Internal energy of a comoving volume increasing as space expands?

[Suekdccia]

TL;DR Summary

Internal energy of a comoving volume increasing as space expands?

I was reading an article by Edward Harrison, which tackles the problems of conservation of energy at cosmological scales.

At some part (point 2.4) he cites several article, including one by Rees and Gott, which he says indicates that the internal energy of a comoving volume (e.g. a cosmic string) increases as the universe expands. However, since cosmic strings are hypothetical objects, I'm not sure if this conclusion is true. Could thus be possible for any kind of known comoving volume (I mean, does this apply to any kind of structure or volume that has been observed or experimentally verified)?

Link to the article: https://adsabs.harvard.edu/full/1995ApJ...446...63H

 

[PeterDonis]

which he says indicates that the internal energy of a comoving volume (e.g. a cosmic string) increases as the universe expands.

Given the fact that he says this hypothetical network of strings "has the same effect as the cosmological term or as a negative-pressure fluid", the "internal energy" he is referring to is functionally equivalent to dark energy. Since the energy density of dark energy is constant, as the universe expands you can view the total energy due to dark energy in some comoving volume (i.e., the dark energy times the spatial volume of some set of comoving worldlines at some instant of cosmological time) as being an "internal energy" of that comoving volume, which will then increase with time due to expansion (since the spatial volume increases and the energy density is constant).

His working out of the dynamics bears this out, since what he obtains are well known equations for a universe dominated by dark energy. None of this is a mystery or is controversial.

Nor is it a mystery or controversial to say that energy is not conserved in an expanding universe. Sean Carroll has a blog post often referenced here at PF that says the same thing and explains what that statement corresponds to in the math. Another way of saying the same thing is that, except in some special classes of spacetimes (asymptotically flat spacetimes and stationary spacetimes), there is no well-defined "total energy" in any volume; more precisely, there is no invariant that corresponds to any such thing. The "total energy" defined in this paper, which I described above, is coordinate-dependent, not an invariant; the "spatial volume" is the volume in comoving coordinates.

 

[PeterDonis]

since cosmic strings are hypothetical objects, I'm not sure if this conclusion is true

It's true for dark energy, and I'm not sure what the point is of introducing a network of hypothetical strings, only to then turn around and say they're functionally the same as dark energy. Why not just use dark energy in the first place? Its properties are well known.

 

[PeterDonis]

Could thus be possible for any kind of known comoving volume (I mean, does this apply to any kind of structure or volume that has been observed or experimentally verified)?

If you are asking about the "tethered body" thought experiment described in the paper, I'm not sure if the paper's analysis is correct. That doesn't mean it is not possible to extract energy in this way in a dark energy dominated universe; it is. I'm just not sure that the detailed calculation given in the paper of how much energy can be extracted is correct. I need to take more time to look at it.

 

[PAllen]

Note that a string attached to comoving bodies can extract energy independent of dark energy. Even without curvature (let alone dark energy), you would extract energy due to relative motion imparted by initial conditions (e.g. the Milne universe). IMO, the more interesting experiment is to posit a string between two bodies such that there is initially zero tension. These bodies could not both be comoving. In the presence of dark energy, you would still extract energy, but the formula would be quite different from that in the linked paper.

 

[Bandersnatch]

Note that a string attached to comoving bodies can extract energy independent of dark energy.

The paper does distinguish between the accelerating/non-accelerating cases.

 

[Suekdccia]

Given the fact that he says this hypothetical network of strings "has the same effect as the cosmological term or as a negative-pressure fluid", the "internal energy" he is referring to is functionally equivalent to dark energy. Since the energy density of dark energy is constant, as the universe expands you can view the total energy due to dark energy in some comoving volume (i.e., the dark energy times the spatial volume of some set of comoving worldlines at some instant of cosmological time) as being an "internal energy" of that comoving volume, which will then increase with time due to expansion (since the spatial volume increases and the energy density is constant).

His working out of the dynamics bears this out, since what he obtains are well known equations for a universe dominated by dark energy. None of this is a mystery or is controversial.

Nor is it a mystery or controversial to say that energy is not conserved in an expanding universe. Sean Carroll has a blog post often referenced here at PF that says the same thing and explains what that statement corresponds to in the math. Another way of saying the same thing is that, except in some special classes of spacetimes (asymptotically flat spacetimes and stationary spacetimes), there is no well-defined "total energy" in any volume; more precisely, there is no invariant that corresponds to any such thing. The "total energy" defined in this paper, which I described above, is coordinate-dependent, not an invariant; the "spatial volume" is the volume in comoving coordinates.

Thank you for your reply, although I have a question:

Even if the increase in internal energy of the system would functionally correspond to that of dark energy, would it manifest into some kind of "conventional" energy? I mean, dark energy is still hypothetical (the universe is accelerating, but we don't really know what is causing it) as far as I know, so, couldn't in any case the internal energy of a comoving volume increase and manifest in an increase of a form of energy that we certainly know it exists?

 

[PeterDonis]

Even if the increase in internal energy of the system would functionally correspond to that of dark energy, would it manifest into some kind of "conventional" energy?

That's sort of what "energy mining" in the paper, in scenarios like the tethered object thought experiment, refers to: extracting conventional energy (the kind we can use to do some sort of useful work, like running a motor) from the tether system as the universe expands.

I say "sort of" because, at least in the dark energy case, the dark energy density does not change (at least, that seems to be the assumption in the paper), so the conventional energy one is "mining" is not being converted from dark energy.

couldn't in any case the internal energy of a comoving volume increase and manifest in an increase of a form of energy that we certainly know it exists?

No. This "internal energy of a comoving volume" just is the dark energy. Unless we figure out what dark energy is and how to detect it, independently of some kind of "mining" process, we have no way of directly knowing it exists; we only know something is there that acts like dark energy by observing accelerating expansion of the universe.

 

[phinds]

I mean, dark energy is still hypothetical (the universe is accelerating, but we don't really know what is causing it)

Saying "dark energy is hypothetical" is wrong. SOMETHING is causing the universal expansion to accelerate. That is not hypothetical, it's known and since we don't know what it is, we've decided to call it "dark energy".

 

[Frabjous]

Saying "dark energy is hypothetical" is wrong. SOMETHING is causing the universal expansion to accelerate. That is not hypothetical, it's known and since we don't know what it is, we've decided to call it "dark energy".

I believe that MOND is an alternative hypothesis, which requires a hypothesis.

 

[PeterDonis]

I believe that MOND is an alternative hypothesis

MOND is not an alternative hypothesis to dark energy. It is an alternative hypothesis to dark matter. They're not the same.

 

[phinds]

I believe that MOND is an alternative hypothesis, which requires a hypothesis.

Good point.

 

[Vanadium 50]

SOMETHING is causing the universal expansion to accelerate.

Sure, but it doesn't have to be Dark Energy. GR could be wrong - put the DE on the other side of the equation. Call it "natural curvature" or something.

 

[PeterDonis]

GR could be wrong - put the DE on the other side of the equation.

That doesn't make GR wrong. The EFE is equally valid with the $\Lambda$ term on either side.

 

[Vanadium 50]

It makes the rquation right, but our interpretation of it wrong./ Conventionally, we have a mysterious energy causing curvature. If you move the equation around, there is no dark energy but stress-energy isn't the only thing causing spacetime to curve: there's also this intrinsic curvature uncaused by stress-energy.

 

[PeterDonis]

It makes the rquation right, but our interpretation of it wrong./ Conventionally, we have a mysterious energy causing curvature. If you move the equation around, there is no dark energy but stress-energy isn't the only thing causing spacetime to curve: there's also this intrinsic curvature uncaused by stress-energy.

Both of these interpretations are found in the literature. The first (the "dark energy" one) is the one that seems to get more pop science mentions nowadays. But the second (the "cosmological constant as intrinsic curvature of spacetime" one) is the one that was used more often in the first couple of decades after GR was discovered.

Also, the difference between these two interpretations is fuzzy. Suppose that the reason for the lambda term in the EFE turns out to be quantum zero point energy. Is this "mysterious energy causing curvature" or is it "intrinsic curvature of spacetime"? After all, if there's a vacuum, the stress-energy tensor is zero. So the zero point energy isn't a property of any "stuff" that's there, it's an intrinsic property of spacetime itself.

Most physicists today would probably favor the first of the two interpretations of quantum zero point energy, but the real message of the math--the fact that the lambda term can be put on either side of the EFE and the equation is still valid (not just mathematically valid but physically valid, the covariant divergence of both sides is still zero so physical conservation of stress-energy is still valid)--is that neither interpretation is "right" or "wrong"; both are valid interpretations and neither one can be "proved" or "disproved" by the theory.

 

[Vanadium 50]

If you really want to be perverse, you can split the difference. There's half as much dark energy as we think, and the remainder is intrinsic curvature. I can't see how this could possibly be right, but don't think there's anything prohibiting it either.

 

[PeterDonis]

There's half as much dark energy as we think, and the remainder is intrinsic curvature. I can't see how this could possibly be right, but don't think there's anything prohibiting it either.

Mathematically, no, there isn't, you can have $\Lambda$ terms on both sides with arbitrary coefficients.

Physically, it would be nice to have some kind of rationale for the terms on either side, but we currently don't have a good one for either, at least not if that requires a derivation from some kind of more fundamental principles.

 

[Haelfix]

If you really want to be perverse, you can split the difference. There's half as much dark energy as we think, and the remainder is intrinsic curvature. I can't see how this could possibly be right, but don't think there's anything prohibiting it either.

Or 99% classical curvature and 1% 'quantum' matter for emphasis with how things normally behave..

Note that before 1998, I would say the majority of physicists thought something like that was the case. The idea being that there was some mechanism that would zero the entire poorly defined <Tuv> object, leaving just some relic classical piece and which would get rid of the headache about radiative instability and so forth..

We even have theoretically minor modifications to gravity that could accomplish such a thing (Unimodular gravity) without too much hassle.

It doesn't quite get you out of the woods, b/c you still have to explain why that number is so much smaller than you would expect by Dirac naturalness and why it happens to coincide with matter domination in the FRW picture (cosmic coincidence), but at least its just one number and not some deep problem at the heart of our best models of the world.

But then 1998 came, and the evidence for inflation started mounting. If you have some static symmetry zero <Tuv> you kill the primary mechanism by which that should works. Worse, you would need new dynamics on the 'intrinsic curvature' side of your equation to achieve the same sort of success.

 

[PeterDonis]

Unimodular gravity

Do you have a reference for this?

then 1998 came, and the evidence for inflation started mounting

Do you mean inflation or the current accelerated expansion of the universe? They're not the same thing.

 

[Haelfix]

Do you have a reference for this?

Unimodular gravity is basically GR with the additional constraint that the determinant of the metric is enforced to be 1. I am not sure if there is a canonical reference of the theory perse, but it is often discussed in reviews on the cosmological constant problem. For instance Weinberg discusses it in his original review paper, and for a (skeptical) take see (eg p8 of arXiv:1502.05296).

Do you mean inflation or the current accelerated expansion of the universe? They're not the same thing.

Both actually. The "old" cosmological constant problem was why the CC was exactly zero (which was ruled out in 98 after the measurements of the type IIa supernova). However right around that time inflation was becoming more and more accepted. The pre-existing but strengthening measurements from COBE as well as the supernova results convinced a lot of my colleagues.. Inflation had passed a nontrivial prediction.