# BRS: The FRW Dust with E^3 Hyperslices

The simplest frame Ansatz consistent with Hubble's discovery is:
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_t - f \, \left( x \, \partial_x + y \, \partial_y + z \, \partial_z \right) \\ \vec{e}_2 & = & \partial_x \\ \vec{e}_3 & = & \partial_y \\ \vec{e}_4 & = & \partial_z \end{array}$$
where f is a function of t only. This says:
• we have a family of observers whose world lines are the integral curves of the timelike unit vector field $\vec{e}_1$,
• the coordinate line x=y=z=0 is one of these world lines,
• the coordinate planes $t=t_0$ are spatial hyperslices orthogonal to the world lines, and they have euclidean geometry,
• the time coordinate t corresponds to elapsed proper time according to clocks carried by our observers,
• the other world lines are curves which expand from the origin at a rate proportional to distance $\rho = \sqrt{x^2+y^2+z^2}$, and also depending on time t via the undetermined function f.
In the last claim, it may help to notice that in polar spherical coordinates in euclidean space
$$\partial_\rho = \frac{1}{\sqrt{x^2+y^2+z^2}} \; \left( x \, \partial_x + y \, \partial_y + z \, \partial_z \right)$$
so that
$$\rho \, \partial_\rho = \left( x \, \partial_x + y \, \partial_y + z \, \partial_z \right)$$
is a vector field pointing away from the origin, with magnitude $\rho$. Needless to say, the rate factor here is Hubble's "constant", which depends on time but is constant over each spatial hyperslice.

Now, computing the Einstein tensor we find we have a perfect fluid, and the condition that the pressure vanish gives a simple ODE which we can easily solve, finding $f = 2/3/t$.

To verify this, we need to first find the dual coframe, which only requires some trivial linear algebra:
$$\begin{array}{rcl} \sigma^1 & = & -dt \\ \sigma^2 & = & dx - f \, x \, dt \\ \sigma^3 & = & dy - f \, y \, dt \\ \sigma^4 & = & dz - f \, z \, dt \end{array}$$
Now, we verify the claims above using Maxima! Here is a file you can run in wxmaxima's batchmode:

Code:
/*
Derive the FRW dust with E^3 hyperslices!
Simplest Anstaz consistent with Hubble law is
e_1 = @_t + f(t) ( x @_x + y @_y + z @_x)
e_2 = @_x
e_3 = @_y
e_4 = @_z
Computing the Einstein tensor, we find this gives a perfect fluid.
We solve a simple ODE to obtain a dust solution.
*/
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [t,x,y,z];
/* variables */
depends(f,t);
/* define background metric */
lfg: ident(4);
lfg[1,1]: -1;
/* define the coframe */
fri: zeromatrix(4,4);
fri[1,1]: -1;
fri[2,1]: -x*f;
fri[2,2]:  1;
fri[3,1]: -y*f;
fri[3,3]:  1;
fri[4,1]: -z*f;
fri[4,4]:  1;
/* 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);
/* solve */
ode2(ein[2,2],f,t);
solve([%],[f]);
subst(0,%c,%);
Next, you can modify the above file to replace f with the solution we just found, $f=2/3/t$.

So we have a dust solution with line element
$$\begin{array}{rcl} ds^2 & = & -\left( 1- \frac{4 \, (x^2+y^2+z^2)}{9 \, t^2} \right) \; dt^2 - \frac{4 \, dt}{3 \, t} \; \left( x \, dx + y \, dy + z \, dz \right) + dx^2+dy^2+dz^3 \\ && 0 < t < \infty, \; -\infty < x, \, y, \, z < \infty \end{array}$$
The frame of observers riding on dust particles is
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_t - \frac{2}{3 \, t} \; \left( x \, \partial_x + y \, \partial_y + z \, \partial_z \right) \\ \vec{e}_2 & = & \partial_x \\ \vec{e}_3 & = & \partial_y \\ \vec{e}_4 & = & \partial_z \end{array}$$
We know this because the Einstein tensor vanishes except for a single component, which also shows that the density of the dust, as measured by our observers, is
$$\mu = \frac{1}{6 \pi \, t^2}$$
The Riemann tensor has two triple eigenvalues, $4/9/t^2, \; -2/9/t^2$, which shows that we have a strong scalar curvature singularity at t=0. This locus is of course familiarly known as "the Big Bang".

The world lines of the dust particles are the integral curves of our timelike unit vector field $\vec{e}_1$, namely
$$x = x_0 \, (t/t_0)^{2/3}, \; y = y_0 \, (t/t_0)^{2/3}, \; z = z_0 \, (t/t_0)^{2/3}$$
It is a good idea to draw some of them and to "decorate" them with some light cones (drawn to scale) at some events on these world lines. Note that the light cones appear to be sheared radially outwards, with (for a given time $t=t_0$, the shear being larger for larger radii $\rho = \sqrt{x^2+y^2+z^2}$. (See sketch below.) Note too that the volume form is
$$dt \wedge dx \wedge dy \wedge dz$$
the same as in flat spacetime, so our chart correctly represents volumes and distances within each constant time slice, but at least some timelike geodesics are not represented as coordinate lines, and due to the shearing of the light cones, Lorentzian "angles" are not correctly represented (we say that this is not a conformal chart). Because of the "cross-terms", is also not an orthogonal coordinate chart.

If you use GRTensorII running under Maple, you can easily find the Killing vector fields of this solution. It turns out that we have a six dimensional Lie algebra of Killing vector fields, consisting of three "infinitesimal rotations"
$$-y \, \partial_x + x \, \partial_y \; -z \, \partial_y + y \, \partial_z, \; -x \, \partial_z + z \, \partial_x$$
and three "infinitesimal translations"
$$t^{2/3} \, \partial_x, \; t^{2/3} \, \partial_y, \; t^{2/3} \, \partial_z$$
In other words, the isometry group of our manifold is the six dimensional euclidean group, a Lie group which acts transitively on each spatial hyperslice t=t_0. In the next post, we'll use this information to solve the geodesic equations.

Again using GRTensorII (or in a pinch, computing by hand as per Flanders, Applications of Differential Forms to the Physical Sciences), you can show that the expansion tensor of the timelike congruence generated by our timelike unit vector field $\vec{e}_1$ is
$${H\left[ \vec{e}_1 \right]}_{ab} = \frac{2}{3 \, t} \; \operatorname{diag} \left( 1,1,1 \right)$$
shows uniform expansion, as expected--- in particular, while in our chart the geodesic $x=y=z=0$ appears to play the role of a dust particle from which all the others are expanding, in fact the expansion tensor and the action by the euclidean group shows that all dust particles are equivalent in this solution. Also, the acceleration and vorticity vectors vanish. (The acceleration vector of a dust particle always vanishes in a dust solution, so this result was expected.) The electroriemann tensor (aka tidal tensor) is
$${E\left[ \vec{e}_1 \right]}_{ab} = \frac{2}{9 \, t^2} \; \operatorname{diag} \left( 1,1,1 \right)$$
which shows uniform tidal compression--- the gravitational attraction of the dust inside a small sphere of dust particles slows the expansion, so this is also, qualitatively, the expected result. The magnetoriemann tensor vanishes, as we should expect because the dust particles are not swirling about each other, so there is no "rotating matter" in that sense.

You might recognize that we have been using a Painleve type chart for the expanding FRW dust with E^3 hyperslices orthogonal to the world lines of the dust particles. It is not hard to immediately write down the analogous chart for the contracting FRW dust with E^3 hyperslices, which can be matched across a contracting sphere of dust particles to a region of the Schwarzschild vacuum to obtain the famous Oppenheimer-Snyder collapsing dust ball solution, which models the formation of a black hole by a dust (pressure-free fluid) which undergoes complete gravitational collapse. This can be generalized to LTB dusts and to various fluid solutions, in addition to solutions exhibiting further possibilities for the stress-energy tensor, such as contributions from a hypothetical minimally coupled massless scalar field or a Lambda term. We can also pass from E^3 hyperslices to S^3 or H^3 hyperslices, and we can take discrete quotients (for example, we can obtain a model with T^3 hyperslices by quotienting the solution we have been discussing).

Figure: Sketch of some world lines of dust particles and some light cones drawn to scale.

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BRS: The FRW Dust with E^3 Hyperslices. II. Geodesics

From the previous post, we know that our solution admits a six dimensional Lie algebra of Killing vector fields. By Noether's theorem on variational symmetries, these should yield six invariants of geodesic motion. Indeed, we obtain three linear momenta
$$A = t^{2/3} \dot{x} - \frac{2 x \, \dot{t}}{3 \, t^{1/3}}, \; B = t^{2/3} \dot{y} - \frac{2 y \, \dot{t}}{3 \, t^{1/3}}, \; C = t^{2/3} \dot{z} - \frac{2 z \, \dot{t}}{3 \, t^{1/3}}, \;$$
and three angular momenta
$$J = -y \, \dot{x} + x \, \dot{y}, \; K = -z \, \dot{y} + y \, \dot{z}, \; L = -x \, \dot{z} + z \, \dot{x}$$
The first three immediately yield three first integrals of the geodesic equations:
$$\begin{array}{rcl} \dot{x} & = & \frac{A}{t^{2/3}} + \frac{2 x \, \dot{t}}{3 t} \\ \dot{y} & = & \frac{B}{t^{2/3}} + \frac{2 y \, \dot{t}}{3 t} \\ \dot{z} & = & \frac{C}{t^{2/3}} + \frac{2 z \, \dot{t}}{3 t} \end{array}$$
Plugging these into the line element
$$\epsilon = -\left( 1- \frac{4 \, (x^2+y^2+z^2)}{9 \, t^2} \right) \; \dot{t}^2 - \frac{4 \, \dot{t}}{3 \, t} \; \left( x \, \dot{x} + y \, \dot{y} + z \, \dot{z} \right) + \dot{x}^2+\dot{y}^2+\dot{z}^3$$
yields a fourth first integral
$$\dot{t} = \sqrt{-\epsilon + \frac{A^2+B^2+C^2}{t^{4/3}}$$
where we choose the positive sign of the square root so that increasing our parameter increases the time coordinate. Up to quadrature this gives a complete solution of the geodesic equations for timelike, null, or spacelike geodesics (choose $\epsilon=-1, \, 0, 1$ respectively).

(See https://www.physicsforums.com/showthread.php?t=380917 for a more general discussion of the procedure for solving the geodesic equations using symmetries encoded in the Lie algebra of Killing vector fields.)

In particular, we have the null geodesic congruence
$$\begin{array}{rcl} \vec{k} & = & \frac{\sqrt{A^2+B^2+C^2}}{t^{2/3}} \; \partial_t + \left( \frac{A}{t^{2/3}} + \frac{2 \, x \, \sqrt{A^2+B^2+C^2}}{3 \, t^{5/3}} \right) \; \partial_x \\ && + \left( \frac{B}{t^{2/3}} + \frac{2 \, y \, \sqrt{A^2+B^2+C^2}}{3 \, t{5/3}} \right) \; \partial_y + \left( \frac{C}{t^{2/3}} + \frac{2 \, z \, \sqrt{A^2+B^2+C^2}}{3 \, t^{5/3}} \right) \; \partial_z \end{array}$$
corresponding to "photons" all having the same linear momenta $A,B,C$ and having angular momenta
$$J = \frac{A \, y - B \, x}{t^{2/3}}, \; K = \frac{B \, z - C \, y}{t^{2/3}}, \; L = \frac{C \, z - A \, x}{t^{2/3}}$$
(The angular momenta are constant on each geodesic, but not constant over the congruence; the linear momenta are however constant over the congruence, by definition of our congruence.) If you have installed GRTensorII running under Maple, you can compute the optical scalars (see D'Inverno's textbook, section 21.9), finding that the optical expansion scalar of this congruence is
$$\theta\left[\vec{k}\right] = \frac{2 \, \sqrt{A^2+B^2+C^2}}{3 \, t^{5/3}}$$
with vanishing shear and twist scalars. For comparison, the analogous congruence in Minkowski spacetime is the plane wave congruence, which has vanishing optical scalars.

In our null geodesic congruence, the integral curve passing through the event $(t, x,y,z) = (t_0, \, x_0, \, y_0, \, z_0)$ is
$$\begin{array}{rcl} x = \frac{3 A \, t}{\sqrt{A^2+B^2+C^2}} + \left( \frac{x_0}{t_0^{2/3}} - \frac{3 A \, t_0^{1/3}}{\sqrt{A^2+B^2+C^2}} \right) \; t^{2/3} \\ y = \frac{3 B \, t}{\sqrt{A^2+B^2+C^2}} + \left( \frac{y_0}{t_0^{2/3}} - \frac{3 B \, t_0^{1/3}}{\sqrt{A^2+B^2+C^2}} \right) \; t^{2/3} \\ z = \frac{3 C \, t}{\sqrt{A^2+B^2+C^2}} + \left( \frac{z_0}{t_0^{2/3}} - \frac{3 C \, t_0^{1/3}}{\sqrt{A^2+B^2+C^2}} \right) \; t^{2/3} \end{array}$$
For some choice of constants A,B,C, every null geodesic has this form, so we have in fact just written down explicitly all null geodesics in our spacetime.

If you use GRTensorII running under Maple, it is easy to find the spherically contracting null geodesic congruence in which the geodesics converge on the world line of the dust particle at the origin. It is
$$\vec{\ell} = t^{-2/3} \, \partial_t + \left( \frac{2}{3 \, t^{5./3}} - \frac{1}{t^{2/3} \, \sqrt{x^2+y^2+z^2}} \right) \; \left( x \, \partial_x + y \, \partial_y + z \, \partial_z \right)$$
The optical expansion scalar of this congruence is
$$\theta\left[\vec{\ell}\right] = \frac{2}{3 \, t^{5/3}} - \frac{1}{t^{2/3} \, \sqrt{x^2+y^2+z^2}}$$
while the shear and twist scalars vanish. This is the congruence consisting of all null geodesics which hit the world line of the observer at the origin, so it extends his past light cones to his absolute past (see Hawking & Ellis, Large Scale Structure of Spacetime). These comprise (some of) the same geodesics we just found, repackaged in a new null geodesic congruence. In the language of geometric optics, the wave vector is $\vec{\ell}$ and the wavefronts are spheres which contract onto the observer at the origin.

For comparison, the analogous congruence in Minkowski vacuum is the spherically contracting congruence
$$\vec{\ell} = \partial_t - \frac{1}{\sqrt{x^2+y^2+z^2}} \; \left( x \, \partial_x + y \, \partial_y + z \, \partial_z \right)$$
which has optical expansion scalar
$$\theta\left[\vec{\ell}\right] = \frac{-1}{\sqrt{x^2+y^2+z^2}}$$
and vanishing shear and twist scalars. Near a given hyperslice $t=t_0$, we can see that, compared to the flat spacetime spherically contracting null geodesic congruence, the expansion scalar of the spherically contracting null geodesic congruence in our FRW dust solution is rescaled and augmented by a positive term.

If we put
$$A = \frac{-x_0}{\sqrt{x_0^2+y_0^2+z_0^2}}, \; B = \frac{-y_0}{\sqrt{x_0^2+y_0^2+z_0^2}}, C = \frac{-z_0}{\sqrt{x_0^2+y_0^2+z_0^2}}$$
in our expression for the general null geodesic trajectory, we obtain the trajectories (unparameterized curves) for our spherically contracting null geodesic congruence in the FRW dust with E^3 hyperslices:
the integral curve passing through the event $(t,x,y,z) = (t_0, \, x_0, \, y_0, \, z_0)$ is
$$\begin{array}{rcl} x = \frac{3 \, x_0 \, t}{\sqrt{x_0^2+y_0^2+z_0^2}} + \left( \frac{x_0}{t_0^{2/3}} - \frac{3 \, x_0 \, t_0^{1/3}}{\sqrt{x_0^2+y_0^2+z_0^2}} \right) \; t^{2/3} \\ y = \frac{3 \, y_0 \, t}{\sqrt{x_0^2+y_0^2+z_0^2}} + \left( \frac{y_0}{t_0^{2/3}} - \frac{3 \, y_0 \, t_0^{1/3}}{\sqrt{x_0^2+y_0^2+z_0^2}} \right) \; t^{2/3} \\ z = \frac{3 \, z_0 \, t}{\sqrt{x_0^2+y_0^2+z_0^2}} + \left( \frac{z_0}{t_0^{2/3}} - \frac{3 \, z_0 \, t_0^{1/3}}{\sqrt{x_0^2+y_0^2+z_0^2}} \right) \; t^{2/3} \end{array}$$
See the figure below and note that in our Painleve chart, these geodesics appear to "roll over", so that the boundary of the absolute past of the event in question looks like the surface of an onion. I stress again that in our chart, although the observer at $x=y=z=0$ appears to play a distinguished role, we know from the expansion and tidal tensors and the Killing flows that this is illusory--- in fact, each dust particle world line is equivalent in the geometry of our spacetime model to any other.

Figure: some null geodesics passing through an event on the world line $x=0, \, y=0, \, z=0$ (blue).

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BRS: The FRW Dust with E^3 Hyperslices. III. The Comoving Chart

You probably noticed in Post I above that, although our frame is very simple, the form of the line element is rather awkward! Even worse, the Painleve chart obscures the fact that the world lines of all the dust particles are geometrically equivalent, in the sense that the six-dimensional Lie group of isometries (which we identified as the euclidean group) can move the world line of any dust particle onto the world line of any other dust particles--- note too that it can move any coordinate line in any slice $t=t_0$ (spacelike geodesic of our spacetime, lying in the slice) onto any other coordinate line in the same slice.

It is often convenient to use a chart which captures the fact that all the dust particles play equivalent roles in our FRW dust solution. We want to straighten out the world lines
$$x/x_0 = y/y_0 = z/z_0 = (t/t_0)^{2/3}$$
so we should put
$$x^\prime = x/t^{2/3}, \; y^\prime = y/t^{2/3}, \; z^\prime = z/t^{2/3}$$
so that in the new coordinates, the world lines of the dust particles are simply
$$x^\prime = {x^\prime}_0, \; y^\prime = {y^\prime}_0, \; z^\prime = {z^\prime}_0$$
The line element becomes (dropping the primes, as we shall have, I think, no further use for the Painleve chart in this thread)
$$ds^2 = -dt^2 + t^{4/3} \; \left( dx^2 + dy^2 + dz^2 \right), \; 0 < t < \infty, \; -\infty < x, \, y, \, z < \infty$$
The frame of the dust particles becomes
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_t \\ \vec{e}_2 & = & t^{-2/3} \, \partial_x \\ \vec{e}_3 & = & t^{-2/3} \, \partial_y \\ \vec{e}_4 & = & t^{-2/3} \, \partial_z \end{array}$$
It is a good idea to immediately draw some light cones to scale at various events along some dust particle world lines. Notice that they maintain constant height in this chart, but appear to narrow as t increases, because the new spatial coordinates are rescaled at each t so that the dust particles maintain constant x,y,z coordinate, but of course the proper distance ("measured" by rulers along a spacelike geodesic in a constant time slice--- note we are speaking of geometrical meaning here, not a physical measurement!) between them is increasing.

The comoving chart is not volume preserving, and it is not conformal, but it does represent two important classes of geodesics as coordinate lines (the world lines of the dust particles, and spacelike geodesics lying in the spatial hyperslices), and it is an orthogonal coordinate chart. Because the coordinate components of the spatial frame vectors only contain t, it is also a spatially isotropic chart; euclidean angles measured in any spatial hyperslice are correctly represented. Like the Painleve chart, this chart correctly represents elapsed proper time measured by observers riding with the dust particles. The simplicity of the line element in the comoving chart is remarkable; very few exact solutions of the Einstein field equation admit such a simple expression!

Since the time coordinate is the same as in the Painleve chart, the eigenvalues of the Riemann tensor keep the form we found earlier; the locus t=0 represents "the Big Bang". Of the six generators of the Lie algebra of Killing vector fields, the three "infinitesimal rotations" keep the same form, while the three "infinitesimal translations" become $\partial_x \, \; \partial_y, \; \partial_z$. Because only t appeared in the components of the expansion tensor, Einstein tensor, and electroriemann tensor when we computed these in the Painleve chart, these components take the same form in the comoving chart.

To find the geodesics in the comoving chart, we can either apply the coordinate transformation to the results we obtained in Post II, or we can apply the same procedure to derive them. Either way, we find the first integrals
$$\dot{x} = A/t^{2/3}, \; \dot{y} = B/t^{2/3}, \; \dot{z} = C/t^{2/3}, \; \dot{t} = \sqrt{ -\epsilon + (A^2+B^2+C^2)/t^{4/3}}$$
The null geodesic congruence with fixed linear momenta A,B,C is
$$\vec{k} = \frac{\sqrt{A^2+B^2+C^2}}{t^{2/3}} \; \partial_t + \frac{1}{t^{4/3}} \; \left( A \, \partial_x + B \, \partial_y + C \, \partial_z \right)$$
which has optical expansion scalar
$$\theta \left[ \vec{k} \right] = \frac{2 \, \sqrt{A^2+B^2+C^2}}{3 \, t^{5/3}}$$
(positive, but small at large times) with vanishing twist and shear scalars. The integral curve through the event $(t,x,y,z) = (t_0, x_0, y_0, z_0)$ is
$$\begin{array}{rcl} x & = & x_0 + \frac{3 \, A\, (t^{1/3} - t_0^{1/3} )}{\sqrt{A^2+B^2+C^2}} \\ y & = & y_0 + \frac{3 \, B \, (t^{1/3} - t_0^{1/3} )}{\sqrt{A^2+B^2+C^2}} \\ z & = & z_0 + \frac{3 \, C \, (t^{1/3} - t_0^{1/3} )}{\sqrt{A^2+B^2+C^2}} \end{array}$$
As before, every null geodesic through that event has this form for some choice of A,B,C, so we have here written down the equation of the general null geodesic in the comoving chart.

The contracting null geodesic congruence with spherical wavefronts contracting onto the world line $x=x_0, \, y=y_0, \, z=z_0$ is
$$\begin{array}{rcl} \vec{\ell} & = & \frac{1}{t^{2/3}} \, \partial_t - \frac{1}{t^{4/3} \, \sqrt{(x-x_0)^2+(y-y_0)^2 + (z-z_0)^2}} \; \cdot \\ && \hspace{0.5in} \left( (x-x_0) \, \partial_x + (y-y_0) \, \partial_y + (z-z_0) \, \partial_z \right) \end{array}$$
(where the case $x_0 = y_0 = z_0 = 0$ is the same congruence we found in Post II, expressed in the comoving chart). The expansion scalar is
$$\theta\left[ \vec{\ell} \right] = \frac{2}{3 \, t^{5/3}} - \frac{1}{t^{4/3} \; \sqrt{(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2}}$$
while the shear and twist scalars vanish.

The null geodesic which passes through the event $(t,x,y,z) = (t_0, x_0, y_0, z_0)$ and later passes through the world line $x=y=z=0$ is given by
$$\begin{array}{rcl} x & = & x_0 - \frac{3 \, x_0\, (t^{1/3} - t_0^{1/3} )}{\sqrt{x_0^2+y_0^2+z_0^2}} \\ y & = & y_0 - \frac{3 \, y_0 \, (t^{1/3} - t_0^{1/3} )}{\sqrt{x_0^2+y_0^2+z_0^2}} \\ z & = & z_0 - \frac{3 \, z_0 \, (t^{1/3} - t_0^{1/3} )}{\sqrt{x_0^2+y_0^2+z_0^2}} \end{array}$$
See the Figure below, and notice that these curves appear rather differently in the comoving chart than they do in the Painleve chart. (Those of you who have read Steven Weinberg's classic, The First Three Minutes, may recognize this figure!) Notice that these null geodesics intersect the locus $t=0$ in a circle (shown in green). Hence "the horizon problem": at any given time, the dust particle at $x=y=z=0$ has not yet experienced any causal influences from sufficiently distant dust particles.

Figures: some null geodesics (red) intersecting the world line of the dust particle at $x=y=z=0$ (blue). Two views.

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BRS: The FRW Dust with E^3 Hyperslices. IV. Observables

On cosmological scales, astronomers are puny, slow, and short-lived. For all practical purposes, their world lines are confined to a tiny neighborhood of an event "here and now", which we may as well identify with the event $t=t_1, x=y=z=0$ in our cosmological model (FRW dust with E^3 hyperslices). We need to say something about how we can assign coordinates to distant events using only observations made near this event.

Notice first that according to our model, the average density of matter near "here and now" should be
$$\mu_{\rm now} = \frac{1}{6 \pi \, t_1^2}$$
so we can in principle roughly estimate $t_1$ from a rough estimate of the density of matter inside a sphere of radius, say, half a billion light years.

(We'll see that while there are numerous distinct operationally significant notions of "distance in the large", they all agree for distances which are small compared to the scale of the ambient curvature near "here and now", so we need not worry too much about how we decide what matter lies within this sphere. Most FRW dust models are defined in terms of a free parameter, but the FRW dust model with E^3 hyperslices separates two such classes of models, having S^3 and H^3 hyperslices, so it has no free parameter, modulo scaling the coordinates. This is why we can express "current time" in terms of "current density" in our model.)

The most important observable quantity in any cosmological model is without doubt the redshifts associated with the "galaxies" (modeled here by the dust particles). To find them, begin by rewriting the line element in the comoving chart in polar spherical form
$$ds^2 = -dt^2 + t^{4/3} \; \left( dr^2 + r^2 \; ( d\theta^2 + \sin(\theta)^2 \, d\phi^2) \right)$$
Then for null geodesics (integrating backwards in time from an event on the world line r=0), we have $dr/dt = t^{-2/3}$, so the null geodesic from $(t,r) = (t_0, r_0)$ through $(t,r)=(t_1,0)$ where $t_1 > t_0 > 0$ is:
$$r_0 = 3 \, \left( t_1^{1/3} - t_0^{1/3} \right)$$
Now consider a second null geodesic from $(t,r)=(t_0+h,r_0)$ through $(t,r) = (t_2,0)$; it is given by
$$r_0 = 3 \, \left( t_2^{1/3} - (t_0+h)^{1/3} \right)$$
Combining,
$$\frac{\lambda_{\rm rec}}{\lambda_{\rm em}} = \lim_{h \rightarrow 0^+} \frac{t_2-t_1}{h} = \left(1 + \frac{r_0}{3 \, t_0^{1/3}} \right)^2 = \left( 1 - \left( \frac{t_1}{t_0} \right)^{1/3} - 1 \right)^2 = \left( \frac{t_1}{t_0} \right)^{2/3}$$
a quantity which, I am embarassed to say, astronomers conventionally write $1+z$ even though this guarantees that this notion of "redshift" will problematic. (Statisticians are very familiar with the conceptual pitfalls of using"fractional change" instead of the logarithm of a ratio!)

We can now define a number of notions of "distance in the large", which can be computed from the redshift, which is observable. Of course, our distance estimates depend upon our model, which is simpler than the ones actually used in contemporary cosmology. However, it is worth going through these computations in order to see how the process works in the context of the simplest reasonable cosmological model.

Probably the most natural notion of distance to a distant galaxy is the "lookback time" (sometimes called "light travel time", although of course the photons recognize no such quantity!)
$$\Delta t = t_1 - t_0 = t_1 \; \left(1 - (1+z)^{-3/2} \right) = \frac{3 \, t_1}{2} \; \left( z - \frac{5}{4} \, z^2 \right) + O(z^3)$$
(called $D_{\rm ltt}$ in Wright, "A Cosmology Calculator for the World Wide Web"; see http://arxiv.org/abs/astro-ph/0609593). This is a coordinate time difference, which in our comoving chart corresponds to the elapsed proper time of any dust particle, so this notion of distance does make geometrical sense, and is physically not terribly unreasonable.

Another simple notion of "distance in the large" arises when we apply the current value of the scale factor $a(t_1) = t_1^{2/3}$ to the comoving coordinate radius $r_0$, which gives
$$D_{\rm now} = 3 \, t_1 \; \left( 1 - (1+z)^{-1/2} \right) = \frac{3 \, t_1}{2} \; \left( z - \frac{3}{4} z^2 \right) + O(z^3)$$
which is slightly larger than the lookback time, but agrees with it approximately for modest values of z. Needless to say, we could just as well apply instead the factor $a(t_0)$, which gives
$$D_{\rm then} = D_{\rm now} \; \left( \frac{t_0}{t_1} \right)^{2/3} = \frac{D_{\rm now}}{1+z} = \frac{3 \, t_1}{2} \; \left( z - \frac{7}{4} z^2 \right) + O(z^3)$$
Or we could apply the scale factor at an intermediate time. Which scale factor is "best"? It depends, but if we extend the Hubble law using the current value of the scale factor to distant galaxies, the appropriate distance to use is $D_{\rm now}$ since (by definition!) this is the one for which the linear relation holds exactly. Geometrically, it is the proper distance measured along a spacelike geodesic segement lying in the hyperslice $t=t_1$ and running from the world line $r=0$ to the world line $r=r_0$, but while this makes perfect sense geometrically, this obviously does not correspond directly to an experiment we can carry out!

In models with E^3 hyperslices, such as the one we are studying, $D_{\rm then} = D_A$, the angular-size distance. (See figure below for a comparative graph.)

Another popular notion of distance often used in cosmology is the luminosity distance, which for models with E^3 hyperslices is related to the others by $D_L = (1+z) \, D_{\rm now}$. (See figure below for a comparative graph.)

Figure: left to right:
• Deriving the redshift factor
• Some " distances in the large": a log-log plot with respect to z (computed using the rather silly value t_1 = 5):
• $D_{\rm L}$ (blue)
• $D_{\rm now}$ (red)
• $D_{\rm ltt}$ (green)
• $D_{\rm A}$ (yellow)

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BRS: The FRW Dust with E^3 Hyperslices: Mechanical Note

I am by no means done with this thread, but I am currently distracted, so there may be an awkward and unfortunate delay of one or two weeks at least before I can finish talking about all the points I wanted to make in this thread :sad:

Partly as a note to myself in case the delay becomes extended: I want to
• compare with the Milne model
• discuss how to plug numbers into these models (in particular, I have, without comment, been using geometric notation in which G=c=1, and I have suppressed mathematically inessential parameters which some cosmologists would include from the beginning to avoid a lengthy discussion of how to plug in numbers),
• say a bit more about generalizations to some closely related solutions:
• FRW dusts with S^3 and H^3 slices, citing a seperate BRS thread on Carter-Penrose diagrams,
• Lambda
• FRW radiation fluids (EOS $p = \mu/3$)
• FRW solutions with mixed equations of state (EOS) which crudely interpolate between radiation fluids and dust (EOS $p = 0$)
• (possibly) adding scalar field contributions to the energy-momentum-stress tensor
• perturbations of FRW dusts to anisotropic or inhomogeneous dust solutions
• reverse time and consider contracting FRW dust with E^3 slices, showing how to match a region of this solution to an exterior region which is locally isometric to (part of) the Schwarzschild vacuum, to form the famous Oppenheimer-Snyder model of a black hole formed by gravitational collapse of an isolated dust ball,
• debunk the notion that FRW dusts (or any reasonably accurate cosmological model) can be considered as "an inside out black hole" [sic],
• (possibly) flowcone charts, beacon congruence in a Coll chart (see the papers in the arXiv on a fully relativistic beaconing navigation system--- Coll and others are looking ahead to the epoch in which humans might actually need to locate in spacetime a spaceprobe which is some light days distant from Earth; GPS and similar systems are in contrast only valid near the surface of the Earth).

Eventually, should fate look kindly upon this venture, I'd like to discuss some other famous dust solutions (probably in a new BRS thread) which illustrate various themes of classical gtr and in some cases can serve as cosmological models:
• Kasner dust, the simplest inhomogeneous dust (a special case is plane symmetric, so "partially isotropic"); this example is needed to discuss the important BKL conjecture,
• LTB dusts, the general spherically symmetric dust, often used in modeling gravitational collapse,
• Van Stockum dust, a cylindrically symmetric dust solution in which dust swirls about a distinguished axis r=0; the density increases exponentially and the solution becomes dubious outside a certain radius (see if you can guess why!), but it can be cut off at a smaller radius and matched to an exterior vacuum solution (the cylindrically symmetric static vacuum solution of Levi-Civita, showing that rotating matter here doesn't always produce exterior gravitomagnetism),
• Goedel lambdadust, a homogeneous dust with Lambda which features a different type of swirling consistent with homogeneity, and which includes awkward CTCs through each event, among other strange features,
• Bonnor dust, a stationary solution which models an isolated dust ball in which the dust is swirling and so maintains its shape by balancing (in Newtonian terms) centrifugal and gravitational forces,
• Stephani dust, a famous inhomogeous perturbation of FRW dust which can be given in closed form,
• Senovilla-Vera dust, notable for its simplicity and its surprising Carter-Penrose diagram,
• a large family of dust solutions constructed using left-invariant or right-invariant frame fields for the various Bianchi groups (three dimensional Lie groups, considered as Riemannian manifolds, functioning here as giving the geometry of the hyperslices up to a scale factor $a(t)$), which are defined in terms of interesting system of nonlinear ODEs which can be studied by the methods of the theory of dynamical systems (understood here to refer to nonlinear ODEs, although the notion of a dynamical system is much more general than that--- in particular, see ergodic theory), and which include the famous Misner dust, aka "mixmaster model", which is essential for discussing the BKL conjecture (often regarded as the most important open problem, with the cosmic censorhip conjecture, in classical gtr); these can serve as homogeneous but not neccessarily isotropic cosmological models.
It goes without saying that I do plan BRS threads on essential background, including
• kinematic decomposition of timelike congruences and Raychaudhuri equation,
• optical scalars of null geodesic congruences,
• interpretations of the Riemann tensor and Bel decomposition wrt a timelike congruence,
• Carter-Penrose diagrams,
• matching conditions,
• Komar integrals for the mass and angular momentum of isolated objects in a stationary spacetime.
Discussing the OS model almost impels me to propose related BRS threads on
• Vaidya models of gravitational collapse and the "teleology" of the event horizon,
• recent discussions of alternative definitions of "horizons" which, it is hoped, will prove better behaved, physically more transparent, and better able to assist in the search for quantum gravity than the standard definition of the event horizon,
• mass inflation and various proposed models of generic black hole interiors.

I also plan BRS threads on other topics in classical gtr, mathematical background for this and classical field theory (particularly stuff arising from Lie's theory of symmetry of systems of DEs), Newtonian gravitation, vibration models, and applicable mathematics such as structors, and maybe even such wonders of pure math as the Erdos-Renyi "universal random graph" and Fraisse theory (which relates models in first order logic to... well, more on that later). And a review of entropies seems more urgent than ever, if only to explain that Shannon entropy was not the first entropy devised by Shannon; his earlier notion, which is completely different, was formulated in terms of graphs and has long fascinated graph theorists and combinatorialists, and has recently been revived in "quantum graphity" (not to mention a huge body of work in CS on social networks, the internet...)

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BRS: The FRW Dust with E^3 Hyperslices. V. Comparison with Milne model

Despite my distraction, I'd like to try to discuss the Milne model in order to strike while the iron is hot, as it were, even though I may be unable to produce all the figures I had in mind.

The Milne model is defined by a certain frame field on the Minkowski vacuum, so it is not a dust solution of the EFE. However, the frame in question represents homogeneously expanding observers (in flat spacetime, to be sure), so it can be considered a kind of crude cosmological model which is even more elementary that the FRW dusts.

In this post, I'll probably have to limit myself to providing an example which, together with the posts so far, should help illustrate some important points concerning frame fields versus coordinate charts:
• frame fields should be thought of as models of a class of observers whose physical experience is of interest for some Gedanken experiment, or possibly even a realizable satellite misson; they consist, in Lorentzian four-manifolds (M,g), of one timelike and three spacelike unit vector fields, all mutually orthogonal, and possibly defined only on some domain (simply connected open neighborhood) $\Omega$ in M,
• coordinate charts are needed to locate events in any spacetime model; charts on some domain $\Omega$ in a smooth n-manifold M can be thought of as n monotonic smooth functions on the domain, $u^1, \, u^2 \, \ldots u^n$, where $du^1, \, du^2, \, \ldots du^n \neq 0$, whose level hypersurfaces are nowhere tangent on the domain, $du^1 \wedge du^2 \ldots \wedge du^n \neq 0$,
• every frame field can be written in terms of any coordinate chart, just as any vector field can be written in terms of any coordinate chart,
• every coordinate chart is associated with a coordinate basis of vector fields $\partial_{u^1}, \, \partial_{u^2}, \ldots \partial_{u^n}$ and a dual cobasis consisting of covector fields $du^1, \, du^2, \ldots du^n$,
• every frame is associated with a (metrically) dual coframe $\sigma^1, \, \sigma^2, \, \ldots \sigma^n$ (one of the few failings of MTW is a slightly munged explanation of this duality in an early chapter),
• coordinate bases of vector fields are holonomic in the sense that their Lie brackets vanish (since partial derivatives commute), but nontrivial frames are anholonomic because their Lie brackets generally do not all vanish,
• components of vectors and tensors expanded wrt a coordinate basis do not in general have coordinate-free significance, but components expanded wrt a frame field are the ones which, in principle, the associated family of observers would measure.

Let's start by working in the usual polar spherical chart on Minkowski vacuum, in which the line element becomes
$$\begin{array}{rcl} ds^2 & = & -dt^2 + dr^2 + r^2 \; \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right), \\ &&-\infty < t < \infty, \; 0 < r < \infty, 0 < \theta < \pi, \; -\pi < \phi < \pi \end{array}$$
The Milne frame is
$$\begin{array}{rcl} \vec{e}_1 & = & \frac{t}{\sqrt{t^2-r^2}} \, \partial_t + \frac{r}{\sqrt{t^2-r^2}} \, \partial_r \\ \vec{e}_2 & = & \frac{r}{\sqrt{t^2-r^2}} \, \partial_t + \frac{t}{\sqrt{t^2-r^2}} \, \partial_r \\ \vec{e}_3 & = & \frac{1}{r} \, \partial_\theta \\ \vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi \end{array}$$
Notice that this is only defined inside the upper half cone with vertex at the origin, $t>0, \, t < r$. The world lines of the Milne observers are the integral curves of $\vec{e}_1$, 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.

See the figure below, and notice that if an observer sends out a subluminal spaceprobe which moves at constant velocity, this will pass some Milne observers, but will never reach others. Notice too that because we are neglecting the mutual gravitational attraction of the galaxies, the expansion does not slow over time, as it does in the FRW dust with E^3 hyperslices.

The kinematic decomposition of the unit timelike vector field (see Hawking and Ellis, Large Scale Structure of Spacetime, or Poisson, A Relativist's Toolkit) runs as follows:
• the acceleration vector $\nabla_{\vec{e}_1} \vec{e}_1 = 0$ (verifying that the Milne observers are indeed in a state of inertial motion),
• the expansion tensor (components expanded wrt the Milne frame, written in the polar spherical chart) is
$${H\left[\vec{e}_1\right]}_{ab} = \frac{1}{\sqrt{t^2-r^2}} \; \operatorname{diag}(1,1,1)$$
• the vorticity tensor vanishes
Since the Milne frame covers a region of Minkowski spacetime, the Riemann tensor vanishes, and thus the electroriemann tensor (tidal tensor), magnetoriemann tensor, and Einstein tensor all vanish identically. The analogue of the spherically contracting null geodesic congruence discussed in Post 3 above is
$$\vec{\ell} = \partial_t - \partial_r = \sqrt{\frac{t-r}{t+r}} \left( \vec{e}_1 - \vec{e}_2 \right)$$
which has optical expansion scalar $\theta\left[\vec{\ell}\right] = -1/r$ (the simple form of this result indicates spherical contraction in more general contexts, for example the Schwarzchild vacuum, so it is worth remembering) and vanishing shear and twist scalars.

All of these results will more closely resemble our results for the frame of observers comoving with dust particles ("galaxies") in the FRW dust solution, expressed in the comoving chart, if we change to a suitable comoving coordinate chart:
$$t^\prime = \sqrt{t^2-r^2}, \; r^\prime = \log \sqrt{\frac{t+r}{t-r}}$$
The line element becomes (dropping the primes)
$$\begin{array}{rcl} ds^2 & = & -dt^2 + t^2 \; \left( dt^2 + \sinh(r)^2 \; \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right) \right), \\ && 0 < t, \, r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi \end{array}$$
In terms of this chart, the Milne frame is:
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_t \\ \vec{e}_2 & = & \frac{1}{t} \, \partial_r \\ \vec{e}_3 & = & \frac{1}{t \, \sinh(r)} \, \partial_\theta \\ \vec{e}_4 & = & \frac{1}{t \, \sinh(r) \, \sin(\theta)} \, \partial_\phi \end{array}$$
The kinematic decomposition of the timelike unit vector field $\vec{e}_1$, in terms of the new chart, is:
• the acceleration vector and vorticity tensor vanish (clearly, any such claim is always a coordinate-free statement!),
• the expansion tensor is
$${H\left[\vec{e}_1\right]}_{ab} = \frac{1}{t} \; \operatorname{diag}(1,1,1)$$
which plainly shows homogeneous expansion, at a rate decreasing with increasing lapse of proper time.
Because the vorticity tensor vanishes, the integral curves of $\vec{e}_1$ (the rays in the figure below) form a hypersurface orthogonal timelike congruence, and the three-dimensional riemann tensor of the hyperslice $t=t_0$ is
$$r_{2323} = r_{2424} = r_{3434} = -1/t_0^2$$
where once again we see the superiority of expanding wrt a frame (had we used a coordinate basis, it would not have been quite so evident that the hyperslices have the geometry of $H^3$, hyperbolic three-space. The spherically contracting null geodesic congruence becomes
$$\vec{\ell} = \frac{1}{t} \, \partial_t - \frac{1}{t^2} \, \partial_r$$
and the expansion tensor, written in the comoving chart, is
$$\theta\left[\vec{\ell}\right] = \frac{1}{t^2} - \frac{\coth(r)}{t^2}$$
which resembles our result in Post III above.

I leave it as a exercise for the reader to transform to a cartesian style comoving chart, or to the upper half cone chart in which the line element becomes
$$\begin{array}{rcl} ds^2 & = & -dt^2 + (t/z)^2 \; \left( dx^2 + dy^2 + dz^2 \right), \\ && 0 < t, \, z < \infty, \; -\infty < x, \, y < \infty$$

Each of these charts provides a welcome (?) opportunity to practice finding generators for the Lie algebra of Killing vector fields (exceptionally rich---ten generators--- in the case of Minkowski vacuum) in various charts, and then using Noetherian magic to obtain a solution of the geodesic equations for timelike, null, and spacelike geodesics (a task which doesn't require knowing all ten generators). See https://www.physicsforums.com/showthread.php?t=380917 and also Post II above.

Figure:
Milne observers inside an upper half cone in Minkowski vacuum

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BRS: The FRW Dust with E^3 Hyperslices. V. Comp with Milne, contd

Now we come to the real point: comparison of some notions of "distance in the large" in the Milne model with the corresponding notions in the FRW dust with E^3 hyperslices.

Let us write both the FRW and Milne model in polar spherical charts comoving with the dust and with the Milne observers respectively. Then, supressing the angular coordinates, our FRW model has line element
$$ds^2 = -dt^2 + t^{4/3} \, dr^2, \; 0 < t, \, r < \infty$$
while the Milne model has line element
$$ds^2 = -dt^2 + t^2 \, dr^2, \; 0 < t, \, r < \infty$$
In the latter case, the null geodesic through the event at $t=t_1, r=0$ is
$$R = \log (t_0/t)$$
The resulting curves have an appearance qualitatively similar to the null geodesics in the FRW model (in the range we are interested), so referring again to the figure in Post above, we deduce that the redshift is
$$1 + z = \frac{t_1}{t_0}$$
Thus, the lookback time is
$$D_{\rm ltt} = t_1 - t_0 = t_1 \; \frac{z}{1+z} = t_1 \, (z - z^2) + O(z^3)$$
And integrating $t \, dr$ with $t=t_1$ fixed gives
$$D_{\rm now} = t_1 \; \log (1+z) = t_1 \; (z - 1/2 \, z^2) + O(z^3)$$
Similarly
$$D_{\rm then} = t_1 \; \frac{\log (1+z)}{1+z} = t_1 \; ( z - 3/2 \, z^2 ) + O(z^3)$$
See the figure below and note that the order relations between these distances are the same as we found in the FRW model:
$$D_{\rm then} < D_{\rm ltt} < D_{\rm now}$$
suggesting that $D_{\rm ltt}$ offers a compromise, if you will, between $D_{\rm now}, \; D_{\rm then}$, which behave quite differently (in particular, as first noticed by none other than Karl Schwarzschild, $D_{\rm then}$ tends to "turn over" at large redshifts.

Because our hyperslices now have hyperbolic geometry, $D_A$ no longer agrees with $D_{\rm then}$, and $D_L$ no longer agrees with $(1+z) \, D_{\rm then}$. I'll defer discussion of these until I have created suitable figures.

The essential point here, I think, is that we require multiple distinct notions of "distance in the large" even in Minkowski spacetime, but the graphs also suggest that the effect of incorporating gravity into our model (by replacing the Milne model with the slightly more sophisticated FRW dust with E^3 hyperslices) is to exacerbate the effect of increasing redshift, and to emphasize the differences between the three notions of "distance in the large" compared here, when plotted against the observable quantity, the redshift z.

But there is one other important point to be made: the null geodesics we found for our comoving chart (comoving with the Milne observers) in Minkowski vacuum show that all the Milne observers are always visible to each other. This is not true, as we saw in Post, for the FRW dust with E^3 hyperslices, which features a cosmological horizon.

Figures: Log-log plots of $D_{\rm now}, \; D_{\rm ltt}, \; D_{\rm then}$ (red, blue, purple curves) wrt z for two models; left to right:
• FRW dust with E^3 hyperslices
• Milne model
Both plots use the silly example $T_1 = 5$. Both plotted with Gnuplot.

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BRS: FRW Dust, E^3 Hyperslices. V. Comparison with Milne. Correction, Mechanics

In Post #6:

Let's start by working in the usual polar spherical chart on Minkowski vacuum.... The Milne frame is
$$\begin{array}{rcl} \vec{e}_1 & = & \frac{t}{\sqrt{t^2-r^2}} \, \partial_t + \frac{r}{\sqrt{t^2-r^2}} \, \partial_r \\ \vec{e}_2 & = & \frac{r}{\sqrt{t^2-r^2}} \, \partial_t + \frac{t}{\sqrt{t^2-r^2}} \, \partial_r \\ \vec{e}_3 & = & \frac{1}{r} \, \partial_\theta \\ \vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi \end{array}$$
Notice that this is only defined inside the upper half cone with vertex at the origin, $t>0, \, t < r$.

For $t>0, \, t < r$ please read $0 < r < t < \infty$. Thanks to User:Ich for the correction; see https://www.physicsforums.com/showthread.php?p=2632110&posted=1#post2632110
It's too bad I can't edit the original, so if you think you spot a typo/oversight, please let me know ASAP since I have only 24 hours from posting to make corrections, and sometimes in my rush to get onto other BRS posts, I goof! As I write (March 2010) no-one, including me, has enough experience with BRS to know the best way to ask questions and correct errors, bearing in mind that I am trying to accomodate several goals which might turn out to be conflicting:
• provide an easy, fun, and topical way for SA/Ms to learn a bit, as time/energy/interest allow, about some of the sophisticated conceptual/technical issues which are often involved in providing "optimal" answers to queries about gtr (and sometimes other mathy topics) in the public areas of PF,
• provide a safe environment for SA/Ms to discuss some tough issues without distractions from complete novices (or even worse, from socks, shills, and trolls) in the public areas,
• provide some readable (?) and ultimately, I hope, interlinked, set of capsule reviews of various notions, with citations to textbooks and review papers, which can provide background material for SA/Ms to draw upon in answering queries in the public areas--- it being understood that in many cases it might not be appropriate to mention advanced citations or conceptually challenging details, but in most cases it would be helpful for SA/Ms to know a bit about them in framing responses for the public areas.
Asking questions in the thread itself is the obvious response, but this might disrupt the flow of a BRS thread like this one and readers who have "dropped in from the future". OTH, readers who "drop in from the future" might well have the very same questions, so they should be able to find the discussion thread, if any. Anway, if someone has a quick minor correction and doesn't want to wrestle with this largely untested conundrum, I try to keep my PM box unfull and check it regularly, so that's always an easy option! I'd also be happy to do what John Baez sometimes does, and say "Someone asked me...".

Also, add to Post #5: note to self: debunk the notion that "the universe is just a black hole" [sic] (or even "an inside-out black hole" [sic]).

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BRS: The FRW Dust with E^3 Hyperslices VI. A spherically symmetric perturbation

Someone recently mentioned in one of the public subforums (I think it was the Cosmology subforum) an interesting recent proposal in cosmology. The idea is this: suppose there is an unusually dense region of galaxies whose world lines lie outside the absolute past of the event "here and now" (give or take a light year in space and a hundred years in time). More prosaically, the universe is too young and these galaxies are too far for Earthbound astronomers to have seen them yet. Then it might happen that galaxies we can see are observably streaming toward the overdensity.

Is this idea theoretically plausible? One way to start studying it would be to find a family of exact solutions which represent spherically symmetric perturbations of the FRW dust with E^3 hyperslices, and to study the expansion tensor of the congruence of world lines of the dust particles (model galaxies); recall that the expansion tensor (wrt this congruence) exhibits the Hubble flow for cosmological dust solutions.

Fortunately, the family of all (neccesarily dynamic, i.e. non-static) spherically symmetric dust solutions of the EFE have been known for a long time; this is the famous LTB (Lemaitre-Tolman-Bondi) family of dust solutions. One nice way of writing them (from my notes, I think I obtained this by rewriting Bondi's way of writing the LTB dusts) is this: the frame of the dust particle is
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_t \\ \vec{e}_2 & = & \frac{\sqrt{1+f}}{R_r} \, \partial_r \\ \vec{e}_3 & = & \frac{1}{R} \, \partial_\theta \\ \vec{e}_4 & = & \frac{1}{R \, \sin(\theta)} \, \partial_\phi \end{array}$$
where f is an arbitrary function of r and R depends only on t,r and obeys the constraint equation
$$R_{tt} = \frac{f - R_t^2}{2 \, R}$$
The line element is
$$\begin{array}{rcl} ds^2 & = & -dt^2 + \frac{R_r^2}{1+f} \, dr^2 + R^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right), \\ && t_{\rm min} < t < t_{\rm max}, \; r_{\rm min} < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi$$
(where I am being deliberately vague about the ranges of the t,r coordinates). Note that R is a Schwarzschild type radial coordinate (so that the area of concentric spheres "centered" at "the origin" r=0 is $A = 4 \pi \, R^2$, and their Gaussian curvature is $1/R^2$), but here plays the role of a metric function. This chart is comoving with the world lines of the dust particles.

Noteworthy special cases include:
• $f=0, \, R = r \, t$ is Minkowski vacuum (written in what chart? to what observers corresponds the given frame field?),
• $f = 0, \, R = r \, t^{2/3}$ is the FRW dust with E^3 hyperslices (written in a now familiar comoving chart),
• (exercise) is the Schwarzschild vacuum written in the Lemaitre chart, with our frame field corresponding to the Lemaitre observers (who fall in freely and radially "from rest at spatial infinity"),
as you can easily check by hand by plugging into the line element.

Haelfix suggested that I elaborate on something I said: I claimed (without so much as a handwaving argument) that all (neccesarily dynamic, i.e. non-static) spherically symmetric dust solutions of the EFE are locally equivalent to some LTB dust. (Note well: "local" in the sense of "local neighborhoods", as usual in manifold theory. Old physics books sometimes use "local" to indicate the "infinitesimally small" scale of tangent spaces or jet spaces; this usage is now strongly deprecated, because physics must conform to standard mathematical terminology or suffer uneccessary and troublous confusion.)

This claim resembles the famous Birkhoff theorem in gtr, which says all spherically symmetric vacuum solutions of the EFE are locally isometric to the Schwarzschild vacuum. Indeed, since vacuum is a special case, zero density, of dust (which is a special case, zero pressure, of perfect fluid), and since the Schwarzschild vacuum is a special case of an LTB "dust", the first claim can be regarded as generalization of the second claim. For a handwaving argument for the second claim, see D'Inverno. For a careful discussion, try the excellent cosmology textbook by Plebanski and Krazinski. Part of the difficulty in making the two claims rigorous is that they haven't yet been stated with sufficient care. I'll defer further discussion to a possible future BRS thread on LTB dusts.

The LTB dust is so useful it deserves its own thread (coming later, perhaps), but for now let me just say this: the congruence of world lines of dust particles is vorticity-free, so admits a unique family of orthogonal hyperslices ("spaces at a time"). 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, which turns out to be the case $f=0$. For this case, we can easily find a Painleve type chart, which I plan to discuss in the next post in this thread. In this post, let me just say that we can write a fairly general solution of
$$R_{tt} = \frac{-R_t^2}{2 \, R}$$
as
$$R = \left( a + \left( b + r^{3/2} \right) \, t \right)^{2/3}$$
where a,b are arbitrary functions of r which grow very slowly if at all; to be safe let's stipulate that $a, \, b$ and all their derivatives are at most $O(1)$ (i.e. bounded, or even decaying as r grows). That means that a,b are not growing, and possibly decaying as r grows, but not too quickly. Then putting a=b=0 recovers the FRW dust with E^3 hyperslices, so this is an exact dust solution which is a spherically symmetric (nonlinear) perturbation of our FRW dust model.

(Regarding the idea in the introduction, we really want to introduce a positive Lambda, but for simplicity we are taking Lambda = 0 here.)

Now you can compute the Einstein tensor, expansion tensor, and electroriemann or tidal tensor. Then you can plug in some simple choices of a,b and expand everything in powers of 1/r and 1/t. To get some idea of what to expect, we expand in a multivariable Taylor series wrt a,b and their derivatives and collect terms in r and t. The result is something like this:

The expansion tensor is diagonal with
$${H\left[\vec{e}_1\right]}_{22} = \frac{2}{3 \, t} \; - \; \frac{2 \, a_r}{3 \, r^{1/2} \, t^2} \; + \; \frac{4 \, a_r \, b_r}{9 \, r \, t^2} \; + \; \frac{a}{3 \, r^{3/2} \, t^2} \; - \; \frac{a b}{3 \, r^3 \, t^2} \; + \; O\left( \frac{1}{t^3} \right)$$
and
$${H\left[\vec{e}_1\right]}_{33} ={H\left[\vec{e}_1\right]}_{44} = \frac{2}{3 \, t} \; - \; \frac{2a}{3 \, r^{3/2} \, t^2} \; + \; \frac{2 \, a \, b}{3 \, r^3 \, t^2} \; + \; O\left( \frac{1}{t^3} \right)$$
The only nonvanishing component of the Einstein tensor is:
$$G^{11} = 8 \, \pi \, \mu = \frac{4}{3 \, t^2} \; - \; \frac{8 \, a_r}{9 \, r^{1/2} \, t^3} \; + \; \frac{4 \, a_r \, b_r}{27 \, r \, t^3} \; + \; \frac{a}{3 \, r^{3/2} \, t^3} \; - \; \frac{a b}{3 r^3 \, t^3} \; + \; O\left( \frac{1}{t^4} \right)$$
The tidal tensor is diagonal with
$${E\left[\vec{e}_1\right]}_{22} = \frac{2}{9 \, t^2} \; - \; \frac{4 \, a_r}{3 \, r^{1/2} \, t^3} \; + \; \frac{8 \, a_r \, b_r}{27 \, r \, t^3} \; + \; \frac{2 \, a}{9 \, r^{3/2} \, t^3} \; - \; \frac{2 \, a \, b}{9 \, r^3 \, t^3} \; + \; O\left( \frac{1}{t^4} \right)$$
and
$${E\left[\vec{e}_1\right]}_{33} ={E\left[\vec{e}_1\right]}_{44} = \frac{2}{9 \, t^2} \; - \; \frac{4 \, a}{9 \, r^{3/2} \, t^3} \; + \; \frac{4 \, a \, b}{9 \, r^3 \, t^3} \; + \; O\left( \frac{1}{t^4} \right)$$
Also of interest is the quadratic curvature scalar invariant $C_{ijk\ell} \, C^{ijk\ell}$, which vanishes for the FRW dust (since the FRW dust models are conformally flat), but which does not vanish for our perturbations:
$$C_{ijk\ell} \, C^{ijk\ell} = \frac{256 \, a_r}{243 \, r \, t^6} \; - \; \frac{256 \, a \, a_r}{81 \, r^2 \, t^6} \; + \; \frac{64 \, a^2}{27 \, r^3 \, t^6} \; + \; O\left( \frac{1}{t^7} \right)$$
(Thus, we should expect to find some optical effects due to the density inhomogeneity, although these might be hard to measure.) You can find a similar expression for the Kretschmann scalar $R_{ijk\ell} \, R^{ijk\ell}$, with leading term $80/27/t^4$ provided by the FRW dust, which I'll leave as an exercise.

In these expressions, the leading term agrees with the results for the FRW dust with E^3 hyperslices and the other terms are correction terms due to the perturbation, which is controlled by choice of a,b. As you can see, the effects of the perturbation die out as t grows and also as r grows.

Notice that the radial component of the expansion tensor is quite different from the tangential components! 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, as reflected in the motion of galaxies wrt any fiducial galaxy (in particular, our own, idealized to be comoving with the Hubble flow). It might help to write the leading terms of the expansion tensor like this:
$${H\left[\vec{e}_1\right]}_{ab} = \frac{2}{3t} \; \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \; - \; \frac{2 \, a_r}{3 \, \sqrt{r} \, t^2} \; \alpha \; \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right] \; + \; \frac{a}{3 \, r^{3/2} \, t^2} \; \beta \; \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2 \end{array} \right] \; + O\left( \frac{1}{t^3} \right)$$
where
$$\alpha = 1 - \frac{2 \, b_r}{3 \, \sqrt{r}}, \; \beta = 1-\frac{b}{r^{3/2}}$$
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. For comparison, the expansion tensor for Lemaitre observers in the Schwarzschild vacuum is
$${H\left[\vec{f}_1\right]}_{ab} = \sqrt{\frac{m}{2 r^3}} \; \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2 \end{array} \right]$$
suggesting that when a > 0, a_r < 0, an unusually dense lump of dust at the origin acts, for a dust particle at coordinate radius r=r_0, like a "massive object" at the origin with time varying mass
$$m= \frac{2 \, a(r_0)^2 \, \beta(r_0)^2}{9 \, t^4}$$
whose gravitational attraction is superimposed on the Hubble flow. For a positive, nonconstant decreasing, for sufficiently large t and for r not too small, the radial streaming term $-2 \, a_r/3/\sqrt{r}/t^2$ should give the dominant correction, and if $a_r < 0$ this is positive, indicating enhanced motion towards the origin (the location of an overdensity). For a, b constant the streaming term vanishes and the effective mass is independent of coordinate radius, but decreasing with time (since the dust density is smoothed out by the expansion). In general, if our universe were well modeled as a small perturbation of FRW with E^3 slices, we could in principle estimate the values at the event "here and now" of the scalar functions multiplying the tensor appearing in the decomposition above.

This suggests that the idea of a cosmological streaming is not madness. It should appear whenever a is a decreasing function of r.

With a bit more work, interested readers should be able to play with some specific models where a,b are carefully chosen bump functions, and this should show that it is possible for an overdensity outside the absolute past of the event "here and now" to induce streaming superimposed on the basic Hubble expansion which literally "points" at the not yet visible unusually dense region of the universe.

We have been discussing an expanding cosmological model. To segue into the next post, note that simply reversing the sign of the time coordinate gives a collapsing dust, which can be used to study spherically symmetric gravitational collapse.

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BRS: The FRW Dust with E^3 Hyperslices. VII. Oppenheimer-Snyder collapsing dust ball

The frame of the Lemaitre observers in the Schwarzschild vacuum is
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_\tau -\sqrt{\frac{2m}{r}} \, \partial_r \\ \vec{e}_2 & = & \partial_r \\ \vec{e}_3 & = & \frac{1}{r} \, \partial_\theta \\ \vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi \end{array}$$
(Recall that the Lemaitre observers are inertial observers who fall "from rest at spatial infinity" freely and radially toward the origin, in the Schwarzschild vacuum. The Painleve chart is partially well suited to studying the case where the massive object is a black hole and we wish to follow the physical experience of the Lemaitre observers as they fall through the event horizon and approach the strong spacelike curvature singularity.)

A very simple perturbation is the frame
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_\tau -\sqrt{\frac{2m}{r}} \, \exp(f) \, \partial_r \\ \vec{e}_2 & = & \partial_r \\ \vec{e}_3 & = & \frac{1}{r} \, \partial_\theta \\ \vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi \end{array}$$
where f is an undetermined function of $\tau, \, r$ (tau, r; sorry that these letters look so similar in VB!). (Note that f=0 recovers the Lemaitre observers in the Schwarzschild vacuum.) Seeking an interesting class of exact solutions using this frame, we might try setting all but one (which one?) of the components of the Einstein tensor to zero. This gives a system of partial differential equations which we can take as constraint equations on f. Because the orthogonal hyperslices are locally flat, this gives the desired Painleve chart for a collapsing LTB dust with locally flat hyperslices orthogonal to the world lines of the dust particles.

The constraint equations can be solved as follows: choose an arbitrary real valued function F and write the equation
$$F(\xi) = \tau + \frac{1}{\sqrt{\frac{9m}{2r^3}} \, \xi}$$
If we can find a root $\xi$ of this equation, then we can take $\exp(f) = \xi$. Well, almost; we need to check that the dust density
$$\mu = \frac{m \, f_r \, \exp(2f)}{2 \pi \, r^2}$$
is positive, i.e. we require $f_r > 0$, but this is not guaranteed by this method. Even if the density is positive for large r, typically we can expect to find some world sheet where the density blows up as we move radially inward.

A remark about dust solutions generally: even if we restrict attention to the simplest case, spherically symmetric dusts, these often develop features resembling two spherical shells of dust collapsing at different speeds, which collide to create a world sheet of infinite dust density. Such world sheets are somewhat analogous to a caustic in geometric optics (a surface where the intensity blows up). In recent years there has been much interest in Einstein-Vlasov models, which avoid such problems by treating the distribution of dust statistically. This involves a statistical mechanical treatment which is motivated by concerns of classical physics.

The simplest example of this Painleve chart for the collapsing LTB dusts with E^3 hyperslices arises when we choose F = 0. This gives the very simple frame
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_\tau - \frac{2 \, r}{3 \, \tau} \, \partial_r \\ \vec{e}_2 & = & \partial_r \\ \vec{e}_3 & = & \frac{1}{r} \, \partial_\theta \\ \vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi \end{array}$$
which is an exact dust solution with
$$G^{11} = \frac{4}{3 \, \tau^2}$$
which we recognize as the FRW dust with E^3 hyperslices. This is a collapsing version in which $-\infty < \tau < 0$. The line element is
$$ds^2 = -\left(1-\frac{4r^2}{9 \, \tau^2} \right) \; d\tau^2 \; - \; \frac{4 r}{3 \, \tau} \; d\tau \, dr \; + \; dr^2 + r^2 \; \left( d\theta^2 \; + \; \sin(\theta)^2 \; d\phi^2 \right)$$
This should look familiar (write $r = \sqrt{x^2+y^2+z^2}$ and compare Post #1). Note that the cross term has positive sign because $\tau < 0$.

To form the Oppenheimer-Snyder model of gravitational collapse, we choose a sphere of Lemaitre observers and cut out the region inside its world sheet, and then match this dust region (a portion of the collapsing FRW dust with E^3 hyperslices) to a suitable region of the Schwarzschild vacuum with a suitable mass parameter. The result is the famous OS model of a collapsing spherically symmetric dust ball (with homogeneous density "at a time" as measured by the dust particles) which forms a nonrotating static black hole. (IOW, this is a piecewise defined Lorentzian manifold, analogus to a piecewise defined function.)

Inside the dust region, the expansion tensor is diagonal:
$${H\left[\vec{e}_1\right]}_{ab} = \frac{2}{3 \, \tau} \, \operatorname{diag}(1,1,1)$$
(Note that since $\tau < 0$, this shows contraction, not expansion.) The electroriemann tensor (tidal tensor) is also diagonal:
$${E\left[\vec{e}_1\right]}_{ab} = \frac{2}{9 \, \tau^2} \, \operatorname{diag}(1,1,1)$$

Alternatively, we can consider an interior surface of a spherically symmetric collapsing shell of dust as $r = r_0 \, \tau^{2/3}$, and we can match across this surface to a Minkowski region. This gives a model in which a Minkowksi bubble inside a collapsing FRW dust collapses to a point (at the moment of the Big Crunch).

And we can consider a spherically symmetric "thick shell" of FRW dust which is matched across the outer surface to a suitable region of the Schwarzschild vacuum (with a suitable mass parameter, corresponding to the mass of the shell) and across the inner surface to a suitable region of the Minkowski vacuum.

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BRS: The FRW Dust with E^3 Hyperslices. Ia. Correction of a typo

In Post #1, for
The frame of observers riding on dust particles is
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_t - \frac{2}{3 \, t} \; \left( x \, \partial_x + y \, \partial_y + z \, \partial_z \right) \\ \vec{e}_2 & = & \partial_x \\ \vec{e}_3 & = & \partial_y \\ \vec{e}_4 & = & \partial_z \\ \end{array}$$
$$\begin{array}{rcl} \vec{e}_1 & = & \partial_t + \frac{2}{3 \, t} \; \left( x \, \partial_x + y \, \partial_y + z \, \partial_z \right) \\ \vec{e}_2 & = & \partial_x \\ \vec{e}_3 & = & \partial_y \\ \vec{e}_4 & = & \partial_z \\ \end{array}$$
(note the sign in the timelike unit vector; the world lines should appear to move away from the origin in this chart as t increases, so $\vec{e}_1$ should "point mostly upwards but also tip outwards").

For a collapsing dust model, the sign should be negative (IOW, in this case, $\vec{e}_1$ should "point mostly upwards but also tip inwards").

And just in case you're not yet completely confused, the coframes work just the other way

Sorry I didn't catch this typo earlier. The smallest sign error can completely munge a discussion like this, unfortunately.

BRS: The FRW Dust with E^3 Hyperslices. VIb. The LTB dust with E^3 hyperslices

Let me try to clarify some important points about the spherically perturbed FRW dust.

Recall that the LTB dust can be written
$$ds^2 = -dt^2 + \frac{{R_r}^2}{1+f} \, dr^2 + R^2 \; \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2$$
where f is an arbitrary function of r only and R is a function of t, r satisfying
$$R_{tt} = \frac{f-R_t^2}{2 \, R}$$
Note that differences in the t coordinate correspond to lapse of proper time measured by a dust particle, and that this chart is comoving with the dust particles; that is, the coordinate r serves only to label a sphere of dust particles, not as a "radial distance" (recall also that in any case, there are infinitely many operationally significant notions of "distance in the large" even in flat spacetime). I stress that the slices t=t0 are characterized geometrically (i.e. in a coordinate-free mannter) as the family of spatial hyperslices everywhere orthogonal to the world lines of the dust particles. Since a change in t corresponds to the lapse of proper time as measured by any dust particle, our coordinates are obviously very nice. Nonetheless, we'll see below that there is a nontrivial issue of the "synchronization" of the clocks of observers riding on the dust particles, which can actually lead us astray if we are not careful.

It turns out that there are two ways in which the Riemann curvature of an LTB dust model can blow up:
• R = 0
• R_r = 0
Each condition gives a surface of form $t=h(r)$, and t must be constrained to lie above both these surfaces. You should think of these conditions as defining the Big Bang initial singularity, and the two cases arise because (in the general LTB dust) the expansion is inhomogeneous.

The case of E^3 hyperslices arises when f=0, and then we saw that we could take
$$R = \left( a + (b+ r^{3/2}) \, t \right)^{2/3}$$
where a,b are functions of r only which control a spherically symmetric perturbation from the FRW dust with E^3 hyperslices. For our purposes it suffices to consider only the case b=0, so that we will take
$$R = \left( a + r^{3/2} \, t \right)^{2/3}$$
(Notice that this approaches $R= r \, t^{2/3}$ when $|a| \rightarrow 0$, i.e the metric will be very close to the metric of the FRW dust with E^3 hyperslices in any region where $|a| \rightarrow 0$.)

Then the conditions for finite Riemann tensor components are simply:
• $t > -a \, r^{-3/2}$ (that's the condition $R \neq 0$ for an expanding dust),
• $t > -2/3 \, a_r \, r^{-1/2}$ (that's the condition $R_r \neq 0$ for an expanding dust).
so the Big Bang initial curvature singularity is at t = h(r) where
$$h(r) = \operatorname{max} \left\{ \frac{-a}{r^{3/2}}, \; \frac{-2 \, a_r}{3 \, r^{1/2}} \right\}$$
(pointwise supremum).

The dust density turns out to be
$$\mu = \frac{r^2}{2 \, \pi \; \left( a + r^{3/2} \, t \right) \; \left( 2 a^\prime + 3 \, r^{1/2} \, t \right) }$$
(notice that it blows up exactly in the places where some component of the Riemann tensor blows up).

For those of you using Maxima, here is a file you can run in batchmode in wxmaxima to compute the density using Ctensor (with the frame flag set):
Code:
/*
Lemaitre-Tolman-Bondi dust with E^3 hyperslices;
special case; comoving chart; nsi coframe

This models the general spherically symmetric dust with E^3 hyperslices.
The given coframe only covers the dust portion;
we can match across collapsing spherical surfaces:
1. match to Minkowski vacuum across interior surface of dust region,
2. match to Schwarzschild vacuum across exterior surface of dust region.

The general case of E^3 hyperslices is the case
f=0
R = (a+(b+r^(3/2)*t)^(2/3)
of the general LTB dust (all spherically symmetric dusts)
This file gives the special case b = 0, which illustrates the main points.

Curvature blows up when either:

1.  R = 0, i.e.
t = -a/r^(3/2)
(big bang initial singularity)

2.  diff(R,r) = 0, i.e.
t = -2/3*diff(a,r)/r^(1/2)
(@_t null)

Try

Can't yet verify this with Ctensor, but the expansion tensor has form
H_(ab) = 2/3/t diag(1,1,1) + 1/t^2 P_(ab) + O(1/t^3)
where
P_(ab) = -2*diff(a,r)/3/sqrt(r) diag(1,0,0)
+ a/3/r^(3/2) diag(1,-2,-2)
Here, the first term gives radial streaming toward overdensity at r=0
while second term gives Coulomb effect due to residual overdensity
at r=r0.

Example: void at r = 0, surrounded by slightly overdense spherical shell,
asymptotic to background FRW dust with E^3 hyperslices as r -> infty:
a = exp(-k*r^2)
b = 0
Density vanishes at r=0 for all time, density homogeneizes over time.
But the sphere of max density at r=r_max has r_max -increasing- with time.
Paradox: no dust particles being "transported" wrt spatial coordinates,
so how can density change?
Solution: inhomogeneous expansion changes concentration near any dust ptcle.
Note too that "distance in the large" has no unique definition.
The coordinate r simply labels concentric spheres of dust particles,
is not a "distance".

*/
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(a,r);
/* define the coframe */
fri: zeromatrix(4,4);
fri[1,1]: -1;
fri[2,2]:  diff((a+r^(3/2)*t)^(2/3),r);
fri[3,3]:  (a+r^(3/2)*t)^(2/3);
fri[4,4]:  (a+r^(3/2)*t)^(2/3)*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(false);
factor(ratsimp(factor(lriem[3,3,1,1])));
factor(ratsimp(factor(lriem[4,4,1,1])));
/* Compute R^(mijk) */
uriemann(false);
/* Compute Ricci componets R_(jk) */
ricci(false);
/* Compute trace of Ricci tensor */
/* 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);
/* NOTE: flip sign to get density */
factor(radcan(factor(-ein[1,1]/8/%pi)));

To make a perturbation model, we should allow a nonzero near r=0, but ensure that $|a| \rightarrow 0$ as $r \rightarrow \infty$. Two particularly simple choices are the Gaussian perturbations $a = \pm k \, \exp(-n \, r^2)$ where k,n are positive real constants. Then you can check that
• $a=\exp(-r^2)$ has Big Bang at $R_r=0$ (see the figure), so the hyperslices t=t0 are disconnected for sufficiently small t0. Furthermore, the proper time from a slice t=t0 back to the Big Bang (measured along the world line of some dust particle) varies with r. The density of the dust vanishes for all time at the center of spherical symmetry r=0, and for as r increases (with t held fixed t=t0), the density increases to a maximum, and then decreases to the (positive) FRW value from above. As t increases, the density profile homogeneizes, and the location of the peak density moves to larger values of r.

We'd like to characterize this as a model in which we have an underdense region near r=0, but obviously such a claim requires more justification than I've yet offered. In particular, we should compare the slices t=t0 with slices of constant lookback time to the Big Bang, where we expect to see an unambiguous underdensity near r=0. Then we can explain the (surprising?) modest overdensity before approaching the FRW value (as r increases) by saying that, as r decreases, the slice t=t0 is "going back in time and approaching the Big Bang", and thus encountering larger densities, before the exponential decrease in density near r=0 wins the struggle and the density falls to zero at r=0 (measured along the slice t=t0).
• $a=-\exp(-r^2)$ has Big Bang at $R=0$ (see the figure), so that each hyperslice t=t0 has the Big Bang at r=r0, where r0>0 depends on t0. Once again, the proper time from t=t0 back to the Big Bang (measured along the world line of some dust particle) varies with r. As r decreases with t held fixed (t=t0), the density of the dust slowly falls from the FRW value to a minimum density, then rises steeply and blows up as we approach the Big Bang.

We'd like to say that this represents an overdense region of dust near the origin; we should compare the slices t=t0 with slices of constant lookback time to the Big Bang, where we hope to see an unambiguous overdensity at near r=0. To encourage reader participation, I haven't yet tried to explain why this doesn't seem to correspond to the graphs as well as the previous case, suggesting that the proffered explanation may be incorrect.
There is another "paradox" here. Can you spot it?

The spatial coordinates unambiguously label dust particles, and no dust particles appear to move in this chart. So if not matter is "transported", how can the sphere of peak density move to larger values at t increases (in the first model)?

Recall that the expansion is inhomogeneous. This means that even though the dust particles are not moving wrt the given coordinates, their world lines are (generally) diverging from nearby neighboring world lines at rates depending upon both t and r. The result is that, for a given dust particle world line C, neighboring dust particle world lines are diverging at different rates depending upon t, r and the direction in which the neighbor lies wrt C. So the spatial density of the dust can vary with t,r.

I encourage anyone interested to play around with some other choices of the perturbation function a, including bump functions of compact support. Unfortunately, Maxima's Ctensor may not yet be sufficiently powerful to compute the expansion tensor of a timelike congruence (the world lines of the dust particles, in this case); I still have not had a chance to try harder to follow the directions in the help file which claim that one can define a tensorial object in Itensor and then, using a certain Maxima script, compute its components for a specific manifold using Ctensor.

I should also try to explain a bit more clearly what I was trying to do with the multivariable Taylor expansion in 1/t, 1/r above. The idea is to decompose the expansion tensor of the congruence of world lines of the dust particles (timelike geodesics) in the form
$$H_{ab} = {H^{\rm FRW}}_{ab} + {H^{\rm Perturb}}_{ab} + O \left( \frac{1}{t^3} \right)$$
where
$${H^{\rm FRW}}_{ab} = \frac{2}{3 t} \; \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$$
and
$${H^{\rm Perturb}}_{ab} = \frac{1}{t^2} \; \left( \frac{-2 \, a_r}{3 \, r^{1/2}} \; \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right] \; + \; \frac{2 \, a}{3 \, r^{3/2}} \; \left[ \begin{array}{ccc} \frac{1}{2} & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array} \right] \right)$$
where the second term can be compared to the expansion tensor for Lemaitre observers in the Schwarzschild vacuum (also timelike geodesics, but in a different spacetime)
$$\sqrt{\frac{2 m}{r^3}} \; \left[ \begin{array}{ccc} \frac{1}{2} & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array} \right]$$
from which one obtains the "effective mass"
$$M = \pm \frac{2 \, a^2}{9 \, t^4}$$
which depends on r through a. (The expansion tensor doesn't "care about" the sign of the effective mass; why not?)

The idea is that we choose a so that $|a| \rightarrow 0$ as $r \rightarrow \infty$, which ensures that the metric of our LTB dust approaches the FRW dust with E^3 hyperslices as r grows. Then the perturbation term in the expansion tensor has two terms, one resembling the expected contribution from a massive object located at r=0, and one representing a radial streaming. The expansion tensor itself, of course, is evaluated at t=t0, r=r0, and describes, for the world line (a timelike geodesic) of any dust particle passing through the two dimesional sphere t=t0, r=r0, how the world lines of nearby dust particles (other timelike geodeiscs) are expanding wrt the fidicial world line. I still haven't had a chance to read any eprints on the so-called KSZ (kinematic Sunayev-Zeldovich) effect to check that this corresponds to the alleged radial streaming. The idea is that if we follow a small spherical shell centered on t=t0, r=r0, at t increases it should become radially displaced (streaming term) and should also distort into a prolate spheroid (Coloumb term). The idea of the effective mass is that if we have an underdense region at r=0, that will among other things act somewhat like an imaginary negative mass object in a FRW dust, but in addition we should have the radial displacement. Which term dominates should depend on the perturbation function a and the value of r.

However, this idea ignores the phenomenon which we unearthed when we considered some simple Gaussian perturbations; we have two distinct families of hyperslices which are of interest
• slices of constant proper time "since the Big Bang", measured along the world lines of the dust particles,
• slices everywhere orthogonal to the world lines of the dust particles,
These happen to coincide for FRW dust but not for the spherically symmetric perturbations (special case of LTB dust with E^3 hyperslices). Thus, the proposed interpretation (not the computation) may be suspect.

Thanks to Haelfix for a useful PM discussion!

Figures (left to right):
• location of the Big Bang (strong spacelike scalar curvature singularity) for two Gaussian perturbations (sketch),
• density of dust versus r for t=1 when $a = \exp(-r^2)$
• density of dust versus r for t=1 when $a = -\exp(-r^2)$

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