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Cosmological Space-Times
Lecture notes compiled by Geoff Bicknell based primarily on: Sean Carroll: An Introduction to General Relativity plus additional material

Thursday, 10 September 2009

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Metric of special relativity
ds
2

= -c dt + dx + dy + dz
2 2 2 2

2

= -dx

02

+ dx + dy + dz
2 2

2

= µ dx dx
µ

where

µ



µ,

=

= Minkowski tensor = diag [-1, 1, 1, 1] 0, 1, 2, 3

This is the metric of four-dimensional flat space time Generalised by Einstein in his 1916 General Theor y of Relativity to:
ds = gµ dx dx
2 µ

Metric tensor
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General relativity field equations
gµ g


µ

Christoffel Symbols 1 = g (g,µ + gµ, - gµ 2 = Inverse of gµ

,

)

The Christoffel symbols appear in the equations of test particles: Geodesics of space time - and also in generalised (covariant) derivatives Riemann cur vature tensor :
R
Thursday, 10 September 2009

µ

=

,µ

-

µ,

+

µ

-

µ

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Tensors derived by contraction over indices
Ricci tensor
R
µ

=R

µ

Ricci scalar
R=g
µ

R

µ

Einstein tensor
G
Thursday, 10 September 2009

µ

=R

µ

1 - gµ R 2
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Einstein's field equations
G = gµ 8 G + 4T c

Newton's Constant of gravitation
µ

µ



Cosmological constant "Dark energy" Matter tensor
T
µ

Matter tensor

= (c2 + p)U µ U + pg

µ

dxµ 1 dxµ µ = 4-velocity of matter U = ds c d
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Metric of the Universe
Homogeneity and isotropy => Geometr y invariant under translations and rotations => Maximally symmetric space time
ds = -c dt + a (t) e
2 2 2 2 2 (r )

dr + r d + r sin d2
2 2 2 2 2

Spatial part of metric:
d = a (t) e
2 2 2 (r )

dr + r d + r sin d
2 2 2 2 2

2

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6


When

e

2

=1

d = a (t) dr + r d + r sin d
2 2 2 2 2 2 2

2

which is the normal metric of flat space modified by the scale factor a(t) The scale factor informs us how the universe is expanding In this space-time metric the coordinates are comoving coordinates, i.e. as the Universe expands the spatial coordinates of galaxies remain constant
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Space-time geometry
Neighbouring world lines

3D hypersurface t = t1 ds2=a2(t1) ijduiduj 3D hypersurface t = t0 ds2=a2(t0) ijduiduj

Comoving observers ui=constant

Geometr y of 3D hypersurfaces
d 2 = a2 (t) e2
Thursday, 10 September 2009

(r )

dr2 + r2 d

2
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Maximally symmetric spaces (consequence of homogeneity and isotropy)
Characterised by
(3)

R

ij kl ij

= k (ik jl - il jk ) = 2k ij
2 (r )

R

Spatial metric

ij = diag(e

,r ,r sin )

2

2

2

Equations for metric tensor (3) -2 2 R11 = e r - 1 + 1 = 2k 11 = 2ke r (3) -2 2 R22 = e r - 1 + 1 = 2kr r 2 2 (3) -2 2 R33 = e r - 1 +1 sin = 2kr sin r
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Solution
e
2

d = d + sin d
2 2 2

2

=

1 1 - kr

2 2

d

2

dr2 2 2 = a (t) + r d 2 1 - kr

Coordinate transformation
r
2

= |k |r

2

r d
2

= |k | r dr 2 a2 (t) = |k | 1 - sgn(k )r

1/ 2

2

+ r 2 d

2

Thursday, 10 September 2009

10


1/2 into a(t); k = -1, 0, 1 Absorb |k|
e
2

=

1 1 - kr

2 2

d

2

k

dr2 2 2 + r d = a (t) 1 - kr2 = -1, 0, +1

k=0
d
2

= a (t) dr + r d

2

2

2

2

Expanding flat space
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k=-1
dr2 2 2 2 d = a (t) + r d 1+ r2
2

New radial variable
d = dr (1 + r2 )1
-1 /2

= sinh r

r

= sinh

Metric of each 3D hypersurface
d
2

= a (t) d + sinh d

2

2

2

2

Thursday, 10 September 2009

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k=+1
dr2 d 2 = a2 (t) + r 2 d 1 - r2
2

New radial variable
d = dr2 (1 - r2 )1
-1 /2

= sin r

r

= sin

Metric of each 3D hypersurface
d
Thursday, 10 September 2009

2

= a2 (t) d2 + sin2 d

2

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Summary of 3D metrics
d 2 = a2 (t) d2 + S 2 ()d
2

S () = k=0 S () = sinh() k = -1 S () = sin k = +1

What geometr y do these metrics represent? k=0 => Metric of an expanding flat space

Thursday, 10 September 2009

14


k=+1
Consider a 3-sphere embedded in a 4-dimensional Euclidean space (not space-time) Let the equation of the sphere in (w,x,y,z) space be:
w +x +y +z =a
2 2 2 2 2

The metric of the 4-dimensional space is:
d 2 = dw2 + dx2 + dy 2 + dz
2

Thursday, 10 September 2009

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Metric of the surface of the 3-sphere
Consider the following set of spherical polars in 4-space; these provide a parametric description of the surface of the 3-sphere which has radius a(t). There are 3 angular parameters.
w = a cos

z = a sin cos x = a sin sin cos y = a sin sin sin

We now determine the metric of the surface of the sphere by determining the differentials of the coordinates w, x, y and z.

Thursday, 10 September 2009

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Euclidean metric restricted to 3-sphere
These are the differentials of w,x,y,z in terms of the polar angles
dw dz = -a sin d

dx = a cos sin cos d + a sin cos cos d - a sin sin sin d dx = a cos sin sin d + a sin cos sin d + a sin sin cos d

= a cos cos d - a sin sin d

This gives:
dw + dx + dy + dz = a (t) d + sin (d + sin d )
2 2 2 2 2 2 2 2 2 2

which is the spatial part of the space-time metric
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Conclusions for k=+1:
1. The 3-space of this metric can be thought of a as a 3-sphere of radius a(t) embedded in a 4 dimensional Euclidean space 2. The 3-sphere is expanding 3. Since a 3-sphere is closed the k=1 metric represents a closed Universe

Thursday, 10 September 2009

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Embedding diagram
w

Consider section y=0: y = a sin sin sin = 0 = 0 or
=0 w z section

= a cos = a sin cos

a

x

x = a sin sin = w z section

= a cos = a sin cos

z Embedding of a 3-sphere in a 4-dimensional Euclidean space
Thursday, 10 September 2009

x = -a sin sin
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The case k = -1
Consider the equation of a 3-hyperboloid embedded in a 4dimensional Euclidean space:
w2 - x2 - y 2 - z 2 = a2

We can parametrically express this in terms of hyperspherical polars
w z y = a cosh = a sinh cos = a sinh sin sin

x = a sinh sin cos

Thursday, 10 September 2009

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Differentials:
dw dz dy = a sinh d = a cosh cos d - a sinh sin d

dx = a cosh sin cos d + a sinh cos cos d - a sinh sin sin = a cosh sin sin d + a sinh cos sin d + a sinh sin cos

Metric restricted to 3-hyperboloid
d
2

= dw2 + dx2 + dy 2 + dz
2 2 2

2 2 2 2

= a (t) d + sinh (d + sin d )

Thursday, 10 September 2009

21


Embedding: y = 0 section
=0 = w =0

y = 0 sin = 0 =0 or

a x

Embedding of a 3-dimensional space of negative curvature in a 4-dimensional Minkowskian space z
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Summary
The metric of the expanding Universe can be expressed in one of the 3 following ways:
ds2 = -c2 dt2 + a2 (t) d2 + 2 d
2 2 2 2 2 2

k=0 Infinite flat Universe
2

ds

2

= -c dt + a (t) d + sin d

k=1 Finite closed Universe
2

ds

2

= -c dt + a (t) d + sinh d
2 2 2 2

2

k=-1 Infinite, open Universe
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Thursday, 10 September 2009