I Inflation theory and General Relativity

[Tio Barnabe]

Does General Relativity predicts that in the early universe vacuum energy was converted into matter? How does it relates to the Inflation Theory by Allan Guth?

I'm asking this because I remember reading in a book on GR that there are ways of calculating the total amount of energy in the universe by adding the matter contribution, etc... including vacuum energy. In fact, I remember reading about this in several books.

But today there's less vacuum out there than there was before, correct? So what happened to the associated vacuum energy?

 

[PeterDonis]

Does General Relativity predicts that in the early universe vacuum energy was converted into matter?

This sort of prediction wouldn't be made using GR. It would be made using quantum field theory, which would model the "vacuum energy" (I'm not sure what you mean by this--see below) and the "matter" (and radiation) as quantum fields and using that to predict how energy might be transferred between them.

What GR would be used for, given the above QFT model, would be to compute the behavior of the spacetime geometry using the effective stress-energy tensor of the quantum fields involved. That's how we reconstruct the expansion history of the universe based on the energy density in various forms--matter, radiation, dark energy, etc.

As I noted above, the term "vacuum energy" is vague. Do you mean the inflaton field (the field that drove inflation)? Or do you mean something else?

 

[Dale]

I remember reading in a book on GR that there are ways of calculating the total amount of energy in the universe by adding the matter contribution, etc... including vacuum energy. In fact, I remember reading about this in several books.

Can you cite one of these books? To my knowledge it is not correct: http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

 

[Tio Barnabe]

As I noted above, the term "vacuum energy" is vague. Do you mean the inflaton field (the field that drove inflation)? Or do you mean something else?

I mean the curvature energy in a region where there's no matter. In some treatments the curvature caused by a source is considered when accounting for all the sources of energy. That is what I've read on a pdf by Prof Mattias Blau from the Bern University I found on web. (Maybe I misunderstood what is said on there.)

Can you cite one of these books?

One of them is actually a pdf, as I say in above.

 

[Dale]

One of them is actually a pdf, as I say in above.

Can you link to it or otherwise identify it. It sounds like a legitimate reference, so I think you are misunderstanding it.

 

[Tio Barnabe]

Can you link to it or otherwise identify it. It sounds like a legitimate reference, so I think you are misunderstanding it.

This is the website http://www.blau.itp.unibe.ch/Lecturenotes.html
The pdf file is named "newlecturesGR.pdf"

 

[Dale]

Um, there are more than 900 pages. Can you please narrow it down a bit? Where does he calculate the total amount of energy in the universe?

 

[Tio Barnabe]

Ok. Just give me some time for finding it.

 

[George Jones]

http://www.blau.itp.unibe.ch/newlecturesGR.pdf

From what I have seen, this is a really excellent, freely available, book.

Um, there are more than 900 pages. Can you please narrow it down a bit? Where does he calculate the total amount of energy in the universe?

The discussion in section 21.6 "Comments on Gravitational Energy" is quite interesting (pages 259 - 465)!

It may not have escaped your attention that in the entire discussion of energy and energy-momentum tensors of fields in a gravitiational field the notion of the energy of the gravitational field itself has not appeared so far, even though clearly there can be an exchange of energy between matter and gravitational fields and one should not expect one to be conserved without taking into account the other. This is evidently a major omission, and I will try to rectify this now, but as you will perhaps understand from the discussion below there is a good reason why I have so far tried to avoid this issue. ...

This is treacherous territory and I cannot guarantee that even these elementary remarks are widely considered to be uncontroversial (in fact, the number of uncontroversial statements one can make about this subject is probably quite small - but is likely to include this parenthetical remark . . . ). ...

Thus it is possible that an appropriate notion of gravitational energy is just sufficiently ‘non-local’ (by depending on the second derivatives of the metric) that it cannot be eliminated by going to a freely falling reference frame along a single worldline. Nevertheless, this provides a first indication that the notion of energy density of the gravitational field is perhaps somewhat more subtle than expected.

 

[Tio Barnabe]

I remember that there was a picture involved in the discussion. I'm still trying to find it.

 

[stoomart]

I mean the curvature energy in a region where there's no matter. In some treatments the curvature caused by a source is considered when accounting for all the sources of energy. That is what I've read on a pdf by Prof Mattias Blau from the Bern University I found on web. (Maybe I misunderstood what is said on there.)

One of them is actually a pdf, as I say in above.

Here's the handful of places the notes discuss "vacuum energy"; the 3rd one seems to call it matter, alongside dust and radiation.

Page 365

...Λ is often said to play the role of a vacuum energy density (more precisely vacuum energy should perhaps be considered as one possible contribution to the cosmological constant - see section 37.4 for further discussion of this issue).​

Pages 376-377

As we saw above, a cosmological constant term can be included by adding a constant term to the Einstein-Hilbert Lagrangian. One can equally well add a constant term to the matter Lagrangian instead (and this clearly reveals its interpretation as a constant shift of the energy, e.g. by a vacuum energy contribution, of the matter fields).​

Page 789

In addition to the vacuum energy (and pressure) provided by Λ, there are typically two other kinds of matter which are relevant in our approximation, namely matter in the form of dust (w = 0) and radiation (w = 1/3).​

Page 829

We have remarked before that the cosmological constant looks like a vacuum energy contribution to the energy-momentum tensor. It is perhaps better to turn this around and to say that vacuum energy is one potential contribution to the cosmological constant...​

Pages 832-833

This is nothing other than the usual Einstein equations with a cosmological constant...the crucial difference being that here Λ is not determined by the matter content and its vacuum energy but arises solely as an integration constant. While this does not explain the observed tiny value of the cosmological constant, it separates this issue from that of the vacuum fluctuations.​