I If energy is relative, is the rest mass also relative?

[Orodruin]

The SET in SR always obeys the vanishing divergence condition you give, at every point of spacetime.

He did mention "for a closed system". While the total SET in SR vanishes, it is not necessarily true for a particular system as it may not be everything that contributes to the total SET. What holds is that $\sum_i \partial_\mu T_i^{\mu\nu} = 0$ where the sum is over all contributions to the SET. In some sense, this is the SR equivalent of Newton's third law. If the divergence of one system (say the EM field) is non-zero, then this must be compensated by the negative of that divergence for another system (say the field of charged matter). This is then essentially summarised as the Lorentz force law.

Edit: In fact, for suitable choices of constants and metric conventions, one finds that
$$
\partial_\mu M^{\mu\nu} = - F^{\nu\mu} J_{\mu},
$$
where $M^{\mu\nu}$ is the SET of the electromagnetic field, $F^{\mu\nu}$ is the EM field tensor, and $J^\mu$ the EM 4-current density. The RHS here is not identically equal to zero. In fact, it is just the negative of the 4-force acting on the 4-current according to the Lorentz force law - just as it should be.

 

[PeterDonis]

He did mention "for a closed system".

But he didn't say what he meant by that. The usual interpretation of "closed system" would be "a region of spacetime such that no stress-energy crosses its boundary". But no such condition is necessary for the vanishing divergence condition to hold.

While the total SET in SR vanishes, it is not necessarily true for a particular system as it may not be everything that contributes to the total SET.

I see what you mean--for example, we might only look at the SET due to matter and not that due to the EM field, and if both are present and the matter is charged, neither SET would obey the divergence condition by itself, only the sum of them would. (You describe this further on in your post.)

However, I don't think the term "closed system" is a good way to say "we're counting all of the SET at this point in spacetime, not just a piece of it". One should always do that anyway; doing the opposite is not "an open system", it's just wrong.

 

[Orodruin]

But he didn't say what he meant by that. The usual interpretation of "closed system" would be "a region of spacetime such that no stress-energy crosses its boundary". But no such condition is necessary for the vanishing divergence condition to hold.I see what you mean--for example, we might only look at the SET due to matter and not that due to the EM field, and if both are present and the matter is charged, neither SET would obey the divergence condition by itself, only the sum of them would. (You describe this further on in your post.)

However, I don't think the term "closed system" is a good way to say "we're counting all of the SET at this point in spacetime, not just a piece of it". One should always do that anyway; doing the opposite is not "an open system", it's just wrong.

I guess I have a more "liberal" view of what a "system" refers to where what you decide to include cannot only be spatially and temporally separated, but also separated by what it is. At the very least I find it useful to talk about things such as the force on a charge exerted by the electromagnetic field, which is in essence drawing a boundary between the two - one system exerting a force on the other and the other exerting an equal but opposite force on the first. We do this in non-relativistic electrostatics too, where we consider the force on an object exerted by the electric field. We do not include the energy of the electric field in the region occupied by the object. A closed system would just be a system not subject to any external forces, whether forces acting across a spatial boundary or within the volume.

 

[Dale]

what you decide to include cannot only be spatially and temporally separated, but also separated by what it is.

I think that is often done. However, if you do that you have to be willing to accept the fact that there may not be a unique separation based on what it is, and that the conservation laws may not apply individually to the separated components.

A good example of this is the famous Abraham Minkowski controversy about the momentum of light in a transparent medium. Abraham and Minkowski disagreed about how to separate the field from the matter. As a result they obtained different expressions for the momentum of the light. But the total momentum is what is conserved and agreed for both approaches.

 

[Orodruin]

I think that is often done. However, if you do that you have to be willing to accept the fact that there may not be a unique separation based on what it is, and that the conservation laws may not apply individually to the separated components.

A good example of this is the famous Abraham Minkowski controversy about the momentum of light in a transparent medium. Abraham and Minkowski disagreed about how to separate the field from the matter. As a result they obtained different expressions for the momentum of the light. But the total momentum is what is conserved and agreed for both approaches.

Sure, I am fine with this. Momentum in classical mechanics also depends on how I choose to define my system. The question is if you can really define anything that is really a closed system. In some sense this will never be the case unless you include everything, at least not in SR as, if you do have two or more completely closed systems, they would just as well be describable on their own.

 

[vanhees71]

I'm not sure what you mean by "for a closed system". The SET in SR always obeys the vanishing divergence condition you give, at every point of spacetime. In GR that condition becomes $\nabla_{\mu} T^{\mu \nu}=0$, and is again always true. Since it applies at each point of spacetime, there is no such thing as a "closed system" vs. "open system" if you're just looking at the SET; it's just a continuous distribution of stress-energy.

For a closed system yes. There's a lot of confusion in the literature by considering only the energy-momentum tensor of the em. field and then wondering, why there is problem with total em. energy and momentum at presence of sources $\rho$ and $\vec{j}$. You have to consider always the total EM tensor of a closed system, obeying the continuity equation, and only then the spatial integral gives an energy-momentum four-vector (for the entire system consisting of the em. field and the charges).

In relativity these conservation laws follow from the presence of Killing vector fields and are different from the divergence condition on the SET.

I didn't want to discuss the GR case here, where it is of course even more complicated. You only have local energy conservation $\nabla_{\mu} T^{\mu \nu}=0$, but you cannot so easily define total energy and momentum in a coordinate independent way. I guess you know this better than I.

 

[PeterDonis]

my own paper

Which is not a valid reference. Personal research is off limits for discussion at PF. (And no, posting it on vixra doesn't mean it isn't personal research. If you get it published in an actual peer-reviewed journal, then we might take a look.)

I'm sure other ones are possible.

I'm afraid no one else shares your belief on this point. Please do not post further about this in this thread; if you do, you will receive a warning and a thread ban.