
General Relativity implies spacetime is curved in the presence of matter
 since universe contains matter, might expect overall curvature (as well as local “gravity wells”)
 how does this affect measurements of large-scale distances?
 what are the implications for cosmology?
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

Two-dimensional curved space: surface of sphere
 distance between ( r , θ ) and ( r+ d r , θ+ d θ ) given by
 d s 2 = d r 2 + R 2 sin 2 ( r / R )d θ 2
 r = distance from pole
θ = angle from meridian
R = radius of sphere
 positive curvature
“Saddle” (negative curvature)
 d s 2 = d r 2 + R 2 sinh 2 ( r / R )d θ 2

(2D surface of constant negative curvature can’t really be constructed in 3D space)
Nick
Strobel’s
Astronomy
Notes
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
Robertson-Walker metric
 d s 2 = − c 2 d t 2 + a 2 ( t )[d x 2 /(1 – kx 2 / R 2 ) + x 2 (dθ 2 +sin 2 θ dφ 2 )]
 note sign change from our previous definition of ds 2 !

 a ( t ) is an overall scale factor allowing for expansion or contraction ( a ( t
0
) ≡ 1) x is called a comoving coordinate (unchanged by overall expansion or contraction)
 k defines sign of curvature ( k = ±1 or 0),
R is radius of curvature
 path of photon has d s 2 = 0, as before
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 comoving proper distance (d t = 0) between origin and object at coordinate x : r
 x
0

1
 d x kx
2
R
2
 for k = +1 this gives r = R sin −1 ( x / R ), i.e. r ≤ 2π R

 finite but unbounded universe, cf. sphere for k = −1 we get r = R sinh −1 ( x / R ), and for k = 0, r = x

 infinite universe, cf. saddle for x << R all values of k give r ≈ x

 any spacetime looks flat on small enough scales this is independent of a
 it’s a comoving distance
PHY306
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 cosmological redshift : change variables in RW metric from x to r:
 d s 2 = − c 2 d t 2 + a 2 ( t )[d r 2 + x ( r ) 2 dΩ 2 ]
 for light d s = 0, so c 2 d t 2 = a 2 ( t )d r 2 , i.e. c d t / a ( t ) = d r
(assuming beam directed radially)
 suppose wave crest emitted at time t e and observed at t o c
 t t e o d t a ( t )
  r
0 d r
 r c t o t e



 o
 e c d t c a ( t )
  r
0 d r
 r first wave crest next wave crest
PHY306
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
Then c t e


 e t e c d t a ( t )
 c t o
  o
 t o c d t a ( t ) but if λ << c/H
0
, a ( t ) is almost constant over this integral, so we can write c a ( t e
) t e
  e t e
 d t c
 c a ( t o
) t o
  o t o
 d t c i.e.
 e a ( t e
)

 o a ( t o
)
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
So expanding universe produces redshift z , where

1
 z
 a ( t o
)

1 a ( t e
) a ( t e
)
Note:
 z can have any value from 0 to ∞

 z is a measure of t e often interpret z using relativistic Doppler shift formula c c
 v
 v but note that this is misleading: the object is not changing its local coordinates
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1
 z



Conclusions
 in general relativity universe can be infinite
(if k = −1 or 0) or finite but unbounded (if k = +1)
 universe can expand or contract (if overall scale factor a ( t ) is not constant)
 if universe expands or contracts, radiation emitted by a comoving source will appear redshifted or blueshifted respectively
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