Playing with perturbed FLRW spacetime

[GR_fan]

TL;DR

Question about what happens with a specific perturbation of FLRW spacetime

I'm trying to understand what sort of perturbation turns a non-singular cosmological spacetime into a singular cosmological spacetime. In other words, given a cosmological spacetime without a Big Bang singularity, what sort of perturbation would introduce a Big Bang singularity? Most likely, someone has already answered that question, but I'm trying to gain some understanding for myself. To pursue the question, I'm playing around with a specific perturbation to an FLRW spacetime represented using the line element:
$$ds^2 = -(1+2\Phi)dt^2 + a^2(1-2\Psi)\delta_{ij}dx^i dx^j$$
where $\Phi$ and $\Psi$ are scalar fields representing the perturbation. Now, let's consider the case where the unperturbed spacetime (where $\Phi=\Psi=0$) is not singular and so includes (for example) complete null geodesics. In that case, in the unperturbed spacetime, along some null geodesic with affine parameter $\lambda$, we can use the geodesic equation to derive:
$$\lambda_f - \lambda_i \propto \int_{t_i}^{t_f} a(t) dt$$
Since we've assumed the spacetime includes complete null geodesics, we can take a limit on $t_i$ where our integral diverges. The fact that this integral diverges shows us that the null geodesic a long which we took the integral is complete in the past direction (i.e., no Big Bang singularity). Now, I'm interested in the conditions under which our perturbed spacetime also includes complete null geodesics. Using a similar procedure, we can derive the following expression:
$$\lambda_f - \lambda_i \propto \int_{t_i}^{t_f} a(t) \exp\left(- \int\frac{\partial_t \Phi}{1+2\Phi}dt - \int\frac{\partial_t \Psi}{1-2\Psi}dt\right)dt$$
The trouble is that I don't know what to do from here. My best guess is that we should pursue the following. Our integral is basically just $\int f(t, x, y, z) a(t) dt$. If there exists some constant M > 0 such that f(t, x, y, z) > M, then we can show that the integral diverges just in case $\int a(t) dt$ diverges. But I have no idea how to think about that condition.

Does anyone here have some ideas about how to approach this?

 

[Matterwave]

Hello and welcome to PF! :)

I'm a little confused by your premise. The FLRW metric does have a big bang singularity, so how are you starting from it and deriving no singularity? A spacetime with complete null geodesics also doesn't imply there is no singularity as you could have incomplete time like geodesics (and vice versa).

For the conditions under which a singularity will arise (provably so), see the Penrose and Hawking singularity theorems. This is in Wald chapter 9 and Hawking and Ellis chapter 8.

 

[renormalize]

I'm a little confused by your premise. The FLRW metric does have a big bang singularity, so how are you starting from it and deriving no singularity?

GR_fan's metric ansatz $ds^2 = -(1+2\Phi)dt^2 + a^2(1-2\Psi)\delta_{ij}dx^i dx^j$ is just flat Minkowski spacetime when $\Phi=\Psi=0$ and thereby represents the unperturbed, singularity-free starting metric. My understanding is that GR_fan seeks to examine what happens as they "turn-up" the perturbation functions $\Phi,\Psi$ to understand if and how the spacetime described by the perturbed metric devlops singularities.

 

[Matterwave]

GR_fan's metric ansatz $ds^2 = -(1+2\Phi)dt^2 + a^2(1-2\Psi)\delta_{ij}dx^i dx^j$ is just flat Minkowski spacetime when $\Phi=\Psi=0$ and thereby represents the unperturbed, singularity-free starting metric.

There's still the $a^2$ left over though?

 

[renormalize]

There's still the $a^2$ left over though?

Good point! Since the poster specifies that the metric is nonsingular when $\Phi,\Psi$ vanish, I presumed that they are taking $a=\text{constant}$ and so it can be absorbed by rescaling the $x^i$. If that's not the case, then maybe they mean to pick a scale-factor $a\left( t\right)$ that is non-zero finite for all $t$. Perhaps the poster can clarify their intent.

 

[GR_fan]

It's not true that the FLRW metric is always singular. We can pick non-singular scale factors. Examples include $a(t) \propto cosh(Ht)$ and $a(t) \propto 1+exp(At)$. These aren't the matter or radiation dominated universes -- which are singular -- but they are FLRW nonetheless. These non-singular scale factors are non-zero for $\forall t \in (-\infty, \infty)$, though that's not a great way to think about completeness/incompleteness. For that, you need to evaluate the affine parameter along causal curves, such as the integral I gave.

My metric ansatz is just the FLRW metric when $\Phi=\Psi=0$, since we have the scale factor left over.

As far as the Hawking-Penrose theorems, those theorems provide sufficient, but not necessary, conditions for a singularity. The theorems do not apply to the previously mentioned non-singular scale factors, because they violate energy conditions the theorems require. In the case I'm thinking about, we start with a non-singular FLRW spacetime -- such as $a(t) \propto 1+exp(At)$ -- and then we introduce some arbitrary perturbation. I'm trying to think about the conditions under which the perturbation renders the null curve I mentioned incomplete.

 

[PeterDonis]

We can pick non-singular scale factors.

In the sense that you can write de Sitter spacetime in FLRW form, with an appropriate slicing, and that de Sitter spacetime is non-singular, yes. That's what the choices of scale factor you describe are doing (the two choices you describe are using two different slicings of de Sitter spacetime).

You can also write a wedge of Minkowski spacetime in FLRW form (this was done by Milne), and the spacetime will be non-singular (since it's flat). The scale factor for this model is just $t$.

But these are particular special cases and don't generalize to other possible FLRW spacetimes, all of which have initial singularities AFAIK. So you really should be looking at perturbations of those particular spacetimes.

 

[GR_fan]

In the sense that you can write de Sitter spacetime in FLRW form, with an appropriate slicing, and that de Sitter spacetime is non-singular, yes. That's what the choices of scale factor you describe are doing (the two choices you describe are using two different slicings of de Sitter spacetime).

You can also write a wedge of Minkowski spacetime in FLRW form (this was done by Milne), and the spacetime will be non-singular (since it's flat). The scale factor for this model is just $t$.

But these are particular special cases and don't generalize to other possible FLRW spacetimes, all of which have initial singularities AFAIK. So you really should be looking at perturbations of those particular spacetimes.

There are infinitely many scale factors that are non-singular. All you need is for the integral condition that I specified to be satisfied. Notice that I mentioned one scale factor that you didn't, i.e., 1+exp(At). What's true is that non-singular FLRW models will generally violate one of the conditions needed for the Hawking-Penrose theorems and so there may be reason to think that such models involve exotic energy conditions or some weird geometry or whatever.

At any rate, if you take any of those scale factors, the perturbations mentioned, and the geodesic equation, you can derive the integral I mentioned, with a specific function plugged in for a(t). I don't see why that's a helpful thing to do, but perhaps you see something that I don't. I think there are definite advantages to considering the more general case, since then whatever we find isn't specific to one particular scale factor.

 

[PeterDonis]

All you need is for the integral condition that I specified to be non-singular.

No, that's not all you need, as @Matterwave already pointed out in post #2. You need to show that all geodesics are complete, not just that null geodesics are complete.

 

[GR_fan]

No, that's not all you need, as @Matterwave already pointed out in post #2. You need to show that all geodesics are complete, not just that null geodesics are complete.

Yes, you are correct. I made a mistake because I posted my reply without proofreading.

I meant to say that an analogous integral condition must be satisfied for all causal geodesics, at least under the assumption that the singularity -- if there is one -- is spacelike. If we don't assume the singularity is spacelike, then the analogous integral condition must be satisfied for all geodesics.

 

[PeterDonis]

1+exp(At)

This looks like a "phantom energy" or "Big Rip" spacetime, with negative pressures of larger magnitude than the positive energy density. I'm not sure such a model is nonsingular.

 

[GR_fan]

This looks like a "phantom energy" or "Big Rip" spacetime, with negative pressures of larger magnitude than the positive energy density. I'm not sure such a model is nonsingular.

It's provably non-singular and used as a stock example in some papers on singularity theorems. You might convince yourself by considering the integral condition I offered, though -- granted -- that condition is necessary and not sufficient for geodesic completeness.

 

[PeterDonis]

used as a stock example in some papers on singularity theorems

Can you give some references to these papers?

 

[GR_fan]

Can you give some references to these papers?

Sure. One example is this recent paper by Kinney, posted to ArXiv this year: https://arxiv.org/pdf/2607.28412

It was Kinney's article that I had in mind, because it's a paper I've studied closely.

A slight variation -- where a square root is taken of the scale factor used by Kinney -- is offered by George Ellis, et al: https://arxiv.org/pdf/gr-qc/0211082. Another variation -- of which Kinney's is a special case -- is offered in this paper from 2024: https://link.springer.com/article/10.1140/epjc/s10052-024-12575-2. And yet another variation -- where Kinney's is again a special case -- is discussed here: https://arxiv.org/pdf/2402.13031