General Relativity
Physics Honours 2006
A/Prof. Geraint F. Lewis
Rm 557, A29
gfl@physics.usyd.edu.au
Lecture Notes 11
Aside: Kerr black hole
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Aside: Charged black hole
All figures from d’Inverno
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Cosmology
As we have seen, it is difficult to determine general analytic
solutions to the field equations. However, applying symmetries
reduces the complexity of the problem. One place this has
proved successful is in cosmology where the symmetries are
Isotropy: The spacetime geometry is spherically symmetric
about any point.
Homogeneity: The spacetime geometry is the same at all
points (although this can change with time).
These symmetries imply that the energy distributions must
also be isotropic and homogeneous (clearly, this breaks down
on small scales in the universe, but appear to apply “on
average” on large enough scales.
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Chapter 18
Robertson-Walker Metric
These symmetries imply that the metric can only take on a
specific form, the Robertson-Walker metric;
Here, k describes the geometry of space and can take on
values of +1 (closed: spherical), 0 (flat) and -1 (open:
hyperbolic). The general function a must be determined, but it
can only depend upon time.
Notice that unlike the previous metrics we have seen, this one
is dynamic and changes with time. Hence, in this metric, there
is no Killing vector associated with time, but there are with
spatial symmetries.
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Cosmological Equations
Suppose the universe is filled with a fluid with a particular
density and pressure. For an observer at rest with respect to
the fluid, the stress-energy tensor is simply
We now need to take the metric, calculate the Ricci tensor
and scalar to form the Einstein tensor. This is then projected
into the orthonormal frame of our observer so that
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Cosmological Equations
How to the pressure and density change with time? We can
use the conservation equation
If we assume our fluid is at rest in our coordinates, then
u=(1,0,0,0) and
So we can write
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Cosmological Equations
The only non-zero Christoffel symbols that matter are
Which gives;
This can be written as
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Flat Universe
Let’s start with a spatially flat (k=0) universe. The metric is
As a(t) changes, the coordinate distance between any two
points remains fixed, but the physical distance changes as
In our Universe, a(t) appears to be increasing with time, and so
the distance between comoving observers is increasing i.e.
the Universe is expanding. This also means that the volume
between comoving observers is increasing and the Universe is
becoming more dilute.
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Redshift
What about photons exchanged between two comoving
observers? Writing the metric in polar form and considering
photons traveling on radial paths, then
Hence, the coordinate distance traveled by the photon is
Remembering that the frequency is  = 2/t then
Where z is the redshift of photon.
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Hubble Law
Suppose we have a nearby object at coordinate separation R,
so the physical separation at the present time is
The time taken by the photon to travel this distance is
So we know te = to – d. Assuming d is small, then
This is the Hubble Law which says we should see a linear
correlation between distance and redshift.
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Hubble Law
We can turn the redshift into a
velocity via z~v/c and plot
distance verses velocity. In the
local universe, this is a linear
trend (but not in the larger
universe). The slope of the line is
Hubble’s constant.
Hubble’s constant has been measured to be 72km/s/Mpc. It’s
natural units are inverse time, and so expressing this value in
years gives us a Hubble time;
This is the natural universal time scale.
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How do fluids evolve?
We can calculate how the density and pressure of the fluids in
the universe evolve with time using
Matter: can be represented as a pressureless fluid and so it’s
density simply scales as
As a increases, the universe expands and the density of
matter drops in proportion to the volume of the expansion.
This is what we expect.
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How do fluids evolve?
Photons: For a blackbody gas of photons, the energy density
and pressure are related via
Plugging this into our equation and integrating gives
The photon gas thins out and loses additional energy! Where
does it go? Additionally, the energy density is
And the blackbody spectrum cools as the universe expands.
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How do fluids evolve?
In general, we can define an equation of state for a fluid;
For matter, w=0, and photons, w=1/3. For a general fluid, this
implies that
The quantum vacuum has w=-1 and its energy density is
constant with time. While the energy density in the vacuum is
unknown, it appears that the dominant energy in our universe
is in this form (known as the cosmological constant).
General fluids with w<-1/3 are known as quintessence or dark
energy, while those with w<-1 are known as phantom
energies.
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Friedmann Equation
Let’s consider a spatially flat universe (k=0), the we can write
the Friedmann equation as
All we need to do is put how the fluids evolve with time into
the equation and solve for a. It is convenient to scale to
present day quantities as we know that
We can define a useful critical density to be
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Friedmann Equation
We can normalize our energy densities by this critical value so
And for our spatially flat universe, the sum of the normalized
energy densities must equal unity. In the flat model, we have
the freedom to define a(t0)=1 and
and
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Friedmann Equation
In this example, at the present time energy is equally
distributed between vacuum, matter and radiation. But the mix
was different at other epochs, with one essentially dominant.
In this picture (as in most models) the universe has a beginning
at a finite time in the past when a=0; this is the Big Bang of
cosmological models.
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Friedmann Equation
Matter dominated: m=1
Photon dominated: r=1
Vacuum dominated: v=1
We can of course generalize these expressions for arbitrary
equations of state. Generally, one component will dominate.
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Spatial Curvature
When k is not 0 then the universe can possess spatial
curvature. If k=+1, then space is closed and has spherical
geometry (and it has finite volume which changes with time).
With k=-1 the space is open, the geometry is hyperbolic and
possesses infinite volume.
Read through section 18.6 to understand this spatially
curvature. What does it mean for geometry?
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Friedmann Equation
In its generalized form, this is
Considering the present epoch, we can write
So the spatial curvature is determined by the total energy
density in the universe. Exceeding the critical value results in
a closed universe, while densities less than critical are open.
Generally, the cosmological equations do not possess analytic
solutions for arbitrary equations of state and mixes of fluids.
These must be tackled numerically (especially for interacting
models).
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Matter Dominated Universes
If we consider universes with only matter, it is possible to
derive a parametric form for the expansion. For positively
curved universes
So the Universe expands to a maximum size (and volume)
before recollapsing back to a point (a big crunch).
We can derive a similar expression for negatively curved
universes, but containing hyperbolic functions. These
universes expand for ever, and so the critical density
differentiate between eventually collapsing and expanding
universes (for those containing matter).
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Generalized Cosmology
We can derive a generalized form of the cosmological equation
where
and
Again, this can be generalized further to arbitrary equations of
state for more complex fluids.
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Our Universe
Our universe appears to be flat, with m~0.3 and v~0.7 and
negligible radiation. The curve on the left shows the effective
potential for our Universe, showing that we are entering a
phase of eternal acceleration. On the right is the effective
potential for m=0.2 and v=0.02. This universe has two
possible evolutions, expanding and then collapsing, or
collapsing and then bouncing. With general fluids, we can get
similar behaviour.
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What happens?
If we assume we only have
vacuum energy and matter
(which is a good model for
our universe as the photon
energy density currently
very small).
Even with these two free
parameters at the present
time, there are a number
of differing evolutions of
the universe.
Observational evidence
(Chapter 19) suggests we
are in a flat universe,
sitting on the Tot = 1 line.
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Lecture Notes 11
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