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On the Topology and Future Stability of the Universe$

Hans Ringström

Print publication date: 2013

Print ISBN-13: 9780199680290

Published to Oxford Scholarship Online: September 2013

DOI: 10.1093/acprof:oso/9780199680290.001.0001

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(p.687) C Spatially homogeneous and isotropic metrics

(p.687) C Spatially homogeneous and isotropic metrics

Source:
On the Topology and Future Stability of the Universe
Publisher:
Oxford University Press

In Chapter 5, we wrote down several forms of the standard metric. The purpose of the present appendix is to justify the statement that they are (roughly) equivalent. Let us begin with one of the representations commonly encountered in the physics literature:

(C.1)
d s 2 = c 2 d t 2 + a 2 ( t ) d r 2 1 K r 2 + r 2 d θ 2 + sin 2 θ d ϕ 2 .

In order to relate this form of the metric with (5.4), it is convenient to introduce

ϱ ( r ) = 0 r 1 1 K s 2 d s = { 1 K arcsin ( K r ) , K 0 r , K = 0 1 | K | arcsinh ( | K | r ) , K 0.

Clearly, if K 0 we implicitly assume | K r | 1 . Defining

(C.2)
S K ( ϱ ) = { 1 K sin ( K ϱ ) , K 0 ϱ , K = 0 1 | K | sinh ( | K | ϱ ) , K 0 ,

we see that

(C.3)
d r 2 1 K r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) = d ϱ 2 + S K 2 ( ϱ ) ( d θ 2 + sin 2 θ d ϕ 2 ) .

This computation justifies the statement that the representations (C.1) and (5.4) are equivalent. However, we still need to relate one of these metrics with (5.2). Note, to this end, that the metric

g ˉ = d ϱ 2 + S K 2 ( ϱ ) ( d θ 2 + sin 2 θ d ϕ 2 )

is a warped product metric, and that d θ 2 + sin 2 θ d ϕ 2 is an expression for the standard metric on the unit 2 -sphere with respect to local coordinates. Using, e.g., [98, Proposition 42, p. 210], one can calculate that g ˉ is a metric of constant curvature K. To conclude, the expression (C.3) can be used to (p.688) define a metric of constant curvature on an open ball in R 3 of radius π / K in case K 0 and of radius ∞ in case K 0 . Due to [98, Corollary 15, p. 223], we conclude that g ˉ is locally the same as one of the standard metrics on S 3 (if K 0 ), R 3 (if K = 0 ) or H 3 (if K 0 ). Thus (5.2) and (C.1) are locally the same. Moreover, the representations can be argued to be the same globally for K 0 . However, for K 0 , coordinate representations of the form (C.3) are inappropriate in that the topology becomes obscure.