Alexander Vilenkin and energy of a closed universe

[Question69]

TL;DR

Why does Alexander Vilenkin(one of the founders of Inflation theory) say the energy of a closed universe equals 0? I thought some spacetimes don't have a well defined energy definition, and in this case a closed one wouldn't meet the well-defined criteria either.

For reference:

 

[ergospherical]

The energy of a spacetime is the value of its Hamiltonian, which is zero so long as the spacetime satisfies the constraint equations (in a closed universe it's permissible to neglect boundary terms which might otherwise appear in the Hamiltonian). Refer to Wald E.2 for details.

 

[Question69]

The energy of a spacetime is the value of its Hamiltonian, which is zero so long as the spacetime satisfies the constraint equations (in a closed universe it's permissible to neglect boundary terms which might otherwise appear in the Hamiltonian). Refer to Wald E.2 for details.

This would mean giving up concepts like positive or negative energy(which Vilenkin uses), and one can argue it's not physically meaningful.(like with the pseudotensors)

 

[ergospherical]

This would mean giving up concepts like positive or negative energy(which Vilenkin uses), and one can argue it's not physically meaningful.(like with the pseudotensors)

In what sense?

 

[Question69]

In what sense?

For pseudotensors in the sense that it is arbitrary where you define 0, and Relativity is perceived best under tensors.

 

[ergospherical]

I don't follow, apologies. What sources are you learning from?

 

[Question69]

I don't follow, apologies. What sources are you learning from?

Well if you inject pseudo-tensors into general relativity in the case of a non-stationary spacetime don't you end up with energy = 0 defined for some picked coordinate systems? At least that's how I've understood it. From a brief search on Wikipedia on how they are installed in GR.

 

[PeterDonis]

From a brief search on Wikipedia

Wikipedia is not a good source for this. The issue of pseudotensors in GR has been a topic of discussion in the peer-reviewed literature for decades. That literature is what you should be reading.

Similar remarks apply, actually, to the video you give in your OP. Vilenkin has published peer-reviewed papers on this topic (and AFAIK so has Tryon, whom Vilenkin refers to in the video). Those are what you should be using as a source.

 

[PeterDonis]

This would mean giving up concepts like positive or negative energy(which Vilenkin uses)

And those concepts as Vilenkin is using them are based on pseudotensors (at least that's the only way I've seen them presented in any literature on the topic), so if you think pseudotensors are not physically meaningful, neither are the concepts of positive and negative energy that Vilenkin is using.

 

[Question69]

And those concepts as Vilenkin is using them are based on pseudotensors (at least that's the only way I've seen them presented in any literature on the topic), so if you think pseudotensors are not physically meaningful, neither are the concepts of positive and negative energy that Vilenkin is using.

Right, but then why do they think this is physically meaningful?

 

[PeterDonis]

why do they think this is physically meaningful?

I couldn't say. That is what you would need to read their peer-reviewed papers to find out.

 

[Question69]

I couldn't say. That is what you would need to read their peer-reviewed papers to find out.

Right, thanks for the insight.Would you say the Hamiltonian formulation suffers from the same problem?

 

[PeterDonis]

Would you say the Hamiltonian formulation suffers from the same problem?

You mean not being physically meaningful? Well, the physical meaning of the Hamiltonian of any system that has a well-defined Hamiltonian at all is clear: it's the system's total energy. So the fact that the Hamiltonian of a closed universe is zero does have a clear physical interpretation, that the total energy of the closed universe is zero.

How to reconcile that obvious physical interpretation with the fact that a closed universe can contain a nonzero stress-energy tensor that, integrated over the closed universe, would appear to have a positive total energy, is a separate question that is discussed in the literature I referred to.

 

[Question69]

Right, but wouldn't this too depend on your set of preferred coordinates?

 

[PeterDonis]

wouldn't this too depend on your set of preferred coordinates?

Wouldn't what depend?

 

[Question69]

Wouldn't what depend?

The total energy.

 

[PeterDonis]

The total energy.

If you mean the Hamiltonian of the closed universe being zero, no, that is an invariant and does not depend on any choice of coordinates.

 

[PeterDonis]

The total energy.

If you mean the integral of the stress-energy tensor, yes, the exact numerical value might depend on your choice of coordinates (although in the simplest closed universe, the FRW one, the natural symmetry of the spacetime picks out a particular foliation by spacelike hypersurfaces that leads to the standard FRW coordinates), but it will be positive no matter what your choice of coordinates is (at least for stress-energy tensors that obey the usual energy conditions, which would have to be true for the universe to be closed to begin with).

 

[Question69]

If you mean the integral of the stress-energy tensor, yes, the exact numerical value might depend on your choice of coordinates (although in the simplest closed universe, the FRW one, the natural symmetry of the spacetime picks out a particular foliation by spacelike hypersurfaces that leads to the standard FRW coordinates), but it will be positive no matter what your choice of coordinates is (at least for stress-energy tensors that obey the usual energy conditions, which would have to be true for the universe to be closed to begin with).

Would it be positive and also increasing?

 

[PeterDonis]

Would it be positive and also increasing?

With the standard FRW foliation, I don't think so, I think it would be positive and constant, at least for the simplest closed matter-dominated model. But I have not done the math to check.

 

[Question69]

With the standard FRW foliation, I don't think so, I think it would be positive and constant, at least for the simplest closed matter-dominated model. But I have not done the math to

Oh alright.In the meanwhile I have checked Tyron's original quantum fluctuation paper but he doesn't seem to give motivation for the usages of the pseudotensors.

 

[PeterDonis]

I have checked Tyron's original quantum fluctuation paper

Can you give a reference?

 

[Question69]

Can you give a reference?

https://cosmo.nyu.edu/andrei/physicsII/ET.pdf

 

[PeterDonis]

https://cosmo.nyu.edu/andrei/physicsII/ET.pdf

Ok, thanks. The model proposed in this paper has some issues which, AFAIK, were not necessarily understood in 1973 when Tryon published the paper, but which are clear now, decades later (and AFAIK have been discussed in multiple papers during the intervening years):

The model requires our universe to have zero values for all conserved quantities. This would include a zero net particle number (i.e., total number of particles minus total number of antiparticles), since in Tryon's model all matter would have to be created by pair production. But our universe has a slight excess of particles over antiparticles. (This was not known in 1973 but is well accepted now.)

The concept of "gravitational potential energy" Tryon uses is only valid in a stationary spacetime. But the spacetime of our universe is not stationary (it's expanding).

One could in principle formulate an alternate definition of "gravitational potential energy" that would work in an asymptotically flat spacetime even if that spacetime was not stationary (for example, such a spacetime could describe the gravitational collapse of a massive star to a white dwarf, neutron star, or black hole). But the spacetime of our universe is not asymptotically flat either.

Tryon's model is based on a closed universe, but our actual universe does not appear to be closed. (Strictly speaking, a closed universe with an extremely large radius of curvature, much larger than the size of our observable universe, is not absolutely ruled out by our data, but it is highly unlikely.) For this discussion I assume we are ignoring this particular issue since the topic is closed universe models, but it is still worth mentioning.

he doesn't seem to give motivation for the usages of the pseudotensors.

From what I can see Tryon does not mention pseudotensors at all, nor does he give any detailed mathematical models. He only uses a few heuristic equations.

 

[PeterDonis]

The model proposed in this paper has some issues

One additional issue: we now do know of possible mechanisms by which a "bounce" could occur, which were not known in 1973. At least one such mechanism involves properties of the quantum vacuum that are better understood now than they were in 1973. So Tryon's rationale given in the paper for ruling out bounce models no longer holds.

 

[Question69]

Ok, thanks. The model proposed in this paper has some issues which, AFAIK, were not necessarily understood in 1973 when Tryon published the paper, but which are clear now, decades later (and AFAIK have been discussed in multiple papers during the intervening years):

The model requires our universe to have zero values for all conserved quantities. This would include a zero net particle number (i.e., total number of particles minus total number of antiparticles), since in Tryon's model all matter would have to be created by pair production. But our universe has a slight excess of particles over antiparticles. (This was not known in 1973 but is well accepted now.)

The concept of "gravitational potential energy" Tryon uses is only valid in a stationary spacetime. But the spacetime of our universe is not stationary (it's expanding).

One could in principle formulate an alternate definition of "gravitational potential energy" that would work in an asymptotically flat spacetime even if that spacetime was not stationary (for example, such a spacetime could describe the gravitational collapse of a massive star to a white dwarf, neutron star, or black hole). But the spacetime of our universe is not asymptotically flat either.

Tryon's model is based on a closed universe, but our actual universe does not appear to be closed. (Strictly speaking, a closed universe with an extremely large radius of curvature, much larger than the size of our observable universe, is not absolutely ruled out by our data, but it is highly unlikely.) For this discussion I assume we are ignoring this particular issue since the topic is closed universe models, but it is still worth mentioning.From what I can see Tryon does not mention pseudotensors at all, nor does he give any detailed mathematical models. He only uses a few heuristic equations.

I apologize for the misuse of language.

However this seems pretty vexing to me, what should we do? Does this conundrum in any way affect the underlying physics of anything? For example in Inflation theory Guth makes use of this "cancelling" between energies too when it comes to the inflaton field expanding.Could we just say the inflaton field violates energy conservation and get away with it?

 

[PeterDonis]

this seems pretty vexing to me, what should we do?

About what? Why do we need to "do" anything? We're talking about proposed physics models that will eventually either be confirmed or falsified by evidence. (Tryon's original model has arguably been falsified by now, for the reasons I gave.) Until that time they are just proposed models. What is there to "do"?

Does this conundrum in any way affect the underlying physics of anything?

I'm not sure what you mean. It doesn't affect the way the physics of anything you deal with in your everyday life works. Water is still wet, you can still breathe air and eat food, gravity still holds you and other things on the Earth.

in Inflation theory Guth makes use of this "cancelling" between energies too when it comes to the inflaton field expanding.

Please give a specific reference. Note that the video you link to in the OP of this thread is not a valid reference; it's not a peer-reviewed paper. I'm sure Guth says similar things in videos too, or in other informal contexts. But in his peer-reviewed papers he needs to be more careful.

That said, as I've already commented, discussion of pseudotensors and whether they are physically meaningful goes back decades in the literature and the issue remains unresolved to this day. One side of the debate (the side I personally favor) says that only actual tensors, geometric objects that don't depend on your choice of coordinates, are physically meaningful. The other side says that there are particular cases where pseudotensors or other coordinate-dependent things can be physically meaningful, if the coordinate choice they depend on is "natural" in some way (but then they have to say what "natural" means, and not all of them agree on that). Vilenkin and Guth appear to be on the latter side of the debate. But at the end of the day, it's a debate about theoretical models, not actual physical observables.

 

[PeterDonis]

Could we just say the inflaton field violates energy conservation and get away with it?

There is no global energy conservation in GR to begin with, so there is no law to be violated. None of the proposals violate local energy conservation (the covariant divergence of the stress-energy tensor being zero), which is the only energy conservation law in GR.

 

[Question69]

About what? Why do we need to "do" anything? We're talking about proposed physics models that will eventually either be confirmed or falsified by evidence. (Tryon's original model has arguably been falsified by now, for the reasons I gave.) Until that time they are just proposed models. What is there to "do"?I'm not sure what you mean. It doesn't affect the way the physics of anything you deal with in your everyday life works. Water is still wet, you can still breathe air and eat food, gravity still holds you and other things on the Earth.Please give a specific reference. Note that the video you link to in the OP of this thread is not a valid reference; it's not a peer-reviewed paper. I'm sure Guth says similar things in videos too, or in other informal contexts. But in his peer-reviewed papers he needs to be more careful.

That said, as I've already commented, discussion of pseudotensors and whether they are physically meaningful goes back decades in the literature and the issue remains unresolved to this day. One side of the debate (the side I personally favor) says that only actual tensors, geometric objects that don't depend on your choice of coordinates, are physically meaningful. The other side says that there are particular cases where pseudotensors or other coordinate-dependent things can be physically meaningful, if the coordinate choice they depend on is "natural" in some way (but then they have to say what "natural" means, and not all of them agree on that). Vilenkin and Guth appear to be on the latter side of the debate. But at the end of the day, it's a debate about theoretical models, not actual physical observables.

I should have been more specific, by "do" I mean: Should we just admit there is no meaning in using pseudotensors? I also believe it makes no sense.

 

[PeterDonis]

Should we just admit there is no meaning in using pseudotensors?

If you can convince all the people who have written papers claiming that pseudotensors, or at least some of them, do have physical meaning, sure. As I noted before, two of the people you would need to convince are Vilenkin and Guth.