General Relativity
Physics Honours 2005
Dr Geraint F. Lewis
Rm 557, A29
gfl@physics.usyd.edu.au
Cosmology
Cosmology studies a on-off phenomenon – the Universe.
The fact that General Relativity can provide a dynamical
description of the evolution of the Universe is seen as one of the
great successes of the theory.
While we live in a single instance of the Universe, the equations of
relativity describe a whole range of potential universes whose
evolution depend upon their energy content.
In this course, we will look at the GR formulation of cosmology
and the potential forms of evolution.
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Cosmology
Western society viewed the Universe as infinite in size, but finite
in age, naturally explaining Oblers’ paradox: why is the sky at
night dark?
If the Universe was infinitely old and unchanging, then the Earth
should receive radiation from an infinite number of stars, and the
night sky should be as bright as the surface of the Sun. With a
finite age, we only receive light from a finite number of stars.
An alternative, however, is that the star light loses energy as it
travels to the Earth, such as through a Doppler shift in an
expanding universe. (Why can’t the energy just be absorbed by
some intervening material?)
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Newtonian Cosmology
Much of cosmology can be expressed in terms of Newtonian
mechanics, although this was realised after GR cosmology.
Consider the energy of a finite number of particles
The final term is a cosmological repulsion force of the form F =
1/3 mi ri (something we will see in more detail later). Does this
imply a preferred origin?
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Newtonian Cosmology
If we assume the cosmological principle (more on this shortly)
the view point from any point within the distribution, and motion
must be purely radial (why?). Hence, at any time t we can write
where S(t) is a scale factor which is the same for all the particles.
With this we also have
where H(t) is the
Hubble parameter.
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Newtonian Cosmology
The energy equation can therefore be written as
where A, B & D are positive constants. If we rescale such that
R(t)= S(t), then this equation can be written as
where
(GR gives the same Friedmann equation)
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Newtonian Cosmology
If E=0 then  is arbitrary (k=0). If E 0 then we can choose
then k only takes on the value of §1.
Hence, we have a simple differential equation to describe the
motions of the particles. If =0, then there are three simple
evolutionary states;
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The Cosmological Principle
Relativistic cosmology is based on several simple principles. The
Cosmological Principle states that
“at each epoch, the universe present the same aspect
from every point, except for local irregularities”
So, there are no preferred locations, and no preferred directions
to look, and so the Universe is homogeneous and isotropic.
Note that this refers to spatial directions; the Perfect
Cosmological Principle states that the Universe appears the
same at all times and it is unchanging (this is the source of the
well known “Steady State” model of the Universe).
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Weyl’s Hypothesis
Simply put, Weyl’s hypothesis states that we can ignore all of the
small scale irregularities in the Universe, and describe the energy
distributions in the Universe as perfect fluids.
The postulate requires particles in the fluid to follow nonintersecting time-like geodesics. We can also assign a population
of observers that ride along with the fluid (i.e. they follow the
same null geodesics).
We will see that the proper time for these observers is the same as
the cosmic coordinate time we use to describe the metric.
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Relativistic Cosmology
Working with coordinates (t,x1,x2,x3), we can slice the universe into
a series of t=constant hypersurfaces. The Weyl hypothesis requires
that the geodesics pass through each hyperslice the same spatial
coordinates (x1,x2,x3) i.e. the geodesics are orthogonal to the
hypersurface. These define comoving coordinates.
The line element is then
where t is the world or cosmic time. Note that hypersurfaces of
constant t are surfaces of simultaneity (but the view from a point is
related to its past light cone!).
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Relativistic Cosmology
The cosmological principle forces us to consider universes with
constant spatial curvature; if curvature varied from place-to-place
then different points would distinguishable.
Furthermore, it also forces h to be of the form
otherwise different directions would be distinguishable.
Given these facts, a general form for the metric for the universe
can be derived.
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Constant Curvature
The curvature tensor for a 3-space of constant curvature is
where K is the curvature. The Ricci tensor is then
The most general spherically symmetric line element in 3-d is
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Constant Curvature
The non-vanishing elements of the Ricci tensor are
For constant curvature, this gives
with the solution
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Robertson-Walker Metric
Combining to get the full relativistic line element, we see
where k=-1,0,+1 (this requires a coordinate change of the form
r’=| K |1/2 r to eliminate K). This is the Robertson-Walker line
element (and associated metric) which is the foundation of
modern cosmology.
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Robertson-Walker Metric
The Robertson-Walker metric describes any homogeneous and
isotropic universe, irrespective of its content (in terms of energy
distributions).
The specifics of the energy densities, however, govern the choice
of k and the evolution of R(t).
We will look at this in more detail with the Friedmann equations.
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3-Space Geometry (22.8)
The value of k determines the spatial geometry. We will consider
the k=+1 case in detail (but read the others in the textbook).
k=+1: The geometry is a 3-sphere (analogous to the surface of a 2sphere in 3-d). Introducing  such that r=sin then
We can embed this 3-d surface into 4-d Euclidean space with the
coordinates (w,x,y,z)
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3-Space Geometry
This yields
Hence we can think of R(t) as the radius of the Universe.
Remember that this radius is only apparent in a 4-d view.
Note that the total volume of the Universe is then V=22R3(to).
Hence the Universe has finite volume, but it contains no edges. In
terms of topology, it is closed and unbounded. When considering
the time axis, the overall space-time is cylindrical.
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3-Space Geometry
k=0: Here the 3-Space is flat. In terms of topology the space is
open and unbounded i.e. you can travel for ever in a straight line,
and never come back to the same point. Hence, the universe is
infinite in extent (and always has been!)
k=-1: Following a similar argument for the spherical space, we
can describe this case as a 3-hyperboloid. Here, however, the
geometry is open and unbounded.
These completely describe the potential geometries of the universe
(given the Robertson-Walker metric).
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Propagation of Light
Our view of the Universe depends upon the propagation of light
through the curve space-time metric. To understand this, we need
to consider the paths of null geodesics.
Suppose an observer sits at r=0 and we consider radial light rays.
Given that ds=0, then
Suppose a light ray is emitted at a time t1 at a distance r1 and is
received today (r=0) at t0
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Propagation of Light
Remembering that for a comoving source at distance r, the
coordinate is fixed, then
Hence, in an expanding universe there will be a redshift
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Hubble’s Law
If we consider a nearby source, then we can write
then
Hence, we can see we can derive Hubble’s law from the
Robertson-Walker metric. Hubble’s constant (which is not a
constant) is now known to be 72 km/s/Mpc.
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Cosmological Distance
Generally, the idea of “distance” in cosmology is rather
complicated. The instantaneous proper distance, d=R(t)r, is of little
practical use. One of the most commonly used is the luminosity
distance, which is akin to the r2 law for radiation.
Unlike Euclidean cosmology, there is an additional term of (1+z)
for the redshifting of photons, plus an additional (1+z) as the
emission rate is time dilated.
The resulting definition is
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Cosmological Distance
Note that the textbook is wrong! The luminosity distance is
related to other distances through
Here dt is the transverse proper distance and equals the proper
distance in a flat (k=0) universe. In a non-flat universe, dt
depends upon the curvature.
While we will not look at the derivation, another common
cosmological distance is the angular diameter distance which
tells us the angle subtended by a object with proper length L.
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Friedmann Equations
We shall not consider the detailed derivation of Friedmann’s
equations, but we will consider universes described by the
Robertson-Walker metric and assuming Weyl’s postulate then;
where  is the proper density in the special coordinate system in
which ua=(1,0,0,0). For normal matter, p=0, meaning that the
energy in motion is negligible to that in the form of rest mass.
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The Cosmological Constant
Einstein found that field equations could be extended such that
Einstein added this term on the left, introducing a curvature that
drives objects apart (i.e. anti-gravity!). This allowed him to make
the cosmos static. The modern interpretation, however, places
this term on the right i.e. a fluid in a universe where p=- will
provide the same term in the field equation.
In modern cosmology, it appear that  is the dominant energy
density in the universe today.
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Friedmann Equations
Taking the field equations, Friedmann derived the differential
equations which govern the evolution of the scale factor. Assuming
a universe with a general fluid and a classical cosmological
constant , then (exercise)
Assuming the fluid is dust-like (p=0) then we can combine these to
give the generalized Friedmann equation.
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Friedmann Equation
Assuming the fluid is dust-like (p=0) then
where C=8 R3 /3. Amazingly, this is the same form as the
Newtonian version we saw earlier.
Hence, the evolution of the Universe can be determined by
specifying the parameters in this equation and integrating to
determine R(t).
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Deceleration
The rate of change of H(t) can be encapsulated in the
Deceleration Parameter such that
If q>0 then the cosmos is slowing down over time.
Hubble’s constant Ho and q can be determined by observations
of distant sources as they modify Hubble’s relation.
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Quintessence
In deriving the Friedmann equation, we have assumed the
Universe is composed of two fluids, dust (p=0) and the
cosmological constant (p=-).
In general, we could put any mix of fluids into the Universe. Each
of these are parameterized by an Equation of State
Where =0 for dust and =-1 for the cosmological constant. It is
found that <-1/3 causes the Universe to accelerate. Determining
 (and its potential evolution) is a major goal of cosmology.
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Cosmological Models
There are several analytical solutions to the Friedmann equations
(but in general they have to be determined numerically), although
a qualitative understanding of the evolution of R(t) can be found
from recasting the Friedmann equation as an energy equation;
Hence a sketch of the form of the potential energy term will
reveal if the universe expands for ever or contracts etc. Various
cosmological models are summarized in Fig 23.1. But which
one do we live in?
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Our Universe
The Friedmann equation can be written as
where c = 3H2/8G is the critical density. With this, the spatial
curvature is related to the energy densities in the Universe.
So we can work out the various energy densities in the Universe
and compare it to this critical value to determine k. Note that a
spatially flat universe (k=0) implies /c = 1.
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Cosmological Supernovae
Different combinations of the
cosmological parameters predict
different relations between
apparent brightness and redshift.
The cosmological supernova
project has found that the Universe
appears to be flat (k=0) with 28%
of the energy in matter and 72% in
a dark energy component (possibly
a cosmological constant). Baryons
are ~5% of the energy.
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Why Flat?
The Big Bang model, as derived from the Friedmann equations,
predicts that the Universe was denser and hotter in the past.
However, it does not predict the values the energy densities in the
Universe.
Worryingly, randomly selecting energy densities result in a nonflat universe which, in general, expands or collapses too quickly.
A solution can be found in the idea that the Universe underwent a
period of rapid inflation in its very early history.
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Why Flat?
The idea is related to GUTs and the symmetry breaking that gave
us the forces we see today. At this time, the energy density in the
Universe is dominated by a vacuum energy density Vo scalar field.
Hence the Friedmann equations can be written as
Integrating this we find R = Ro exp(H t) and we see the Universe
expands exponentially. This occurred ~10-35secs and inflated the
Universe by ~1040. Hence if the Universe is curved, it is so large
that this curvature is impossible to measure. Inflation also
explains why the Universe appears so homogeneous.
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Into the Future
Given the current make-up of the Universe, it will expand for ever
with accelerated expansion. We can, therefore, chart the future
• 1014 years, all stars extinguished
• 1030 years, protons decay, leaving only black holes
• 10100 years, the black holes evaporate
After this, the Universe is thin sea of electrons, positrons, neutrinos
and photons that hardly ever interact. It will become a sad, cold,
lonely place, a little bit like Canberra.
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Anthropic Principle
Weak AP: The observed values of all physical and cosmological
quantities are not equally probable but they take on values restricted
by the requirement that there exist sites where carbon-based life can
evolve and by the requirements that the Universe be old enough for it
to have already done so.
Strong AP: The Universe must have those properties which allow
life to develop within it at some stage in its history.
Final AP: Intelligent information-processing must come into
existence in the Universe, and, once it comes into existence, it will
never die out.
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