A One-Term (1 Quarter) Undergraduate Course on General Relativity with Applications

by George W. Rainey[*]

Abstract:  A one-quarter undergraduate course on General Relativity with Applications is outlined and described.  The course employs tensor mathematics, but in a somewhat non-rigorous 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

1.  Brief Review of Special Relativity  (about 1½ weeks)

   1. Postulates

   2. Transformations

   3. Dynamics

   4. Geometry
      
       

2. The General Theory of Relativity  (about 4 weeks)

   1. The Principle of Covariance

   2. The Principle of Equivalence

   3.  The Einstein Field Equations

   4. The Schwarzschild Solution

   5. The Tests of General Relativity
      
       

3. Gravitational Waves and Gravitational Collapse  (about 2 weeks)

   1. Weak Fields and Gravitational Waves

   2. Strong Fields and Gravitational Collapse
      
       

4.   Cosmology  (about 2 ½ weeks)

   1. The Cosmological Principle

   2.  Redshifts

   3. Cosmological Models 

A.     The Special Theory of Relativity (SR) – pertains to measurements made in reference frames (coordinate systems) which are in un-accelerated 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 energy-momentum relation.

 4. Geometry
The quadratic form (or line element)  ds² = c²dt² – dx² – dy² – dz²  is invariant under a Lorentz transformation, and represents the (infinitesimal) interval between events. 

If ds² > 0, the interval is time-like; if ds² < 0, the interval is space-like; if ds² = 0, the interval is light-like.

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

ds² = g_ijdxⁱ dx^j (contravariant) = g^ij dx_i dx_j (covariant) = dx^j dx_j (mixed).

Now extend this to the four-dimensional spacetime geometry of SR; i.e.consider a four-vector  ds = (dx¹,dx²,dx³,dx⁴) = (idx,idy,idz,cdt), so that ds² =  – dx² – dy² – dz² + c²  dt².

Similarly an energy-momentum four-vector 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 (= 4⁰) 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  ds² = g_mndx^mdxⁿ  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, Riemann-Christoffel tensor, geodesic equation, raising and lowering of indices, contraction, covariant differentiation, etc.]

In the case of SR we may use the so-called Galilean coordinates, (x, y, z, ct), in which case the metric tensor has components g₁₁ = g₂₂ = g₃₃ = -1, g₄₄ = +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 4-dimensional Riemannian metric space: ds² = g_mn dx^m dxⁿ ; 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 non-rectangular coordinates, the g_mn will no longer all be constants, even though the quadratic form of ds² is preserved.  In spherical coordinates the Minkowski metric is

ds² = c² dt² – dr² – r² dq² – r² sin² q dj² ,

and thus g_tt = c², g_rr = -1, g_qq = -r², g_jj = -r² sin² 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 non-uniform 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  Ѳy = 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 stress-energy 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 4-th 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 Ѳy = 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 stress-energy tensor T_mn.  As a consequence of the conservation laws for energy and momentum, the divergence of the stress-energy tensor must vanish.  Unfortunately the Ricci tensor does not have zero covariant divergence; however, the Einstein tensor  G_mn = R_mn – ½ g_mnR  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/c⁴.] 

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

ds² =  – B dr² – C r² dq² – D r² sin² q dj²  + A c² dt² + 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

ds² =  – B dr² – r² dq²  – r² sin² q dj² + A c² dt² . 

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 eⁿ 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 non-vanishing.  [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 stress-energy tensor are all zero, which requires that the solution be static (Birkhoff’s Theorem).  Solving the remaining field equations (non-linear) results in eⁿ = 1 – 2m/r  – Lr² /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 one-body 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/c².  Also note that in this case the Schwarzschild line element is singular in the coefficient of dr² at a radius r = 2m = 2GM/c²; 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 non-vanishing 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 3mu², where u = 1/r and m = GM/c²).  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 – e²)c².  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 3mu², and the total asymptotic deflection at distance R from the gravitating body is Q = 4GM/Rc². 

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/Rc².  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 (four-dimensional) 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/c².  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 pseudo-singularity 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).

Reissner-Nordstrom Black Hole (1918) – spherically symmetric mass M, with net charge Q.  The metric [given] has g_tt = (1 – 2m/r + q²/r²) and g_rr = - (1 – 2m/r + q²/r²)^-1, where

m = GM/c² = 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; X-ray 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 Robertson-Walker metric:

ds² = c²dt² – R²(t)du²

where  du² = (1 – kr²)^-1dr² + r²(dq² + sin² q dj²)

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 4-dimensional 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 = c²dt² – R²(t)du² , or

c dt/R(t) = du = (1 – kr²)^-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(t₀)/R(t₁), where t₁ is the time of emission, and t₀ 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 magnitude-redshift relation (1929).  The Hubble Law  V = Hd, where H is the “Hubble constant”, is more properly referred to as redshift-magnitude 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 t₀, hence obtaining z as a power series expansion about t₀, where (t₀ – t) is the “look-back” time.  Define the Hubble parameter H = (1/R)dR/dt (present value denoted H₀), and the deceleration parameter  q = -R(dR/dt)^-2d²R/dt² (present value denoted q₀).  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/H₀ + 1.086 (1 – q₀)z + ….. (higher order terms in z).

Observational problems:  Aperture affect, bolometric corrections, K-terms, 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 H₀ (approximately 70 km/s/Mpc) or q₀ (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/c⁴.

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 Robertson-Walker metric to determine the non-vanishing Christoffel symbols; then determine the components of the Ricci tensor.  The Einstein equations reduce to the following system of two equations:

8pGr/c² = - L +  [3k/R²(t) + 3(dR/dt)²/c²R²(t)]

            8pGP/c⁴ = L + [k/R²(t) + (dR/dt)²/c²R²(t) + 2(d²R/dt²)/c²R(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 re-express 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 = dR²/dt²) 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/R², which is the closed, finite Einstein universe (1917).

Non-static models allow for a spectral shift, and redshifts imply an increasing R(t) near the present time t₀.  In the case L = 0, then dR²/dt² < 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, q₀ > ½, R(t) is a cycloid -- the oscillating universe
          k = -1, q₀ < ½, R(t) is monotonically increasing -- expansion forever,
           with dR/dt ® c
          k = 0, q₀ = 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 t₀ » 13 x 10⁹ years; it also provides a simple relation between look-back time and redshift, viz. t₀ – t = t₀ [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 {q₀² [q₀z + (q₀ – 1)((1 + 2q₀z)^1/2 – 1)]} + 25 + 5 log c/H₀.

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 = 3H²/c².  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 non-zero cosmological constant, but these, as well as spinning black holes, provide attractive and fruitful topics for which undergraduates may pursue research in the literature.

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