Notes on Relativity and Cosmology - Physics Department, UCSB

290 CHAPTER 10. COSMOLOGY
-1 -2 0
1 2
t = const
x = 0
Note that this is not the k = 0 Universe which has flat space. Instead, the
entire spacetime is flat here when viewed as a whole, but the slice representing
space on the above diagram is a hyperboloid, which is most definitely not flat.
Instead, this hyperboloid is a constant negative curvature space (k = −1). Since
the spacetime here is flat, we have drawn the limit of the k = −1 case as we
take the matter density to zero. It is not physically realistic as a cosmology, but
I include it here to give you a diagram that illustrates the co-moving coordiante
system used in cosmology. In addition, for k = −1 the matter density does
become vanishingly small in the distant future (if the cosmological constant
vanishes; see below). Thus, for such a case this diagram does become accurate
in the limit t → ∞.
Shown here in the reference frame of observer #0, that observer appears to
be the center of the expansion. However, we know that if we change reference
frames, the result will be:
0 -1 1
2 3
t = const