A Assumptions of Hawking-Penrose 1970 Singularity Theorem

[Cerenkov]

I've been thinking some more about the Hawking - Penrose Singularity theorem and was wondering if you could help me gain a better understanding of the assumptions they made when they wrote it, in 1970.

In Hawking's book, A Brief History of Time (chapter 3, page 25) he writes....


In 1965 I read about Penrose’s theorem that any body undergoing gravitational collapse must eventually form a singularity. I soon realized that if one reversed the direction of time in Penrose’s theorem, so that the collapse became an expansion, the conditions of his theorem would still hold, provided the universe were roughly like a Friedmann model on large scales at the present time. Penrose’s theorem had shown that any collapsing star must end in a singularity; the time-reversed argument showed that any Friedmann-like expanding universe must have begun with a singularity.

For technical reasons, Penrose’s theorem required that the universe be infinite in space. So I could in fact, use it to prove that there should be a singularity only if the universe was expanding fast enough to avoid collapsing again (since only those Friedmann models were infinite in space).

During the next few years I developed new mathematical techniques to remove this and other technical
conditions from the theorems that proved that singularities must occur. The final result was a joint paper by
Penrose and myself in 1970, which at last proved that there must have been a big bang singularity provided
only that general relativity is correct and the universe contains as much matter as we observe.

Reading this, here are the assumptions that I think they made.

1. That the universe is roughly like a Friedmann model on large scales at the present time.
2. That the universe is infinitely large.
3. That the universe was expanding fast enough to avoid re-collapse.
4. That general relativity is correct.
5. That the universe contains as much matter as we observe.

So, for now I have two questions. Is my list correct regarding what Hawking says was assumed by them? Have any findings made since 1970 rendered the above assumptions false?

Thank you in advance for help.


Cerenkov.

 

[PeterDonis]

Reading this, here are the assumptions that I think they made.

1. That the universe is roughly like a Friedmann model on large scales at the present time.

Not in the 1970 theorem, no.

2. That the universe is infinitely large.

Not in the 1970 theorem, no.

3. That the universe was expanding fast enough to avoid re-collapse.

Not in the 1970 theorem, no.

4. That general relativity is correct.

Yes--the technical assumption is that the Einstein Field Equation is correct.

5. That the universe contains as much matter as we observe.

This was Hawking being typically vague in a pop science context. The actual assumptions connected with matter-energy content are, first, that the universe is dominated by ordinary matter and/or radiation, so that the energy conditions are satisfied, and second, that the universe is expanding.

Have any findings made since 1970 rendered the above assumptions false?

Dark energy does not satisfy the energy conditions, so the singularity theorem does not apply to a universe that is dark energy dominated at all times. Our universe is dark energy dominated now, but it wasn't before about 2 billion years ago, so dark energy being in our best current model in itself does not make the singularity theorem inapplicable to our universe.

However, inflation also violates the energy conditions, so the singularity theorem is inapplicable to most inflation models, since inflation occurs in the very early universe.

 

[Cerenkov]

Not in the 1970 theorem, no.


Not in the 1970 theorem, no.


Not in the 1970 theorem, no.

Then these three assumptions were proper to papers by Hawking and Penrose prior to the 1970 paper?

Yes--the technical assumption is that the Einstein Field Equation is correct.

Thank you.

This was Hawking being typically vague in a pop science context. The actual assumptions connected with matter-energy content are, first, that the universe is dominated by ordinary matter and/or radiation, so that the energy conditions are satisfied, and second, that the universe is expanding.

Ok, let's be specific here, so that I get this right.

Are you talking about these four energy conditions described in the 1970 paper's abstract?

(i) Einstein’s equations hold (with zero or negative cosmological constant),
(ii) the energy density is nowhere less than minus each principal pressure nor less than minus the sum of the three principal pressures (the ‘energy condition’),
(iii) there are no closed timelike curves,
(iv) every timelike or null geodesic enters a region where the curvature is not specially aligned with the geodesic. (This last condition would hold in any sufficiently general physically realistic model.)

Dark energy does not satisfy the energy conditions, so the singularity theorem does not apply to a universe that is dark energy dominated at all times. Our universe is dark energy dominated now, but it wasn't before about 2 billion years ago, so dark energy being in our best current model in itself does not make the singularity theorem inapplicable to our universe.

Ok then.

What you say seems to suggest that the singularity theorem does apply to a universe that is dark energy dominated at certain times, but not all times. Is that what you meant to imply Peter?

With this new type 1a supernovae finding, the dark energy-driven acceleration of expansion during the last 2 billion years is called into question. If this holds up then there is no need to invoke dark energy to account for this acceleration. It simply didn't happen.

Which would satisfy the condition of our universe not being dark energy dominated at all times. Which prompts the following questions.

Was there ever a time in the history of our universe when it was dark energy dominated?

And given that our universe was not dark energy dominated at all times, does the theorem become applicable to our universe?

However, inflation also violates the energy conditions, so the singularity theorem is inapplicable to most inflation models, since inflation occurs in the very early universe.

Could you please explain (in terms that I can understand) how inflation violates the energy conditions?

Moreover, which ones?


I realise that this is a lot of questions Peter, but with your help I believe that I am gaining a better understanding of this.

Thank you,


Cerenkov.

 

[javisot]

Dark energy does not satisfy the energy conditions, so the singularity theorem does not apply to a universe that is dark energy dominated at all times. Our universe is dark energy dominated now, but it wasn't before about 2 billion years ago, so dark energy being in our best current model in itself does not make the singularity theorem inapplicable to our universe.

However, inflation also violates the energy conditions, so the singularity theorem is inapplicable to most inflation models, since inflation occurs in the very early universe.

Both inflation and dark energy violate the energy conditions and refer to the accelerated expansion of the universe. Is this a coincidence, or could some variable dark energy model explain inflation and the subsequent deceleration?

 

[PeterDonis]

Then these three assumptions were proper to papers by Hawking and Penrose prior to the 1970 paper?

As far as I know, Hawking and Penrose didn't publish any papers together prior to the 1970 paper. Penrose published a paper in 1965, which is referred to in the excerpt you quoted from Hawking's popular book.

That paper's assumptions were more restrictive than the ones in the Hawking-Penrose 1970 paper, but only in one key respect: the 1965 theorem only works in spacetimes that are spatially infinite (the technical term is spacetimes that have a non-compact Cauchy surface). The theorem is not limited to FRW spacetimes. (Hawking's excerpt that you quoted doesn't say it is, but it can give that impression--which is an example of why you shouldn't try to learn actual physics from pop science sources.)

 

[PeterDonis]

Are you talking about these four energy conditions described in the 1970 paper's abstract?

(i) Einstein’s equations hold (with zero or negative cosmological constant),

The "Einstein's equations hold" part is "general relativity is valid". The "zero or negative cosmological constant" is part of the energy conditions.

(ii) the energy density is nowhere less than minus each principal pressure nor less than minus the sum of the three principal pressures (the ‘energy condition’),

Yes, this is also part of the energy condition.

(iii) there are no closed timelike curves,
(iv) every timelike or null geodesic enters a region where the curvature is not specially aligned with the geodesic. (This last condition would hold in any sufficiently general physically realistic model.)

Neither of these have anything to do with energy conditions. They are more technical conditions that basically amount to making sure the spacetime is physically reasonable.

 

[PeterDonis]

What you say seems to suggest that the singularity theorem does apply to a universe that is dark energy dominated at certain times, but not all times. Is that what you meant to imply Peter?

Sort of. It can apply to an expanding universe that is only dark energy dominated after a certain time, but not before that.

With this new type 1a supernovae finding, the dark energy-driven acceleration of expansion during the last 2 billion years is called into question. If this holds up then there is no need to invoke dark energy to account for this acceleration. It simply didn't happen.

That is one possible way it could turn out, yes.

Which would satisfy the condition of our universe not being dark energy dominated at all times.

If we only consider the accelerated expansion of the last few billion years, our universe already satisfies that condition. That was my point in the previous post of mine that you quoted.

However, none of that has anything to do with inflation, which is in the very early universe, and which is not called into question at all by the Type 1a supernovae re-analysis that was discussed in the other thread.

Was there ever a time in the history of our universe when it was dark energy dominated?

Inflation has the same implications, and our best current model says the very early universe was in a period of inflation.

given that our universe was not dark energy dominated at all times, does the theorem become applicable to our universe?

Not if our best current model of inflation in the very early universe is correct.

Could you please explain (in terms that I can understand) how inflation violates the energy conditions?

The same way dark energy does: it violates item (ii) in your post #3.

 

[PeterDonis]

For reference, this paper looks like it has a good discussion of the 1965 Penrose singularity theorem:

https://arxiv.org/abs/1410.5226

 

[Cerenkov]

As far as I know, Hawking and Penrose didn't publish any papers together prior to the 1970 paper. Penrose published a paper in 1965, which is referred to in the excerpt you quoted from Hawking's popular book.

That paper's assumptions were more restrictive than the ones in the Hawking-Penrose 1970 paper, but only in one key respect: the 1965 theorem only works in spacetimes that are spatially infinite (the technical term is spacetimes that have a non-compact Cauchy surface). The theorem is not limited to FRW spacetimes. (Hawking's excerpt that you quoted doesn't say it is, but it can give that impression--which is an example of why you shouldn't try to learn actual physics from pop science sources.)

Thank you for this, Peter.


In my defence I'm only using Hawking's book as a starting point for my line of questions.


Cerenkov.

 

[Cerenkov]

The "Einstein's equations hold" part is "general relativity is valid". The "zero or negative cosmological constant" is part of the energy conditions.

Thank you.

Yes, this is also part of the energy condition.

Thank you.

Neither of these have anything to do with energy conditions. They are more technical conditions that basically amount to making sure the spacetime is physically reasonable.

Thank you again.

I suspected this was the case. But thank you for confirming it.


Cerenkov.

 

[Cerenkov]

Sort of. It can apply to an expanding universe that is only dark energy dominated after a certain time, but not before that.

Thank you.

That is one possible way it could turn out, yes.

Thank you.

If we only consider the accelerated expansion of the last few billion years, our universe already satisfies that condition. That was my point in the previous post of mine that you quoted.

Now this needs unpacking.

If the type 1a finding holds, then our universe doesn't satisfy the condition of accelerated expansion during the last 2 billion years. They imply that accelerated expansion never happened. So dark energy is not the cause of that. It never was.

But if our universe satisfies the condition of accelerated expansion during some other epoch in it's evolution
then is inflation the cause?

However, none of that has anything to do with inflation, which is in the very early universe, and which is not called into question at all by the Type 1a supernovae re-analysis that was discussed in the other thread.

I understand this. Inflation occurred in the very earliest moments of the universe, long before matter could form the stars that would go on to become type 1a supernovae.

Inflation has the same implications, and our best current model says the very early universe was in a period of inflation.

These implications being accelerated expansion? I know from reading Alan Guth's book that inflation, once initiated, accelerates. Exponentially, I seem to recall.

Not if our best current model of inflation in the very early universe is correct.

Because inflation violates the energy conditions of the theorem, as you say below.

The same way dark energy does: it violates item (ii) in your post #3.

(ii) the energy density is nowhere less than minus each principal pressure nor less than minus the sum of the three principal pressures (the ‘energy condition’),

Is it worth me asking how it violates item (ii) or is that a bridge too far for my Basic-level understanding? I defer to your knowledge and expertise here, Peter. If you think it's better that I just take your word that this violation happens, then so be it. But if you would like to take a stab at explaining the 'how' to me then I'm willing to be guided by you.


Thanks again,


Cerenkov.

 

[PeterDonis]

If the type 1a finding holds, then our universe doesn't satisfy the condition of accelerated expansion during the last 2 billion years.

Yes.

They imply that accelerated expansion never happened.

Not "never". Just not in the last two billion years.

if our universe satisfies the condition of accelerated expansion during some other epoch in it's evolution
then is inflation the cause?

Inflation does cause accelerated expansion, yes. In our best current model, that occurred in the very early universe.

These implications being accelerated expansion?

Yes.

Is it worth me asking how it violates item (ii)

In cosmology, we model matter, radiation, dark energy, and the field that causes inflation as fluids with an equation of state $p = w \rho$. The different words I just used are simply labels for different values of $w$:

"Matter": $w = 0$

"Radiation": $w = 1/3$

"Dark Energy": $w = -1$

The field that causes inflation, in the simplest inflation model, is just like dark energy: $w = -1$. In more complicated models, $w$ might not be quite constant and might vary in some range, but it won't be very different from $-1$.

Any fluid for which $w < - 1/3$ will violate the energy conditions. From the above, we can see that dark energy does, and inflation does as long as $w$ doesn't vary enough to be greater than or equal to $- 1/3$ (and it comes nowhere close to that in any inflation model I know of).

 

[Cerenkov]

With apologies Peter, I'm not feeling very well right now and will get back to you asap.


Cerenkov.

 

[PeterDonis]

With apologies Peter, I'm not feeling very well right now and will get back to you asap.

No worries, post as you can. I hope you feel better soon.