Conserved Stress-Energy in Gravitational Theories

[gerald V]

My post https://www.physicsforums.com/threa...tional-field-wei-tou-ni-stress-energy.969033/ got no replies. So I formulated those statements on the stress-energy tensor (SET) which are of most interest to me. I would be very grateful if you either confirmed them or explained when and why they are wrong:

- In any theory - be it on gravitation or otherwise - where the action is an integral over space-time, there exists a locally covariantly conserved SET

- The SET can always be derived from the variation of the Lagrangian (including the metric determinant) w.r.t. the space-time metric

- General Relativity is a very distinguished theory of gravity. Namely, the covariantly conserved SET is those of the matter alone, while there is none for the gravitational field (except strange pseudo-tensors). Any other theory of gravitation would have the property that there is a nonvanishing SET of the gravitational field, and only the sum of the gravitational plus the matter SET is covariantly conserved.Thank you very much in advance.

 

[PeterDonis]

- In any theory - be it on gravitation or otherwise - where the action is an integral over space-time, there exists a locally covariantly conserved SET

How can you even know what the stress-energy tensor is if you're not talking about a theory of gravitation?

For example, consider Maxwell electrodynamics. It has an action which is an integral over spacetime of a Lagrangian density. But how would you derive a stress-energy tensor from it, considering only that Lagrangian density alone?

- The SET can always be derived from the variation of the Lagrangian (including the metric determinant) w.r.t. the space-time metric

Varying the Lagrangian with respect to the metric gives you the field equation for the theory, if it's a theory of gravitation (i.e., the Lagrangian has to include a term or terms that relates to the metric and its derivatives). Picking out which term in the field equation is the SET will, I think, depend on the theory (but I'm only really familiar with GR, where doing that is simple, as you note).

Any other theory of gravitation would have the property that there is a nonvanishing SET of the gravitational field, and only the sum of the gravitational plus the matter SET is covariantly conserved.

I don't know whether this is true or not.

 

[pervect]

I believe the Hilbert stress-energy tensor can be defined as a functional derivative of the matter Lagrangian

https://en.wikipedia.org/wiki/Stress–energy_tensor#Variant_definitions_of_stress–energy
Apparently there are a number of different concepts lumped into the general category of the "stress-energy tensor".

Wald has a bit about this too, but I'm hazy about the details, even after reviewing the appropriate section in section E about the Lagrangian (and Hamiltonian) formulations of GR.

 

[PeterDonis]

the Hilbert stress-energy tensor can be defined as a functional derivative of the matter Lagrangian

With respect to the metric. And the "matter Lagrangian" here is really $\sqrt{-g} \mathcal{L}_{\text{matter}}$, which includes the metric determinant. So this only really makes sense if we are talking about a theory that includes gravitation, so the complete Lagrangian includes both the metric and its derivatives, and "matter" (which really means "everything else").

Apparently there are a number of different concepts lumped into the general category of the "stress-energy tensor".

Yes, the Wikipedia article mentions three. The one that I think has the best claim to having an interpretation in a theory that doesn't include gravity is the canonical SET, i.e., the conserved current associated with spacetime translations.

 

[gerald V]

Thank you very much.

Your replies helped me to see my problem clearer. I have always struggled with the definitions of the SET. The corresponding conservation law is at the heart of physics, it cannot be subject to definition. Rather, all the definitions of the SET eventually must have the same meaning.

For the Einstein-Hilbert plus matter action, in union with the postulate that the metric itself is the dynamic field, the situation is absolutely transparent. That the Hilbert SET is in line with what you call the „canonical SET“, I interprete in the way that any reasonable theory of gravity has to be metric.

I looked up the passage of the Itzykson-Zuber where they explain internal symmetries, in particular U(1). From the invariance of the action one can immediately see that there is a conserved electromagnetic current. Then they say „We can, of course, directly verify this conservation using the equations of motion“. This again is completely transparent and the corresponding situation in gravitation theory are the Einstein equations, which because of the Bianci identies directly allow to verify the covariant conservation of the SET.

Here is the problem: Theories of gravity like those of Ni are still metric, however the metric is not the dynamic field itself. So the source is not the stress-energy directly, rather it has to be contracted with the derivative of the metric w.r.t.\ the dynamic field (a scalar in Ni’s theory). Hence, the equations of motion do not allow to directly verify the conservation of the SET.

My conjecture is that in such theories the SET nevertheless is conserved (covariantly?), but one has to verify this by hand, like in the case of classical electrodynamics. But I am not completely sure. I think the symmetries behind theories like Ni’s go beyond the freedom to parametrize space-time.

 

[pervect]

I usually don't use the Lagrangian approach, and I generally try to motivate the stress-energy tensor using a swarm-of-particles approach that I've seen used in some textbooks (MTW, I believe), and a few papers (that I'd have to look up).

The basic idea is that particle density can be characterized by the number-flux 4-vector - we see the number-flux 4-vector for the case of charge in the charge-current 4-vector. The number-flux 4-vector is the equivalent to the charge-current 4-vector, but it uses particle density rather than charge. The energy-momentum of a particle can be characterized by it's energy-momentum 4-vector, and the stress-energy tensor is just the tensor product of the energy-momentum 4-vector and the number-flux 4-vector. We divide the swarm-of-particles into groups that have the same energy-momentum 4-vector, find the appropriate densities of this subset of the swarm, multiply the density (the number-flux 4-vector) for this group of particles by its shared energy-momentum 4-vector, then sum over all possible values of energy-momentum. The result is a tensor, and we call it the stress-energy tensor.

The need for the stress-energy tensor arises as early as special relativity. It seems to get a lot of pushback from people who don't quite see why we need it. And, unfortunately, much of the early literature on the topic (why we need a stress-energy tensor) is written in German.

Going from the swarm-of-particles idea to a field may also be a bit of a leap. I suppose we can argue that a swarm-of-particles is an example of a field, and then argue that approaches that work for a swarm-of-particles work for at least this one example of a field, though we may need additional arguments to say that the approach always works for all fields.

However, I haven't found much written about the connection between swarms-of-particles and fields. And I suspect that the swarm-of-particles apprroach, though it's one I use and has been written about, might not actually be the best approach here. Though I do find the very "physical" nature of the model appealing.

I think the goal of the Lagrangian formulation is simply to assume that we have a Lagrangian density which is "independent of coordinates". Then we need to argue that the vanishing of the functional derivative of the Lagrangian density with respect to the metric is what we mean mathematically by the more physical goal of "being independent of coordinates". I don't recall reading any such argument, unfortunately, though I'd like to.

I do see people using the functional derivative as the defintion of the stress-energy tensor, for instance Wald, and I suspect one can make some argument that using this defintion winds up satisfying the end goal of making the formulation of physics "independent of coordinates". But I can't quote any such arguments, and I might be letting my intuition out on too long a leash here.

 

[PAllen]

The need for the stress-energy tensor arises as early as special relativity. It seems to get a lot of pushback from people who don't quite see why we need it. And, unfortunately, much of the early literature on the topic (why we need a stress-energy tensor) is written in German.

I want to emphasize this point. The GR SET can be seen as the obvious generalization of the SR SET. You need the SR SET if you want to consider continuous matter and fields.