COSMOLOGY
Want to look at
— space part of metric of universe
— how do we join it onto a time part?
— some consequences: the redshift of galaxies
But first:
C OORDINATES VERSUS PROPER INTERVALS
Illustrate with the Schwarzschild metric:
ds
2
=
−
µ
2GM
1− 2
c r
¶
1
dt − 2
c
2
µ
2GM
1− 2
c r
¶−1
dr2
r2 2 r2 sin2 θ 2
dθ −
dφ .
c2
c2
Here ds (written dτ when it is meant to be a time) is the
proper interval: this is physically measurable by local
measurements.
As r
→ ∞ and with dr = dθ = dφ = 0 then clearly
ds = dτ → dt.
Thus dt is the proper time of an observer at rest at infinity.
Note therefore gravitational redshift from an emitter at position
1
r in the Schwarzschild metric, to ∞ is given by
1+z = q
1
1−
(1)
2GM
c2 r
(Note also binomial theorem on this would give z
correct to first order.)
≈
GM
c2 r —
Elsewhere t is just a coordinate label.
Same is true for r for radial distances.
c ds =
µ
1−
2GM
c2 r
¶−1/2
dr
(2)
is the physical (measurable) length element radially (will look
at this again more generally in a moment).
But, in this Schwarzschild case:
4πr2 is the physical area of a sphere at coordinate label r.
K ∝ 1/r3 is the physical curvature (measurable by tidal
forces).
So r is a physical variable but is not equal to the radial proper
distance.
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C IRCULAR O RBITS
An important application of the orbit formula in the context of
high energy astrophysics, is what it tells us about circular
orbits in Schwarzschild geometry. These will be approximately
the orbits of material accreting onto black holes, since infalling
material nearly always has angular momentum, and we would
not generally expect direct radial infall.
If r is constant, then our equation for u (1.13 in Handout 19):
GM
3GM 2
d2 u
+
u
=
+
u .
2
2
2
dφ
h
c
(1)
GM r2
h =
.
r − 3GM/c2
(2)
yields
2
Putting ṙ
= 0 in equation (1.8) (Handout 19) gives us
µ
¶
³
´
2
2GM
GM m
1
1
1− 2
m rφ̇
−
= mc2 (k 2 − 1).
2
c r
r
2
(3)
Putting both these last two results together (and using
r2 φ̇ = h) yields an equation for k in terms of r alone. We
derive:
2GM
rc2
− 3GM
rc2
1−
k=q
1
1
.
(4)
Now what is k ? Use the argument given in Handout 19:
Take our function G2 as being a Lagrangian for the particle
— then the energy has to be proportional to ∂L/∂ ṫ
— i.e.
E ≡ pt ∝ (1 − 2GM/c2 r)ṫ = k
— Evaluate the constant of proportionality by demanding that
at infinity have ṫ
= 1 and E = mc2
Shows that we should identify k
= E/mc2 , where E is the
particle energy. Thus we have found that the energy of a
particle in a circular orbit is
2GM
rc2
− 3GM
rc2
1−
Ecirc = mc2 q
1
.
(5)
Obvious check on this equation, is whether it can reproduce
the Newtonian expression for the total energy of a circular orbit
in the limit of large r . Using the binomial theorem we see that
the first two terms in an asymptotic expansion in r are
Ecirc ∼ mc2 −
GM m
+ ...,
2r
(6)
Now usual Newtonian expression for a circular orbit is derived
via
2
Etot
= K.E. + P.E. =
=
where we used
GM m
1
mv 2 −
2
r
−GM m
2r
mv 2
r
=
GM m
r2
Thus get agreement provided we realise that Newtonain
energy enters as a correction to the rest mass energy mc2 ,
which is the dominant term.
The equation we have just found for the energy of a circular
orbit, provides us with useful information about the nature of
such orbits.
First we see that in the limit m
→ 0, the orbit r → 3GM/c2
is of interest, since the singularity in the denominator can
cancel the zero at the top. In fact this is the circular photon
orbit at r
= 3GM/c2 , commented on earlier, and which we
can see immediately is possible from equation (4.29) in
Handout 19, i.e.
d2 u
3GM 2
+
u
=
u .
2
2
dφ
c
(7)
Secondly, we can see which orbits (for particles of (non-zero
rest mass) are bound. This will occur if Ecirc
< mc2 , since
then we have less energy than the value for a stationary
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particle at inifinity. The condition for Ecirc
µ
2GM
1−
rc2
This happens for r
range 4
¶2
= mc2 is that
3GM
=1−
rc2
(8)
= 4GM/c2 or r = ∞. Thus over the
< r < ∞ the circular orbits are bound.
(However, turns out that only orbits with r
= 6GM/c2 are
stable, so this is likely to be inner edge of any accretion disc
around a Schwarzschild black hole.)
4
Now know a version of the Metric for the universe, and want to
work out consequences for things we can observe —
distances, angles etc. — the redshift.
Our information about the universe comes primarily from
photons moving radially. Thus would be nice if the radial part
of the metric was simple. To achieve this define a new
comoving radial coordinate χ via

−1

sin
σ


dσ
, i.e. χ =
dχ = √
σ
2

1 − kσ


sinh−1 σ
k = +1,
k = 0,
k = −1.
(3)
The overall FRW metric then becomes
n
o
2
R
(t)
2
dχ2 + [S(χ)] (dθ2 + sin2 θdφ2 ) ,
ds2 = dt2 − 2
c
(4)
where
S(χ) =



 sin χ



k = +1,
χ
k = 0,
sinh χ
k = −1.
(5)
This form of the metric, which we have arrived at after two
radial coordinate transformations (first r
→ σ , then σ → χ),
is essentially our final form, and is probably the form to
remember it in.
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