# Metric to describe an expanding spacetime from coordinates reflecting the perspective of a local observer

The FLRW metric describes the metric expansion of spacetime from the perspective of comoving coordinates. Given the way this metric is usually formulated, comoving distances stay constant, and the size of the unit ball "shrinks" as you translate it along the time axis (but not the spatial axes).

I'm interested in seeing how to reformulate this metric given a choice of coordinates that lines up with the perspective of an observer who is in the space. From the observer's perspective, everything is falling along timelike geodesics that are moving away from it, and the further objects are from the observer, the faster they seem to be falling, and the size of the unit ball around the observer remains constant with time.

To keep it simple, I only care about the case where curvature is 0 and we set $c=1$. So the FLRW metric in those circumstances is

$$\mathrm{d}s^2 = -\mathrm{d}t^2 + a(t)^2 \cdot (\mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2)$$

Where $a(t)$ is the time-dependent scale factor. I thought that to flip this on its head, you'd want a metric that looks like this:

$$\mathrm{d}s^2 = -b(x,y,z)^2 \cdot \mathrm{d}t^2 + \mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2$$

Where we now have the space-dependent scale factor of $b(x,y,z)$, and it affects the time coordinate. I was thinking that for some such suitable function $b$, which is presumably monotonic, you'd end up with a metric which stays constant in time rather than in space, and timelike geodesics that are all constantly moving away from one another. (They're also moving away from the line $(t,0,0,0)$, representing the observer I was talking about.)

Am I on the right track here? I'm looking for something which is basically exactly the same as FLRW, but just reflects the different choice of coordinates.

• There is an underlying misconception being displayed, which is that coordinate systems in GR relate to observers. That's not true. – Ben Crowell Feb 2 at 17:00

(Two and a half years late -- guess this fell through the cracks.)

No, you can't do that, except in special cases. And the FLRW model of our actual universe is not one of those special cases.

The short proof is: your second metric has a timelike Killing vector whereas FLRW generally does not.

To explain:

There are scalar quantities that are invariant under any spacetime co-ordinate transformation. One of these quantities is $\rho c^2 - 3p$, where $\rho$ is the local mass density and $p$ the local pressure (if pressure is isotropic, otherwise, replace $3p$ with the sum of the three principal pressures).

Now in the usual FLRW model of our universe, this scalar decreases with comoving time as matter (ordinary and dark) dilutes. You may wonder if there is a frame of reference where it doesn't. Well, we can pick out the subspaces over which this scalar is constant. But it turns out these are spacelike. (In fact, this characterizes "comoving time".) There can be no timelike observer which sees this scalar as constant.

Your proposed transformed metric is independent of $t$. That means the observer $(t,0,0,0)$ sees the scalar I mentioned as constant. So, we know it cannot be a co-ordinate transformation of the usual FLRW model of our universe.

For simplicity you only considered the flat case of the original FLRW metric, and your proposed transformed metric is also flat. But "flatness" is not an invariant. (The curvature scalar of space is an invariant under spatial co-ordinate transforms; the curvature of spacetime is invariant under spacetime transforms; but the curvature of space is not generally invariant under spacetime transforms.) This point is relevant to the special cases I mentioned.

The usual FLRW model of our universe does actually converge on one special case, called de Sitter space. In the distant future, the cosmological constant trumps everything, and (flat) space becomes exponentially expanding:

$$d\tau^2 = dt^2 - c^{-2}e^{2ct/r_0} [ dr^2 + r^2d\Omega^2 ]$$

I've written space in polar co-ordinates, $d\Omega^2$ is some metric for the surface of a sphere.

de Sitter space has a well-known static frame of reference $(t^\star,r_c,\Omega)$:

$$d\tau^2 = (1 - \frac{r_c^2}{r_0^2})d{t^\star}^2 - c^{-2}(1 - \frac{r_c^2}{r_0^2})^{-1}d{r_c}^2 - c^{-2}r_c^2d\Omega^2$$

which looks like a metric for a black hole, except turned inside-out. Exponentially expanding flat space has been transformed into static curved space!

Or, my favourite, but less well-known static frame of reference $(t^\star,r^\star,\Omega)$:

$$d\tau^2 = (\cosh\frac{r^\star}{r_0})^{-2} [ d{t^\star}^2 - c^{-2}d{r^\star}^2 - c^{-2}r_0^2(\sinh\frac{r^\star}{r_0})^2d\Omega^2 ]$$

The part in square brackets is a standard metric for static hyperbolic space, with curvature radius $r_0$; the overall metric scales this by a conformal factor (that doesn't depend on time, and that is equal to 1 at $r^\star=0$).

By the way, the current estimate of the ultimate $r_0$ for our universe is about ten billion light years.

You can transform any FLRW metric into

$$d\tau^2 = A^2 [ d{t^\star}^2 - c^{-2}d{r^\star}^2 - c^{-2} B^2 d\Omega^2 ]$$

such that $A=1$ when $r^\star=0$, and such that

$$\lim_{r^\star \to 0} \frac{B}{r^\star} = 1$$

Generally the factors $A$ and $B$ will depend on both $t^\star$ and $r^\star$.

These are called radar coordinates, because the radially outgoing and incoming speeds of light are constant $c$, and, for an observer with constant $r^\star=0$, the time coordinate is proper time. That means the events at $(t^\star,r^\star,\Omega)$ are the intersection of the past and future light cone of the reference observer at times $t^\star \pm \frac{r^\star}{c}$. In that sense, you might say that it "lines up with the perspective of an observer who is in the space."

By the way, the only other special case I'm aware of is of a linearly expanding hyperbolic space. This is called the Milne cosmology. The radar coordinates for that are static and flat -- the Minkowski metric!