Ok, from astronomical observations we can tell that the observable matter is separating – so rewind the clock about 13.7 billion years and it was all at a single point.
However, how do we distinguish between the following two options:
- Universe is expanding
- Matter distribution is increasing into infinite void
Clarification 1:
(My notion of) The traditional notion is that all time/space/matter was created at the instant of the big bang.
I.e. BB was inital conditions of: $t=0$, $V=0$, $E=very big$
People say "the universe is expanding", rather than "observable matter is separating".
Why is this?
How do we know that the big bang event wasn't started by all matter condensed at a single point within a larger (otherwise empty) universe?
How do we know that BB wasn't: $t=0$, $V(universe)>0$ but $V(matter)=0$, $E=very big$?
Best Answer
I will assume that E in the question is energy density. First we have to distinguish between two sizes:
size of the observable universe = current proper distance to particle horizon = current proper distance to infinite redshift, and
size of the whole universe.
Assuming trivial global topology, the size of the entire universe is infinite if space is flat or open (hyperbolic), and finite if space is closed (spherical). To note, if the magnitude of the current density parameter for curvature $\Omega_k(to)$ is smaller than a certain threshold, it is impossible to know whether the universe is flat and infinite or closed and finite. (Vardanyan et al (2009) How flat can you get?) Clearly the total size of an infinite universe was infinite at t=0, while the total size of a finite universe was zero.
Now, the alternative view of matter travelling into pre-existing empty space (Minkowski spacetime) is precisely the model proposed by Edward Milne in 1932. For that precise model to be valid we have to assume:
either that GR is not applicable to cosmology (Milne's original position),
or that the universe is gravitationally empty at large scales (an unlikely but still open possibility).
Now, if you stay within GR, then matter distribution determines spacetime, so that a non-neglibible matter density implies a spacetime different from Minkowski (so that if the matter moves, the spacetime changes over time). Having cleared that, as I said in this answer, any spacetime described by the FLRW metric, where objects are static in comoving coordinates and redshift is due to expansion of space, can also be described, via an appropriate change of coordinates, by a spherically-symmetric (SS) metric, where objects move along radial timelike geodesics and redshift is due to positional (gtt) and Doppler factors. Since both metrics describe the same physical system, they are observationally equivalent in all respects.
Two notes on the equivalent SS metric:
usually (but not in the Milne case) has a cosmological horizon (gtt & grr -> $\infty$), located at the Hubble distance in flat FLRW models,
is static (gtt & grr do not depend on time), only in the empty (Milne) and lambda-vacuum (de Sitter) cases,
even in the next simplest case, the flat matter-only FLRW model known as Einstein-de Sitter model, gtt & grr cannot be expressed algebraically in terms of non-comoving time (t' below), as I mentioned in this answer.
I expanded this below to practice a bit of LaTex:
Any homogeneous and isotropic spacetime described by the FLRW metric in terms of a set of coordinates $(t, r, \theta, \phi)$:
$$\begin{equation} ds^2 = - c^2 dt^2 + \frac {a(t)^2 dr^2} {(1 - k r^2)} + a(t)^2 r^2 d\Omega^2 \end{equation}$$
where:
t = comoving time = proper time for all observers at constant $(r, \theta, \phi)$
r = comoving radial coordinate = radial coordinate enclosing constant proper mass
$d\Omega^2 = d\theta^2 + sin^2(\theta) d\phi^2$
can be described by a spherically symmetric (SS) metric in terms of a different set of coordinates $(t', r', \theta, \phi)$, where the ' does not mean a derivative:
$$\begin{equation} ds^2 = - c^2 gtt(t', r') dt'^2 + grr(t', r') dr'^2 + r'^2 d\Omega^2 \end{equation}$$
where: $\Omega$, $\theta$ and $\phi$ are the same as in the FLRW metric, and obviously $r' = a(t) r$
Expressing $t' = f(t, r)$ and its partial derivatives as $ft$ and $fr$:
$$\begin{equation} grr = \frac {1} {[1 - k r^2 - \left( \frac {r} {c} \frac {da(t)} {dt} \right)^2]} \end{equation}$$
$$\begin{equation} gtt = \frac {grr} {ft^2} (1 - k r^2) \end{equation}$$
The problem lies in expressing gtt & grr in terms of $(t', r')$, which can be done only in the Milne and de Sitter cases.