Zero-Energy Universe: Stephen Hawking Explains

[Mircea Golumba]

Let me quote Stephen Hawking who sadly just passed away, in his book A Brief History of Time:

The total energy of the universe is exactly zero. The matter in the universe is made out of positive energy. However, the matter is all attracting itself by gravity.

Two pieces of matter that are close to each other have less energy than the same two pieces a long way apart, because you have to expend energy to separate them against the gravitational force that is pulling them together.

Thus, in a sense, the gravitational field has negative energy. In the case of a universe that is approximately uniform in space, one can show that this negative gravitational energy exactly cancels the positive energy represented by the matter. So the total energy of the universe is zero.​

This is surely something you've heard about. I understand the concept of negative gravitational potential energy. What I don't know is how one can show it precisely compensate the matter energy, given by mc² in a relatively uniformly filled Universe.

[Moderator's note: corrected the spelling of Stephen Hawking's name.]

 

[rkolter]

Remember that the zero-energy universe was suggested when the universe was believed to be flat, we didn't think the universe's expansion could be accelerating, etcetera. There are specific situations in which the zero-energy universe may apply and we may or may not be in such a situation.. Having said that I'll take a stab at an answer:

If the universe started as a quantum fluctuation then the net energy of the universe is zero. Inflation caused the universe to grow instead of cancel itself out. You cannot change the net energy of the universe - but can add equal amounts of positive and negative energy to the universe because that won't change the net total energy of the universe. So it is inevitable in a zero-energy universe, that the amount of negative energy exactly and precisely equals the amount of positive energy.

 

[kimbyd]

Remember that the zero-energy universe was suggested when the universe was believed to be flat, we didn't think the universe's expansion could be accelerating, etcetera.

That isn't really accurate. The zero-energy universe only applies to a closed universe, not a flat or open universe. It's based upon the Hamiltonian formulation of General Relativity, which contains a sort of potential energy (in the typical formulation of GR, there is no potential energy). This calculation works just fine in the context of an accelerated expansion.

 

[Mircea Golumba]

On a sidenote, is it OK to assert that an Universe in accelerated expansion, as it was proved to be the case, is still flat?

 

[Ibix]

On a sidenote, is it OK to assert that an Universe in accelerated expansion, as it was proved to be the case, is still flat?

The two things aren't connected in general, as far as we are aware. We can model a universe that is spatially flat, closed, or open and has dark energy. And we can model one that is spatially flat, closed, or open and does not have dark energy.

The real universe appears to be spatially flat as near as we can measure, and with dark energy.

 

[Mircea Golumba]

Thank you all. Just found this on another Q/A board:

"There are a number of motivations for saying the total mass-energy of the universe is zero. One of them is the Hamiltonian constraint for ADM general relativity that has N H = 0, for N the lapse function." source

This seem to resonate with kimbyd's reply. I understand I must keep digging yet GR is unfortunately a little out of my league.

 

[Cerenkov]

I hope it's ok to reawaken an old(ish) thread like this.

It's the apparent difference between kimbyd's reply and what this Wiki page says that's of interest to me. Kim says that Hamiltonian GR only applies to a closed universe and not to a flat or open one. Whereas, Wiki says that a zero energy universe must be a flat one.

https://en.wikipedia.org/wiki/Zero-energy_universe
"Due to quantum uncertainty, energy fluctuations such as an electron and its anti-particle, a positron, can arise spontaneously out of vacuum space, but must disappear rapidly. The lower the energy of the bubble, the longer the duration it can exist. A gravitational field has negative energy. Matter has positive energy. The two values cancel out provided the universe is completely flat. In that case, the universe has zero energy and can theoretically last forever.[4][8]"

Edward Tryon and John Gribbin are cited.

Berkeley Lab, Smoot Group - http://aether.lbl.gov - Inflation for Beginners, JOHN GRIBBIN "Quantum uncertainty allows the temporary creation of bubbles of energy, or pairs of particles (such as electron-positron pairs) out of nothing, provided that they disappear in a short time. The less energy is involved, the longer the bubble can exist. Curiously, the energy in a gravitational field is negative, while the energy locked up in matter is positive. If the universe is exactly flat, then as Tryon pointed out the two numbers cancel out, and the overall energy of the universe is precisely zero. In that case, the quantum rules allow it to last forever." archived, 2014

So, given the following...

1. My understanding of these matters is at the pop science level.
2. That I don't know what the Hamiltonian reformulation of GR is.
3. That I won't be able to follow any reply unless it is couched in the most basic way.

...is it possible that a kind member of PF can help me understand the apparent difference between kimbyd and Wikipedia?

Please note that I do understand and appreciate that I may be asking the wrong question here. Any help in suggesting a better question would be very much appreciated. Thank you.

Cerenkov.

p.s.
In case any of the relevant parties missed it, I posted this... AN APOLOGY: A COOLER HEAD FINALLY PREVAILS ...here on the 21st June.

https://www.physicsforums.com/threads/is-there-any-way-forward-for-me.949787/page-2

 

[PeterDonis]

is it possible that a kind member of PF can help me understand the apparent difference between kimbyd and Wikipedia?

Yes, they are using the word "energy" to refer to different properties of different models.

In the model @kimbyd was referring to, the "energy" is given by the GR Hamiltonian (more precisely the Hamiltonian constraint in the ADM formulation of GR), which only has a well-defined value for a closed universe; and for any closed universe, that value is zero. This can be viewed as negative "gravitational potential energy" just cancelling out positive "energy contained in matter and radiation".

In the model Edward Tryon was describing, as I understand it, the "energy" is given by the average energy contained in flat Minkowski spacetime, which, as I understand it, is the "background" spacetime that was assumed in his model of quantum fluctuations creating particle-antiparticle pairs out of nothing. The average energy contained in Minkowski spacetime is by definition zero. (I'm not clear, however, how Tryon reconciled this with the presence of "gravitational potential energy", since there is none in flat Minkowski spacetime. I have not been able to find a copy of his original paper online.)

 

[Chronos]

Berman may have best summed up the case for the zero energy universe concept as discussed here; https://arxiv.org/abs/gr-qc/0605063v3, On the Zero-energy Universe.

 

[Cerenkov]

Thanks very much for your help, guys.

PeterDonis,

Yes, now that I read your reply I recall Guth writing that Tryon's conjecture involved quantum fluctuations in some preexisting continuum. Presumably the flat Minkowski 'background' spacetime?

Being used to only pop science books and websites like this one... https://map.gsfc.nasa.gov/media/990006/index.html ...I've only known about these three Friedmann solutions (Open, Flat and Closed) to GR. Discovering that there are (many?) more models of the universe is quite an eye-opener. So is there anything like a classification system of these models that I can pore over? Or something that visually displays their general characteristics and/or properties?
This diagram of the Standard Model of particle physics does a fine job in that regard. Thank you.

Chronos,

I've bookmarked your link and will be taking a look there soon.

Thank you both,

Cerenkov.

 

[PeterDonis]

I've only known about these three Friedmann solutions (Open, Flat and Closed) to GR.

Those three categories are fine as far as describing the possible spatial geometries. However, there is a second way in which universes meeting the general FRW conditions (homogeneity and isotropy) can vary: they can have a cosmological constant that is zero, negative, or positive.

Prior to about the 1990s, cosmology textbooks usually treated only the case of zero cosmological constant, since that's the simplest one and since up until the 1990s there was no evidence of any nonzero cosmological constant so the possibility was basically ignored. With a zero cosmological constant, the three spatial geometries match up nicely with the possibilities for how the universe will evolve in the future:

Open -> expands forever, expansion rate never goes to zero

Flat -> expands forever but expansion rate goes to zero in the limit as t -> infinity

Closed -> recollapses to a Big Crunch.

However, in the 1990s cosmologists found that the expansion of the universe had been accelerating for the past few billion years, indicating that the actual cosmological constant is not zero, but has a small positive value. With a small positive cosmological constant, it becomes possible for a closed universe to expand forever instead of recollapsing, and the expansion rate never goes to zero as t -> infinity (it approaches a certain positive value that depends on the cosmological constant). So basically the future evolution of the universe looks like the "open" case above (though with some details different), regardless of the spatial geometry.

 

[PeterDonis]

Berman may have best summed up the case for the zero energy universe concept as discussed here; https://arxiv.org/abs/gr-qc/0605063v3, On the Zero-energy Universe.

Unfortunately, I think this paper is more hand-waving than rigorous or even semi-rigorous argument. It basically picks particular coordinates and particular coordinate-dependent quantities (such as pseudo-tensors) that make the answer come out the way the author wants it to, but gives no argument for why those particular choices are the "right" ones (much less why we should be looking at coordinate-dependent quantities in the first place, when GR is supposed to be a theory that describes all of the actual physics with invariants, quantities that are independent of your choice of coordinates).

 

[Cerenkov]

Thanks Peter,

That's very helpful and you've conveyed it to me succinctly.
Btw, this link will take you to what I believe is Tryon's original paper concerning a zero energy universe emerging from a flat Minkowski spacetime. You'll need to subscribe and then read it to verify that for yourself though.

https://www.nature.com/articles/246396a0

Thank you,

Cerenkov.

 

[kimbyd]

Unfortunately, I think this paper is more hand-waving than rigorous or even semi-rigorous argument. It basically picks particular coordinates and particular coordinate-dependent quantities (such as pseudo-tensors) that make the answer come out the way the author wants it to, but gives no argument for why those particular choices are the "right" ones (much less why we should be looking at coordinate-dependent quantities in the first place, when GR is supposed to be a theory that describes all of the actual physics with invariants, quantities that are independent of your choice of coordinates).

Right, but that's the only way to introduce the concept of a potential energy into General Relativity. Because the energy within a specific volume in General Relativity is a coordinate-dependent quantity, the potential also has to be. I don't see it as being inherently worse than the Friedmann equations in terms of its applicability.

But it's very true that any coordinate-specific conclusion within General Relativity needs to be taken with a huge grain of salt: it's frequently the case in GR for features to appear in specific coordinate systems which don't have any physical basis. For instance, if you use latitude/longitude coordinates while describing the curvature of the Earth, and try to calculate the Earth's curvature at each point on the surface in that coordinate system, you end up a singularity at the North and South poles of the coordinate system! Thus it's very natural that experts in GR would tend to be suspicious of any conclusion which depends upon a specific coordinate system.

In this case, I think the most you can say is that it's possible to write down your equations such that it looks like the total energy of the universe is always zero (given a closed universe). That's not nothing, as it isn't expected to be possible to do in any universe. But it isn't necessarily saying something fundamental about energy in the universe.

 

[PeterDonis]

that's the only way to introduce the concept of a potential energy into General Relativity

No, it isn't. In a stationary spacetime, you can use the timelike Killing vector field to define an invariant notion of "potential energy".

The problem, for people who are trying to wave their hands in the way I described, is that this only works in a stationary spacetime, and the spacetime that describes our universe as a whole is not stationary. There are basically two ways of dealing with this, which I would describe as: (1) accept that the notion of "potential energy" is not well-defined for the universe as a whole; (2) refuse to accept that and wave your hands about pseudotensors instead.

 

[kimbyd]

No, it isn't. In a stationary spacetime, you can use the timelike Killing vector field to define an invariant notion of "potential energy".

The problem, for people who are trying to wave their hands in the way I described, is that this only works in a stationary spacetime, and the spacetime that describes our universe as a whole is not stationary. There are basically two ways of dealing with this, which I would describe as: (1) accept that the notion of "potential energy" is not well-defined for the universe as a whole; (2) refuse to accept that and wave your hands about pseudotensors instead.

One that results in energy conservation, or one that results in energy conservation with an additional curvature term?

 

[PeterDonis]

One that results in energy conservation, or one that results in energy conservation with an additional curvature term?

I'm not sure what distinction you are trying to make here. Can you give an example of energy conservation without an additional curvature term?

 

[kimbyd]

I'm not sure what distinction you are trying to make here. Can you give an example of energy conservation without an additional curvature term?

The reason why I'm objecting a little bit to this claim that you can get potential out of a killing vector field is because I just doubt it will behave like a classical notion of potential energy at all. With the Killing Field case, are you referring to the fact that there is a conserved quantity related to the field and the motion of objects along geodesics?

If so, that's fine when considering trajectories of individual objects. But I don't think it elucidates more the more generic notion of energy in a region (let alone global energy). In that case, I just don't think you can represent that in a coordinate-independent way.

I'll totally buy that it makes sense to think of a potential energy in the case of objects moving along geodesics, though.

 

[PeterDonis]

are you referring to the fact that there is a conserved quantity related to the field and the motion of objects along geodesics?

That's one application, yes.

don't think it elucidates more the more generic notion of energy in a region (let alone global energy). In that case, I just don't think you can represent that in a coordinate-independent way.

You can, but I'll agree that it is a matter of interpretation. The Komar mass of a stationary spacetime is not, in general, just the integral of the stress-energy over a spacelike slice; there is a correction factor due to spacetime curvature that can be viewed (because the timelike Killing vector field appears in the integrand) as the (negative) gravitational binding energy, or potential energy, of the system.

 

[cliffhanley203]

I've heard Lawrence Krauss, when talking about a universe from nothing, talk about the total energy of the universe being zero which shows that you can get a universe from nothing; is net zero energy the same as zero energy? I can have £100 in my pocket but be £100 overdrawn at the bank giving me a net of £0; but there's still money in the system, no?