Observational difference between OS and FLRW

[wabbit]

Reading
http://www.aei.mpg.de/~rezzolla/lnotes/mondragone/collapse.pdf

(p. 19) In spherical symmetry, the dynamical spacetime of a collapsing (expanding) region occupied by homogeneous matter is a Friedmann-Robertson-Walker (FRW)-Universe (...)
There is an important difference between the FRW Universe and the spacetime of an OS collapse since in the latter case not all of the spacetime is occupied by matter (the dust sphere has initially a finite radial size R0) and the vacuum portion (i.e. for R > R0) will be described by a Schwarzschild spacetime.

The reasonning looks like it should generalize from pressureless dust to a matter-radiation mix, and presumably also works with a (small) cosmological constant thrown in, as in a Schwarzschild-de Sitter solution - is that a correct assumption?
If so, assuming R0 is (significantly) larger than the radius of the observable universe, are there in principle observational differences between this (a variant of the interior of a white hole or time-reversed black hole) and our expanding universe?
Of course this would violate the cosmological principle since it would, in this simple case, imply a homogeneous distribution out to a certain radius and nothing after that - but other than that, is there a way to distinguish the two?

 

[PeterDonis]

The reasonning looks like it should generalize from pressureless dust to a matter-radiation mix

Yes, it does.

and presumably also works with a (small) cosmological constant thrown in, as in a Schwarzschild-de Sitter solution

I believe so but I am not really familiar with what work (if any) has been done on the Schwarzschild-de Sitter case.

assuming R0 is (significantly) larger than the radius of the observable universe, are there in principle observational differences between this (a variant of the interior of a white hole or time-reversed black hole) and our expanding universe?

Good question. In principle I think there should be, because the white hole model will have a past horizon inside the expanding matter, whereas the standard FRW model (where the matter occupies the entire universe) will not. That difference should have observable consequences, but I'm not sure off the top of my head what they would be.

 

[wabbit]

Thanks. I don't see the past being an issue, as the region concerned is the interior of the white hole (time reversed OS-like collapse) which has no (classical) past e the initial singularity).

If this picture is correct, and if Rovelli's Planck star account of black hole evolution is correct, then the interior of the expanding white hole looks like it would be very hard to distinguish from an expanding universe - except that after a finite proper time, observers eventually leave the interior. If that interpretation were to make sense for our observable universe, the mass of the past black hole would have to be larger than the mass of the observable universe, and the exit time larger than the current age of the universe. It is unclear to me whether those constraints are reasonable at all, though if we take the LQC bounce picture and replace the collapsing universe by a large finite region thereof, it might again be hard to distinguish the two.

However in such a picture, if we are not to assume that we just happen to be in a very special place, R0 and the exit time should be much larger than the minimum required for it to be just possible.

 

[PeterDonis]

I don't see the past being an issue

It's not just "the past". It's the fact that, in the white hole version, there is a past horizon inside the matter (i.e., inside our observable universe). This is because of the presence of the asymptotically flat exterior region, which is not present in the standard FRW model. In principle, this difference ought to be observationally detectable, because it is inside the matter region, i.e., inside the region we can observe. Observing it would not require seeing past the (classical) initial singularity; the past horizon is after the initial singularity, just as the (future) horizon of a black hole comes before the final singularity.

the interior of the expanding white hole looks like it would be very hard to distinguish from an expanding universe - except that after a finite proper time, observers eventually leave the interior.

They wouldn't necessarily ever have to leave the interior. A comoving observer would stay inside the expanding matter forever (or until it recollapses into a Big Crunch).

I have thought of another issue with the white hole cosmology, btw: even though we are inside the expanding matter, we should still be able to see causal influences coming in from the vacuum exterior region, since part of that region is in our past light cone. Put another way, we should still be able to see complete emptiness out beyond some distance from us--i.e., we should be able to see the boundary between the expanding FRW region and the exterior vacuum region. We do not observe anything like this.

 

[wabbit]

It's not just "the past". It's the fact that, in the white hole version, there is a past horizon inside the matter (i.e., inside our observable universe).

Not sure I get this. The white hole (well, what I am calling a white hole here) is just a time-reversed black hole - or rather here, a time-reversed OS collapse. within the collapse, all future wordlines points to the singularity so in the time reversed version all past wordlines including any photons come from the singularity

This is because of the presence of the asymptotically flat exterior region, which is not present in the standard FRW model.

Right, but in the collapse that is the exterior region, and so it must be for the white hole.

In principle, this difference ought to be observationally detectable, because it is inside the matter region, i.e., inside the region we can observe. Observing it would not require seeing past the (classical) initial singularity; the past horizon is after the initial singularity, just as the (future) horizon of a black hole comes before the final singularity.

But in this scenario we are inside the interior region, and the horizon is in our future, as it is in the past of the black hole interior region. On the other hand we do see the initial singularity, it is radiating towards us.

They wouldn't necessarily ever have to leave the interior. A comoving observer would stay inside the expanding matter forever

I was assuming that an observer in the interior region of a black hole must have crossed the horizon a finite proper time ago in his past - I can't really see a way around that but I may be missing something... If that is true then an observer in the interior of the white hole must cross the horizon outward also in finite proper time.

I have thought of another issue with the white hole cosmology, btw: even though we are inside the expanding matter, we should still be able to see causal influences coming in from the vacuum exterior region, since part of that region is in our past light cone.

Isn't this the same issue? As I understand it this is impossible in the same way as causal influences escaping from the black hole interior.

I thought of another issue : what we don't see from the interior, namely anything that has already escaped - anything close to the horizon. It seems to me this requires the homogeneous region to be quite large: we see the CMB as a sphere, but if we were close to the horizon, there would be a big disk missing from the CMB, corresponding to the part that has already exited the interior region.
So we must be deep within the horizon that the universe still looks homogeneous. I don't quite see how this constraint evolves looking back farther in time to neutrino of gw background - but possibly there would be a missing disk from either if the FRW content was not uniform all the way, but only uniform within a large sphere then zero outside (in which case it matches the reverse OS - I think?)

Another issue I'm unclear about is radiation - matter is comoving and allways stays nicely put at a fixed comoving radius from the center - but radiation isn't so nicely behaved, it must flow out of the boundary and this complicates thing.

Another way to look at this in principle would be to ask about an extreme case - is there an observational difference between a standard flat FLRW universe and one which is identical out to 10^100 times our Hubble radius, but then transitions to an empty vacuum farther out.

 

[PeterDonis]

The white hole (well, what I am calling a white hole here) is just a time-reversed black hole

Exactly; and that means it has a horizon which is the time reverse of the black hole's horizon. The black hole's horizon is the boundary of a spacetime region from which light cannot escape; the white hole's horizon is the boundary of a spacetime region into which light cannot enter.

in the collapse that is the exterior region, and so it must be for the white hole

Yes, that's what I was saying; I was pointing out that in the standard FRW model, there is no such exterior region; the expanding matter fills the entire universe.

in this scenario we are inside the interior region, and the horizon is in our future, as it is in the past of the black hole interior region

No; you're confusing two different meanings of the term "interior region". I am using it to mean "the interior of the region of spacetime occupied by the expanding matter", not "the interior of the region of spacetime inside the horizon". The region of spacetime occupied by the expanding matter has a portion inside the horizon and a portion outside the horizon (just as the collapsing matter region does in a black hole spacetime). If we are in the expanding matter region outside the horizon, then the horizon is in our past. It is possible that we could be in the expanding matter region inside the horizon, in which case both senses of "interior" would be equivalent. But that is not required by the "white hole" model, as far as I can see; all that is required is that we are inside the expanding matter region, which could be either inside or outside the horizon.

an observer in the interior of the white hile must cross the horizon outward also in finite proper time.

Yes, if "interior" means "inside the horizon". If we are inside the expanding matter but outside the horizon, then we must have crossed it a finite proper time in our past. See above.

Isn't this the same issue? As I understand it this is impossible in the same way as causal influences escaping from the black hole interior.

No. Once again, you are confusing two different senses of "interior". The expanding matter region has a boundary, outside of which there is vacuum; this is true both inside and outside the horizon. Therefore, the past light cone of any event inside the expanding matter region includes a portion of that boundary, and a portion of the exterior vacuum region (where "exterior" means "outside the expanding matter", not "outside the horizon"), just as the future light cone of any event inside the collapsing matter of a black hole spacetime include a portion of the boundary of the collapsing matter and a portion of the vacuum outside it, even if the event is inside the horizon. So w should be able to see a portion of the boundary and the vacuum beyond it if the "white hole" model is true.

what we don't see from the interior, namely annything that has already escaped - anything close to the horizon.

If you're inside the horizon of a white hole, you can't see anything outside the horizon; this is just the time reverse of the fact that someone outside the horizon of a black hole can't see in.

If you are now using "interior" in my sense, i.e., to mean "inside the expanding matter", then you're talking here about the same issue I'm talking about--we can't see a boundary or a vacuum region beyond it (which would include anything that "escaped" the boundary in the past).

we see the CMB as a sphere, but if we were close to the horizon, there would be a big disk missing from the CMB, corresponding to the part that has already exited the interior region

There is a possible problem with the CMB, but it's not quite this. Remember, we're talking about a white hole horizon, not a black hole horizon; it keeps light from getting in, not from getting out. The CMB light, or at least the part of it that has escaped the expanding matter, is going out anyway, whether we're inside or outside the horizon; the CMB light we see is just the part that hasn't escaped yet, and there isn't any discontinuous change in that that I can see if we are near the horizon.

The possible problem with the CMB is simply that, since it can only have originated from events inside the expanding matter, there will be a region of spacetime--the region outside the boundary of the expanding matter, i.e., the "exterior vacuum" region that I said above we should be able to see but can't--from which no CMB originated. So any observer inside the expanding matter in the white hole model should only see CMB radiation for a finite proper time after the surface of last scattering, since there is only a finite portion of that spacelike slice of the spacetime from which CMB radiation originated. (In the standard FRW model, that is not true: CMB radiation originated from the entire spacelike slice that marks the surface of last scattering, so we should continue seeing it forever.)

Also, I don't think that the CMB radiation, even if we are still within the finite proper time for which we should be seeing it, should be isotropic for all observers inside the expanding matter; that should only be true for one particular "comoving" observer, the one exactly at $r = 0$, i.e., at the center of the expanding matter. (The fact that there is such a center, btw, is another key difference between this model and the standard FRW model.) So the observed isotropy of the CMB would only be possible, in this model, if we were at the center.

 

[wabbit]

Right I was using interior in the sense of interior region but at the same time mixing it with the boundary of the homogeneous region concerned. In the OS collapse all matter ends inside the interior region after a finite time and I was thinking about this period, where we are inside both radii, and being careless in not distinguishing them, hence my misreading your first reply, sorry. We were actually saying the same thing as you point out.

Now I think I'm finally getting your point. That ball in inside the horizon, we can see other places outside the ball but also inside the horizon.
Hmm.. Back to the black hole, this means the collapsing ball emits radiation that is received by observers outside the ball but within the interior. For some reason I was thinking that is impossible and that the boundary of the ball acts like an effective horizon in this respect - i.e. an observer inside the black hole cannot receive signals from one "further in" - but that seems to be again the curse of the feet falling in, so I need to look at a diagram and understand this whole situation better - thanks.

One thing though, by assumption there is only a vacuum outside the ball, so there would be nothing to see. But as you say it becomes difficult not to see this absence as some anisotropy... something to chew on :)

 

[wabbit]

Regarding the CMB, the disk I described is indeed because light isn't getting in - if part of the currently observed CMB sphere (the 42 M ly one) is outside the white hole horizon, we see no light from that part and this means a disk-shaped hole in the CMB.

But I still think it is isotropic if the whole 42 M ly sphere was inside the horizon at the time of emission, or perhaps the condition is stricter - but for a large enough horizon.

 

[PeterDonis]

the disk I described is indeed because light isn't getting in

But that won't have the effect you're describing. The light can't get in because, for any event inside the white hole horizon, the region outside the horizon is outside the past light cone. It's no different from not being able to receive light from outside the past light cone anywhere else.

if part of the currently observed CMB sphere (the 42 M ly one) is outside the white hole horizon, we see no light from that part

No; we see no light from that part yet. But that's true for any point outside our past light cone. At any instant on our worldline, we are only seeing CMB light from the portion of the CMB emission surface that exactly intersects our past light cone at that instant. This is true in the standard FRW model; we don't see light from the entire CMB emission surface at the same instant.

I still think it is isotropic if the whole 42 M ly sphere was inside the horizon at the time of emission

No, it won't be if we're not exactly at the center at $r = 0$, because, once again, the CMB emission surface in this model is only a finite portion of the entire spacelike surface at that instant of "comoving" time.

Look at it this way. Suppose we are exactly at the center of an expanding ball of matter with a boundary. At some instant, every piece of matter in the ball emits light. We see the light emitted by the matter right next to us right away; then we see light from matter a little further away; then a little further; then...and so on, until we see light from matter right at the boundary. Then we see no more light.

The light we see in what I just described will be isotropic, including when it stops coming, but that's only true because we are at the exact center of the ball. If we were off center, we would see light stop coming from one direction before it stopped coming from other directions (because we would be closer to the boundary in one direction and further in others). So we would expect to see a "hole" in the CMB in some direction that would gradually widen until the entire sky was empty of CMB.

 

[PeterDonis]

Back to the black hole, this means the collapsing ball emits radiation that is received by observers outside the ball but within the interior.

Yes.

an observer inside the black hole cannot receive signals from one "further in"

That's true; but that's not what's happening in what's described in the quote above. What's happening is that an observer inside the collapsing ball emits radiation at some radius $r_1$, and an observer outside the collapsing ball receives the radiation when he is at some smaller radius $r_2$. The light moves "inward" in the sense of decreasing $r$, just slower than the collapsing matter does, so it escapes the ball and can be seen outside.

 

[wabbit]

I tried this again the other way round, with a flat matter-only FLRW $ds^2=dt^2-a(t)^2(dr^2+r^2d\Omega^2),a(t)=(\frac{3}{2}H_0t)^{2/3}$
At $t_0$ , we only see signals emitted from within the finite region $0\leq t \leq t_0, r \leq r(t)=(\frac{12}{H_0^2})^{1/3}(t_0^{1/3}-t^{1/3})$, which starts at $r(0)=(\frac{12t_0}{H_0^2})^{1/3}$
Anything that was at some time inside this region is now within a ball of radius $2r(0)$ centered at our position.

The alternative assumption are (a) standard FLRW vs (b) we are now within a large spherical homogenous region $H$ that fully contains that ball or radius $2r(0)$, and outside $H$ there is now only a vacuum.

In case (b) going back in time this seems to be an OS collapse.
Is (b) the exact same metric and matter content as (a), within the observable region ?
If so the CMB (or earlier backgrounds) should look the same to us in (a) and (b). The currently seen CMB surface, of radius r(t_{em}) in particular is fully enclosed within the region where (b) and (a) are identical.

As time passes however, in (b) we will eventually see a hole in the CMB - but only when our $r(0)$ gets large enough that this (expanding) initial comoving ball starts to cross the boundary of $H$ , which could be a long time to wait if that region is large enough and we are not close to its edge.

I lost a white hole horizon along the way, it seems to be playing no role in this reasonning, so something might be amiss.

 

[PeterDonis]

In case (b) going back in time this seems to be an OS collapse.

Basically, yes; but you have made a slightly different assumption about the boundary condition. See below.

Is (b) the exact same metric and matter content as (a), within the observable region ?

Yes, if the boundary condition is the same. Since you have assumed a flat matter-only FLRW metric for the matter region, you have imposed the boundary condition that the matter region will expand forever at exactly the "critical" rate--i.e., the rate of expansion will get slower and slower, approaching zero, but never quite reach it; more precisely, the limit of the expansion rate as $t \rightarrow \infty$ is zero. The time reverse of this would be a collapsing matter region that was always collapsing in the past, just slower and slower the further back into the past you go; i.e., the limit of the collapse rate as $t \rightarrow - \infty$ is zero. In both of these cases, in the "white hole" or "black hole" version of the model, i.e., your (b) and its time reverse, the radius of the expanding/collapsing matter increases without bound as you go forward/backward in time.

The Oppenheimer-Snyder model was actually of a collapse that started from rest at a finite radius at a finite time. In this model, the collapsing matter region actually has a closed FLRW geometry, i.e., the spatial slices of constant comoving time are 3-spheres, not flat. The time reversed version of this would be an expanding matter region that reaches maximum expansion at a finite time, then starts collapsing, i.e., a closed FLRW universe with density greater than the "critical density". Spatial slices in this model would be 3-spheres as well.

One other thing I wonder about with your version, in which the spatial slices inside the matter region are flat, is that in the (a) case, i.e., the standard FLRW flat expanding model, the universe is spatially infinite. In a "white hole" model, your (b) case, the matter region would have to be spatially finite; but I'm not sure that actually works, because I don't think the Einstein Field Equation can be satisfied at the boundary between the matter region and the vacuum region. The technical reason for this involves what are called "junction conditions", which must be looked at whenever you want to build a full spacetime by "stitching together" regions with different properties, as this model does. Intuitively, it seems to me that the junction conditions could only be satisfied if the spatial slices in the matter region were curved like a 3-sphere, as they are in the OS model. However, I haven't had time to work out the details to see if my intuitive guess here is correct. If it is, though, it would provide another way to observationally test the "white hole" model; if that model were true, our universe would have to be spatially curved (though possibly very slightly). Our best current data indicates that the universe is spatially flat, but there is still enough error to leave open the possibility that it is actually spatially curved like a 3-sphere.

As time passes however, in (b) we will eventually see a hole in the CMB - but only when our $r(0)$ gets large enough that this (expanding) initial comoving ball starts to cross the boundary of $H$

Yes.

which could be a long time to wait if that region is large enough and we are not close to its edge.

Yes, but I think there are also constraints on how large the region can be, based on the observed density of matter and the expansion history. However, I haven't had time to work that out in detail either.

I lost a white hole horizon along the way, it seems to be playing no role in this reasoning

It doesn't in the reasoning about the CMB and when a "hole" in it would appear. I'm still not sure what observational consequences we should expect based on the presence of the horizon.

 

[wabbit]

Thanks for your comments. I see your point about the boundary conditions. Indeed an initially static ball looks much more like a finite region of a closed FRW model (static at the time of maximum expansion before recollapse) than of a flat one, so if there is an exact match this might be with that, I'll try some more struggling with coordinate transforms to see how this goes - in any case, exact or not, the struggling is instructive : )

The initial question is more whether a finite region expansion can be distinguished from an infinite one - but the matter-only OS case is a toy model of that and easier to work with so I'll stay with it for now.

I don't know the junction conditions, I was just waving my hands and assuming that this would work here in the same way as in collapse models with outer Schwarzschild solution glued to inner region.

 

[PeterDonis]

I don't know the junction conditions, I was just waving my hands and assuming that this would work here in the same way as in collapse models with outer Schwarzschild solution glued to inner region.

It will if you use a closed FRW model for the matter region; that's just the time reverse of the OS collapse model. The potential issue with junction conditions would be with using a flat or open FRW model for the matter region.

 

[wabbit]

Ah OK thanks - as long as the curvature can be arbitrarily low that's not a problem then.
Also, just came across the Lemaitre-Tolman metric and the related articles about inhomogeneous spherically symmetric timespaces, which look like a nice framework in which to compare a finite patch collapse with a global one.

 

[PeterDonis]

as long as the curvature can be arbitrarily low that's not a problem then

In principle it can be. However, the amount of spatial curvature is not an independent variable in this model; as I mentioned in post #12, there are constraints on this model based on the observed expansion history to date. Those constraints should determine what the spatial curvature at present must be for the model to be consistent. I have not done any calculations to see if that spatial curvature is small enough to be within the current observational uncertainty. (I don't see any reason why it couldn't be; I just haven't done the calculations.)

 

[wabbit]

The other option is to stay flat - then this is not OS but something a bit different, with the ball static at infinity as you mentionned in a previous post (quite similar to the raindrops described in http://en.wikipedia.org/wiki/Lemaître_coordinates for the Schwarzschild vacuum, which have proper time intriguigly similar to FLRW cosmological time). This sounds doable as a Lemaitre-Tolman model.

This is fine too, maybe even better - after all in a flat FLRW we don't expect recollapse so why try to match it with a recollapsing model - and OS must do that, if we solve from the static starting point in either direction it should be the same, so really it must describe an (expanding then) collapsing ball. The point here isn't so much OS per se as a finite patch expansion - OS was just the one to prompt the question, and the only one I knew of then.

 

[PeterDonis]

The other option is to stay flat - then this is not OS but something a bit different, with the ball static at infinity as you mentionned in a previous post (quite similar to the raindrops described in http://en.wikipedia.org/wiki/Lemaître_coordinates for the Schwarzschild vacuum, which have proper time intriguigly similar to FLRW cosmological time). This sounds doable as a Lemaitre-Tolman model.

The Lemaitre-Tolman metric allows a flat solution, yes, but it still describes the entire universe, as far as I can see; it has the same potential issue with junction conditions if it is "cut off" at a finite radius as the flat FLRW model. (In fact, from what I can see, the L-T metric with flat spatial slices is the flat FLRW model; I don't think the spatial slices will remain flat if the relevant parameters are allowed to vary from the constant values that define the flat FLRW case.)

 

[wabbit]

Right but since it handles non-homogeneous spherically symmetric distributions, it seems a finite homogeneous ball+vacuum (perhaps with a small transition region if smoothness is required) could fit right in, and in a way allowing direct comparison with the homogeneous /infinite ball (FLRW) case.

Or you re saying the inhomogenous case won't be spatially flat? Hmmm that would be bad then, in that case it may be getting too complicated for me...

Still, E seems to play a role similar to the curvature parameter in FLRW so I am pinning my hopes on the E=0 case with M(r) appropriately chosen (in the wikipedia notation).

 

[PeterDonis]

you re saying the inhomogenous case won't be spatially flat?

Yes.

 

[wabbit]

OK thanks.

Edit: Tolman seems to treat this case in www.pnas.org/content/20/3/169.full.pdf, p. 175

e. Combination of Uniform Distributions.- [...]We may consider an initial distribution at t = 0 which corresponds in a given zone, say from 0 to ra, to the instantaneous conditions in a particular Friedmann model [...] This can then be surrounded by a transition zone from ra to rb where the values change to a new set which correspond for a further range from rb to r, to a different Friedmann model [...] and this can be followed by such additional transition zones and Friedmann zones as may be desired. [...] the dust in each Friedmann zone will then behave as in some particular completely homogeneous model without reference to the behavior of other parts of the model.

 

[PeterDonis]

Tolman seems to treat this case

But in the "white hole" model, we are not talking about a transition zone between two different Friedmann models. We are talking about a transition zone between a Friedmann model and a vacuum region with a past null infinity, and which also contains a past horizon in both the vacuum region and the matter region. As far as I can tell, the Lemaitre-Tolman metric, like the standard FRW metric, does not have a past null infinity; if there is a past singularity, that singularity covers the entire universe, i.e., the entire spacetime described by the model. The "white hole" model does not have that property.

 

[wabbit]

Getting somewhere maybe? Something rather weird here so maybe not - but I was missing a key point about inhomogeneous collapses in the Tolman picture : their big bang is not synchronous, and taking this into account seems to provide a more consistent picture.

Staying in the flat case here, using comoving coordinates. We can parametrize our (Tolman-Lemaitre) expanding dust spacetimes by their density profile $\rho(r)$ at a given time $t_0$ and we can take $a(t_0)=1$ in all FRW cases.

The FRW case corresponds to $\rho(r)=\rho=constant$. Solving the equation $(\frac{\dot a}{a})^2=\frac{8\pi\rho}{a^3}$ we get $\frac{2}{3}(a(t)^{3/2}-1)=\sqrt{8\pi\rho}(t-t_0)$ and the singularity is at $a(t_B)=0$ i.e. $t_B(\rho)=t_0-\frac{2}{3\sqrt{8\pi\rho}}$ - it gets pushed farther in the past as we lower the density. The vacuum case is just Minkowski spacetime, with the singularity pushed back to past infinity.

In the inhomogenous case, the singularity lies at $\{(t_B(\rho(r)),r),r\geq 0\}$.

The finite ball expansion case here has a FRW interior but a Minkowski - not Schwarzschild - exterior. If we use a small transition zone where $\rho(r)$ goes smoothly down from the homogeneous value $\rho_H$ for $r<R-\epsilon$ to $0$ for $r>R$, the singularity in comoving coordinates consists of a vertical segment $t=t_B(\rho_H),0\leq r \leq R-\epsilon$, and then a curve asymptotic for $t<t_B(\rho_H)$ to the horizontal line $r=R$.

An observer at $(t,r<R-\epsilon)$ inside the expanding ball $H$ sees, for $t_B(\rho_H)<t<t_{max}(r)$, a past that is indistinguishable from what he would see in the corresponding FRW case. He does not receive any information from the Minkowski exterior.

This is a little strange as the exterior region is not Schwarzschild but Minkowski - i.e. it doesn't "see" the mass lying in the interior region. Also the region $t<t_B(\rho(r)),r<R$ is not part of the solution. Minkowski (non-comoving) observers can enter the region H, but not the observable region of our (early) inside observer above.

Possibly the whole forbidden region should be identified with the half-line $r=0,t<t_B$, gluing together seemingly distant regions of the past Minkowski exterior into a single Minkowski past - this seems fairly natural if swapping into proper spatial coordinates from comoving ones. Comoving geodesics starting in the Minkowski past just flow around the boundary of H - but I still cannot understand how the exterior for $t>t_B$ can be Minkowski and not Schwarzschild, this must be an error in my reading of Tolman. Bondi has a more readable treatment but doesn't quite treat this case.

 

[PeterDonis]

I still cannot understand how the exterior for $t>t_B$ can be Minkowski and not Schwarzschild, this must be an error in my reading of Tolman. Bondi has a more readable treatment but doesn't quite treat this case.

I don't understand that either; I need to read through this stuff in more detail.

 

[wabbit]

I don't quite follow Bondi's solution of the EFE in this case, but his last equation (p.425) seems pretty much to imply that the exterior vacuum has non-zero curvature as expected - so my bet is on my having misunderstood Tolman here. The t coordinate here is the comoving proper time, while the r coordinate is orthogonal to that, so this looks a bit like a region of Schwarzschild spacetime in something related to Lemaitre coordinates. Well I don't know, just confused : )

 

[wabbit]

OOK I think I found the error in my "Minkowski" post above. Tolman' bands are each FRW-like but each band has the geometry, not of the usual FRW space of the same density, but of one with that density and a "central mass". In the case of a flat homogeneous region surrounded by a vacuum, the outside FRW region is indeed a Schwarzschild vacuum, in (shifted) Lemaitre coordinates.

 

[wabbit]

Not quite sure about this, but I get the following for a flat homogeneous dust ball of mass $M$ and radius $R$ in a vacuum :
$r\leq R : ds^2=dt^2-\left(\sqrt{\frac{9M}{2}}\cdot t\right)^{4/3}\frac{1}{R^2}dr^2- \left(\sqrt{\frac{9M}{2}}\cdot t\right)^{4/3}\frac{r^2}{R^2}d\Omega^2$
$r\geq R : ds^2=dt^2-\frac{2M}{\left(\sqrt{\frac{9M}{2}} (t+r-R)\right)^{2/3}}dr^2-\left(\sqrt{\frac{9M}{2}} (t+r-R)\right)^{4/3}d\Omega^2$
Singularity : $r\leq R : t_B(r)=0 ; \quad r\geq R : t_B(r)=r-R$
Horizon : $r\leq R : t_{\mathcal{H}}(r)=\frac{M}{6}(3-\frac{r}{R})^3\leq \frac{9M}{2}; \quad r\geq R : t_{\mathcal{H}}(r)=\frac{4M}{3}+R-r$

The $dr^2$ coefficient of the metric is discontinuous at $r=R$ due to the discontinuity in $M'(r)$ - this can be cured with a small transition region, though as it stands it is not severe - light rays are bent at the surface in the same way as they are at the interface between two materials with different indices of refraction.

The conclusion about invisibility of the outside for early observers stands however as it does not depend on the details of the outside geometry. This applies here for $r\leq R, t\leq t_{\mathcal{I}}(r)=\frac{M}{6}(1-\frac{r}{R})^3$

 

[wabbit]

An interesting aspect of this parametrization is that if I read it correctly you can glue two of them together at the singularity and get a classical (or even a smoothed out toy-QG) version of a Planck star (or an LQG bounce if R is infinite). I tried it with paper and scissors, it looks nice : )