# BRS: The FRW Dust with E^3 Hyperslices - Discussion

## Main Question or Discussion Point

Ok, I've looked a bit into your Posts. The problem is, I suck at math (i.e. differential geometry).

So please tell me if I got the basics right (your post #6):

The Milne frame is
$$\begin{array}{rcl} \vec{e}_1 & = & \frac{t}{\sqrt{t^2-r^2}} \, \partial_t + \frac{r}{\sqrt{t^2-r^2}} \, \partial_r \\ \vec{e}_2 & = & \frac{r}{\sqrt{t^2-r^2}} \, \partial_t + \frac{t}{\sqrt{t^2-r^2}} \, \partial_r \\ \vec{e}_3 & = & \frac{1}{r} \, \partial_\theta \\ \vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi \end{array}$$
The Minkowski basis vetors are $\partial_t,\partial_r,\partial_\theta,\partial_\phi$, with length $1,1,r,r \, \sin(\theta)$, respectively. (Q: is it ok to think of $\partial_t$ as implicitly acting on the interval s? i.e. change in interval per change in coordinate value?) So we construct an orthonormal basis at each event with scaled vectors
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_t\\ \vec{e}_2 & = & \partial_r \\ \vec{e}_3 & = & \frac{1}{r} \, \partial_\theta \\ \vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi \end{array}$$
(Q: This is no longer a coordinate basis? Is there some explanation for dummies as to what "nonholonomic" means in this context? If not, don't bother, I'll come back to it later)

Boosting with v=r/t yields the Milne frame.

Correct so far?

Two corrections for your subsequent paragraph:
The frame is defined for r<t, and the Hubble 'constant' scales as 1/t.

Now for something completely different:
#5 said:
debunk the notion that FRW dusts (or any reasonably accurate cosmological model) can be considered as "an inside out black hole" [sic],
You would disagree with de Sitter space looking like an "inside out black hole"?

Thanks

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Chris Hillman
In the interests of usability in the future, we should probably give a link to

The Minkowski basis vetors are $\partial_t,\partial_r,\partial_\theta,\partial_\phi$, with length $1,1,r,r \, \sin(\theta)$, respectively.
(You mean: the coordinate basis vectors.) Correct.

(Q: is it ok to think of $\partial_t$ as implicitly acting on the interval s? i.e. change in interval per change in coordinate value?)
I am tired, which may explain why I don't follow. Can you explain?

So we construct an orthonormal basis at each event with scaled vectors
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_t\\ \vec{e}_2 & = & \partial_r \\ \vec{e}_3 & = & \frac{1}{r} \, \partial_\theta \\ \vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi \end{array}$$
Yes, when the coordinate basis for our chart happens to consider of mutually orthogonal vector fields, we can obtain an ONB (frame field) by simply normalizing by dividing by the norms (lengths).

If the line element written in our chart is not orthogonal, we will need to do a bit of linear algebra (eventwise) to obtain frame fields. There are always infinitely many frame fields, of course, corresponding to our freedom to choose a great variety of families of ideal observers (and to choose smoothly varying spatial vectors for them). Generally one wants to focus on frame fields which can be used to directly study the physical experience of physically interesting observers. Often but not always this will involve choosing a nonspinning inertial congruence (the acceleration vector of the unit timelike vector field vanishes, so our observers are in a state of inertial motion, and the Fermi derivatives wrt same of the unit spacelike vector fields also vanish, so "nonspinning" wrt gyroscopes carried by our observers).

In practice, choosing an "interesting" frame field is a lot easier than I am probably making it sound. Usually it is pretty obvious which timelike unit vector field to choose, and usually it is then not very hard to obtain orthogonal spacelike unit vector fields. Often the simplest approach, as here, is simply to apply a suitable boost at each event to some simple frame field. Typically the most involves an undetermined function of one or more coordinates, and then computing and setting to zero the acceleration and or other quantities gives us an inertial (or rigid, or hypersurface orthogonal) frame field.

Q: This is no longer a coordinate basis?
Correct. An easy way to tell: the coordinate basis vector fields and the frame vector fields are simply vector fields, so we can compute commutators with abandon. A coordinate basis is holonomic because the commutators all vanish; a frame is (generally) anholonomic because the commutators will (generally) not all vanish.

Any scalar f on a domain $\Omega$ which is monotonic in the sense that $df \neq 0$ on $\Omega$ can be a coordinate, and it can belong to many local coordinate charts on the domain (simply connected open neighborhood). If we have four coordinates in this sense, they make a chart on the domain if their level surfaces are nowwhere tangent inside the domain, which is guaranteed if a certain four-form is nonzero
$$df^1 \wedge df^2 \wedge df^3 \wedge df^4 \neq 0$$
on $\Omega$. This is manifestly a coordinate-free notion, and if it holds, the volume form is a nowwhere zero scalar multiple of this four-form.

So we can check using coordinate-independent computations whether or not we can make a local chart on some domain using a specific list of coordinates.

Boosting with v=r/t yields the Milne frame.

Correct so far?
Exactly.

Two corrections for your subsequent paragraph:
The frame is defined for r<t, and the Hubble 'constant' scales as 1/t.
I can't find the place you must be looking at, but if you are talking about the Milne observers in Milne chart, that sounds right.

Now for something completely different:

You would disagree with de Sitter space looking like an "inside out black hole"?
This might be context dependent, since in some contexts there are both mathematical and physical similarities between cosmological horizons and event horizons. I was thinking of comments from various posters, frequently seen in the public areas, claiming that "cosmological models are equivalent to black hole models", which is absurd. This is an important point which I forgot to mention in my "note to self". I am too distracted to give a very coherent explanation right now, but very briefly:
• black holes are compact massive objects; models of an isolated black hole sans Lambda term will generally employ an asymptotically flat spacetime, in which the curvature is very strong near the hole, and vanishes as we move far away from it,
• cosmological models are typically perturbations of simple FRW dust models, so "almost homogeneous" and "almost isotropic", and on large scales, using some suitable (and not neccessarily obvious) notion of "averaging", there should be roughly speaking no huge differences between here and very distance regions

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Thank you.
I'll have more questions, but I need some time to think it through.

my question
Ich said:
is it ok to think of $\partial_t$ as implicitly acting on the interval s? i.e. change in interval per change in coordinate value?
I'm not good at abstract thinking, so I don't like a partial being there without something to do. I think that $\partial s / \partial t$ is the length of the t basisvector, namely 1?

Chris Hillman said:
Ich said:
Two corrections for your subsequent paragraph:
The frame is defined for r<t, and the Hubble 'constant' scales as 1/t.
I can't find the place you must be looking at, but if you are talking about the Milne observers in Milne chart, that sounds right.
Right below the https://www.physicsforums.com/showpost.php?p=2622276&postcount=6":
Chris Hillman said:
Notice that this is only defined inside the upper half cone with vertex at the origin, LaTeX Code: t>0, \\, t < r . The world lines of the Milne observers are the integral curves of LaTeX Code: \\vec{e}_1 , namely rays from the origin which lie inside this upper half cone. So the Milne observers are inertial observers. In terms of a cosmological model, we can consider them as model galaxies which obey a strict Hubble law, in which the Hubble constant is not only homogeneous on spatial hyperslices but not varying in time.

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Chris Hillman
I don't like a partial being there without something to do.
Oh! Sure, that is a very common reaction, and I have always planned to try to address it in a BRS post on vector fields, which would focus above all on trying to explain why writing a vector field as a first order linear partial differential operator (with variable coefficients) is a good idea in manifold theory, so good in fact that it is well worth getting used to what initially seems like a very odd way of thinking. So, let me get back to you on that with a BRS thread "What is a VF?" or something like that, and remind me if I don't start on that. At the moment I am distracted so unfortunately the rate/quality of BRS posts may be adversely affected while I battle hostile forces.

I think that $\partial s / \partial t$ is the length of the t basisvector, namely 1?
Sorry, I still don't get it.

Oh right, yes, should be $0 < r < t < \infty$, thanks! I am making a correction.

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It's great that you added the LTB dust, that's something I'd like to understand. I still haven't, and there are some places where I don't make progress. I'm still in an early phase, so most probably I'm struggling with some basic misunderstandings.

The line element is
$$\begin{array}{rcl} ds^2 & = & -dt^2 + \frac{R_r^2}{1+f} \, dr^2 + R^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right), \\ && t_{\rm min} < t < t_{\rm max}, \; r_{\rm min} < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi$$
[...]
Noteworthy special cases include:

* $f=0, \, R = r \, t$ is Minkowski vacuum (written in what chart? to what observers corresponds the given frame field?)
From my understanding, this should be an FRW chart in reduced-circumference coordinates (as Wiki calls it):
$$\Sigma}^2 = \frac{dr^2}{1-k r^2} + r^2 \Omega^2$$
But empty FRW-space is hyperbolic, so I get f = -kr² = r² (or r²/t²). Is this correct?
I think I understood the other two cases.

Now for the spatially flat, perturbed LTB dust:
There's also something that bothers me. After writing down the tensors, you say:
As you can see, the effects of the perturbation die out as t grows and also as r grows.
Ok, that's obvious from the maths, but it's the opposite of what I expected - structure formation, i.e. self-enhancing perturbations.
And it seems to be at odds with your identifications later:
If $a_r < 0$ , we have a spherically symmetric overdensity, and the Hubble flow shows a radial streaming toward the origin superimposed on the basic Hubble expansion
and
Here, the first term is the bare Hubble flow and the last term has the form of a Coulomb term induced by an "effective massive object" at the origin.
So we have two corrections to the Hubble flow, both inwards, and still the overdensity is vanishing?

Maybe a related issue: Your starting point is
In general, the three-dimensional Riemann tensor of each of these slices is nonzero. But we can consider the special case when it vanishes identically
so we have flat space. Flat space in the FRW case means $\rho \propto H^2$, so I'd expect a positive correction to the expansion tensor in an overdense region. This would also explain the decaying of the perturbation.
This "streaming" has the right sign (it's not necessarily infalling matter, it could also be expanding?), but acts only on r, so things seem to be more complicated.
A question: could it be that some kind of streaming (away from an overdensity) is built in the constraint f=0? Or did I severly misunderstand what you're doing here and what the math means?

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Chris Hillman
It's great that you added the LTB dust, that's something I'd like to understand. I still haven't, and there are some places where I don't make progress. I'm still in an early phase, so most probably I'm struggling with some basic misunderstandings.
That's the best kind, because they are shared by others (most likely). In this case, maybe even by me, since after a PM conversation with Haelfix I decided the "simple" LTB spherical perturbation of FRW dust with E^3 hyperslices models are trickier than I realized at the time I wrote the first two posts on them in this thread! From my understanding, this should be an FRW chart in reduced-circumference coordinates (as Wiki calls it):
$$\Sigma}^2 = \frac{dr^2}{1-k r^2} + r^2 \Omega^2$$
But empty FRW-space is hyperbolic, so I get f = -kr² = r² (or r²/t²). Is this correct?
I think you mean a line element for the spatial hyperslices (related to the RW chart?)
$$d\sigma^2 = \frac{dr^2}{1 - k \, r^2} + r^2 \, d\Omega^2$$
The key point about that is not what is different from euclidean space (the coeffiecient of dr^2) but what is the same (the angular part, showing that r is a Schwarzschild radial coordinate, aka areal radial coordinate, since the surface area of the Riemannian two-sphere r=r0 is $A = 4 \pi \, r_0^2$ aka curvature radial coordinate, since the curvature of the Riemannian two-sphere r=r0 is 1/r0^2. It seems that if you take k=0 you recover the polar spherical chart on E^3.

I wrote the LTB dust using a comoving chart of form
$$ds^2 = -dt^2 + \frac{R_r^2}{1+f} \, dr^2 + R^2 \, d\Omega^2$$
where R is a function of t,r only and f is a function of r only. Here, f=0 gives the case of E^3 hyperslices; otherwise the hyperslices are in general inhomogeneous Riemannian three-manifolds. In general, f can be arbitrary but R must satisfy the PDE
$$R_{tt} = \frac{f-R_t^2}{2 \, R}$$
• The case $R=rt, \; f=0$ gives the line element
$$ds^2 = -dt^2 + t^2 \; \left( dr^2 + r^2 \, d\Omega^2 \right)$$
which is locally flat, so a vacuum (special case of dust). Here, the world lines of the dust particles becomes world lines of timeline congruence of world lines of observers who are none other than the Milne observers. Or so I said--- now I think I need to check that, but my notes are not at hand [EDIT: sure enough, that was a brain blip on my part: the Milne chart for the Minkowksi vacuum is comoving with observers who are linearly and homogeneously expanding, and it resembles (by design) expanding cosmological models with the world lines of the galaxies corresponding to the world lines of the Milne observers (test particles, not dust particles). The line element of the Milne chart is
$$ds^2 = -dt^2 + t^2 \; \left( dr^2 + \sinh(r)^2 \, d\Omega^2 \right)$$
which is the H^3 hyperslice version of the line element (E^3 hyperslices!)
$$ds^2 = -dt^2 + t^2 \; \left( dr^2 + r^2 \, d\Omega^2 \right)$$
What tipped me off was that I know that the Milne observers have H^3 hyperslices, while the observers in the second model have E^3 hyperslices. Sorry, everyone!]

Now for the spatially flat, perturbed LTB dust: ...it's the opposite of what I expected - structure formation, i.e. self-enhancing perturbations.
Yes, I know. From dynamical systems theory we expect that perturbations will be smoothed out by expansion; a good reference is the review paper by Ellis, Cosmological Models (in the arXiv). So structure formation appears to involve something more clever than the simple LTB dusts I explored.

And it seems to be at odds with your identifications later:

So we have two corrections to the Hubble flow, both inwards, and still the overdensity is vanishing?
Yes, I messed up my first try at explaining that, and said I'd have to come back later when I'd had a chance to fix it, which I haven't yet had the chance to do.

Flat space in the FRW case means $\rho \propto H^2$, so I'd expect a positive correction to the expansion tensor in an overdense region. This would also explain the decaying of the perturbation.
I'll have to think and get back to you, but off the top of my head I think that is only valid for certain charts, and not valid for the chart I am working with.

Part of the confusion is that it turns out to be not entirely straightforward to distinguish an overdense region in this chart, because the proper time to the BB taken along the world line of each dust particle varies with r in the chart I used. You can have slices t=t0 which are orthogonal to the world lines, or you can have slices with constant proper time since the BB, but not both (in general)!

This "streaming" has the right sign (it's not necessarily infalling matter, it could also be expanding?), but acts only on r, so things seem to be more complicated.
Yes, things are more subtle than I thought when I wrote the first two posts on the perturbation (working from old notes, and assuming that the younger me had been making things too complicated--- apparently not!).

A question: could it be that some kind of streaming (away from an overdensity) is built in the constraint f=0?
I'll have to think about that. IIRC, f=0 is precisely the condition for E^3 hyperslices, so I think you would be guessing that in this case, a positive overdensity is associated with streaming away from the origin. I think everyone (including me) needs to think harder about what we really want the term "region of overdensity" to mean, due to the subtlety mentioned above about the variable "distance" along the world lines of various dust particles from BB to a slice t=t0.

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I think I've made some progress with the LTB dust. In fact, when I combined the knowledge from this thread with that in the http://en.wikipedia.org/wiki/LTB_dusts" [Broken]article, everything suddenly made so much sense that I got suspicious again. Maybe you can check what I'm writing here, it'd really be a pity if I misunderstood something or had wrong information - I know you're wary of Wikipedia.

First important thing: the constant f equals 2E, with E the total specific energy of a particle in a given shell. That makes sense, as it corresponds to the open/flat/closed - ever expanding/recollapsing thing for matter dominated FRW models. Remarkably, in LTB coordinates, the expression for energy is formally identical to the Newtonian one:
$$E=\dot R^2 /2 - M/R$$
where M is all the mass enclosed by said shell.
$$R_{tt} = \frac{f - R_t^2}{2 \, R}$$
we get the extraordinarily simple equation
$$\ddot R = -M/R^2$$
So we see three things:
1. We're dealing with a formally Newtonian model (some subtleties aside)
2. The universe outside a given shell is completely irrelevant (Birkhoff)
3. The universe inside a given shell is completely described by one number, its mass (also Birkhoff)

If there is a local overdensity in a spatially flat (i.e. zero total energy) universe, we can see from the energy equation that indeed (itex]\rho\propto H^2[/itex], and one suspects that this is the reason for the counterintuitive behaviour (dilution) of said inhomogeneity for two reasons:
1. The faster expansion will naturally dilute the overdensity.
2. A spatially flat universe is described by one number, its age. An overdensity means a universe that is some dt younger compared to the outside universe far away. Both follow FRW laws and keep being synchronized, so said dt becomes less important as the universes evolve, and the overdensity vanishes.

I made some simulations for a (initially) flat universe 10 Gy old, with a 10% gaussian overdensity of 1 Gly radius at that time. I started it with two different initial conditions:
1. Overall spatial flatness, E=f=0. That means, H larger in the overdensity.
2. Same Hubble parameter everywhere (that's a tricky one, btw, as H is defined on proper distance, not LTB's R). That means, E<0 in the overdensity.
From E<0 in the second case, one sees that the inhomogeneity will finally collapse, which is the expected behaviour.
I attached the respective diagrams.

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Chris Hillman
(mumble, mumble...)

(...hours later:)

That sounds basically correct.

the constant f equals 2E, with E the total specific energy of a particle in a given shell
This energy is constant on a specific shell r=r_0, but (for a generic LTB dust solution) varies with r.

• The faster expansion will naturally dilute the overdensity.
• A spatially flat universe is described by one number, its age. An overdensity means a universe that is some dt younger compared to the outside universe far away. Both follow FRW laws and keep being synchronized, so said dt becomes less important as the universes evolve, and the overdensity vanishes.
I think that is what I was saying? Except that the LTB dusts representing spherical perturbation of an FRW dust with E^3 hyperslices include perturbations which retain the locally flat hyperslices property, and then the age varies from shell to shell.

So we see three things:
• We're dealing with a formally Newtonian model (some subtleties aside)
• The universe outside a given shell is completely irrelevant (Birkhoff)
• The universe inside a given shell is completely described by one number, its mass (also Birkhoff)
Agreed. And very nice to see someone is really on the ball! Can you make png images instead of pdf?

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Chris Hillman
Of course there's a long version...

I tried, I really did, but I just cannot write a short post... sigh... so please bear with me, I have lots more to say.

I remarked that the LTB dusts deserve a thread of their own. Excellent background reading for cosmology generally: Ellis & Elst, Cosmological Models, http://arxiv.org/abs/gr-qc/9812046. And an excellent chapter (!) on LTB dusts can be found in the textbook by the late Jerzy Plebański and Andrzej Krasiński, An Introduction to General Relativity and Cosmology, Cambridge University Press, 2006. Unfortunately, as I write, I do not have this book at hand.

First, your constraint equation is just a first integral of the one I gave, and it is pretty standard. The Wikipedia article appears to be based upon the discussion near (15.33) in the monograph by Stephani et al., Exact Solutions of Einstein's Field Equations, Cambridge University Press, 2003. The form of the metric is due to Tolman, IIRC, but AFAIK the first integral is due to Bondi, as I think I mentioned.

Second, the LTB dusts include exact dust solutions for which roughly speaking we can envisage the matter as expanding/contracting S^2, E^2, and H^2 shells of dust. The case I discusssed is of course the one in which the matter consists of expanding or constracting S^2 shells. Here, the expanding case gives cosmological models (with a well defined family of hyperslices orthogonal to the world lines of the dust particles) which can thought of as spherical perturbations of FRW dusts (which are, in terms of the spatial hyperslices orthogonal to the world lines of the dust particles, homogeneous and isotropic). The contracting case can be thought of as models of spherically symmetric gravitational collapse of pressureless matter (somewhat dubious at large densities, of course). I focused in particular on the case when the hyperslices are locally flat.

Third, in more detail, the form of the line element given in Stephani et al. for the comoving chart is basically this:
• case of S^2 shells of dust:
$$ds^2 = -dt^2 + \frac{R_r^2}{1+2\,\epsilon} \, dr^2 + R^2 \; \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right)$$
where R is a function of t,r only and subject to a first order constraint,
• case of E^2 shells of dust
$$ds^2 = -dt^2 + \frac{R_z^2}{2\,\epsilon} \, dz^2 + R^2 \; \left( dr^2 + r^2 \, d\phi^2 \right)$$
where R is a function of t,z only and subject to a first order constraint; this can be rewritten
$$ds^2 = -dt^2 + \frac{R_z^2}{2\,\epsilon} \, dz^2 + R^2 \; \left( dx^2 + dy^2 \right)$$
and if we like, we can then replace the planar shells with cylindrical shells, which probably looks more perspicuous:
$$ds^2 = -dt^2 + \frac{R_r^2}{2 \, \epsilon} \, dr^2 + R^2 \; \left( dz^2 + d\phi^2 \right)$$
• case of H^2 shells of dust:
$$ds^2 = -dt^2 + \frac{R_r^2}{-1+2\,\epsilon} \, dr^2 + R^2 \; \left( d\theta^2 + \sinh(\theta)^2 \, d\phi^2 \right)$$
where R is a function of t,r only and subject to a first order constraint,
These are comoving charts, so $r=r_0$ simply labels of shell of dust having constant density, not a "radial distance". In all cases, $\epsilon$ is a function of r only which can be chosen more or less arbitrarily, and $\epsilon(r_0)$ is the specific kinetic energy of a dust particle at r=r_0, and in all cases the constraint should then have the Newtonian form you mentioned, with $m(r_0)$ interpreted as the total mass "below" the shell r=r_0 (or z=z_0). The Lie algebra of Killing vector fields respectively corresponds to SO(3), E(3), SO(1,2) acting on "nested" S^2, E^2, H^2 shells, confirming that the matter consists of shells of dust with constant mass density (on shells r=r0) with respectively S^2, E^2, H^2 geometry, "nested" about some point (in the hyperslice t=t0).

For the first case, denoting a derivative wrt t by an overdot, and a derivative wrt r by a prime, the constraint can be written in the form
$$\dot{R}^2/2 = m/R + \epsilon$$
Here, we should think of choosing m and \epsilon (two functions of r) and then solving for R. We should choose the positive sign of $\dot{R}$ for a model with expanding shells (as in a cosmological model of an expanding universe), and the negative sign for a model of spherically symmetric gravitational collapse, i.e.
$$\dot{R} = -\sqrt{m/R + \epsilon}$$

The Riemann tensor has two simple eigenvalues
$$\frac{2m}{R^3}, \; \; \frac{2m}{R^3} - \frac{m^\prime/R^\prime}{R^2}$$
and two double eigenvalues
$$\frac{-m}{R^3}, \; \; \frac{-m}{R^3} + \frac{m^\prime/R^\prime}{R^2}$$
which shows again the importance of the loci $R=0$ (initial or final singularities) and $R^\prime=0$ (shell-crossing singularities).

The frame of the dust particles is of course, in the spherical shell case,
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_t \\ \vec{e}_2 & = & \frac{\sqrt{1+2\, \epsilon}}{R^\prime} \; \partial_r \\ \vec{e}_3 & = & \frac{1}{R} \, \partial_\theta \\ \vec{e}_4 & = & \frac{1}{R \, \sin(\theta)} \, \partial_\phi \end{array}$$
and similarly for the other cases. In all three cases, the sole nonvanishing component (expanded wrt the frame field) of the Einstein tensor shows that the matter density is
$$\mu = \frac{m^\prime/R^\prime}{4 \pi \, R^2}$$
The tidal tensor (wrt the timelike geodesic congruence consisting of the world lines of the dust particles) has the memorable diagonal form
$${ E \left[ \vec{e}_1 \right] }_{ab} = \frac{m^\prime/R^\prime}{R^2} \; \operatorname{diag}(1,0,0) + \frac{m}{R^3} \; \operatorname{diag}(-2,1,1)$$
At a shell-crossing singularity, only the radial component of the tidal tensor, $E_{22}$, blows up.

In the case of spherical shells, this last expression is probably the easiest way to see why we interpret m(r_0) as the total mass inside the spherical shell of dust at r=r0. This shell has surface area $A(r_0) = 4 \pi \, R(t_0,r_0)^2$, so the mass inside this shell doesn't change as t increases--- which makes perfect sense, since the chart is comoving with the dust particles and, naively, if no mass enters or leaves the shell, the mass inside shouldn't change--- but its surface area shrinks (gravitational collapse case) or expands (expanding cosmological model case)!

The three-dimensional Riemann tensor of the hyperslice t=t0 has
$$r_{2323} = r_{2424} = \frac{-\epsilon^\prime/R^\prime}{R}, \; \; r_{3434} = \frac{-\epsilon/2}{R}$$
so we have locally flat hyperslices when we choose $\epsilon=0$.

If m = m_0 is constant at some r=r_0, the density vanishes at r=r_0, and the tidal tensor shows we have pure Coulomb tidal accelerations produced by a spherically symmetric object of mass m=m_0.

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Chris Hillman
A better way of writing the perturbation of the FRW dust with E^3 slices

Ich said:
If there is a local overdensity in a spatially flat (i.e. zero total energy) universe, we can see from the energy equation that indeed (itex]\rho\propto H^2[/itex]
I assume that by H you mean the diagonal components of the expansion tensor in the FRW background. But in the LTB dust the radial component of the expansion tensor differs from the tangential components. The density is
$$\mu = \frac{m^\prime/R^\prime}{4 \pi \, R^2} = \frac{m^\prime/R^\prime}{A}$$
where A(t_0, r_0) is the surface area of the spherical shell r=r_0 at time t=t_0. The tangential part of the expansion tensor (the part which isn't affected by the perturbation) is
$${H\left[\vec{e}_1\right]}_{33} = {H\left[\vec{e}_1\right]}_{44} = \pm \sqrt{2m/R^3 + 2 \epsilon/R^2}$$
(choose the sign for expanding or collapsing dust).

[EDIT: oh, I see, you were saying you want to take H as a function of R, not r. We agree, I guess, that for large t, $R \approx k \, r \, t^{2/3}$ for some positive constant k, but for small r, the relation between r and R is seen to be nonlinear.]

From E<0 in the second case, one sees that the inhomogeneity will finally collapse, which is the expected behaviour.
Agreed:
• shells $r=r_0$ with $\epsilon(r_0) < 0$ (the gravitationally bound case) should collapse,
• shells with $\epsilon(r_0) = 0$ should expand but "asymptotically coast to a halt" (marginally bound),
• shells with $\epsilon(r_0) > 0$ should be unbound
We can make LTB models in which nested shells having each of these characteristics are dispersed in various ways. If we have collapsing shells outside expanding shells, we can expect to see shell-crossing singularities, for example. And we can obtain crude models showing how primordial gas clouds can collapse to form protostars, a physical process proceeding upon a grand Hubble background expansion. And decades ago, Gautreau applied the theory of LTB dusts to the old conundrum about whether or not the orbits of planets participate in the Hubble expansion (answer: not appreciably). These issues are all worth exploring in detail because they actually come up (usually in garbled or disguised form) in the public areas at PF.

In the case of E^3 hyperslices, $\epsilon$ vanishes identically for all r, and we should expect the expansion to smooth out any initial (spherically symmetrical!) inhomogeneities in the density. We can rewrite the solution I gave previously in the more perspicuous form
$$R = \left( \frac{9m}{2} \right)^{1/3} \; ( t-b)^{2/3}$$
where m(r_0) is again the mass inside the spherical shell labeled by r=r_0, and where b(r_0) is the time of the final singularity, which depends on r as I discussed previously. Plugging this into the constraint equation as given by Bondi,
$$\dot{R}^2/2 = m/R + \epsilon$$
we can verify that in order to obtain E^3 hyperslices we should take $\epsilon=0$. (That is, the "kinetic energy" of each dust particle vanishes.)

Then the tangential parts of the expansion tensor (of the timelike geodesic congruence of world lines of dust particles) and the tidal tensor (taken wrt the same congruence) are:
$$H_{33} = H_{44} = 2/3/(t-b), \; \; E_{33} = E_{44} = 2/9/(t-b)^2$$
which agrees with the FRW dust with E^3 hyperslices. But the radial components are different from the FRW background:
$$H_{22} = \frac{ m/R^2 - m^\prime/R^\prime/R - \epsilon^\prime/R^\prime }{\sqrt{2} \, \sqrt{m/R+\epsilon}}$$
and
$$E_{22} = -\frac{2 m}{R^3} + 4 \pi \, \mu$$

To obtain general perturbations of the FRW dust with E^3 hyperslices, we need to allow the spatial geometry of the hyperslices to vary as well, i.e. allow nonzero $\epsilon$.

This is actually well known, so I guess I really messed up my attempt to explain clearly. (I was working from old notes because I was having system problems, which alas have just reappeared.) My mistake was trying to explain just the FRW perturbation without creating a separate thread on the LTB dusts, which clearly are of independent interest. However, I think I inadvertently created a good learning experience for us all! When I get a chance, I hope to reobtain the textbook by Krasinski & Plebanski, and to work up a BRS on the LTB dusts, explaining better things like shell-crossing singularities. For those who know about the Lichernowicz matching conditions and don't want to wait, try making two LTB dust solutions with two different Rs and matching them across a sphere. Recalling what we said about the Birkhoff theorem for spherically symmetric dusts--- that is, consider what happens when the dust density vanishes on between two shells r0 < r < r1--- and see if you can make a model in which an expanding and contracting shell of dust collide. The surface area of the shell generally varies with t, r, but should develop an inflection point at a shell-crossing singularity R^\prime = 0.

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Chris Hillman
Spherical perturbation of FRW Dust with E^3 Hyperslices, refactored

I'm probably only restating what Ich said, but let me try again (when I'm sure I've got a really good way of explaining this, I'll try posting in the original thread):

Returning for a moment to the general LTB dusts for S^2, E^2, H^2 shells, the constraint equation
$$\dot{R}^2/2 = m/R + \epsilon$$
works for all of them; we should think of this as advising us to choose m, \epsilon (two suitable functions of r) and then determining R from these (up to a third function of r, whose role we can study). Furthermore, for all three types of shell geometry, we obtain the same expressions for the mass density of the dust
$$\mu = \frac{m^\prime/R^\prime}{4 \pi \, R^2}$$
expansion tensor, and tidal tensor (electroriemann tensor, not electroweyl tensor!) (taken wrt the timelike geodesic congruence consisting of the world lines of the dust particles). The form of the metric tensor (and the abstract structure of the Lie algebra of Killing vector fields, which in general is three dimensional) unambiguously distinguishes between the case of spherical, cylindrical, and pseudospherical shells, however.

But notice that the we must assume something different about $\epsilon$ (the kinetic energy per dust particle, as a function of r--- remember that r simply labels the shells of dust, so it is constant for a given particle) depending on whether we are using spherical, planar/cylindrical, or pseudospherical shells respectively. Specifically, $\epsilon > -1/2, \; \epsilon > 0, \; \epsilon > 1/2$ respectively.

But we always have a vacuum case for each of these. If m is constant on $r_1 < r < r_2$, we obtain a portion of the Levi-Civita AI, AIII, or AII static type D vacuums respectively, i.e. static spherically symmetric vacuum (Schwarzschild vacuum), static plane or cylindrically symmetric vacuum (plane symmetric Kasner vacuum, aka plane symmetric Taub vacuum), and static H^2 symmetric vacuum respectively. Furthermore, choice of $\epsilon$ simply amounts to a choice of a certain family of inertial observers in the vacuum region. So, once we have proven that the LTB dusts are the unique dusts with nested spherical, cylindrical, and pseudosphericall shells respectively, we will obtain a generalized Birkhoff theorem for dusts (which can be further generalized to charged dusts and so forth).

In addition, for suitable choice of epsilon, we have some FRW cases, but in the case of cylindrical shells we cannot obtain the bound case (i.e. recollapsing case, for which we would need epsilon < 0), and for H^2 shells, we cannot obtain the bound or marginally bound cases.

Next, let's consider spherical perturbations of the FRW dust with E^3 slices in which we keep the property that the family hyperslices orthogonal to the world lines of the dust particles are locally flat. These are precisely the LTB dusts with spherical shells and with kinetic energy $\epsilon=0$, as we saw above. Then we can solve the constraint equation in closed form, and after some manipulation, we can put the solution in the form
$$R = (9m/2)^{1/3} \; (t-b)^{2/3}$$
where m, b are functions of r only, and where we already know that m(r_0) shall be interpreted as the mass contained inside the shell r=r_0. Then the initial or final curvature singularity as the locus where R=0 corresponds to $t=b$, so b also has a vivid interpretation.

The vacuum case is obtained when m is constant, and then b=0 corresponds to the case of expanding Lemaitre observers (flipping the sign of our time coordinate gives the case of contracting dust matching to a contracting Lemaitre observers in the vacuum region).

The case of the FRW dust (with E^3 slices) is obtained when m is proportional to r^3, with b constant (without loss of generality we can take b=0).

The simplest way to obtain a nontrivial spherically symmetric perturbation (maybe too simple!) of the FRW dust with E^3 hyperslices is to keep m proportional to r^3 but to allow b to be a Gaussian (asymptotic to zero as r increases):
$$m = r^3, \; \; b = k \, \exp(-c \, r^2/2)$$
where k, c > 0 are real constants controlling the amplitude and standard deviation of the Gaussian perturbation. We'll see that k > 0 will correspond to an overdensity near r=0 (and also to a shorter "proper time since the Big Bang") while k < 0 will correspond to an underdensity near r=0 (and also to a longer "proper time since the Big Bang").

(Some of you may recall, BTW, that I once suggested developing a fringe theory based on the idea of interpreting probability densities in E^3 as gravitational potentials, so that we can imagine probability densities which are nonzero on different domains as interacting gravitationally, to be interpreted in faux-Jungian fashion, as a way of showing that with mathematics one can come up with much more interesting fringe theories than the cranks can!)

You are probably wondering: how is this different from simply desynchronizing the clocks carried by observers riding on the FRW dust particles, so that the initial or final singularity "occurs earlier or later depending on r"? The density on the spatial hyperslice t=t_0 is no longer homogeneous, but you might well say that if we redefined the time coordinate in some r dependent way, simply because the particle at r=0 now things t=1 is earlier or later than in the original FRW chart, the density should appear to vary as an artifact of our desychronization. But the family of hyperslices which is everywhere orthogonal to the world lines of the dust particles is unique, and defined by a coordinate-free proprerty. And in our alleged perturbation, these slices are simply t=t_0. But now we have inhomogeneous density, whereas previously we had homogeneous density on the spatial hyperslices, and this is also a coordinate free property. Thus our alleged perturbation must give a different spacetime from the original FRW dust. So we really do have a nontrivial perturbation.

Taking the case of an expanding dust for simplicity, here are some points to bear in mind:
• the matter consists of expanding shells of dust,
• each shell has constant density at a given time t=t0,
• the coordinate r labels the shells of dust; that is, the sheets r=r0 are the world sheets of the shells of dust,
• the slices t=t0 are the ones orthogonal to the world lines of the dust particles,
• the slices t-b = t0 are the ones having constant proper time (taken along the world lines of the dust particles) to the initial singularity t=b, but these are not orthogonal to the world lines of the dust particles!,
• the sheets R = R_0 are the sheets of spheres of constant surface area $A = 4 \pi \, R_0^2$,
• for large R, we have $R \approx r\, t^{2/3}$, i.e. the sheets r=r_0 are expanding in FRW fashion except near r=0,
• the sheets r=r_0 appear static in our comoving chart, but are actually expanding in surface area, like $A \approx 4 \pi \, r_0^2 \, t^{4/3}$ for large r_0, in the case of a Gaussian perturbation, and for small r, expanding less quickly than expected by this rule, in the case of a Gaussian overdensity.

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Chris Hillman
Comparision with Chapter 18 in the textbook by Plebanski and Krasinski

OK, I reobtained the textbook by Plebanski and Krasinski.

I forgot to say that Krasinski wrote an earlier monograph on the formation of voids in LTB dust models, so it is not surprising that this excellent textbook contains an entire chapter (chapter 18) on LTB dusts!

I think what I said in the last few posts is consistent with what Krasinksi says in chapter 18 (if not, of course, I should defer to him), with one apparent exception: he stresses (section 18.5) that FRW models are unstable against density perturbations (i.e. they should grow). I am pretty sure that refers to mostly expanding LTB dusts in which we allow shells with negative energies. The expanding LTB dusts containing only zero energy shells (the ones with E^3 hyperslices, the ones I discussed in the other thread) should indeed smooth out density inhomogeneities over time.

BTW, one thing I like about this textbook is that the authors use frame fields quite a bit, and often give components of important tensors wrt a frame field rather than a coordinate basis. Of course, this is what all savvy researchers do (as appropriate).

Another useful snippet: contrary to my recollection, it seems that everything I said in the previous two posts is actually due to Lemaitre 1933, who even considered the extension to nonzero Lambda, which has the constraint
$$\dot{R}^2/2 = m/R + \epsilon + \Lambda/6 \, R^2$$
Tolman 1934 then used LTB dust models to study inhomogeneities in collapsing FRW dusts, and concluded these are generally unstable against density perturbations. (As I said, I don't think that contradicts what I said, that the expanding LTB dusts with E^3 hyperslices have the opposite behavior of smoothing out density perturbations over time.) Then Bondi 1947 reinvigorated the study of LTB dusts by applying the kinematic decomposition (acceleration vector, expansion tensor, vorticity tensor) of a timelike congruence to the world lines of the dust particles in LTB dusts.

As I said, the case of the LTB dusts with spherical shells admits the possibility of including shells r=r_0 with negative, zero, or positive energies $\epsilon(r_0)$. In particular, we can recover the corresponding FRW dusts (the ones with hyperslices orthogonal to the world lines of the dust particles which have respectively S^3, E^3, H^3 geometry, or are discrete quotient manifolds of same) using
• $\epsilon = -r^2/2, \; m = r^3$; then R is a cycloid given parametrically by
$$R = \frac{m}{|\epsilon|} \; (1-\cos(\eta))/2, \; \; \eta - \sin(\eta) = \frac{(-2\epsilon)^{3/2}}{m} \; (t-b)$$
• $\epsilon = 0, \; m = r^3$; then we can solve the constraint in the form $R = (9m/2)^{1/3} \; (t-b)^{2/3}$,
• $\epsilon = r^2/2, \; m = r^3$; then R is given parametrically by
$$R = \frac{m}{\epsilon} \; (\cosh(\eta)-1)/2, \; \; \sinh(\eta) - \eta = \frac{(2\epsilon)^{3/2}}{m} \; (t-b)$$
which is the same result as found in Newtonian gravitation for a particle attracted by a point mass.

To orient yourself when studying a particular LTB dust solution, I recommend starting with the comoving chart we have been using in this thread, suppressing the angular coordinates, and plotting the initial/final singularities $R=0$, then drawing some world lines of dust particles (vertical straight line segments in this chart), then plot contours of R as a function of t,r. This should show how each shell $r=r_0$ expands and/or contracts as time increases. Then, look for and plot shell-crossing singularities (where $R^\prime/m^\prime$ vanishes).

I promised to say more about shell-crossing singularities. Recall that (look at the expressions I gave for the eigenvalues of the Riemann tensor of the LTB dusts with spherical shells) there are two ways the Riemann curvature can blow up:
• $R = 0$ (initial or final singularity)
• $R^\prime = 0$ but $m^\prime \neq 0$ (shell-crossing singularity)
In the first case, the density, expansion tensor and tidal tensor (of the dust) all blow up, so these are "strong" curvature singularities. In the second case, the density and tidal tensor diverges but the expansion tensor does not (in general); verifying this requires taking a suitable limit of $H_{22}$. Thus, shell-crossing singularities do not rip apart or crush small configurations of dust, although they do feature briefly infinite densities and tidal accelerations. Thus, they are "weak" curvature singularities. The comoving chart suffers a coordinate singularity at a shell-crossing singularity, but by changing coordinates we can extend through this locus.

At a shell-crossing singularity $t=t_0, \; r=t_0$, we have $R^\prime = 0$. That means that two shells at $r=r_0-\delta, \; r=r_0+\delta$ have the same Schwarzschild radial coordinate $R(r_0)$, to first order, at $t=t_0$ so shells are colliding at $t = t_0, \; r = r_0$. As I said, in an LTB dust with S^2 slices, we can arrange to have negative energy shells outside positive energy shells, and then such collisions are inevitable, somewhere, somewhen. I said that dust models generally are a bit unrealistic because we insist on zero pressure even at very high densities; really one should think of shell-crossing singularities as places where matter forms acoustic waves with high density and pressure, which could lead to more interesting physics than simply having two shells pass through one another. But the idea of two shells passing through each other is not ipso facto absurd; if you have seen the movie Gravitas you know that this is perfectly acceptable in various Newtonian models, and likewise for gtr.

Another important topic I have yet to discuss are the principle null geodesic congruences in LTB dusts with S^2 slices. These are the ones with spherical wavefronts which are respectively expanding or contracting about r=0. At some events the optical expansion scalar of one of the principle null geodesic congruences might change sign; that happens, by definitiion, at an apparent horizon. We can also consider event horizons, which are of course defined globally, and this is particularly relevant in models in which some of the dust shells collapse to form a black hole. But apparent horizons are interesting more generally, because it can happen that a pair of "white" and "black" apparent horizons intersect, forming a neck, which gives two regions of our spacetime which cannot communicate, analogously to the two exterior sheets in the maximal analytic extension of the Schwarzschild vacuum. Here, at a "white" AH, the signs change from ++ (as in the past interior region of Schwarzschild vacuum) to +- (as in an exterior region), whereas at a "black" AH, the signs change from +- to -- (as in the future interior region of Schwarzschild vacuum). In an exterior region of the Schwarzschild vacuum, or anywhere in Minkowksi vacuum, the optical expansion scalars of the two principle null geodesic congruences are opposite; one is expanding and the other contracting. In the interior of a black hole, both are expanding, or both are contracting.

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Hi Chris,

I'm a bit slow digesting all your contributions, sorry. I'll pick out some interesting points.

Ich said:
* The faster expansion will naturally dilute the overdensity.
* A spatially flat universe is described by one number, its age. An overdensity means a universe that is some dt younger compared to the outside universe far away. Both follow FRW laws and keep being synchronized, so said dt becomes less important as the universes evolve, and the overdensity vanishes.
I think that is what I was saying? Except that the LTB dusts representing spherical perturbation of an FRW dust with E^3 hyperslices include perturbations which retain the locally flat hyperslices property, and then the age varies from shell to shell.
Yes, you had a decaying overdensity. The reason for the decay was not so clear, as you identified two contributions of inward streaming matter (gravitational attraction and a "radial streaming"), but none outward. It seems now that the "streaming" is rather outward, due to the E=const condition.

Can you make png images instead of pdf?
Yes, attached.

Second, the LTB dusts include exact dust solutions for which roughly speaking we can envisage the matter as expanding/contracting S^2, E^2, and H^2 shells of dust.
Does this mean there are LTB models w/o spherical symmetry?

In all cases, $\epsilon(r_0)$ is a function of r only which can be chosen more or less arbitrarily, and $\epsilon(r_0)$ is the specific kinetic energy of a dust particle at r=r_0, and in all cases the constraint should then have the Newtonian form you mentioned, with m(r_0) interpreted as the total mass "below" the shell r=r_0 (or z=z_0).
$\epsilon$ is the total energy, not the kinetic energy.

Ich said:
If there is a local overdensity in a spatially flat (i.e. zero total energy) universe, we can see from the energy equation that indeed $\rho\propto H^2$
I assume that by H you mean the diagonal components of the expansion tensor in the FRW background.
"H" means - in my numerical solution - velocity difference times distance in radial direction. For consecutive shells i-1, i:
$$H = \frac{\dot R_i-\dot R_{i-1}}{R_i-R_{i-1}}\sqrt{1+2\epsilon}$$
Now that I'm writing this I'm no longer sure this is correct. It was meant to reproduce locally the operational definition of H, therefore reproducing the FRW values in a homogeneous universe. But I think I can drop the sqrt-factor?

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Chris Hillman
Simple examples of LTB dusts with zero energy spherical shells

Continuing my Post #11 (Ich, I'll look at what you wrote when I'm done with this), let me add some figures for some simple examples of the LTB dusts with zero energy spherical shells (and thus, with E^3 spatial hyperslices), written in the Lemaitre comoving chart
$$ds^2 = -dt^2 + (R^\prime)^2 \, dr^2 + R^2 \; \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2$$
In a subsequent post, I'll look at examples of LTB dusts in which various shells have positive or negative energies.

[EDIT: I've been calling that the Tolman chart, following numerous previous authors, but it is due to Lemaitre. I am always struck by just how good Lemaitre's instincts have turned out to be--- he included Lambda, which I would no doubt have considered excessive generality in 1933! And in 1933 he was one of the very few who understood that r=2m was merely a coordinate singularity in the familiar Droste-Weyl chart covering the static exterior region, and he understood this in the context of what we now call LTB dusts. Recently, some knowledgable and historically sensitive authors have been wondering aloud why Lemaitre has traditionally been given no credit for his numerous achievements in classical gtr/cosmology. I think the best guess one can make now is that Einstein himself was deeply unnerved in some visceral way by mathematical cosmology--- a subject pretty much founded by Lemaitre, with the essential first step taken by Friedmann--- and in addition, I am pretty sure I would have been deeply suspicious of his motives in 1933, although I hope I would have appreciated that he was doing very good work. There is a revealing photographic record of the first meeting of Lemaitre and Einstein a few years after Lemaitre published his work on the LTB dusts. In the photograph, Lemaitre, who sincerely admired Einstein's work and who had long wanted to meet him, is dressed very carefully and is beaming with joy; Einstein is characteristically rumpled and appears uncharacteristically rattled.]

• suppressing the angular coordinates,
• finding and plotting the loci $R=0$ but $m >0$ (initial and final singularities),
• plotting world lines of dust particles as vertical segments between these curves,
• finding and plotting the loci $R^\prime=0$ but $m^\prime \neq 0$ (shell-crossing singularities),
• plotting contours of R (the Schwarzschild radical coordinate, a positive function of t,r)
Recall that for the case when all the shells have zero energy (a special case, but general enough to display some interesting possiblities, as it turns out!), we can solve the constraint in the form
• expanding dust: $R = (9m/2)^{1/3} \; (t-b)^{2/3}$
• collapsing dust: $R = (9m/2)^{1/3} \; (b-t)^{2/3}$
where m,b are functions of r only.

Recall that values of r just label shells of dust; these can even be negative, as long as R (the corresponding Schwarzschild radial coordinate) is nonegative. The world sheets of the shells appear as vertical segments.

The simplest interesting example is a collapsing Oppenheimer-Snyder dust ball, for which we can take,
$$\begin{array}{rcl} ds^2 & = & -dt^2 + \frac{(R^\prime)^2}{1 + 2 \, \epsilon} \; dr^2 + R^2 \, d\Omega^2, \\ && R = \left( \frac{9m}{2} \right)^{1/3} \; (b-t)^{2/3}, \\ && -\infty < t < t_{\rm max} \end{array}$$
where m,b are given by
$$\begin{cases} m = m_0 \, r^3, \; b = 0 & 0 < r < r_0 \\ m = m_0, \; b = r & r_0 < r < \infty \end{cases} \right.$$
where $r=r_0$ is the surface of the collapsing dust ball.

Notice that R is continuous but not C^1 at r=r_0. Because the density suddenly jumps from zero to a nonzero value at the surface of the dust ball, so the curvature tensors will jump there too, while the metric tensor is continuous there. This is fine. From the figure (see below), you can see that
• the shells are collapsing (as time increases, each world sheet crosses contours corresponding to decreasing R),
• shells of dust in the dust region collapse slower than shells of Lematire observers in the vacuum region, because these shells only enclose a portion of the total mass of the dust ball (and no shell cares about shells further out), so the contours of R bend upward as we enter the dust region,
• all the shells collapse to R=0 at the same time, t=0.
The OS examples are characterized by the fact that the density is constant on each t=t_0 slice in the dust regions, and each hyperslices is locally flat, so the dust region is locally isometric to the FRW dust with E^3 hyperslices.

A slightly more elaborate example, with density varying over the spherical shells, was given by Yodzis, Seifert, and Muller zum Hagen:
$$R = (9 \,m/2)^{1/3} \; (b-t)^{2/3}, \; \; m = r, \; \; b = (r-r_0)^2$$
This example illustrates a shell-crossing singularity; see the figure below and note these features:
• different shells collapse at different times to the final singularity at R=0 (red curve),
• the contours of R show that the dust particle at r=0 lies at the geometric center R=0 for all time,
• shells with small r fall in rather slowly,
• the shell crossing singularity (green curve) is the locus where $R^\prime = 0$, i.e. where the contour lines go horizontal.
The shell-crossing singularity arises here because the outer shells are more dense, so they actually collapse faster and eventually catch up with inner shells and form a "density caustic", which is the shell-crossing singularity.

An interesting variant of this example is obtained when we match a dust region to a Schwarzschild exterior. If we do this with a well chosen choice for the surface of the dust ball (see figure below), a sphere of infinitely dense dust is visible to exterior observers at a certain time in their history; this infinitely dense sphere (notice that it has positive surface area!) occurs at the moment when the outermost shell catches up with neighboring inner shells. Krasinski offers a good discussion of why this example should not be considered a genuine counterexample to the Cosmic Censorship Hypothesis.

Dyer 1979 gave a self-similar example (that is, an LTB dust with E^3 hyperslices which possesses a conformal Killing vector field), defined by
$$m = m_0 \, r, \; \; b = b_0 \, r$$
in which for suitable choices of the positive constants, the Big Crunch singularity can be "naked", but I've exceeded the limit of three figures, so see the textbook by Krasinski and Plebanski, where you find a discussion of why this is not really a counterexample to the Cosmic Censorship Hypothesis either.

I still haven't said anything about principle null geodesic congruences and their expansion scalars, but Krasinski offers a good discussion and plots the apparent horizons (which I omitted in the figures below).

[EDIT: Now something is worrying me about the leftmost and rightmost figures, something which I know how to check but was too lazy to check When I have more energy I'll try to check and correct as needed.]

[EDIT: The clock is running out and I haven't fixed th e suspect figures, so I have removed them and will have to put the corrected figures in a later post. Sorry!]

Figures: some LTB dusts plotting in Lemaitre-Tolman comoving chart; left to right:
• Expanding OS dust ball (oops!--- reverse time to get collapsing OS dust ball), removed for rennovation!
• YSM example of a collapsing LTB dust with spherical shells and E^3 hyperslices,
• the same, matched across a surface to a Schwarzschild exterior. removed for rennovation!
In the leftmost figure, dust region in cyan, vacuum region in pink, initial singularity at t=0. In the rightmost two figures, final singularity in red, shell crossing singularity in green, contours of R in blue (dust region) and pink (vacuum region). In the rightmost figure, the world sheet of the surface of the collapsing dust ball is shown in black. In all three figures, we also plot some world lines of dust particles (blue) and Lemaitre observers (gold).

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Chris Hillman
The reason for the decay was not so clear, as you identified two contributions of inward streaming matter (gravitational attraction and a "radial streaming"), but none outward. It seems now that the "streaming" is rather outward, due to the E=const condition.
I still haven't had a chance yet to re-examine the post in the other thread on "streaming", so I don't yet know whether I want to retract anything. I am confident I didn't make any computational errors, but I might have erred in the nontrivial function of interpretation of the results of my computations! So stay tuned...

Yes, attached.
Mebbe try again? I am not seeing the attachments.

Does this mean there are LTB models w/o spherical symmetry?
Yes. The ones with dust in nested spherical shells have spherical symmetry, and we can arrange shells with various energy values. The ones with dust in nested cylindrical shells have cylindrical symmetry, and we can give them various energies, except that now all the shells must have positive energy (for an expanding LTB dust). The ones with dust in nested H^2 shells have H^2 symmetry, and again we must give them all sufficiently positive energy.

The FRW dusts involve shells all having suitable negative, zero, or positive energies (chosen as a function of r to yield homogeneous density at each time), so these arise as special cases of the LTB dusts with spherical shells. But I would not be surprised if with a bit of further work, the LTB dust with E^2 shells can be rearranged to include the FRW dust with H^3 hyperslices, with the shells appearing as horospheres in the each slice t=t0. Also
• vacuum regions in the LTB dusts with spherical shells must be locally isometric to a region of Schwarzschild vacuum,
• vacuum regions in the LTB dusts with cylindrical shells must be locally isometric to a region of the plane symmetric Kasner vacuum (Petrov type D, a special case of Levi-Civita's cylindrical symmetric static vacuums, and not to be confused with the plane symmetric Taub vacuum with timelike singularities, sorry!),
• vacuum regions in the LTB dusts with H^2 shells must be locally isometric to a region of the Levi-Civita AII vacuums (static H^2 symmetric).

$\epsilon$ is the total energy, not the kinetic energy.
Yes, I misspoke, sorry. (For others: these terms are based upon the formal Newtonian analogy, in which energy = KE + PE is constant for a particle attracted by a Coulomb type gravitational potential.)

"H" means - in my numerical solution - velocity difference times distance in radial direction. For consecutive shells i-1, i:
$$H = \frac{\dot R_i-\dot R_{i-1}}{R_i-R_{i-1}}\sqrt{1+2\epsilon}$$
Now that I'm writing this I'm no longer sure this is correct. It was meant to reproduce locally the operational definition of H, therefore reproducing the FRW values in a homogeneous universe. But I think I can drop the sqrt-factor?
H should not be a single number in an inhomogeneous dust! The Hubble expansion is correctly described by giving the expansion tensor (components taken wrt a nonspinning inertial frame field corresponding to the dust particles).

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Chris Hillman
Some examples of LTB dusts with spherical shells having various energies

Once we drop the requirement that the hyperslices be locally flat, we can really go to town.

One thing we can immediately do is to take any FRW dust and excise a number of world tubes, each bounded by the world sheet of a spherical shell of dust, and then match across the world sheets of each shell to a suitable LTB dust (each spherically symmetrical within its own little region), to obtain a variety of "Swiss cheese" type models.

Returning to dust solutions with global spherical symmetry, Hellaby and Lake 1985 gave two particularly interesting examples of LTB dusts with spherical shells. The first, which we can call "HL two center LTB dust", is a (mostly) expanding LTB dust defined by taking
$$m = m_0 \, r^3 \, (\ell-r)^3, \; \; \epsilon = \epsilon_0 \, r^2 \, (\ell-r)^2 \, (\lambda \, r - 1) \, (\lambda \, (\ell-r) - 1), \; \; 0 < r < \ell$$
(all constants positive with $\lambda > 2/\ell$) in the constraint equation
$$\dot{R}^2/2 = \frac{m}{R} + \epsilon$$
Note that the energy of the shells is given by a sextic polynomial with two negative minima flanking a positive maximum, so we have a region of positive energy shells which expand forever, and two regions of negative energy shells each recollapsing to a different center R=0, corresponding to the labels $r=0, \; r=\ell$. I'll leave it as an exercise to study the geometry of the spatial hyperslices t=t_0 to see how we can have the two regions filled with negative energy spherical shells of dust each collapse to its own "center of spherical symmetry".

The second example, which we can call "HL two-sheet LTB dust", is given by taking
$$m = (m_0 - m_1 \, r^3)/2, \; \; \epsilon = -(1-\lambda \, r^2)/2, \; \; -\infty < r < \infty$$
in the constraint equation. This includes
• a region containing shells which collapse from infinite surface area to a Big Crunch, with $r=-\infty$ corresponding to a sphere of infinite surface area,
• a region containing shells which expand forever from a Big Bang, with $r=\infty$ corresponding to a sphere of infinite surface area,
• an interpolating region containing shells of negative energy which expand from a Big Bang, halt, and recollapse to a Big Crunch
Again, I'll leave it as an exercise to study how the geometry of the spatial hyperslices t=t_0 evolve as time increases.

Hellaby gave a third interesting example, which we can call "the Hellaby string of beads LTB dust". It is defined by taking
$$m = m_0 + m_1 \, \exp(-a r^2) \, \cos(br), \; \; \epsilon = -(1-\lambda \, \exp(-a r^2) \, \sin(br)^2 )/2, \; \; -\infty < r < \infty$$
The hyperslices show that we have a bunch of initially disjoint dust filled regions which expand from their own Big Bang singularities, then join up and continue to expand, then halting, then recollapsing, and eventually pinching off into disjoint regions again which soon collapse to their own Big Crunch singularities. Because differences in t taken along a vertical line segment (world line of a dust particle) correspond to elapsed proper time, some dust particles really do exist longer than others, in this model.

Figures: left to right:
• Carter-Penrose diagrams (heavy lines correspond to strong curvature singularities) illustrating two examples of Hellaby and Lake 1985
• top: HL two-center LTB dust,
• bottom: HL two-sheet LTB dust,
• the Hellaby "string of beads" LTB dust; schematic figure depicting the Lemaitre-Tolman comoving chart with world lines of some shells of dust shown as solid vertical line segments and hyperslices t=t_0 shown as horizontal dotted lines.

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Mebbe try again? I am not seeing the attachments.
Strange. I'm sure I uploaded them. Anyway, I try again.

H should not be a single number in an inhomogeneous dust! The Hubble expansion is correctly described by giving the expansion tensor (components taken wrt a nonspinning inertial frame field corresponding to the dust particles).
H is H_rr. That's the most interesting, as I set it as an initial condition. The tangential components are relatively simple, if I'm not mistaken (maybe you can check?):
$$H_{\theta}=\frac{\dot R}{R}\sqrt{1+2\epsilon}$$
I'll write more on H_rr soon, I think I'm about to understand a bit more again.

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Chris Hillman
Hi, Ich, when you wrote
H is H_rr.
you did mean $H_{22}$? I.e., the component taken wrt the frame field (written in the Lemaitre chart)
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_t \\ \vec{e}_2 & = & \frac{\sqrt{1 + 2 \, \epsilon}}{R^\prime} \; \partial_r \\ \vec{e}_3 & = & \frac{1}{R} \; \partial_\theta \\ \vec{e}_4 & = & \frac{1}{R \, \sin(\theta)} \; \partial_\phi \end{array}$$
and its dual coframe? Or did you mean $H_{rr}$, the component taken wrt the coordinate basis and its dual cobasis?

Did you forget to apply the constraint equation to eliminate time derivatives?

$$H_{\theta}=\frac{\dot R}{R}\sqrt{1+2\epsilon}$$
Partial derivative wrt theta? Oh, I see, you must mean $H_{\theta \theta}$ (a component taken wrt the coordinate basis).

The tangential components are relatively simple, if I'm not mistaken (maybe you can check?):
Yes, that's right; I gave them above but for convenience: the LTB dust (zero Lambda) with spherical shells of dust, written in the Lemaitre comoving chart, is
$$\begin{array}{rcl} ds^2 & = & -dt^2 + \frac{(R^\prime)^2}{1+2\, \epsilon} \; dr^2 + R^2 \; \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right), \\ && \dot{R}^2/2 = m/R + \epsilon \end{array}$$
where R is a function of t,r only and $m, \epsilon$ are functions of r only (solving the constraint for R will give a third function of r only), and where I wrote out the angular part of the line element in order to emphasize the S^2 symmetry. (The angular part is different for the $E^2$ (or $R \times S^1$) and the H^2 symmetric cases; we always have, for a general LTB dust, a three-dimensional Lie algebra of Killing vector fields, which in the case of S^2 shells are linear combinations of the three familiar Killing vector fields for the round two-sphere. In special cases of an LTB dust, e.g. Schwarzschild or Minkowski vacuum, we might acquire further metrical symmetries.)

The eigenvalues of the Riemann tensor consist of two simple eigenvalues $2m/R^3, \; -m/R^3$ and two double eigenvalues $2m/R^3 - m^\prime/R^\prime/R^2, \; -m/R^3+m^\prime/R^\prime/R^2$, from which we see that the curvature blows up at two loci
• $R=0$ and $m > 0$ (initial or final strong curvature singularity)
• $R^\prime = 0$ and $m^\prime \neq 0$ (shell-crossing singularity)

Let us write the constraint equation as
$$\dot{R} = \begin{cases} \sqrt{2} \, \sqrt{m/R+\epsilon} & \rm{expanding} \; \rm{shells} \\ -\sqrt{2} \, \sqrt{m/R+\epsilon) & \rm{contracting} \; \rm{shells} \end{cases}$$
Then we can eliminate all the time derivatives of R, and wrt the frame of the dust particles (the frame field given above, which is inertial nonspinning), the only nonzero components of the expansion tensor are
$$\begin{array}{rcl} H_{22} & = & \frac{ -\frac{m}{R^2} + \frac{m^\prime/R^\prime}{R} + \epsilon^\prime/R^\prime}{\dot{R}} \\ &&\\ H_{33} & = & H_{44} = \frac{\dot{R}}{R} \end{array}$$
or using the constraint:
• in a region of expanding shells
$$\begin{array}{rcl} H_{22} & = & \frac{ -\frac{m}{R^2} + \frac{m^\prime/R^\prime}{R} + \epsilon^\prime/R^\prime}{\sqrt{2} \, \sqrt{m/R + \epsilon}} \\ H_{33} & = & H_{44} = \frac{\sqrt{2} \, \sqrt{m/R + \epsilon}}{R} \end{array}$$
• in a region of contracting shells
$$\begin{array}{rcl} H_{22} & = & \frac{ \frac{m}{R^2} - \frac{m^\prime/R^\prime}{R} - \epsilon^\prime/R^\prime}{\sqrt{2} \, \sqrt{m/R + \epsilon}} \\ H_{33} & = & H_{44} = - \frac{\sqrt{2} \, \sqrt{m/R + \epsilon}}{R} \end{array}$$
That is, after using the constraint equation to eliminate the time derivatives, the sign of the expansion tensor flips when we move from a region containing expanding shells into a region containing contracting shells. As a check, in the vacuum case $m^\prime = 0$, these reduce to the expressions for outgoing or ingoing generalized Lemaitre observers in the Schwarzschild vacuum (then $H_{22}$ and $H_{33} = H_{44}$ have opposite signs, whereas in the FRW case they have the same sign and even agree).

The following statements are valid regardless of whether we are in a region of expanding or contracting shells:

The vorticity tensor vanishes, so we have a unique family of hyperslices everywhere orthogonal to the timelike geodesic congruence with tangent vector field $\vec{e}_1$, namely the slices t=t_0. The density of the dust (measured in the frame of the dust) is
$$\mu = \frac{m^\prime/R^\prime}{A}$$
where $A(t_0,r_0) = 4 \pi \, R(t_0,r_0)^2$ is the surface area of the shell r=r_0 at time t=t_0. Then, the only nonvanishing component of the Einstein tensor is $G^{11} = 8 \pi \; \mu$, so we have an exact dust solution. The only nonvanishing components of the electroriemann tensor (tidal tensor) are
$$\begin{array}{rcl} E_{22} & = & \frac{-2m}{R^3} + \frac{m^\prime/R^\prime}{R^2} \\ E_{33} & = & E_{44} = \frac{m}{R^3} \end{array}$$
and the magnetoriemann tensor vanishes (the dust particles aren't spinning or swirling). The only nonvanishing components of the three-dimensional Riemann tensor of the spatial hyperslices t=t_0 are
$$\begin{array}{rcl} r_{2323} & = & r_{2424} = - \frac{\epsilon^\prime/R^\prime}{R} \\ r_{3434} & = & - \frac{2 \, \epsilon}{R^2} \end{array}$$

For those of you who use Maxima, try this loading this file in batch mode:
Code:
/*
Lemaitre-Tolman-Bondi dust; comoving Lemaitre chart; nsi coframe

CONSTRAINT (in a region of expanding shells; flip signs for a region of contracting shells):
subst( diff(sqrt(2)*sqrt(m/R+epsilon),r,1,t,1),diff(R,r,1,t,2),%);ev(%,diff);
subst( diff(sqrt(2)*sqrt(m/R+epsilon),r),diff(R,r,1,t,1),%);ev(%,diff);
subst( diff(sqrt(2)*sqrt(m/R+epsilon),t),diff(R,t,2),%);ev(%,diff);
subst( sqrt(2)*sqrt(m/R+epsilon), diff(R,t),%);ev(%,diff);
ratsimp(%);

This models a spherically symmetric dust.
*/
cframe_flag: true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [t,r,theta,phi];
/* define background metric */
lfg: ident(4);
lfg[1,1]: -1;
depends(R,[t,r]);
depends([m,epsilon],r);
/* define the coframe */
fri: zeromatrix(4,4);
fri[1,1]: -1;
fri[2,2]:  diff(R,r)/sqrt(1+2*epsilon);
fri[3,3]:  R;
fri[4,4]:  R*sin(theta);
/* setup the spacetime definition */
cmetric();
/* display matrix whose rows give coframe covectors */
fri;
/* compute a matrix whose rows give frame vectors */
fr;
/* metric tensor g_(ab) */
lg;
/* compute g^(ab) */
ug: invert(lg);
christof(false);
/* Compute fully covariant Riemann components R_(mijk) = riem[i,k,j,m] */
lriemann(true);
/* Compute R^(mijk) */
uriemann(false);
/* Compute Ricci componets R_(jk) */
ricci(true);
/* Compute trace of Ricci tensor */
tracer;
/* Compute R^(jk) */
uricci(false);
/* Compute and display MIXED Einstein tensor G^a_b */
/* For (-1,1,1,1) sig Flip sign of top row to get G^(ab) */
einstein(false);
cdisplay(ein);
/* WARNING! leinstein(false) only works for metric basis! */
Remember that ein[1,1] is the component ${G^1}_1$ so you must flip the sign to get $G^{11}$. Paste
Code:
EX[2,2] = lriem[2,2,1,1];
EX[2,3] = lriem[2,3,1,1];
EX[2,4] = lriem[2,4,1,1];
EX[3,2] = lriem[3,2,1,1];
EX[3,3] = lriem[3,3,1,1];
EX[3,4] = lriem[3,4,1,1];
EX[4,2] = lriem[4,2,1,1];
EX[4,3] = lriem[4,3,1,1];
EX[4,4] = lriem[4,4,1,1];
into the wxmaxima command window to see the components of the electroriemann tensor defined wrt the given $\vec{e}_1$ vector field (i.e. the tidal tensor defined for the dust particles, not to be confused with the electroweyl tensor, which is a different quantity in a dust solution!). Paste in
Code:
BX[2,2] = lriem[2,4,3,1];
BX[2,3] = lriem[2,2,4,1];
BX[2,4] = lriem[2,3,2,1];
BX[3,2] = lriem[3,4,3,1];
BX[3,3] = lriem[3,2,4,1];
BX[3,4] = lriem[3,3,2,1];
BX[4,2] = lriem[4,4,3,1];
BX[4,3] = lriem[4,2,4,1];
BX[4,4] = lriem[4,3,2,1];
to see the magnetoriemann components (all vanish for this Lorentzian manifold and this frame). To apply the constraint to any tensor component or scalar quantity, paste in
Code:
subst( diff(sqrt(2)*sqrt(m/R+epsilon),r,1,t,1),diff(R,r,1,t,2),%);ev(%,diff);
subst( diff(sqrt(2)*sqrt(m/R+epsilon),r),diff(R,r,1,t,1),%);ev(%,diff);
subst( diff(sqrt(2)*sqrt(m/R+epsilon),t),diff(R,t,2),%);
subst( sqrt(2)*sqrt(m/R+epsilon), diff(R,t),%);ev(%,diff);
ratsimp(%);
All but the last are redundant, but Maxima isn't (yet) smart enough to realize it can differentiate a given constraint to eliminate higher order partial derivatives, so I started with high order partials containing t and worked my way down. I have given the constraint for a region of expanding shells, but you can flip all the signs!

I still haven't taken the time to figure out how to coax Maxima into computing, for a given Lorentzian manifold and a given frame, the components of the accelleration vector, expansion tensor, vorticity tensor, which is currently by far the biggest lack in existing Ctensor routines. I confess I have been hoping someone will do the work for me and will then contact the Maxima development team to get them to put debugged code into the next edition.

Strange. I'm sure I uploaded them.
When you click on "Manage Attachments", another window probably pops up, where you do the "Upload Attachments" thing. But then you need to return to the original edit window, scroll down to the "Manage Attachments" section, and click on "Save Changes". That second step is easy to forget.

Your left figure, the case $\epsilon=0$, looks like mine, which may or may not be good (more in a moment). Re your right figure, in your Post #7, you wrote
H is defined on proper distance, not LTB's R
Not sure what you mean by "proper distance"; as I have tried to emphasize (it can never be repeated too often) there is no single prefered notion of distance in the large for a Riemannian or Lorentzian manifold, in general. R is the Schwarzschild radial coordinate, so A = 4 \pi R(t_0,r_0)^2 is the surface area of the spherical shell r=r_0.

I seem to keep uncovering more goofs in my initial approach, which I hope is proving more illuminating than confusing. The latest: I am now convinced I must be confused regarding the issue of whether perturbations should grow over time regardless of whether nearby shells are expanding or contracting, or (as I thought) should be smoothed out by expansion and magnified by contraction. Look at p. 302 in the textbook by Krasinski and Plebanski if you have that at hand.

A better approach would no doubt be to modify the FRW choices
$$m = m_0 \, r^3, \; \; \epsilon = -k \, r^2/2$$

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Or did you mean $H_{rr}$, the component taken wrt the coordinate basis and its dual cobasis?
Well, there's a problem: I don't know what I mean - let me explain what I meant to mean:
Ich said:
H is defined on proper distance, not LTB's R
Not sure what you mean by "proper distance"
I'm trying to follow the operational definition of H, that is dv/dx as seen by a comoving observer. As we're using proper time for these observers anyway, I thought it was necessary to use the according proper distance also (which is unproblematic, as I evaluate H in the limit dx->0). The local orthonormal frame.
However, since v and x scale the same way, it's ok to use $d\dotR/dR$. By this reasoning, it should make no difference in this case whether I use the frame field or the coordinate basis. If it does, I haven't understood something.

Ich said:
The tangential components are relatively simple, if I'm not mistaken (maybe you can check?):
Yes, that's right; I gave them above but for convenience
Sorry, I think there's a misunderstanding. I meant the tangential parts of the expansion tensor, which I think should be $\frac{\dot R}{R}\sqrt{1+2\epsilon}$. wrt the frame field, I guess, so that \theta was a misnomer. But I see that you don't have the \sqrt term, so I better rethink what I'm doing here.
$$H_{22}=\frac{ -\frac{m}{R^2} + \frac{m^\prime/R^\prime}{R} + \epsilon^\prime/R^\prime}{\sqrt{2} \, \sqrt{m/R + \epsilon}}$$
Yes, that's the interesting formula. I gained some more understanding from expressing it in terms of specific kinetic energy ke:
$$H_{22}=\frac{dv}{dke}{dke}{dr}=1/\dotR(\frac{d\epsilon}{dR}+\frac{d(M/R)}{dR})$$
For \epsilon=0 (your scenario), the derivatives of the potential are the key to understanding the behaviour of an overdensity:
$$\frac{\partial(M/R)}{\partial M}= M^\prime/R\\ \frac{\partial(M/R)}{\partial R}= - M/R^2$$
Near the center of the overdensity (where it's almost FRW), both add to the first Friedman equation, so H is increased there, diluting the center.
In the border area, where the overdensity is significantly less, the first term loses wrt the second, so you have decreased expansion there due to the gravitating potential of the inhomogeneity. So here, the density inceases relative to the pure FRW case, expanding the boundary.
Far away, even the second term vanishes(i.e. is no longer different from the undisturbed one), and the regular FRW behaviour is recovered.
I also made a diagram showing this behaviour, but I'll have to attach it tomorrow.
Your left figure, the case \epsilon=0 , looks like mine, which may or may not be good (more in a moment).
I'd say it's good, because it behaves as you said (except the inward streaming, which was an issue with your interpretation only).
I seem to keep uncovering more goofs in my initial approach, which I hope is proving more illuminating than confusing.
Those "goofs" are exactly what made me think about it in more detail. Trying to resolve the issue is far more illuminating than superficially absorbing spoon-fed knowlede. I really do appreciate your seminar, and learn from it. Please don't be annoyed that I follow my path here, and not all the details and exercises you kindly provide. I almost don't find the time for this "slimmed-down" version. You day must be longer than mine .
I am now convinced I must be confused regarding the issue of whether perturbations should grow over time regardless of whether nearby shells are expanding or contracting, or (as I thought) should be smoothed out by expansion and magnified by contraction.
It's a matter of initial conditions. The \epsilon=0 condition seems to be nonstandard, while the H=const condition seems to be what is normally used when modeling perturbations.

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Chris Hillman
Just to emphasize, in what I wrote above, I was discussing the general LTB dust with S^2 shells, including the case where the shells of dust can have nonzero energies varying with r.

I'm trying to follow the operational definition of H, that is dv/dx as seen by a comoving observer.
That's one component of the expansion tensor.

(Note to self: must wean SA/Ms from the fixed idea that Hubble expansion is always homogeneous in cosmological models.)

As we're using proper time for these observers anyway,
yes, for the observers riding with the dust particles (clearly the ones we want for cosmology!).

I thought it was necessary to use the according proper distance also (which is unproblematic, as I evaluate H in the limit dx->0).
I think you are saying that you want to use the basis vector $\vec{e}_2$ as a differential operator applied to functions to find rates of change wrt "radial proper distance in the small". OK, that's fine.

However, since v and x scale the same way, it's ok to use $d\dotR/dR$.
Now I think you are saying that you want to use R, which I was using as a metric function, as a new radial coordinate (i.e. you want to write the LTB dusts in the Schwarzschild chart, which is known to be a bad idea). I think you are saying that
$$\vec{e}_2 = \frac{R'}{\sqrt{1+2\, \epsilon}} \; \partial_r = \partial_R$$
when you change coordinates. But is that true?

By this reasoning, it should make no difference in this case whether I use the frame field or the coordinate basis. If it does, I haven't understood something.
I clearly don't understand what you are trying to do, but it almost always makes a difference whether you use a frame field vector or a coordinate basis vector to differentiate something.

Sorry, I think there's a misunderstanding. I meant the tangential parts of the expansion tensor,
That's what I thought you meant.

which I think should be $\frac{\dot R}{R}\sqrt{1+2\epsilon}$. wrt the frame field,
That doesn't look right.

Yes, that's the interesting formula. I gained some more understanding from expressing it in terms of specific kinetic energy ke:
$$H_{22}=\frac{dv}{dke}{dke}{dr}=1/\dotR(\frac{d\epsilon}{dR}+\frac{d(M/R)}{dR})$$
ke? Not sure either of us really know what we mean by v either. (Yes, I know you are thinking of that as "the velocity of a nearby dust particle" or something like that, but... you can probably see you have something to explain there.)

Only in some portions of what I said. That's precisely the case where all the shells have zero energy, in which case we can solve for R expliclity in terms of m and a new function of r only, b.

the derivatives of the potential are the key to understanding the behaviour of an overdensity:
$$\frac{\partial(M/R)}{\partial M}= M^\prime/R\\ \frac{\partial(M/R)}{\partial R}= - M/R^2$$
Your M is my m, I think. So now you are using M as yet another new radial coordinate? OK, if so (since my m(r) is an increasing function of r, so we can use it as a radial coordinate if we like), but just trying to understand.

Near the center of the overdensity (where it's almost FRW), both add to the first Friedman equation, so H is increased there, diluting the center.
In the border area, where the overdensity is significantly less, the first term loses wrt the second, so you have decreased expansion there due to the gravitating potential of the inhomogeneity. So here, the density inceases relative to the pure FRW case, expanding the boundary.
Far away, even the second term vanishes(i.e. is no longer different from the undisturbed one), and the regular FRW behaviour is recovered.
I also made a diagram showing this behaviour, but I'll have to attach it tomorrow.

I'd say it's good, because it behaves as you said (except the inward streaming, which was an issue with your interpretation only).
Hmm...

Those "goofs" are exactly what made me think about it in more detail.
Good, we'reboth learning Trying to resolve the issue is far more illuminating than superficially absorbing spoon-fed knowlede. I really do appreciate your seminar, and learn from it. Please don't be annoyed that I follow my path here, and not all the details and exercises you kindly provide. I almost don't find the time for this "slimmed-down" version.
Back and forth is just what I hoped to see when I started the BRS. Hopefully at the end we'll understand each others approaches, if there are any remaining differences.

So you are using the frame field for the dust, not the coordinate basis, right? That's good, if so.

It's a matter of initial conditions. The \epsilon=0 condition seems to be nonstandard, while the H=const condition seems to be what is normally used when modeling perturbations.
That doesn't make any sense at all, because uniform expansion tensor takes us right back to the FRW solutions (by definition).

Are you using another source for this H that I don't know about?

Hi Chris,

That's one component of the expansion tensor.
I'm aware of that. That's the component I'm interested in.
Ich said:
I thought it was necessary to use the according proper distance also (which is unproblematic, as I evaluate H in the limit dx->0).
I think you are saying that you want to use the basis vector$\vec{e}_2$ as a differential operator applied to functions to find rates of change wrt "radial proper distance in the small". OK, that's fine.
Actually, I don't know what I'm doing. But what you said sounds fine. I'm trying to get from the coordinates (and metric functions) to the local orthonormal system, where the radial basis vector is e_2. (Right?)
However, since v and x scale the same way, it's ok to use $\dot R/dR$.
Now I think you are saying that you want to use R, which I was using as a metric function, as a new radial coordinate (i.e. you want to write the LTB dusts in the Schwarzschild chart, which is known to be a bad idea). I think you are saying that
$$\vec{e}_2 = \frac{R^\prime}{\sqrt{1+2\\, \epsilon}} \; \partial_r = \partial_R$$
when you change coordinates. But is that true?
Ok, I want to use R like a coordinate. My line of thinking:
$$dR=\frac{\partial R}{\partial r}dr+\frac{\partial R}{\partial t}dt$$
so, at a certain moment for the observers (dt=0),
$$ds=\frac{1}{1+2\epsilon}dR$$
is a (small) proper radial distance, expressed in terms of R. So ds is dR times a scaling factor.
Whatever, I'm handwaving.
Could you help me?
I want to know the local radial Hubble parameter. How do I obtain it formally correct, using frame fields or coordinates? I'm assuming it's $d\dot R/dR$.
I clearly don't understand what you are trying to do, but it almost always makes a difference whether you use a frame field vector or a coordinate basis vector to differentiate something.
Yes, but in this special cace, if I divide $d\dotR$ by dR, isn't the result independent of $1+2\epsilon$? (Not important if you don't understand, I better learn to do the maths without handwaving).
ke? Not sure either of us really know what we mean by v either.
ke: kinetic energy, v: $\dot R$. The formula is formally Newtonian, that's why I use Newtonian names for the variables, to help intuition.

So now you are using M as yet another new radial coordinate?
Sorry, I garbled the tex. It should be:

Ich said:
Yes, that's the interesting formula. I gained some more understanding from expressing it in terms of specific kinetic energy ke:
$$H_{22}=\frac{dv}{dke}{dke}{dr}= (\frac{d\epsilon}{dR}+\frac{d(M/R)}{dR})/\dot R$$
For $\epsilon=0$ (your scenario), the derivatives of the potential are the key to understanding the behaviour of an overdensity:
$$\begin{array}{rcl} \frac{\partial(M/R)}{\partial M} & = & M^\prime/R = 4\pi\rho R \\ \frac{\partial(M/R)}{\partial R} & = & - M/R^2 \end{array}$$
Near the center of the overdensity (where it's almost FRW), both add to the first Friedman equation, so H is increased there, diluting the center.
In the border area, where the overdensity is significantly less, the first term loses wrt the second, so you have decreased expansion there due to the gravitating potential of the inhomogeneity. So here, the density inceases relative to the pure FRW case, expanding the boundary.
Far away, even the second term vanishes(i.e. is no longer different from the undisturbed one), and the regular FRW behaviour is recovered.
I also made a diagram showing this behaviour, but I'll have to attach it tomorrow.
Ich said:
It's a matter of initial conditions. The \epsilon=0 condition seems to be nonstandard, while the H=const condition seems to be what is normally used when modeling perturbations.
That doesn't make any sense at all, because uniform expansion tensor takes us right back to the FRW solutions (by definition).
Are you using another source for this H that I don't know about?
I'm writing H for the Hubble parameter in the radial direction ($H_{rr}$, obviously). So, H=const. means an initial condition where H_rr is initially the same for every shell.

#### Attachments

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Chris Hillman
Continuation of Post #14

I have fixed the figures, and sorry for the delay.

I have yet to explain the Darmois matching conditions, so I'm getting ahead of myself, but here are two simple examples in which we have matched an LTB dust interior to a vacuum exterior (a region of the Schwarzschild vacuum, which is itself a special case of the LTB dusts, of course). It turns out that the matching condition is very simple, especially for the case of zero energy slices, which includes both examples: R(t,r) should be continuous across the surface of the dust ball r=r_s, for all t.

Here is how the examples are defined (my notation here is slightly different from Post #14, for convenience in verifying the matching condition):

The OS collapsing dustball can be written
$$R = \begin{cases} (9 \, m/2)^{1/3} \; (r/r_s) \; (r_s - t)^{2/3} & 0 < r < r_s \\ (9 \, m/2)^{1/3} \; (r-t)^{2/3} & r_s < r < \infty \end{cases}$$
The exterior region is asymptotically flat and static and has Komar mass m, so the event horizon (in the vacuum region) is located at R=2m. The apparent horizon (inside the dust region) meets the event horizon at the surface.

The Yodzis-Seifert-Muller zum Hagen example can be written
$$R = \begin{cases} (9 \, r/2)^{1/3} \; \left( (r- r_s-\sqrt{r_s})^2-t \right)^2^{2/3} & 0 < r < r_s \\ (9 \, r_s/2)^{1/3} \; (r-t)^{2/3} & r_s < r < \infty$$
The exterior region is asymptotically flat and static and has Komar mass r_s, so the event horizon (in the vacuum region) is located at R=2 r_s. The apparent horizon (in the dust region) meets the event horizon at the surface.

Figures (left to right):
• The OS collapsing dust ball
• The YSMzH example of an LTB dust with a shell-crossing singularity visible to exterior observers
In both figures, contours $R(t,r) = 1, 2, \ldots$ are plotted with respect to t (vertical) and r (horizontal); dust region shown in cyan, vacuum region shown in pink; final singularity at top; event horizon at R=2. In the OS plot, m=1, r_s = 3, and the world lines of two dust particles (blue) and two Lemaitre observers (gold) are shown. In the YSMzH example, r_s = 1, and the shell-crossing singularity is shown as the green curve (note the contours of R are horizontal there, and that the endpoint of the SC singularity is visible to exterior observers who hover outside the event horizon).

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Chris Hillman
Hi, Ich,
I'm aware of that. That's the component I'm interested in.
OK, I guess I'm quibbling over notation.
I'm trying to get from the coordinates (and metric functions) to the local orthonormal system, where the radial basis vector is e_2. (Right?)
OK so far.
Ok, I want to use R like a coordinate. My line of thinking:
$$dR=\frac{\partial R}{\partial r}dr+\frac{\partial R}{\partial t}dt$$
OK
so, at a certain moment for the observers (dt=0),
$$ds=\frac{1}{1+2\epsilon}dR$$
is a (small) proper radial distance, expressed in terms of R. So ds is dR times a scaling factor.
It turns out that Schwarzschild charts (using R as radial coordinate) are not very useful in this context. If you use r as radial coordinate, then look at the line element and set $dt=0, \; d\Omega=0$ to obtain
$$ds = \frac{R^\prime}{\sqrt{1+2\, \epsilon}} \, dr$$
for proper distance along a radius (spacelike curve) $t=t_0, \theta=\theta_0, \phi = \phi_0$.
I want to know the local radial Hubble parameter. How do I obtain it formally correct, using frame fields or coordinates?
Given a timelike congruence with unit tangent vectors $\vec{U}$, the expansion and vorticity tensors of that congruence are
$$H_{ab} = {h_a}^m \; {h_b}^n \; U_{(m;n)}, \; \; \Omega_{ab} = {h_a}^m \; {h_b}^n \; U_{[m;n]}$$
(symmetrization and antisymmetrization respectively, projected onto a spatial hyperplane orthogonal to $\vec{U}$). If you like to compute covariant derivatives wrt the coordinate basis, that is fine; you can then convert from coordinate components to frame components the same way to change a matrix representation to a new vector basis.

I highly recommend Chapter 14 of the textbook by Krasinski and Plebanski, Introduction to General Relativity and Cosmology; this has everything you want and much more!

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Chris Hillman
Recapitulation: Overview of the LTB dusts

This will be a bit sketchy; for details see Chapter 18 of Jerzy Plebanski and Andrzej Krasinski, An Introduction to General Relativity and Cosmology, Cambridge University Press, 2006. This book is IMO essential reading for anyone studying cosmology at the graduate level, not to mention working cosmologists. I can't recommend it too highly!

The basic idea of the LTB dusts is due to Lemaitre 1933: consider a dust (pressureless perfect fluid) which consists of nested spherical shells, such that each shell has homogeneous mass density, but shells can shrink over time, and the density can vary between shells. This led Lemaitre to introduce the comoving chart
$$ds^2 = -dt^2 + \frac{(R^\prime)^2}{1+2\,\epsilon} \; dr^2 + R^2 \; \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right)$$
where \epsilon is a function of r only and R is a function of t,r, and where it will be convenient to denote time derivatives by overdots, and r derivatives by primes. Then computing the Einstein tensor wrt the frame I gave earlier and requiring all but $G^{11}$ to vanish makes this the frame of the dust and also leads to a constraint equation
$$\ddot{R} = \frac{2 \, \epsilon - \dot{R}^2}{2 \, R}$$
which can be integrated once to give a first order PDE:
$$\dot{R}^2/2 = \frac{m}{R} + \epsilon$$
where m is a new function of r only. This is formally identically to the equation of radial motion of a test particle moving in a Coulomb field in Newtonian gravitation, and then $\epsilon(r_0)$ is the specific energy (energy per unit mass) of the test particle, an invariant of the motion for any particle in the shell r=r_0, and the first term is (the negative of) the Newtonian gravitational potential. Thus, for a given shell $r=r_0$ we make the following interpretations:
• $m(r_0)$ is the active mass inside the shell,
• $\epsilon(r_0)$ is the energy of the shell;
• negative energy shells are bounded (expand and recollapse),
• zero energy shells are marginally bound
• positive energy shells are unbound
• $A = 4 \pi \; R(t_0, r_0)^2$ is the surface area of the shell at t=t_0
The coordinate r simply labels the shells, so it can go negative as long as R remains nonnegative. In many LTB dust solutions, r=0 will correspond to the world line of an axis of spherical symmetry, but this typically does not exist for all times. Also, differences in t along the world line of a dust particle (vertical constant lines, i.e. constant angular and radial coordinates) correspond to elapsed proper time, so R and the angular coordinates have immediate geometric significance, and t has direct physical significance.

Later workers noticed that one can easily generalize to other shell geometries, obtaining the same constraint equation, with the same interpretations! (Except for the bit about surface area of the shells, which need no longer be compact.) In particular, we can consider cylindrical shells (with $\mathbb{R} \times S^1$ geometry):
$$ds^2 = -dt^2 + \frac{(R^\prime)^2}{2\,\epsilon} \; dr^2 + R^2 \; \left( dz^2 + d\phi^2 \right)$$
and pseudospherical shells (with $H^2$ geometry):
$$ds^2 = -dt^2 + \frac{(R^\prime)^2}{-1+2\,\epsilon} \; dr^2 + R^2 \; \left( d\theta^2 + \sinh(\theta)^2 \, d\phi^2 \right)$$
(Notice that we require $\epsilon > -1/2, \; \epsilon > 0, \; \epsilon > 1/2$, for spherical, cylindrical, and pseudospherical shells respectively.) Then the spacetime globally organizes the geometry of the "nested" shells such that we have a three-dimensional Lie algebra of spacelike Killing vector fields, corresponding respectively to the Lie groups
• SO(3) acting on the nested spherical shells,
• E(3) acting on the nested planar shells (or, locally, on nested cylindrical shells),
• SO(2,1) acting on the nested pseudospherical shells
In all cases, the density of the dust is simply
$$\mu = \frac{m^\prime/R^\prime}{4 \pi \, R^2}$$
where in the case of spherical shells, we can interpret the denominator as the surface area of a shell.

We should think of the constraint like this: we choose two functions of one variable, $m, \; \epsilon$ and then solve to find R. (Indeed, from the interpretation, we should think of m as a given non-negative function of r.) From the form of the constraint, we should expect that the solution should involve specifying a third function of r only, b. Indeed, we can solve for R implicitly in the form
$$t - b = \frac{\pm R^{3/2}}{\sqrt{2 m}} F\left( \frac{-\epsilon \, R}{m} \right)$$
where we choose the minus sign for expanding shells and the plus sign for contracting shells, and where
$$F(\xi) = \begin{cases} \xi^{-3/2} \; \operatorname{arcsin}(\sqrt{\xi}) - \frac{\sqrt{1-\epsilon}}{\xi} & 0 < \xi \leq 1 \\ 2/3 & \xi = 0 \\ -(-\xi)^{-3/2} \; \operatorname{arcsinh}(\sqrt{-\xi}) - \frac{\sqrt{1-\epsilon}}{\xi} & -\infty < \xi < 0 \end{cases}$$
Here, F is positive, increasing, convex, bounded, smooth function defined on $(-\infty, 1]$, with range$(0,\pi/2]$ (see the plot below), so the inverse function exists. In fact, we put $G(\xi) = \xi^{3/2} \; F(\xi)$ and consider its inverse as a special function with known properties, we can solve for R in closed form as a function of the three (almost) arbitrarily chosen functions $m, \, \epsilon, \, b$. Thus, the Riemann wealth of LTB dusts is, crudely, "three functions of one variable"; this might include some gauge transformations among different LTB representations of the same spacetime plus matter density/flow. However, the form of the Lemaitre chart is quite restrictive, as noted above, so it is reasonable to hope that it is essentially a canonical chart for LTB dusts.

The eigenvalues of the Riemann tensor turn out to consist of
• two simple eigenvalues $2 \, m/R^3, \; 2 \, m/R^3 - m^\prime/R^\prime/R^2$
• two double eigenvalues $-m/R^3, \; -m/R^3 + m^\prime/R^\prime/R^2$
From these expressions, we can see that there are two ways the curvature can blow up:
• $R = 0$ when $m > 0$,
• $R^\prime = 0$ when $m^\prime \neq 0$
The first of these possibilities corresponds to something familiar from FRW dusts: initial and/or final singularities, which are strong and spacelike. The second is something new: shell-crossing singularities, where the density of the dust blows up--- more about that later.

Suppose you have found a solution R(t,r) to the constraint equation. To visualize the dynamics, use the spherical symmetry to suppress the angular coordinates, and plot contours of R wrt t,r, with the boundaries of the singular locus $R=0$. Also plot the shell-crossing singularities, if any. Plot some world sheets of shells of dust particles as vertical line segments
• between initial and final singularities (if the shell has negative energy)
• above an initial singularity or below a final singularity (if it has zero or positive energy)
Then you can see how each shell expands and contracts by looking at which contours of R it crosses. (See the figure below.)

Suppose that R=0 and m=0 at r=0, so that r=0 is a center of spherical symmetry. Then the principle outgoing null geodesic congruence consist of all null geodesics which go out from r=0; it has spherical wavefronts which "want" to expand (but in a black hole interior, wind up contracting). The principle ingoing null geodesic congruence consist of all null geodesics which go into r=0. From the line element you can see how to plot the world sheets of these expanding or contracting spherical wavefronts in our picture, so you can plot some of them too for additional insight.

In more detail, the principle outgoing null geodesic congruence has the form
$$\vec{k} = f \, \left( \partial_t + \frac{\sqrt{1+2 \, \epsilon}}{R^\prime} \, \partial_r \right)$$
where f is an undetermined function of t,r. Computing the acceleration and requiring this to vanish gives an equation on f which cannot be solved easily, but computing the expansion scalar using this equation as a constraint, we can verify that the shear and twist scalars vanish identically, so that this congruence really is expanding or contracting spherically. Even better, f factors out from the expression we find for the expansion, so we can compute the locus where the expansion scalar vanishes, which turns out to be $R = 2 \, m$, which defines implicity a locus, a certain submanifold. That is, if our dust includes region containing negative energy shells which collapse to form a black hole, the world sheets encounter a future apparent horizon where the expansion scalar of the principle outgoing null geodesic congruence changes sign from positive to negative (the expansion scalar of the ingoing congruence is remains negative). This is, by definition, an apparent horizon. Pictorially, it is the locus where the contours of R become tangent to the world sheets of the spherical wavefronts of the outgoing null geodesic congruence (increasing surface aera, so expanding, below the AH; decreasing surface area, so contracting above it, with the wavefronts ultimately collapsing to a Big Crunch final singularity).

Similar statements hold for the principle ingoing null geodesic congruence.

We can study the physical experience of observers riding on dust particles by computing some tensors wrt the frame of the dust (case of an LTB dust with spherical shells):
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_t \\ \vec{e}_2 & = & \frac{R^\prime}{\sqrt{1+2 \, \epsilon}} \; \partial_r \\ \vec{e}_3 & = & \frac{1}{R} \; \partial_\theta \\ \vec{e}_4 & = & \frac{1}{R \, \sin(\theta)} \; \partial_\phi \end{array}$$
The acceleration vector vanishes (as it must, since the dust particles feel no forces, in particular, no pressure). So does the vorticity tensor, so the timelike geodesic congruence defined by $\vec{U} = \vec{e}_1$ is hypersurface orthogonal, and the unique family of hyperslices orthogonal to the world lines of the dust particles are the coordinate planes t=t_0 in the Lemaitre chart. The expansion tensor is diagonal with
$$\begin{array}{rcl} H_{22} & = & \frac{-m/R^2 + m^\prime/R^\prime/R + \epsilon^\prime/R^\prime}{\dot{R}} \\ H_{33} & = & H_{44} = \frac{\dot{R}}{R} \end{array}$$
The electroriemann tensor is diagonal with
$$\begin{array}{rcl} E_{22} & = & \frac{-2m}{R^3} + \frac{m^\prime/R^\prime}{R^2} \\ E_{33} & = & E_{44} = \frac{m}{R^3} \end{array}$$
and the magnetoriemann tensor vanishes identically (no spinning sources). The three dimensional Riemann tensor of the hyperslices is
$$\begin{array}{rcl} r_{2323} & = & r_{2424} = \frac{-\epsilon^\prime/R^\prime}{R} \\ r_{3434} & = & \frac{-2\, \epsilon}{R^2} \end{array}$$
The only nonzero component of the Einstein tensor is of course $G^{11} = 8 \pi \; \mu$.

If we set $\epsilon=0$, as a special case of the implicit solution above we obtain the explicit solution
$$R = \begin{cases} ( 9\,m/2 )^{1/3} \; ( t - b )^{2/3} & \rm{expanding \; shells} \\ ( 9\,m/2 )^{1/3} \; ( b - t )^{2/3} & \rm{contracting \; shells} \end{cases}$$
where b gives the coordinate time at which the initial or final singularity occurs, for each shell. This defines the class of LTB dusts in which all the shells have zero energy, i.e. the hyperslices are locally flat. This class of LTB dusts turns out to be too restrictive, in the sense that doesn't exhibit any of the really interesting behavior! (The wealth of this class is only two functions of one variable.) As a special case, $m = k \; r^3$ and $b = b_0$ recovers the (expanding) FRW dust with E^3 hyperslices. Similarly, we can ask when the spatial hyperslices are uniformly curved with S^3 or H^3 geometry. This leads to an equation which can be solved to recover the FRW dusts with S^3 or H^3 geometry. And when $\epsilon$ is constant, we should have slices with $\mathbb{R} \times S^2$ topology (not geometry, unless R has the form $R=r \, f(t)$).

Another important special case: any region $r_1 < r < r_2$ where m is constant is a vacuum region, and because it is spherically symmetric, this must be a portion of the Schwarzschild vacuum. Now the dust congruence is no longer priviliged, but it is still an inertial congruence corresponding to a spherically symmetric family of radially ingoing or outgoing observers. If we choose $\epsilon = 0$, so that the orthogonal hyperslices are locally flat, we recover the Lemaitre observers.

If $R^\prime = 0$ at some r=r_0, t=t_0, two nearby shells $r= r_0-\delta, \; r= r_0 + \delta$ will have almost the same surface area, suggesting that shells are colliding, hence the name "shell-crossing singularity". The Lemaitre chart has a coordinate singularity at such loci, but we can change to a Gautreau chart (a generalized Painleve chart) and this clearly shows that, indeed, shell-crossing singularities occur when either
• outer shells are contracting faster than inner shells and overtake them, piling up to create a "density caustic",
• inner shells are expanding faster than outer shells and overtake them, again forming a density caustic.

(...to be continued)

Figures (left to right):
• Schematic picture of a region in an LTB dust containing a center of spherical symmetry surrounded by negative energy shells of dust, with angular coordinates suppressed. The center of spherical symmetry r=0 at left; initial singularity at bottom; final singularity at top. Contours R=1, R=2 shown with some world sheets of shells (dotted line segments), showing how some of the shells expand to larger sizes than others before recollapsing. Also, recalling that t corresponds to proper time along world lines of dust particles, some particles exist longer than others.
• Plot of the function F(\xi)

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Recapitulation: Overview of the LTB dusts (contd)

The class of LTB dusts (spherical shells) with E^3 hyperslices contains entirely zero energy shells, and the evolution of initial data is fairly uninteresting: if we specify the density on some hyperslice $t=t_1$, this determines the density at future times. Perturbations in expanding shells are smoothed out; perturbations in contracting shells grow over time. This behavior is too simple to allow for phenomena such as overdense regions collapsing to form a black hole in an expanding dust background.

Things get interesting when we consider LTB dusts in which some shells have negative and others positive energy. In a region with negative energy shells, we have three functions of one variable to play with, so we can hope to specify the densities in two hyperslices $t=t_1, \; t=t_2, \; t_1 < t_2$ and interpolate an LTB solution between them. In a series of papers (since 2002, so fairly recent), Krasinski and coworkers have shown that if $\dot{R} \neq 0$ on each slice (there is a hard-to-spot overdot in that last), this can be done! Something similar is true for a region with positive energy shells, but is a bit trickier. In general, we can expect to have zero energy shells functioning as "sepatrices" between regions with negative and positive energy shells. The current "state of the art" in this process seems to be that at the end of the "density to density evolution", one must still check that no shell crossing singularities occur and that the bounds on the allowed values of the energy are obeyed. Krasinski et al. have also studied defining an initial "velocity" distribution using $m^{-1/3} \, \dot{R}$, which is evolved to a given density distribution on a second hyperslice. On e important conclusion drawn in the recent papers of Krasinski's school is that perturbations in both "velocity" and density are required for physically reasonable LTB models of structure formation.

It is useful to bear in mind here that one can feed a numerical solution of the constraint equation to theoretically defined quantities and plot them, so using Maple or Mathematica it should be practical to follow the advice I gave above for visualizing an LTB dust solution using its Lemaitre chart by plotting initial/final/shell-crossing singularities, apparent horizons, and contours of R.

To construct LTB dust models "to order", it is also useful to recognize that we can define R piecewise on different regions. Even better, because the FRW dusts are special cases of LTB dusts and are spatially homogeneous, we can cut out "world tubes" with topology $[0, 1) \times S^2$ and replace the FRW dust there with more interesting (spherically symmetric but inhomogeneous) LTB dust regions,. to create a "Swiss cheese" model (see figure below). The boundaries of these world tubes should be the world sheets of spheres of dust particles in the FRW dust (and also, in the limit, of the interior LTB dust). It is not supposed to be obvious, I think, but to obtain a legal LTB dust it is only neccessary that the three functions of r be continuous across the boundary (a condition independent of time). To see this, you need to know a bit about the Darmois matching conditions, which say that you can match two spacetimes across some world sheet (a submanifold with -++ signature) provided that both the metric tensor (projected to the world sheet) and the extrinsic curvature tensor of the world sheet are continuous across the sheet. (The extrinsic curvature tensor is the negative of the expansion tensor where we formally allow a spacelike rather than timelike normal.) Applying this to the world sheet of a common shell for two LTB dust solutions, we find that the continuity of the three functions $m, \; \epsilon, \; b$ is all that is needed.

Another important special case involves matching an LTB dust (with spherical shells) interior to a Schwarzschild vacuum exterior (recall that this is a very special case of an LTB dust!); then the exterior region is static asymptotically flat, so the Komar mass is always defined, and can be understood as the total mass contained in the dust region. See my Post #22 for two examples (belonging to the class of E^3 hyperslices) of such an asymptotically flat LTB dust soluton. Note that the contours of R are continuous but not C^1 across the boundary.

Regarding the matching to FRW dusts, note for future reference that we can write a collapsing FRW (zero energy shells) dust ball in the form
$$\begin{array}{rcl} R(t,r) & = & (9 \,m(r)/2)^{1/3} \; (b(r)-t)^{2/3} \\ m(r) & = & m_{\rm tot} \, (r/r_s)^3 \\ b(r) & = & r_s \\ \epsilon(r) & = & 0 \end{array}$$
where $r = r_s$ is the surface of the ball and $m_{\rm tot}$ is its mass, and similarly for an expanding FRW dust ball.

While I have been focusing here on the LTB dusts constructed using spherical shells, much of what I have said holds true with minimal changes for the other shell geometries. One change needed is that the vacuum case of the LTB dust with cylindrical shells is the Levi-Civita AIII vacuum, aka the plane symmetric Kasner vacuum, an important and useful example of a homogeneous but anisotropic static vacuum solution; it is in fact the unique cylindrically symmetric static vacuum of Petrov type D. Similarly, the vacuum case of the LTB dust with pseudospherical shells is the Levi-Civita AII vacuum, the H^2 symmetric static vacuum of Petrov type D. In the case of cylindrical shells, I have already tacitly refered to the possibility of unwrapping them to obtain E^2 shells (the plane being the universal covering space of the cylinder). We can go in the other direction too, replacing H^2 shells with shells all having the geometry of the same discrete quotient manifold of H^2 (as in those funky pictures of Escher), and likewise for S^2, and we can use flat toroidal shells instead of cylinders or planes in the E^2 symmetric LTB dusts.

As Ich pointed out, because the constraint equation shared by all the LTB dusts is simply the Newtonian equation governing radial motion in a spherically symmetric gravitational field, we know that the motion of each shell is unaffected by the motion of exterior shells (at least, in the case where we have a center of symmetry r=0, and where R is increasing with r). Thus we obtain a number of generalizations of Birkhoff's theorem on spherically symmetric vacuums to other dusts solutions containing vacuum regions having local spherical, cylindrical/planar/flat-toroidal, or pseudospherical symmetry: the vacuum region must actually be static and locally isometric to LC AI, AIII, or AII vacuums respectively. (I am speaking somewhat loosely; Krasinski offers a good explanation of some subtle points involved in a rigorous proof of the original Birkhoff theorem on spherically symmetric vacuum regions.)

When we allow both positive and negative energy shells we can have some interesting global structure, because some of the shells exist for only finite proper times (according to an ideal clock carried by an ideal observer riding on some dust particle in some shell) while others exist indefinitely, so the boundary of a final singularity may "bend up to plus infinity" at some r=r_0 in the Lemaitre chart, and likewise an initial singularity may "bend down to minus infinity". See my Post #16 for two simple examples illustrating these possibilities. It will be instructive to embed (at least qualitatively) the spatial hyperslices (with one angular coordinate suppressed)! These examples show that we while at least locally we must have spherical symmetry, our LTB dust model might have no center of spherical symmetry (which happens only when $R=0$ on $r=0$ for $t_1 < t < t_2$), or more than one center of symmetry, even in the case where we have global spherical symmetry. If we have two centers of symmetry, we may find that there are two exterior sheets which cannot communicate with each other (i.e. no timelike or null curve passing through one region passes through the other region).

Clearly, even though the LTB dusts are defined by only three functions of one variable, they allow for a great deal of variety, and can be used to study many interesting possibilities. In particular, they allow for models in which black holes are formed by the collapse of inhomogeneous configurations of dust (see figures below) in various ways.

LTB dusts can be generalized to include nonzero Lambda; indeed Lemaitre 1933 already did so. The generalized constraint equation is
$$\dot{R}^2/2 = \frac{m}{R} + \epsilon - \Lambda/6 \, R^2$$
There are important further generalizations of LTB dusts to include charged dusts (with charges stuck to the dust particles), or dusts with an imposed magnetic or electric field (with sources located outside the dust region), and so on. But the most important desired generalization, to include regions where the congruence of world lines of the dust particles acquires nonzero vorticity (so that the orthogonal hyperslices are no longer defined), is currently lacking. "Swirling" dust solutions (in which the dust particles have nonzero angular momentum about some axis of axial symmetry) are known, but there is no known general construction including the LTB dusts as a special case.

The LTB dusts are defined using nested shells with either spherical, cylindrical, or pseudospherical geometries, where you must choose one of these alternatives for each LTB model. Thus, they have a three dimensional Lie algebra of Killing vectors (defined at least locally; in the Swiss Cheese models we certainly do not have a global SO(3) action!) There are dust solutions, such as the Szekeres dust, which have no Killing vector fields at all, so they are completely inhomogeneous and anisotropic. Whese these include world tube with a spherical surface, we may be able match to an LTB interior, obtaining even more general Swiss Cheese models.

Figures (left to right):
• schematic illustrating the idea of Swiss Cheese models, with inhomogenous regions constructed "to order" in world tubes inside a suitable FRW background,
• schematic of a very simple two component Swiss Cheese model (depicted using Lemaitre chart), consisting of an LTB dust region, with a center of spherical symmery at r=0 and nested negative energy (bound) spherical shells of dust, matched to a vacuum exterior, showing apparent horizons; at bottom, the ingoing principle null geodesic congruence changes from expanding below the AH to contracting above, while at top, the outgoing principle null geodesic congruence changes from expanding above the AH to contracting above it.
• schematic of a more elaborate Swiss Cheese model, showing how matching various world tubes containing suitable vacuum regions, or inhomogeneous LTB dust, to homogeneous FRW dust allows one to construct arbitrarily detailed models of gravitational collapse, not neccessarily globally spherically symmetric.

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