous principle for momentum is that to add more momentum to a system, we need stress/pressure, which describes the flow of momentum the same way momentum describes the flow of mass-energy.

Not necessarily, this would be very "mechanistic" view/analogy of looking at transport of momentum through space. That is, this view requires pushes and pulls impacted through some contact surface.

In general there are no such mechanical contact forces; there is empty space, and in this space there are fields: matter density field and EM field and possibly other fields, and we can define some auxiliary fields, such as spatial density of momentum. If momentum is increasing in some region of space, we assume there is a flow of momentum from the surrounding space to that region.

We can visualize this flow in the following way: in any small region of space there is a vector that says how much momentum is there, and there is also a tensor (matrix) that says, for any given oriented surface in that small region, how much momentum crosses the surface per unit time. This surface can be in vacuum, and then there is no material body and no forces there, but momentum may still be imagined to flow in that region.

For this momentum field and tensor field to exist, it is enough that some force field exists there, such as EM field, or gravity field or fictitious gravity field (due to the fact that the observer is accelerating in inertial system). Presence of this field then allows us to define the energy-momentum tensor of that field (sometimes misleadingly called stress-energy tensor, as if field in vacuum could be stressed as some kind of elastic material medium).

So in general everything balances and the amount of stress-energy-momentum flowing into a point in space time is the same as the amount flowing out even when observed from a non-inertial frame.

Not in general. There can be accumulation of momentum or loss of momentum in progress. We can only say that increase in momentum in some region is due to transport of this momentum through the surface that separates the region from the surrounding space.