A OneTerm (1 Quarter) Undergraduate Course
on General Relativity with Applications
by George
W. Rainey[*]
Abstract: A onequarter undergraduate course on
General Relativity with Applications is outlined and
described. The course employs tensor mathematics, but in a
somewhat nonrigorous manner due to time constraints. The
first half of the course is devoted to theoretical
development, while the latter half involves applications.
Course Title:
Relativity, Gravity, Black Holes (offered Spring Quarter in
alternate years)
Overview of Course Content

Brief Review of Special Relativity (about 1½ weeks)

Postulates

Transformations

Dynamics

Geometry

The General Theory of Relativity (about 4 weeks)

The
Principle of Covariance

The Principle of Equivalence

The Einstein Field Equations

The Schwarzschild Solution

The Tests of General Relativity

Gravitational Waves and Gravitational Collapse (about 2
weeks)

Weak Fields and Gravitational Waves

Strong Fields and Gravitational Collapse

Cosmology (about 2 ½ weeks)

The Cosmological
Principle

Redshifts

Cosmological Models
A. The
Special Theory of Relativity (SR)
– pertains to measurements made in reference frames
(coordinate systems) which are in unaccelerated relative
motion and in the absence of gravitation.
1. Postulates
First Postulate – Principle of Relativity: No
absolute reference frames (i.e. no experiments by which
inertial motion with respect to an absolute reference frame
can be detected); equivalently the laws of physics are the
same for all inertial observers. Einstein’s departure was
to extend this principle to electromagnetic phenomena.
Second Postulate – Constancy of the Speed of Light:
The speed of light in free space is an absolute constant,
independent of any motion of source or observer.
2. Transformations
An event in spacetime is specified by spatial
coordinates x,y,z and temporal coordinate t. Applying the
Postulates to two inertial observers with a constant
relative velocity v leads to the Lorentz
Transformations. (Compare with Galilean
Transformations.) Abandonment of the notion of space and
time as independent concepts; replace with the notion of a
3+1 dimensional spacetime continuum.
Lorentz transformations in differential form can be used to
obtain velocity transformations and acceleration
transformations.
Length contraction
(in the direction of relative motion) and time dilation.
3. Dynamics
Apply conservation of momentum principle to obtain
the mass transformation (i.e. moving mass increases
relative to rest mass).
Force, kinetic energy, rest energy, and energymomentum
relation.
4.
Geometry
The quadratic form (or line element) ds^{2}
= c^{2}dt^{2}
– dx^{2}
– dy^{2}
– dz^{2}
is invariant under a Lorentz transformation, and
represents the (infinitesimal) interval between
events.
If ds^{2}
> 0, the interval is timelike; if ds^{2}
< 0, the interval is spacelike; if ds^{2}
= 0, the interval is lightlike.
Vectors: In a Euclidean vector space let unit vectors e_{j}
(j = 1,2,3) define Cartesian coordinates (not necessarily
orthogonal); then the contravariant components of a
vector are defined as the projections parallel to the
coordinate axes, while the covariant components are
the projections perpendicular to the axes. The
scalar products of the base vectors constitute the elements
of a symmetric matrix g_{ij}
whose inverse is denoted g^{ij}
. (For orthogonal Cartesian coordinates this is just the
unit matrix.) The line element, or metric, can be
expressed as
ds^{2}
= g_{ij}dx^{i}
dx^{j}
(contravariant) = g^{ij}
dx_{i}
dx_{j}
(covariant) = dx^{j}
dx_{j}
(mixed).
Now extend this to the fourdimensional spacetime geometry
of SR; i.e.consider a fourvector ds = (dx^{1},dx^{2},dx^{3},dx^{4})
= (idx,idy,idz,cdt), so that ds^{2}
= – dx^{2}
– dy^{2}
– dz^{2}
+ c^{2
}
dt^{2}.
Similarly an energymomentum fourvector can be defined with
components ip_{x},
ip_{y},
ip_{z},
E/c .
Tensors: In a 4 dimensional space with generalized
coordinates x_{a}
(a
= 1, 2, 3, 4) a tensor of rank r is defined as a
collection of 4^{r} quantities associated with a
given point in the continuum, and whose values transform in
a certain prescribed manner when a new set of coordinates (x’)^{a}
are introduced.
A tensor of rank zero, a scalar S, is defined as a
single (= 4^{0})
quantity unaltered by the transformation; i.e. S’ = S.
Definition of contravariant and covariant tensors of rank
one (vectors).
Definition of contravariant, covariant, and mixed tensors of
rank two.
Definition of tensors of higher order.
Now suppose the continuum (or manifold) has a
metric property ds^{2}
= g_{mn}dx^{m}dx^{n}
for the scalar measure of the line element. Here g_{mn}
is called the metric tensor (covariant of rank two),
with inverse g^{mn}
(contravariant of rank two), and g = det(g_{mn});
the mixed tensor g^{m}_{n}
is defined as the Kronecker delta (i.e. 4x4 unit matrix).
These fundamental tensors can be used to define the method
of raising, lowering, and changing indices.
The great and useful advantage of tensor analysis is that it
provides a very condensed means of expressing
physical laws, and moreover a tensor equation, (tensor) = 0
expressing a physical law must have exactly the same form in
all coordinate systems.
[A handout of some useful tensor formulae is distributed,
including summation convention for repeated indices, metric
tensor properties, Christoffel symbols, RiemannChristoffel
tensor, geodesic equation, raising and lowering of indices,
contraction, covariant differentiation, etc.]
In the case of SR we may use the socalled Galilean
coordinates, (x, y, z, ct), in which case the metric
tensor has components g_{11}
= g_{22}
= g_{33}
= 1, g_{44}
= +1, and all other g_{mn}
= 0.
Alternatively, we may use the Minkowski coordinates,
(ix,iy,iz,ct), for which the g_{mn}
= +1 if
m
=
n
, and g_{mn}
= 0 if
m
¹
n
.
B. The
General Theory of Relativity
– a generalization of the Special Theory in which we
remove the restriction of the principle of relativity to
inertial frames, and allow for the presence of
gravitational fields.
1. The Principle of Covariance
The laws of nature should be expressible in a (generally
covariant) form which is independent of any particular
choice of spacetime coordinates. The justification for this
is that the laws of physics are ultimately a codification of
experimental results, and that the physical
reality represented by these laws can in no way be
affected by the arbitrary choice of a coordinate system.
Actually finding the appropriate covariant
expressions may, of course, be an enormously difficult
problem. Since tensor equations are covariant, we might
seek to express these laws in tensor form.
As a first example consider the line element in a
4dimensional Riemannian metric space: ds^{2}
= g_{mn}
dx^{m}
dx^{n}
; here the metric tensor is assumed symmetric, with the
matrix elements differentiable to any order desired. As a
specific example, the Minkowski metric of SR has
metric tensor components all constants in rectangular
coordinates, and the spacetime geometry is said to be
flat. But for transformations to accelerated axes or to
nonrectangular coordinates, the g_{mn}
will no longer all be constants, even though the quadratic
form of ds^{2}
is preserved. In spherical coordinates the Minkowski metric
is
ds^{2}
= c^{2}
dt^{2}
– dr^{2}
– r^{2}
dq^{2}
– r^{2}
sin^{2}
q
dj^{2}
,
and thus g_{tt}
= c^{2},
g_{rr}
= 1, g_{qq}
= r^{2},
g_{jj}
= r^{2}
sin^{2}
q
.
As a second example of covariance we consider the equations
for the motion of free particles and light rays. In SR the
“generalized velocity” components are all constants, the
“generalized acceleration” components are all zero, and thus
the trajectory is a straight line. A straight line is a
geodesic in a flat geometry. We now desire to express
the geodesic condition in the curved spacetime of General
Relativity (GR). Since a geodesic represents a minimal path
between two events, we use the variational principle
ds
= 0 to derive the geodesic equation.
[A handout is distributed using the calculus of variations
to derive the geodesic equation, and introducing Christoffel
symbols of the first and second kind.]
2. The Principle of Equivalence
This principle gives a correspondence between results
obtained by an observer at rest in a gravitational field,
and results obtained by a second observer in an accelerated
reference frame in the absence of gravity.
Restricted form: An observer in free fall in a uniform
gravitational field is equivalent to an unaccelerated
observer in free space. Thus a uniform gravitational field
may be “transformed away” by a suitable choice of
coordinates.
General form: A nonuniform gravitational field may
be transformed away locally, but not universally;
thus we must use the more general metric, where the g_{mn}
are not all constants. The unrestricted form of the
Principle of Equivalence provides a connection between
gravitation and the metric tensor. Spacetime may be taken
as locally flat, i.e. first derivatives of g_{mn}
vanish, but second derivative of g_{mn}
are not, in general, zero (analogous to replacing a curved
surface by a tangent plane).
We must be somewhat more careful in justifying the Principle
of Equivalence than in accepting as axiomatic the Principle
of Covariance, since the Principle of Equivalence makes a
definite physical statement which ought to be
testable. Supporting evidence is the apparent equivalence
of inertial and gravitational mass (experiments of Galileo,
Eötvös, Dicke), and of course the predicted results
of GR.
The geodesic equation relates the generalized acceleration
to the first derivatives of the metric tensor (which occur
in the Christoffel symbols); this is somewhat analogous to
the situation in Newtonian gravitational theory in which the
acceleration is related to the first derivatives of the
gravitational potential
y.
However, in Newtonian theory there is a single gravitational
potential
y,
while in Einsteinian theory the metric tensor consists of
ten independent quantities g_{mn}
(since the metric tensor is a 4x4 symmetric matrix),
which we may now consider as “gravitational potentials”
(note plural) as well as geometric quantities.
[Handout on Classical Gravitation, Gauss’ Law, Poisson’s
equation
Ñ^{2}y
= 4pGr.]
3. The Field Equations
The metric tensor g_{mn}
serves a dual role: a metric aspect in determining
the geometry of spacetime, and a gravitational aspect
in determining motion. We need field equations which
will relate the metric and gravitational field to the
distribution of matter and energy. The task is to find a
covariant analogue of Poisson’s equation. Since they
must reduce to Newtonian results (Poisson’s equation) in a
first approximation, they must be second order partial
differential equations in the gravitational potentials g_{mn},
and in the absence of gravitation the results must reduce to
those of SR (i.e. the field equations must admit the
Minkowski metric as a particular solution). Finally there
is the mathematically desirable property that the field
equations, in order to have a unique solution, should be
linear in the second derivatives of g_{mn}.
Accordingly we seek a second rank tensor equation of the
form:
(tensor representing geometry of spacetime) = (tensor
representing energy content).
The tensor on the left involves the gravitational potentials
g_{mn}
up to and including their second derivatives; the tensor on
the right includes all forms of energy except
gravitational energy, and is referred to as the
stressenergy tensor (usually in units of energy density).
By taking second order covariant derivatives of a
contravariant vector, and requiring that the order of
covariant differentiation be interchangeable, we obtain
the result R^{t}_{mns}
= 0.
This 4th rank mixed tensor is the Riemann curvature
tensor. This equation is both a necessary and
sufficient condition that spacetime be flat, so we may take
these as the field equations of SR.
For field equations in empty space, but in the
vicinity of a gravitating body, we need a less stringent
condition than R^{t}_{mns
}
= 0. The only meaningful contraction of the Riemann
tensor is between
t
and
s
(i.e. setting
t
=
s
and summing). The contracted Riemann tensor is the Ricci
tensor, R_{mn},
which is symmetric and of second rank. Einstein suggested
using R_{mn}
= 0 as the field equation (actually 10 independent
equations) in empty space. This is the covariant analogue
of Laplace’s equation
Ñ^{2}y
= 0 .
We still need to generalize Poisson’s equation, that is,
find a covariant field equation interior to a
distribution of matter and energy, with a nonzero
stressenergy tensor T_{mn}.
As a consequence of the conservation laws for energy
and momentum, the divergence of the stressenergy
tensor must vanish. Unfortunately the Ricci tensor does not
have zero covariant divergence; however, the Einstein
tensor G_{mn
}
= R_{mn}
– ½ g_{mn}R
does have zero covariant divergence, where R is the
contracted Ricci tensor, known as the Riemann
scalar. The most general second rank tensor which has
zero divergence, and is constructed from the metric tensor
and its first and second derivatives, and is linear in the
second derivatives, is of the form G_{mn}
+
Lg_{mn},
where
L
is a constant (called the cosmological constant). Thus
Einstein suggested field equations of the form
R_{mn}
– ½ R g_{mn}
+
Lg_{mn}
= –
kT_{mn}
, where
k
is a constant.
Note that the field equations are not really “derived”, but
rather postulated. Their validity lies in the extent
to which they describe nature. The field equations
postulated by Einstein lead to the geometric theory of
gravity known as the General Theory of Relativity (allgemeine
Relativitätstheorie). Other (different) postulations of
field equations, but which employ the principles of
Covariance and Equivalence, yield alternative theories of
gravity.
[Finally we consider weak, static fields and test particles
moving at speeds u << c to show that the Einstein theory
reduces to Newtonian theory, and with the constant
k
= 8pG/c^{4}.]
4. The Schwarzschild Solution
The Einstein field equation is nonlinear, and therefore
difficult to solve even for a fairly simple distribution of
matter and energy. For a given problem it is usually more
feasible to consider the general form the metric must
take in accordance with the appropriate physical
restrictions pertaining, and then use the field equations to
solve for the relevant g_{mn}.
Thus for a spherically symmetric mass distribution we
consider a spherical coordinate system with origin at the
center of symmetry, and an isotropic metric of the
form
ds^{2}
= – B dr^{2}
– C r^{2}
dq^{2}
– D r^{2}
sin^{2}
q
dj^{2}
+ A c^{2}
dt^{2}
+ E dr dt
where A, B, C, D, E are functions of r and t to be
determined. Without loss of generality, and defining new
radial and time variables, the metric can be made to take
the form
ds^{2}
= – B dr^{2}
– r^{2}
dq^{2}
– r^{2}
sin^{2}
q
dj^{2}
+ A c^{2}
dt^{2}
.
The problem is to determine the functions A = A(r,t) and B =
B(r,t). Since these are intrinsically positive, it is
convenient to denote them e^{n}
and e^{l}
respectively. This is called the standard form of the
spherically symmetric metric, with 14 of the 16
components of the metric tensor already determined. There
are 40 Christoffel symbols, of which 17 are nonvanishing.
[These are simply listed.] Then the components of the Ricci
tensor can be computed and substituted into the field
equations.
Exterior
solution: The components of the stressenergy tensor are
all zero, which requires that the solution be static (Birkhoff’s
Theorem). Solving the remaining field equations
(nonlinear) results in e^{n}
= 1 – 2m/r –
Lr^{2}
/3 = e^{l},
where m is called the “gravitational radius”.
[For simplicity we solve for the case
L
= 0, and just quote the result for
L
¹
0.] The case
L
= 0 corresponds to the classical onebody problem of
celestial mechanics, and from the Newtonian gravitational
potential
y
= GM/r for a mass M we find that the gravitational radius m
= GM/c^{2}.
Also note that in this case the Schwarzschild line element
is singular in the coefficient of dr^{2}
at a radius r = 2m = 2GM/c^{2};
this is the Schwarzschild radius, and falls well
inside any body even up to nuclear densities, and thus
the singularity is not encountered in the exterior solution
except in the case of gravitational collapse.
[The Schwarzschild interior solution for a sphere of
incompressible perfect fluid at rest is covered in a
handout.]
5. The Three Tests of General Relativity
Einstein suggested three observational tests of his General
Theory: (a) precession of the line of apsides for bound
orbits; (b) Deflection of light rays in a gravitational
field; (c) Gravitational redshift of emitted spectra. Since
the first two of these involve spacetime trajectories (of
free particles in test (a), of photons in test (b)), we make
use of the geodesic equations. The tests will be applied in
the gravitational field of the Sun, which we take as
spherically symmetric since it is rotating rather slowly.
Thus the Schwarzschild exterior solution applies, and there
are 13 nonvanishing Christoffel symbols in the static
case. The equations of motion can be simplified by choosing
a coordinate system such that the body is moving in the
plane
q
=
p/2,
with dq/ds
= 0 (i.e. “orbits” in equatorial plane). The equations of
motion can be integrated to yield relativistic versions of
conservation of angular momentum, and conservation of
mechanical energy.
(a) Precession of line of apsides for bound orbits: As in
the classical Kepler problem, describe the orbit in space
rather than in time, obtaining an equation which differs
from the Newtonian result by a quadratic term (specifically
3mu^{2},
where u = 1/r and m = GM/c^{2}).
The Newtonian solution for a bound orbit is an ellipse, and
the relativistic solution is approximately a precessing
ellipse. For each revolution there is a perihelic
shift Dj
= 6pGM/a(1
– e^{2})c^{2}.
For planet Mercury the predicted perihelic advance is 43.0
seconds of arc per century compared with the observed value
of 43.1 + 0.5 seconds of arc per century.
(b) Deflection of light rays in a gravitational field:
Again the relativistic equation of motion differs from the
Newtonian equation (light not interacting with a
gravitational field) by 3mu^{2},
and the total asymptotic deflection at distance R from the
gravitating body is
Q
= 4GM/Rc^{2}.
For light just grazing the limb of the Sun this predicted
deflection is 1.75 seconds of arc, compared with observed
values (during eclipses) of 1.6 to 2.2 seconds of arc.
(c) Gravitational redshift of spectra: This is actually a
time dilation effect, derivable from the Equivalence
Principle alone, and so not really a test of the Einstein
field equations. The predicted redshift is
Dl/l
= GM/Rc^{2}.
For the Sun this is only about 2 x 10^{6},
but is about 100 times greater for a white dwarf star.
Greater precision is obtainable in terrestrial experiments,
with good agreement between theory and observation.
C. Gravitational
Waves and Gravitational Collapse
– The Einstein field equations are difficult to solve
because of their nonlinearity in the derivatives of
the g_{mn};
this is because the gravitational field of a body can do
work, therefore constitutes energy (effective mass), and
hence serves as a further source of gravitational field,
i.e. feedback. The superposition principle does not apply.
1. Weak Fields and Gravitational Waves
In the general case of weak fields the feedback can
be neglected, so that the field equations will become
linearized. We know that in the limit of a vanishing
gravitational field the geometry becomes Lorentzian, and the
metric tensor in Minkowski coordinates is the 4x4 identity
matrix, which we shall denote
h_{mn}.
Then in a weak field we may represent the metric tensor
components by g_{mn}
=
h_{mn}
+ h_{mn
}
, where h_{mn}
may be regarded as a small perturbation; thus we
shall retain only terms of first order in the h_{mn}
and their derivatives. The field equation, neglecting
cosmological term, reduces to a (fourdimensional)
inhomogeneous wave equation for a wave propagating at
speed c.
Do such gravitational waves actually exist? Joseph Weber
and others have attempted to detect gravitational waves, but
their properties (amplitudes, frequencies, pulse shapes,
polarization, etc.) are still a matter of considerable
conjecture. The source of such gravitational radiation
would have to be a transverse disturbance in the
geometry of spacetime.
2. Strong Fields and Gravitational Collapse
Consider again the Schwarzschild exterior solution in the
case of a spherically symmetric collapsing mass. We
have (with
L
= 0) g_{tt}
= (1 – 2m/r) and g_{rr}
=  (1 – 2m/r)^{1},
where m is the “gravitational radius” GM/c^{2}.
Note that the quantity 2m/r can be regarded as a measure of
the strength of the gravitational field; as r decreases the
spacetime becomes more highly “curved”. In fact at the
Schwarzschild radius r_{S}
= 2m the metric becomes “singular”. Also at the
Schwarzschild radius the escape velocity equals the velocity
of light, and as the object collapses through its
Schwarzschild radius, the escape velocity exceeds c; neither
matter nor energy may escape, and the collapsar has
become a black hole.
Can such objects actually exist? Apparently the answer is
yes. Although the pressure of degenerate electrons
(maximally packed according to the Exclusion Principle) may
support a white dwarf star (up to 1.4 solar masses), and the
pressure of degenerate neutrons may support a neutron star
(up to approximately 2 ½ to 3 solar masses), there are no
known forces to withstand gravitational collapse of a mass
greater than about 3 solar masses.
Schwarzschild Black Hole
– spherically symmetric mass undergoes complete collapse,
with a physical singularity at r = 0, and a
pseudosingularity at r = r_{S},
the event horizon.
[Discussion of photon sphere, exit cone, comparison of
collapse for an observer in the reference frame of the
collapsar vs. observer in a distant outside reference frame,
etc.]
Black holes are physically simple objects, theoretically
having only three independent properties: Mass (M), Charge
(Q), and Angular Momentum (L).
ReissnerNordstrom Black Hole
(1918) – spherically symmetric mass M, with net charge Q.
The metric [given] has g_{tt}
= (1 – 2m/r + q^{2}/r^{2})
and g_{rr}
=  (1 – 2m/r + q^{2}/r^{2})^{1},
where
m = GM/c^{2}
= mass in “distance” units,
q = charge in “distance” units.
[Discussion of two event horizons, extreme case, naked
singularity.]
Kerr Black Hole (1963) – axially symmetric (rotating) mass
M, with angular momentum L.
[The metric is given, and the following discussed: static
limits, event horizons, singularity in the equatorial plane
(“ring” singularity), extreme case, naked singularity,
ergosphere, Penrose mechanism.]
Hawking radiation and evaporation of quantum black holes.
Observational evidence for black holes: Close binary
systems with massive (greater than 3 solar masses) unseen
components; Xray emission from accretion disk.
D. Cosmology
– The problem of determining the overall structure and
behavior of the universe as a whole (the totality of all
matter and energy, all space and time) is difficult because
the observational evidence is hard to come by, and any
theoretical hypotheses cannot be tested in generality, but
only on a single system. We begin by assuming the
universality of physical laws as we know them
(otherwise there is no basis on which to proceed). Of the
four known forces or interactions in nature
(gravitational, electromagnetic, strong, weak) the strong
and weak forces are of extremely short range, and large
scale units of matter (clusters of galaxies) appear to be
electrically neutral. Thus on a cosmological scale the
gravitational force is the only one of importance. We shall
adopt GR as our theory of gravity.
1. The Cosmological Principle (CP)
This is the rather strong statement that the universe is the
same everywhere; that is, on the large scale the
universe is homogeneous and isotropic. While consistent
with observations on the large scale, it obviously does not
hold on a small, or even moderate, scale. Its implication
is that if we were to observe the universe from any other
location, we would see the same general large scale
distribution.
The most general metric consistent with the CP is the
RobertsonWalker metric:
ds^{2}
= c^{2}dt^{2}
– R^{2}(t)du^{2}
where du^{2}
= (1 – kr^{2})^{1}dr^{2}
+ r^{2}(dq^{2}
+ sin^{2}
q
dj^{2})
and R(t) is an unspecified function of “cosmic” time t,
called the scale factor. Here k is a trichotomic
constant which, suitably normalized, takes on the values +1,
0, 1; it is referred to as the curvature index, and
determines the nature of the spatial geometry as a subspace
of the 4dimensional spacetime geometry as, respectively,
elliptical (closed), parabolic (Euclidean), or hyperbolic
(open).
For convenience we place the coordinate origin at the
observer; typical objects (e.g. galaxies) have constant
spatial coordinates (r,q,j).
Light travels on null geodesics (ds = 0), and, we assume,
radial tracks (dq
= 0 = dj).
Thus 0 = c^{2}dt^{2}
– R^{2}(t)du^{2}
, or
c dt/R(t) = du = (1 – kr^{2})^{1/2}
dr ,
where the quantity on the left depends only on time, while
the quantity on the right depends only on space.
Note that u = sin^{1}
r for k = +1, u = r for k = 0, and u = sinh^{1}
r for k = 1 (which are all about the same for small r).
The quantity R(t) u is the instantaneous proper distance
to the object at coordinate r, but is not observationally
relevant due to the finite speed of light. In fact if the
scale factor is not constant, there will be a spectral
shift 1 + z = 1 +
Dl/l
= R(t_{0})/R(t_{1}),
where t_{1}
is the time of emission, and t_{0}
is the time of reception. Hence if R(t) is an increasing
function, then a redshift is predicted.
2.
Redshifts
Astronomical observational data can be placed in one of
three categories:
Positional – coordinate
q,j
Photometric – apparent magnitudes m
Spectroscopic – dispersion of radiation
(spectral features and shifts)
Edwin Hubble: Identification of “white nebulae” as external
galaxies (1924), and discovery of the magnituderedshift
relation (1929). The Hubble Law V = Hd,
where H is the “Hubble constant”, is more properly referred
to as redshiftmagnitude relation, since radial velocity V
and distance d are not directly observed, but inferred from,
respectively, redshift z and magnitude m.
What can be inferred from the m – z relation? One
approach is to consider a Taylor expansion of the scale
factor R(t) about the present time t_{0},
hence obtaining z as a power series expansion about t_{0},
where (t_{0}
– t)
is the “lookback” time. Define the Hubble parameter
H = (1/R)dR/dt (present value denoted H_{0}),
and the deceleration parameter q = R(dR/dt)^{2}d^{2}R/dt^{2}
(present value denoted q_{0}).
Also we can define a “luminosity distance” (bolometric
distance) in terms of the inverse square law of light and
which relates apparent magnitude m and absolute magnitude
M. Finally we obtain (after much algebra) the distance
modulus m – M = 25 + 5 log cz/H_{0}
+ 1.086 (1 – q_{0})z
+ …..
(higher order terms in z).
Observational problems: Aperture affect, bolometric
corrections, Kterms, galactic absorption, and, most
importantly, absolute magnitudes to be assigned to
luminosity indicators (“standard candles”). Unfortunately
scatter in the data (z vs. m) precludes a definitive
determine of H_{0}
(approximately 70 km/s/Mpc) or q_{0}
(seems to lie somewhere in the range 0 to 2). Even if they
could be precisely determined, we would still not know
R(t) exactly. We need to put more physics (i.e.
dynamics) into the problem, namely the field equations.
3.
Cosmological Models
We make use of the Einstein field equations R_{mn}
– ½ R g_{mn}
+
Lg_{mn}
= 8pGT_{mn}/c^{4}.
Assume an ideal fluid of galaxies, with average density
r
and an average internal pressure P (these may be functions
of time, but not position). To reduce these field equations
we proceed as follows: Use the RobertsonWalker metric to
determine the nonvanishing Christoffel symbols; then
determine the components of the Ricci tensor. The Einstein
equations reduce to the following system of two equations:
8pGr/c^{2}
= 
L
+ [3k/R^{2}(t)
+ 3(dR/dt)^{2}/c^{2}R^{2}(t)]
8pGP/c^{4}
=
L
+ [k/R^{2}(t)
+ (dR/dt)^{2}/c^{2}R^{2}(t)
+ 2(d^{2}R/dt^{2})/c^{2}R(t)].
The task is to solve for R(t) in terms of the constant
parameters k and
L,
provided an equation of state P = P(r)
is specified. It is also convenient to reexpress these
field equations by linear combinations yielding an equation
in which k and the first derivative of R do not appear, and
an equation in which the cosmological constant does not
appear.
Static models
(dR/dt = 0 = dR^{2}/dt^{2})
are briefly considered for their historical importance, and
there are only two cases: If
L
= 0, then P = 0 =
r,
and k = 0, which is the empty, flat universe of Special
Relativity  the Minkowski universe; if
L
¹
0, then k = +1 and
L
~
1/R^{2},
which is the closed, finite Einstein universe (1917).
Nonstatic models
allow for a spectral shift, and redshifts imply an
increasing R(t) near the present time t_{0}.
In the case
L
= 0, then dR^{2}/dt^{2}
< 0, i.e. deceleration of the expansion. Neglecting
the pressure P we obtain the Friedmann models (1922),
for which there are three cases:
k = +1, q_{0}
> ½, R(t) is a cycloid  the oscillating universe
k = 1, q_{0}
< ½, R(t) is monotonically increasing  expansion forever,
with dR/dt
®
c
k = 0, q_{0}
= ½, R(t) = (constant) t^{2/3}
 continuous expansion, with dR/dt
®
0.
In all three cases R(t)
®
0 as t
®
0, thus they represent exploding universes (Big Bang
models), differing in the degree to which the expansion is
being decelerated. The limiting case k = 0 separating the
open models from closed models yields the time since the Big
Bang of about t_{0}
»
13 x 10^{9}
years; it also provides a simple relation between lookback
time and redshift, viz. t_{0}
– t = t_{0}
[1 – (1 + z)^{3/2}].
In all three cases there is the same closed form
relation between distance modulus and redshift:
m – M = 5 log {q_{0}^{2}
[q_{0}z
+ (q_{0}
– 1)((1 + 2q_{0}z)^{1/2}
– 1)]} + 25 + 5 log c/H_{0}.
The case
L
¹
0 is the most general case. One special case of historical
interest is the de Sitter universe (1917), which
model assumes that H = (1/R)dR/dt = constant throughout the
evolution; this results in an empty (r
= 0 = P), flat (k = 0) universe with
L
= 3H^{2}/c^{2}.
In the general case the cosmological constant represents an
antigravity term, referred to as “dark energy”, and
producing an acceleration of the expansion.
Summary
In summary, the course
outlined above is sufficiently elementary and
straightforward as to be readily grasped by advanced (upper
division) undergraduate physics students. Tensors are used
as tools, but without rigorous mathematical
development. Rather the course emphasis is on astrophysics
applications, especially black holes and cosmology. Time
constraints (this is a 10 week course) preclude extensive
development of inflationary cosmology, or cosmological
models with nonzero cosmological constant, but these, as
well as spinning black holes, provide attractive and
fruitful topics for which undergraduates may pursue research
in the literature.
