Does potential energy curve spacetime?

[Leureka]

TL;DR Summary

Is potential energy included in the momentum energy tensor? My guess is that it is not, because there appears to be no way to uniquely define it.

Hi there,
I looked around on the net but I didn't quite find the answer to my question. I preface that I don't have training in GR, even though I know about the basics (like what tensors are, geodesics, a bit about topology and differential geometry...). So I wasn't sure if to put this question as High School level or Undergrad.

It's commonly said that the energy density which goes into the momentum-energy tensor accounts for all kinds of energy. Does it also include potential energy, which is a relative notion? I thought specifically about gravitational potential energy: this is usually defined by setting it as zero at infinity, hence it being negative at all points, but it appears to me that negative energy would create a repulsive gravitational field, so my hunch is that this is not defined like this in GR.

Another problem I have with potential energy in this context is this: imagine an infinite ladder. As I climb this ladder, my potential energy increases (it would all turn to kinetic energy if I decided to jump from the ladder). If I climbed high enough, I could accumulate enough potential energy to become a black hole. Again, this makes no sense, and is obviously an artifact of the fact that potential energy is a relative concept.

 

[phinds]

this

potential energy is a relative concept.

 

[Orodruin]

You cannot take the concept of classical Newtonian potential gravitational energy and apply it to general relativity. It does not exist as such as gravity is not a force.

As for the potential energy related to things like the electromagnetic field, such as the potential energy for some charge configuration, this is essentially covered by the energy terms of the electromagnetic field - which has a local stress energy tensor. This does act as a source in the Einstein field equations.

 

[Leureka]

You cannot take the concept of classical Newtonian potential gravitational energy and apply it to general relativity. It does not exist as such as gravity is not a force.

As for the potential energy related to things like the electromagnetic field, such as the potential energy for some charge configuration, this is essentially covered by the energy terms of the electromagnetic field - which has a local stress energy tensor. This does act as a source in the Einstein field equations.

So the position of charges is relevant, but those of masses aren't? What about other forms of potential energy, like elastic potential energy?

 

[Orodruin]

The positions of masses is highly relevant. They contribute to the stress energy tensor!

 

[Orodruin]

elastic potential energy?

Ultimately, yes. On a microscopic level that is nothing but electromagnetic fields.

 

[PeterDonis]

If I climbed high enough, I could accumulate enough potential energy to become a black hole.

No, you couldn't, for two reasons.

First, whether you are a black hole is not determined by your total energy, but by packing your total energy compactly enough that it can be enclosed by a 2-sphere whose area is the area of the horizon of a black hole with your total energy. Climbing up in a gravity well doesn't make you any more compact.

Second, climbing up in a gravity well means you're part of an overall system, not an isolated object, so the real question is whether that overall system is a black hole. But it can't be, because if it were, you wouldn't be able to climb. And your climbing does not change the total energy of that system at all; it just transfers some of it from the internal chemical energy in your body to your potential energy. So no amount of climbing can make the overall system a black hole.

 

[cianfa72]

As for the potential energy related to things like the electromagnetic field, such as the potential energy for some charge configuration, this is essentially covered by the energy terms of the electromagnetic field - which has a local stress energy tensor. This does act as a source in the Einstein field equations.

You said that the local stress energy tensor of electromagnetic field acts a source in EFE. My question is: may we reasonably assign a potential energy to the gravitational field itself (i.e. to spacetime geometry) ?

 

[PeterDonis]

You said that the local stress energy tensor of electromagnetic field acts a source in EFE.

Yes.

My question is: may we reasonably assign a potential energy to the gravitational field itself (i.e. to spacetime geometry) ?

If you are asking if the "gravitational field" acts as a source in the EFE, the answer is no.

 

[Vanadium 50]

two reasons.

The third of which is that you are carrying the energy you need to climb the hill up with you as you climb it. You actually have less (or the same, if perfectly efficient) energy than when you started.

 

[cianfa72]

If you are asking if the "gravitational field" acts as a source in the EFE, the answer is no.

No, my question is whether we may/makes sense assign potential energy to gravitational field (like potential energy assigned to electromagnetic field).

 

[Orodruin]

No, my question is whether we may/makes sense assign potential energy to gravitational field (like potential energy assigned to electromagnetic field).

No, the metric has no own contribution to the stress energy tensor. You need a non-zero matter Lagrangian to provide that.

 

[PeterDonis]

(like potential energy assigned to electromagnetic field).

Meaning, coming from a stress-energy tensor. That is where the "potential energy assigned to electromagnetic field" comes from. So you are asking whether that works for the "gravitational field". No, it doesn't, because there is no stress-energy tensor for the gravitational field. As has already been said.

 

[PAllen]

There are special situations where you can talk about gravitational potential energy curving spacetime in GR, in an unambiguous way.

Consider a massive ball, with a bunch of smaller balls held above it by telescoping supports. Compare the state where all the small balls are h above the big ball versus 2h above the big ball, all having been brought to identical temperature. The assembly will produce greater average curvature (effective gravitational mass) measured at large distance around the system for the 2h case than the 1h case.

This static situation is made precise in GR using Komar mass. The contribution of the small balls to the total mass will be greater when they are at lower gravitational redshift relative to infinity, which is sort of GR speak for potential energy in the special cases where it can be define.

 

[cianfa72]

There are special situations where you can talk about gravitational potential energy curving spacetime in GR, in an unambiguous way.

Sorry, I've not a clear understanding of the disconnect between the two energy terms we're talking about.

In post#13 @PeterDonis said there is no stress-energy tensor for gravitational field (i.e. for spacetime geometry).

Here you are claiming there is an unambiguously concept of gravitational potential energy at least in some cases (as far as I can tell Komar mass/energy applies to any stationary spacetimes -- it is basically the coniugate of the KVF timelike symmetry).

Which kind of potential energy is this ? Can we conceive it as distributed over the spacetime or it is just localized where the "source" masses are ?

 

[PAllen]

In post#13 @PeterDonis said there is no stress-energy tensor for gravitational field (i.e. for spacetime geometry).

This is true, there is no term in this tensor corresponding to gravitational field energy.

Here you are claiming there is an unambiguously concept of gravitational potential energy at least in some cases (as far as I can tell Komar mass/energy applies to any stationary spacetimes -- it is basically the coniugate of the KVF timelike symmetry).

Yes, gravitational potential energy becomes well defined for stationary spacetimes. This is basically because all the nonlinear effects in GR become stationary - there is no dynamics.

Which kind of potential energy is this ? Can we conceive it as distributed over the spacetime or it is just localized where the "source" masses are ?

It is not localized. Effectively, the contribution to total average curvature at a distance of some local stress energy becomes weighted by the gravitational potential difference between that location and infinity. It is only in this sense that one can talk about the potential energy contributing to curvature. And you can only talk about position based potential energy because of the timelike kvf.

 

[PeterDonis]

There are special situations where you can talk about gravitational potential energy curving spacetime in GR, in an unambiguous way.

I don't think "curving spacetime" is an appropriate description of this. See below.

Consider a massive ball, with a bunch of smaller balls held above it by telescoping supports. Compare the state where all the small balls are h above the big ball versus 2h above the big ball, all having been brought to identical temperature. The assembly will produce greater average curvature (effective gravitational mass) measured at large distance around the system for the 2h case than the 1h case.

The total mass (i.e., Komar mass) of the system will be greater in the 2h case vs. the 1h case. That reflects the fact that you would have to add energy to the 1h system to bring it to the 2h state.

However, I don't think the increased Komar mass means "greater average curvature". The increased Komar mass is spread over a larger volume. The average density of the system, and hence the average curvature, might well be lower.

The underlying issue here is that "gravitational potential energy", like any kind of "energy of the gravitational field", can't be localized. As we have agreed, there is no tensor corresponding to it. The total Komar mass of the system is not a local quantity, and neither is the "gravitational" contribution to it.

(Note that in the actual Komar mass integral, there is no separate "gravitational" term.)

 

[cianfa72]

The total mass (i.e., Komar mass) of the system will be greater in the 2h case vs. the 1h case. That reflects the fact that you would have to add energy to the 1h system to bring it to the 2h state.

Sorry for the basic question: where is "stored" that added energy (let me say that added "binding energy") ?

 

[PeterDonis]

Sorry for the basic question: where is "stored" that added energy (let me say that added "binding energy") ?

Nowhere. "Gravitational energy" can't be localized; there is no tensor corresponding to it, so you can't point to anywhere that it is "stored", because any such "storage" would mean there was a tensor corresponding to it.

 

[cianfa72]

Nowhere. "Gravitational energy" can't be localized; there is no tensor corresponding to it, so you can't point to anywhere that it is "stored", because any such "storage" would mean there was a tensor corresponding to it.

Ok, so what does actually mean that "gravitational potential energy" of post#16 (apart from that it is the coniugate of timelike KVF) ?

 

[PeterDonis]

what does actually mean that "gravitational potential energy" of post#16 (apart from that it is the coniugate of timelike KVF) ?

"Gravitational potential energy" is not the conjugate of the timelike KVF. The total Komar energy (or Komar mass) is the conjugate of the timelike KVF.

There is no way to separate out the "gravitational potential energy" part of the total Komar energy. That was my point when I said that there is no separate "gravitational term" in the Komar energy.

The integral that is done to obtain the Komar energy includes a factor for gravitational time dilation; that is what @PAllen meant when he talked about "weighting" in that integral.

 

[PAllen]

I don't think "curving spacetime" is an appropriate description of this. See below.

...

However, I don't think the increased Komar mass means "greater average curvature". The increased Komar mass is spread over a larger volume. The average density of the system, and hence the average curvature, might well be lower.

I disagree in that the curvature I am discussing is as averaged over a distant sphere around the system. Komar mass is exactly 'mass measured from afar' for the stationary system, and mass in this sense, is total curvature measured from afar.

 

[PeterDonis]

the curvature I am discussing is as averaged over a distant sphere around the system

The Komar mass can be computed as a surface integral over a distant sphere, but it's a surface integral of the covariant derivative of the timelike KVF, not an integral of curvature.

The ADM mass can also be computed as a surface integral over a distant sphere, but it's an integral of the derivatives of the induced metric of a spacelike hypersurface.

 

[PAllen]

The Komar mass can be computed as a surface integral over a distant sphere, but it's a surface integral of the covariant derivative of the timelike KVF, not an integral of curvature.

The ADM mass can also be computed as a surface integral over a distant sphere, but it's an integral of the derivatives of the induced metric of a spacelike hypersurface.

It is still a measure of curvature. The Weyl curvature at large distance from the system, normalized by the distance, measures its Komar mass. I am assuming the system is isolated in empty space.

 

[cianfa72]

The Komar mass can be computed as a surface integral over a distant sphere, but it's a surface integral of the covariant derivative of the timelike KVF, not an integral of curvature.

You mean the integral over a "distant" 2-sphere on a spacelike hypersurface of constant coordinate time associated/adapted to the timelike KVF ?

 

[PeterDonis]

The Weyl curvature at large distance from the system, normalized by the distance, measures its Komar mass.

No, the Komar mass, as I said, is a surface integral of the covariant derivative of the timelike KVF. If you convert it to a volume integral, it's an integral of Ricci curvature (or the trace reversed stress-energy tensor, by the EFE), not Weyl curvature. See Wald, Section 11.2.

I am assuming the system is isolated in empty space.

This is required for the Komar mass integral to converge, yes--the spacetime needs to be asymptotically flat.

 

[PeterDonis]

You mean the integral over a "distant" 2-sphere on a spacelike hypersurface of constant coordinate time associated/adapted to the timelike KVF ?

Yes.

 

[PAllen]

No, the Komar mass, as I said, is a surface integral of the covariant derivative of the timelike KVF. If you convert it to a volume integral, it's an integral of Ricci curvature (or the trace reversed stress-energy tensor, by the EFE), not Weyl curvature. See Wald, Section 11.2.

We don't seem to be communicating. Assuming a stationary system with vaccuum beyond some surface, the way you measure effective mass is to look at behavior of test bodies at a large distance. What determines the motion of these test bodies is Weyl curvature (since that's all there is in a vaccuum region). This is separate from the way you define Komar mass. The significance of Komar mass is precisely that these agree - accumulated stress/energy weighted by red shift factor matches mass determined by test bodies from afar. Thus, as I look at this, potential energy in a stationary system is contributing to distant Weyl curvature.

This is required for the Komar mass integral to converge, yes--the spacetime needs to be asymptotically flat.

Actually it doesn't. That is a requirement for ADM mass, not Komar mass. The latter only requires timelike KVF. However, to establish that what is computed makes sense to be regarded as mass, you do need vaccuum embedding as described above. It is possible for the integral to converge without asymptotic flatness (trivially, for a closed or otherwise finite spacetime).

 

[PeterDonis]

This is separate from the way you define Komar mass.

Ah, I see; by "effective mass" you mean something like "measure orbital parameters of a test body".

The significance of Komar mass is precisely that these agree - accumulated stress/energy weighted by red shift factor matches mass determined by test bodies from afar.

For an isolated stationary body, all of the masses--Komar, ADM, Bondi, and your "effective mass"--will agree, yes. But this is a very special case.

Thus, as I look at this, potential energy in a stationary system is contributing to distant Weyl curvature.

In the sense that "gravitational potential energy" makes a (negative) contribution to the total mass, yes.

That is a requirement for ADM mass, not Komar mass. The latter only requires timelike KVF.

For defining the Komar mass, yes, you only need a timelike KVF. But I said converge. For example, Godel spacetime is stationary but not asymptotically flat, and IIRC its Komar mass integral does not converge. That's the kind of example I was thinking of.

However, I have thought of one example where I think the Komar mass will converge for a stationary but not asymptotically flat spacetime: the Einstein static universe. So my original claim was too strong. I can't think of any other examples besides that one, though.