Department of Physics, University of Maryland, Prof. T. Jacobson

Physics 675

Introduction to

Relativity, Gravitation and Cosmology

Notes from Fall 2004

These notes are only intended to indicate what was covered in class, and occasionally to include some extra info. We'll see if and how they evolve.

Tu 8/31:
(flat) spacetime structure; relativistic kinematics; energy-momentum 4-vector

Th 9/2:
twin effect; massless particles; GZK cutoff on cosmic ray proton spectrum;
possible role of Lorentz symmetry violating shift of the cutoff;
relativistic Doppler effect.

Hartle invoked the form of a Lorentz transformation to write the frequency
in the source frame in terms of the frequency and wavevector in the observer frame,
Eq. (5.72). Instead I wrote that the frequency in the source frame is w = - k.u,
where k is the 4-wavevector of the photon and u is the 4-velocity of the source,
and the dot on the line means the Minkowski inner product. Then I evaluated
the right hand side using the components of k and u in the observer frame,
k = (w', w' cos(alpha'), w' sin(alpha'), 0) and u = (gamma, gamma v, 0, 0),
so that w = w' gamma (1 - v cos(alpha')). I suppose this just amounts to
making the Lorentz transformation of the source 4-velocity to the observer frame,
rather than transforming the observed 4-wavevector to the source frame
(as did Hartle). It seemed simpler to me, since the source 4-velocity is so simple
in the source frame, (1,0,0,0), that I don't really think of writing it as
(gamma,gamma v,0,0) in the observer frame as invoking the Lorentz
transformation formula.

For the homework problem Hartle suggests using a Lorentz transformation
on the 4-wavevector, but another way to see the result is to equate the frequency
w' found above to w' = -k.u', evaluating the right hand side expressing k and u'
in terms of their components in the source frame.

Tu 9/7:
The basic idea of general relativity (GR): gravity is the curvature of spacetime.

Newtonian gravity:

F = m_inertial a (in an inertial frame)
F_grav = m_passive g, g = -grad phi
nabla^2 phi = 4 pi G rho_active

which defines inertial mass, passive gravitational mass, active gravitational mass.

Equality of the latter two implied by total momentum conservation, and comfirmed
up to 4 x 10^-12 by Bartlett and van Buren (1986) via analysis of the motion of the
moon in view of the inhomogeneity of the moon. (See discussion in C.M. Will,
Experimental Gravitation from Newton's Principia to Einstein's general relativity, in
300 Years of Gravitation, editors S. Hawking and W. Israel.)

Equality of inertial and passive gravitational masses confirmed to 1 part in 10^12
by Eotvos experiments, leads to idea that gravity is a fictitious force, like those
produced by adopting a non-inertial frame of reference (eg. centrifugal and Coriolis
forces). The distinguishing feature of such a "force" is that it is proportional to the mass
of the object: Suppose a is the acceleration of a mass relative to an inertial frame S, a_fr
is the acceleration of a non-inertial frame S_ni relative to S, and a_ni is the acceleration
of the mass relative to S_ni. Then a = a_ni + a_fr, so Newton's 2nd law
implies F_ext - m a_fr = m a_ni. That is, all masses feel an extra (fictitious) force
of the form -m a_fr, which is proportional to the inertial mass. F_grav can be interpreted
this way, if g is uniform. This leads to the equivalence principle, as discussed in
the textbook.

So is gravity just a fictitious force? Yes for uniform gravitational fields, no
for non-uniform ones. Illustrated this with relative accelerations of nearby particles
freely falling in the gravitational field of the earth. But over a small enough region
of space and time, the non-uniformity is negligible.

Curvature Analogy:

free-fall <=> straight line in space
relative acceleration of free-fall trajectories <=> curvature of space

Illustrated with great circles on a sphere.

Description of curvature: line element with non-constant coefficients. But how to
distinguish curvature from curved coordinates? Will answer later.

Spacetime metric ds^2 = g_ab dx^a dx^b, Einstein summation convention.

Th 9/9:

More precisely, gravitational tidal acceleration is due to the curvature of spacetime.

Next topics:
Which metrics g_ab are allowed?
How is curvature described?
How do free (i.e. "freely falling") particles move in a curved spacetime?
How does mass affect the metric?

For the material in the next two or three lectures, here is the suggested reading:

Hartle:
Chapter 6
Sections 7.1,2,3,4
Section 8.1, first three pages

Primer (Spacetime Primer notes; see the course syllabus):
Sections 2.1,2,3 (and 2.4,5 if you like)
Chapter 3

At each point p, the line element ds^2 must define a light cone. This is equivalent
to saying that in the nhd of each pt p there exist coordinates for which, at p,
g_ab = eta_ab = diag(-1,1,1,1). This is equivalent to saying that g_ab as a matrix
has three positive and one negative eigenvalue, which is equivalent to saying that
the determinant of g_ab is negative and it has at least two positive eigenvalues.
(The latter two conditions are in fact independent of the coordinate system.)
Such a metric is called Lorentzian, and is said to have Lorentzian signature.
The signature of the metric is defined as the number of positive minus the number of
negative eigenvalues.

The transformation law for g_ab under a coordinate change follows from the fact that
ds^2 is invariant, so g_ab dx^a dx^b = g'_ab dx'^a dx'^b (see problem 7-7).
The Jacobian of the coordinate transformation has 16 components, whereas g_ab has only
10 independent components. Thus a six dimensional set of linear transformations leaves
g_ab invariant at a point. This is the local Lorentz group.

One can further choose the coordinates so that g_ab,c=0 at any given point p.
(The ,c denotes partial derivative wrt x^c.)
This is because the transformation rule for the first partial derivatives of the
metric is linear in the second partials of the coordinates, which are the same in
number (see problem 7-9). Such coordinates are called local inertial coordinates at p.

One can set some, but not all, of the second derivates at a given p to zero. The
reason is that the equations g_ab,mn=0 are 10x10=100 in number, while the third derivatives
of the coordinates are 4x20=80 in number. Thus, in general, there are at least
100-80=20 non-zero second partials of the metric. These characterize the curvature
of the metric. If at each p one can find coordinates that make all the second partials
of the metric vanish, then the curvature vanishes at each point, so the metric is flat.

Two-dimensional sphere example: ds^2 = da^2 + (sin a)^2 db^2. These coordinates
have Euclidean form at each point on the equator sin a =1, and the first partials of the
metric components vanish everywhere on the equator. Since the sphere is not flat, some of
the second partials of the metric must be necessarily non-zero at any point, for example
points on the equator. We can check: g_bb,aa = 2(cos^2 a - sin^2 a) = -2.

A world line x^a(s) has 4-veclocity dx^a/ds. Under coordinate change, this changes linearly
with the Jacobian del x'^a/del x_b. By contrast, the metric transformation rule involves the
inverse of this Jacobian. The index placement convention links upper
indices to the Jacobian transformation property and lower indices to the inverse Jacobian.
Upper indices are called contravariant, and lower ones are called covariant (seems kinda
backwards to me). When an upper-lower index pair is contracted, i.e. summed over, the
result is invariant, i.e. independent of coordinates, since the Jacobian and inverse Jacobian
meet and make the indentity. For example, the line element ds^2 = g_ab dx^a dx^b is invariant
for this reason. (Actually, the transformation behavior of g_ab was inferred from precisely the
requirement that the line element be invariant!).

Tu 9/14:

- Invariant contraction of co- and contra-variant tensor indices.
- B^m a scalar for all vectors B^m implies A_mis a co-vector, and vice versa.
- d²x^m/dl² is not a vector
- free-fall = inertial = unaccelerated = geodesic motion is characterized by
d²x^m /dl² = f(l) dx^m/dl in l.i.c. (local inertial coordinates) at a point.
- example: equator of sphere
- Impractical because requires a different l.i.c. at each point. Need a characterization
that works in any coordiante system. To find this, turn to an invariant characterization.
Euclidean analogy: arc length is stationary for variations of a geodesic. Lorentzian timelike
geodesics: proper time is stationary. That is, variations of \integral L dl are zero, where
L = (-g_mn dx^m/dl dxⁿ/dl )^1/2 . The stationarity condition is the Euler-Lagrange equations
(which we re-derived).

Th 9/16:

- For the Lagrangian for the proper time, the Euler-Lagrange equations become
(d/dl)(L^-1g_an dxⁿ/dl) - ½ L^-1g_mn,_a dx^m/dl dxⁿ/dl = 0.
If the curve parameter l is chosen to be the proper time, then L=1, so this simplifies to
(d/dl)(g_an dxⁿ /dl) - ½ g_mn,_a dx^m/dl dxⁿ/dl = 0.
This is called the geodesic equation for affinely parameterized geodesics.
In a l.i.c. at a point this geodesic eqn. reduces to d²x^m /dl² = 0, what we had before.
- Example: used this to show that a line of longitude on the sphere is a geodesic. The equation
reduced to d²theta /dl² = 0, which expresses the fact that the affine parameter l must be linearly
related to the angle theta, or equivalently to the arc length.

- Since the geodesic eqn was derived from an invariant condition, if it holds in one coordinate
system it must hold in any. In hw3 it is verified that the lhs transforms as a co-vector, ensuring
that this is true.

- This geodesic equation also follows directly from the variational condition with Lagrangian
L = ½ g_mn dx^m/dl dxⁿ/dl, without the square root. This action is not reparametrization
invariant, which explains why the resulting geodesic eqn is for a special class of paramters,
i.e. affine parameters. Any two affine parameters are linearly related.

- Newtonian limit of the geodesic equation (v << c, g_{mn =} eta_{mn +}h_{mn ,} h_mn<< 1): Using the
parametrization-invariant form, and the coordinate t as the parameter, the Lagrangian is approximately
given by L = 1 - ½ v² + ½ h_tt .The constant 1 affects nothing. This gives the same equations of
motion as the Newtonian Lagrangian L^Newton = ½ mv² - m Phi if the metric component g_tt is related
to the Newtonian potential Phi by h_tt= -2 Phi = 2 GM/r , so that g_tt = -(1 - 2GM/r).

- Relation between Newtonian and EInsteinian gravity theories, form of Einstein equation.

Tu 9/21:

some of the material in today's class is covered in Chapter 12.

- Coordinate transformation between Eddington-Finkelstein (EF) and Schwarzschild line elements:
v = t + r + 2M ln(r-2M)

- EF spacetime diagram: interpreting the coordinates, light cones, and light rays (see Fig. 12.2).
Horizon at r=2M, curvature singularity at r = 0.

- v-translation symmetry timelike for r > 2M, lightlike for r = 2M (the horizon of the black hole), and
spacelike for r < 2M (inside the horizon). This means that the conserved quantity associated with
this symmetry is momentum-like inside the horizon, so can take negative values, This signals the
existence of an instability whereby pairs of particles with opposite values of this conserved
quantity are created: Hawking radiation.

- outer trapped surfaces, and Penrose singularity theorem

- t-coordinate pile-up at the horizon: a coordinate singularity

Th 9/23:

- coordinate basis vectors: given a coordinate system, the vectors (1,0,...,0) etc, whose components are all zero except for one which is 1, are called the "coordinate basis vectors". It is useful to have a name for these that refers to the coordinates themselves. For example, in (t,r,theta,phi) coordinates, the vector with components (1,0,0,0) is sometimes called d/dt, where these "d"'s should really be written as "del"'s, i.e. partial wrt t, but I don't have the html code for del. The reason for this name is that the directional derivative in the direction of this vector is nothing but the partial derivative wrt t. (This duality between vectors and directional derivative operators is sometimes used to define a vector as a derivative operator on functions, i.e. a linear operator that satisfies the Leibniz rule.) Note that the meaning of a coordinate basis vector is not fixed until all the coordinates have been specified. For example, I showed in class how even though the r coordinate is the same in EF and Schwarzschild coordinates, the vector d/dr is different.

- conserved quantities and symmetries: the momentum conjugate to coordinate not appearing in the Lagrangian---a symmetry coordinate---is conserved. For free particles, this means that if the metric components do not depend on a coordinate w, the corresponding momentum g_wbu^b is conserved.

- Killing vectors: the coordinate vector field corresponding to a symmetry coordinate is called a Killing vector or (by me) a symmetry vector. Later we may learn how to recognize symmetry vectors even when the coordinate system is not adapted to the symmetry.

- The momentum conjugate to a symmetry coordinate can also be expressed as the inner product between the 4-velocity of the worldline and the corresponding Killing vector.

- Timelike orbits of Schwarzschild spacetime: conserved quantities, effective potential, Newtonian limit, circular orbits, innermost stable circular orbit (ISCO), binding energy of the ISCO.

Tu 9/28:

- relation between Killing energy in Schwarzschild and energy measured by a static observer: E_Killing = - p.(d/dt) = -p.u_stat ||d/dt|| = E_stat ||d/dt||, where ||d/dt|| = Sqrt[-g_tt] = Sqrt[1 - 2M/r] is the norm of the Killing vector. Expanding ||d/dt|| = 1 - M/r + ... and expanding in slow velocities E_stat = m + 1/2 mv_stat^2 + ... we get E_Killing = m + 1/2 mv_stat^2 - Mm/r + ..., i.e. just the Newtonian conserved energy plus the rest energy.

- gravitational extraction of rest energy by black holes

- quasars

- gravitational redshift

- effective potential for radial motion of photons, photon orbit at r = 3M.

Th 9/30:

- wavevector k_m and conservation of Killing frequency along a null geodesic: I claimed that in the WKB approximation the wavevector is related to the tangent vector of an affinely parameterized null geodesic via k_a = g_an dxⁿ/dl , so that the geodesic equation can be written (d/dl)(k_a) - ½ g_mn,_a dx^m/dl dxⁿ/dl = 0. When the metric is independent of a given coordinate, the corresponding component of the wavevector is conserved along the null geodesic light rays. Here I will explain something of what is behind this. For a more complete explanation see, e.g. Wald's textbook.

First note that without any approximation, if the source has a given Killing-frequency, i.e. frequency wrt the ignorable t-coordinate, then the radiation field will have the same t-frequency, since the metric is invariant under t-translation. If the source is for example an iron atom making a transition, it really has a given frequency with respect to its proper time. If while the atom is emitting a photon the relation between proper time and Killing time t is a fixed proportionality factor, then the source will have a fixed Killing frequency. In practice one can usually use the WKB approximation to relate the wave propagation to null geodesics as follows.

In the WKB approximation the wavefronts---i.e. constant phase surfaces---of electromagnetic radiation are null hypersurfaces---i.e. 3-surfaces everywhere tangent to the light cone. That is, at each point there is one null tangent vector, and the rest of the vectors tangent to the surface are spacelike. The null tangent vector is orthogonal to itself and to all other vectors in the tangent space to the surface, hence it is orthogonal to the surface, even though it lies in the surface. It is called the (null) normal to the surface. The null curves tangent to these null directions are called generators of the null hypersurface. One can easily show that generators of a null hypersurface are always null geodesics. (Moreover, all null geodesics are generators of some null hypersurface, so null geodesics can be defined directly as such, without reference to the geodesic equation. Note however that this defines just the null geodesic curve itself, without specifying a parametrization.)

For a wave of the form A exp(iS), where A is a slowly varying amplitude and S is a relatively rapidly varying phase, the gradient of S defines the wavevector, k_m := S,_m . If the index of the wavevector is raised with the inverse metric one obtains a vector, u^m := g^mn k_n, that is orthogonal to all vectors vn in the surface of constant S: g_mn u^m vⁿ = k_n vⁿ = S,_m vⁿ = 0. This vector must therefore coincide with the normal vector, hence it is tangent to the null geodesic generators. Using the fact that S,_m is null everywhere one can easily show that in fact the vector u^m is not only tangent to the null generators but it corresponds to the 4-velocity dx^m/dl for an affine parametrization.

- redshift from an accretion disk (cf. Hartle, section 11.2)

- binary pulsar systems (cf. Hartle, section 11.3)

Tu 10/5:

- astrophysical black holes:
- stellar mass: star collapse leads to white dwarfs (electron degeneracy pressure, ~ earth sized, under Chandrasekhar limit) , neutron stars (neutron degeneracy pressure, ~ 10 km), or black holes (R_S = 3km (M/M_sun)), according mostly to the initial mass.
- collapse of star cluster: intermediate mass (~100 -1000 M_sun?, form by star scattering) or supermassive (10^6-10^9 M_sun, form by collapse of star clusters and gas at galactic centers, rotational energy believed to power AGN's like quasars)
- high density not needed: R < 2M ~ rho R^3 satisfied when R > rho^-1/2 = 10^9 km for density of 1gm/cc (water).
- primordial- I forgot to mention these: may have formed in early universe from violent inhomogeneities.

- collapse of shell to a black hole: EF diagram, formation of horizon, spacelike singularity, growth of horizon when more mass is added.

- Cosmic censorship hypothesis: a naked singularity will not form from generical nonsingular initial data.

- BH uniqueness theorem: stationary, vacuum black holes are unique: only the Kerr family, labeled by mass M and angular momentum J.
Called the "no-hair theorem". Black holes have no hair, i.e. no distinguishing features left over from whatever gave rise to them. This greatly simplifies the problem of understanding astrophysical black holes!

- Generally collapsing systems spin rapidly due to angular momentum conservation, so we encounter spinning black holes.

- The solution is known exactly. First found by Kerr (1963) looking for stationary axisymmetric solutions to the Einstein eqn of the form g_ab = eta_ab + l_a l_b, where l_a is null wrt eta_ab. Hartle's book give the solution in Boyer-Lindquist coordinates. I wrote this down and started discussing it.

Th 10/7:

+ null hypersurfaces: A null hypersurface is a 3d submanifold of spacetime that is everywhere tangent to the light cone. This means that if you look at the 3d space of vectors at a point on the surface, it includes one null vector n and no timelike vectors. In this case the null vector must be orthogonal (wrt the spacetime metric) to all the other vectors, otherwise a timelike vector could be constructed by adding a non-orthogonal spacelike vector to the null vector. (Proof: let s be a spacelike vector. Then (n + s).(n + s) = 2s.n + s.s. The second term is positive but the first term can be made arbitrarily negative by choosing the scaling and direction of n, as long as n.s is non-zero. Hence if there are no timelike tangent vectors n.s must vanish for all vectors s in the surface.) Since the null vector is also orthogonal to itself, it is orthogonal to all vectors in the surface, i.e. it is the normal vector (determined up to scaling). Conversely, if the normal vector to a 3d hypersurface is null, then the hypersurface surface is null.

Now consider a surface defined by a level set of a function f. The gradient f,a is a covariant vector, but from it one can define a contravariant vector n^a by contraction with the inverse metric gab: n^a = g^ab f,b. This vector is orthogonal to all vectors tangent to the surface, hence the surface is null if this vector is null. Using the definition of the inverse metric it is easily seen that n.n = g^ab f,a f,b.

+ Kerr horizon: From the above it follows that the normal vector to a surface of constant r-coordinate in the Kerr metric has squared norm with the sign of g^rr. This inverse metric component is positive at large r, so the surface is timelike, but it is zero when Delta = 0, i.e. when r = M +/- Sqrt[M^2 - a^2]. The plus sign defines the Kerr event horizon: light rays can cross this surface in one direction only.

- If a > M there is no horizon, in which case the singularity at rho = 0 is naked, i.e. visible from far away. Collapse easily has J > M^2, hence in order to form a Kerr black hole rather than a naked singularity the system must shed angular momentum. It can do so by ejecting matter and/or gravitational radiation. It is not known whether the Einstein equation ensures that enough angular momentum is always shed to ensure cosmic censorship, i.e. the clothing of all singularities.
- Intrinsic geometry of the horizon, shape and area.

+ Ergoregion: The time-translation Killing vector becomes null, i.e. g_tt is zero, outside the horizon everywhere off the axis of rotation, and at the horizon on the axis. Inside this surface is the ergoregion where the time-translation becomes spacelike. A nice article describes the intrnsic geometrie of a t = 0 slice of the even horizon and ergosurface. I showed an embedding diagram from that article showing the intrinsic geometries of the horizon and ergosurface as 2d surfaces embedded in 3d Euclidean space, for a Kerr spin parameter a = 0.9. A region around the poles of the horizon is not embeddable, which can only happen since it has negative curvature (however negative curvature does not imply non-embeddabilty). The ergosurface has conical singularities at the poles. The polar distance is infinite on the ergosurface as a approaches 1, but the polar distance on the horizon remains finite. I am puzzled about how this can happen in view of the fact that the two surfaces coincide at the poles! Figure this out for extra credit!

Tu 10/12:

+ astrophysical accretion disks: Guest lecture by Prof. Chris Reynolds of the UM Astronomy Department. Under the Research link at his homepage you can read and see what he talked about, and more.

Th 10/14:

+ ergoregion: dragging of inertial frames, time-translation Killing vector becomes spacelike.

+ Penrose process: extraction of rotational energy from black hole, first law of black hole mechanics, maximal efficency when area is unchanged, area cannot decrease. This leads to black hole thermodynamics: Area is analogous to entropy, surface gravity to temperature.

+ Black hole thermodynamic analogy:
0th law: surface gravity constant on horizon,
1st law: dM = (kappa/8\pi G)dA + Omega_H dJ
2nd law: dA is never negative
3rd law: surface gravity cannot be brought to zero (extremal Kerr solution)

+ Black hole entropy and generalized second law:
- Bekenstein proposed area really is entropy, S_BH = C A/(L_P)^2 where L_P = L_Planck = (hbar G/c^3)^1/2 = 10^-33 cm and C is an unknown constant of order unity. He gave information-theoretic reasoning, e.g. how many ways could one form the black hole.
- From the 1st law this implies a black hole temperature T_BH = (kappa L_P^2/8\pi G C).
- Bekenstein proposed a generalized second law holds (GSL): d(S_outside + S_BH) is never negative. The idea is that you can throw away entropy into a black hole, but then it gets bigger, hence S_BH increases. However, this law cannot hold: put a black hole in a heat bath at temperature T_outside. Then if T_BH > T_outside the decrease of outside entropy is more than the increase of black hole entropy.

+ Hawking effect: black holes radiate like hot bodies at temperature T_BH with C = 1/4, i.e. T_BH = hbar kappa/2\pi. Then S_BH = A/4 L_P^2. Since nothing can get out of a black hole, the radiation can carry no information, it must have a maximum entropy, reflecting only the aspects of the black hole that can be seen on the outside: mass, angular momentum and charge if the hole is charged. This explains the fact that it is thermal radiation. The existence of this readiation restores validity of the generalized zecond law! Discovery of Hawking radiation: Misner (and others?) suggested a wave analog of the Penrose process: superradiant scattering. The analogy with stimulated emission led to the expectation that in quantum field theory a rotating black hole would spontaneously radiate. Working on the theory of this effect, Hawking noticed that even a non-rotating black hole would spontaneously radiate. It is ironic, since he had been a vociferous critic of Bekenstein's proposal that balck holes really have entropy.

Tu 10/19:

+ Hawking effect: pair creation, information loss, black hole evaporation

+ de Sitter space: Eddington-Finkelstein diagram from homework 5, Gibbons-Hawking temperature and entropy of de Sitter horizon, observer dependence of the horizon (I recently coauthored an article on the concept of horizon entropy in general, which you can find here). One finds both outgoing and ingoing Eddington-Finkelstein coordinates, and they seem to differ. In fact, they cover different overlappping patches of the extended spacetime.

+ maximal analytic extension of Schwarzschild metric: Kruskal-Szekeres coordinates. This is treated in section 12.3 of Hartle. The figures there are useful, but the derivation as I motivated it in class is not given. Warning: Hartle's definition of U and V is different from ours. Our U is his 4M(V-U), our V is his 4M(V+U). For these notes, let 2M = 1, so the Schwarzschild radius is 1 and the surface gravity is kappa = 1/4M = 1/2. Also, suppress the angular coordinates. The steps are then:

ds² = -(1-1/r) dt² + (1-1/r)^-1 dr²
= -(1-1/r) [dt² - dr_*²] r_* = r + ln(r - 1), "tortoise coordinate"
= -r^-1(r-1) du dv u = t - r_*, v = t + r_*
= -r^-1e^-r (e^-u/2 du) (e^v/2 dv) r - 1 = e^-r e^r_* = e^-r e^v/2e^-u/2
= -r^-1e^-r dU dV U = -2e^-u/2, V = 2e^v/2

The relation between t,r and U,V:
4(1-r)e^r = UV (can't be solved explicitly for r(U,V))

t = ln(-V/U)

Comments:
- Make a diagram with U = constant lines with slope 1 and V = constant lines with slope -1. Then the light cone is everywhere of the standard form.
- Past and future horizons: v = -infinity corresponds to V = 0, the past horizon, while u = + infinity corresponds to U = 0, the future horizon. Both of these horizons have radial coordinate r=1.
- A constant r curve other than r = 1 is a hyperbola in the U-V plane. For each value of r there are two such hyperbolae.
- The singularity at r = 0 corresponds to the double hyperbola UV=4. The one with both U and V positive is the black hole singularity. The one with both U and V negative is the white hole singularity.
- t = constant curves are straight lines through the origin of the U-V plane. t = infinity corresponds to the future horizon U = 0, while t = -infinity corresponds to the past horizon V=0. The t = 0 slice has U + V = 0, it is a horizontal slice through the diagram. The intrinsic 3-geometry of this (or any other) constant t slice has a minimal 2-sphere of radius 1 at the center, with 2-spheres of increasing radii out to both sides. This is the "Einstein-Rosen bridge", or "Wheeler wormhole". No observer can pass through the wormhole: once at r=1 any timelike observer must proceed to smaller values of r.
- If the black hole forms from collapse, the negative V region together with part of the positive V region is not relevant, since the collapsing matter produces a different metric on the inside.

Th 10/21:

+ comments on maximal extension of Schwarzschild: This is also called an eternal black hole, since it is not formed at any time but rather has existed forever. However, as we saw, this implies that also there is a white hole in the past, and a past horizon. The same construction works as well for de Sitter spacetime. In fact, it works for flat spacetime, in the following sense, which may demystify the construction:

+ Rindler spacetime:
ds² = -x²dt² + dx²
= -x² [dt² - dx_*²] x_* = ln x
= -x² du dv u = t - x_* , v = t + x_*
= -(e^-u du) (e^v dv) x = e^x_* = e^v/2e^-u/2
= - dU dV U = -e^-u, V = e^v
= -dT² + dX² T = (U + V)/2, X = (V - U)/2

So the initial metric actually describes flat spacetime in funny coordinates. We could have seen this from the beginning, by the analogy with polar coordinates in the Euclidean plane, x playing the role of the radius and t the angle. In fact the relation is just replacing the angle by an imaginary angle. Then the rotation symmetry becomes a Lorentz transformation (boost) symmetry, constant t lines are radial from an origin, and constant x curves are hyperbolae. Comparing with the Schwarzschild case, this shows that near the horizon, the Schwarzschild time coordinate is like a hyperbolic angle, and the time translation is like a Lorentz transformation. Far from the horizon on the other hand, at large r, the Schwarzschild time translation is like a standard time translation in Minkowski spacetime.

+ gravitational waves: I basically discussed material from Chapter 16. I gave a very crude estimate of the maximum strength of a gravitational perturbation from a system with size r. (The article by Scott Hughes from HW7 gives a better estimate, in terms of the non-spherical kinetic energy producing the waves.) The perturbation is h ~ phi (r/d), where phi is the gravitational potential at the source. The maximum phi can be is of order unity, in order not to be inside a black hole, so we get h ~ r/d. Putting in 1km for r, appropriate for a solar mass black hole, and 100 Mpc for d, we get h ~ 10^-21. This is around the amplitude that ground based interferometers are trying to detect. Note that if we increase the mass a millionfold, corresponding to a merger of we can increase the distance a millionfold, by which point we have far more than included the entire observable universe. The corresponding frequencies are much lower (since the gravitational orbits are much larger), suitable for LISA, the proposed space based interferometer.

To find the geodesics in the gravitational wave background, I used the Lagrangian approach we have been using, whereas Hartle referred to the geodesic equation expressed with the Christoffel symbols. The odd thing is that with the coordinate choice of (16.2b), freely falling particles with zero initial coordinate velocity remain at fixed values of the spatial coordinates as the wave passes. Nevertheless, the proper distance between the particles changes. This means that a Weber bar would be set into vibration, and an interference pattern would be set up between laser beams having travelled in different directions.

Tu 10/26:

+ Geometrical units: I explained how to convert from geometrical units (G = c = 1) to general units. The only thing you need remember is the formula for the Schwarzsculd radius, R_s = 2GM/c^2. This implies that the relations between dimensional quantities are L = [G/c^2] M, M = [c^2/G] L, E = [c^4/G] L, etc. Hartle explained that energy density is thus 1/L^2 or equivalently 1/T^2 in geometrical units, which motivated the expression for effective energy density of gravitational waves (w^2 a^2)/32\pi where w is angular frequency. To convert this to general units we must multiply by a combination of G and c with dimensions of energy x time^2/length^3, i.e. c^2/G.

+ Cosmology: Chapter 17 explained the evidence for homgeneity and isotropy of the universe above scales of about 100 Mpc. It also explained the Hubble law of recession of galaxies, v = H_0 d. In practice v is measured from the resdhift z = (lambda_o/lambda_e) -1. In the special relativistic interpretation of the redshift as a Doppler effect, one has for the case of pure recession (no transverse motion) lambda_o/lambda_e = \sqrt{(1+v/c)/(1-v/c)} = 1 + v/c + O(v^2/c^2), hence z is approximately v/c. Hubble's law can thus be re-expressed as z = (H_0/c) d or, with numbers, approximately z = d/4000 Mpc. This is only valid for z<<1.

In general relativistic cosmology, the metric is assumed to have a foliation (layering) by spacelike slices each of which is homogeneous and isotropic. Up to a scale, there are only three possibilities: R^3, S^3, and H^3, i.e. flat 3-space, a 3-sphere, and a 3-hyperboloid.

(A 3-sphere can be constructed as the surface w^2 + x^2 + y^2 + z^2 = R^2 in flat 4d Euclidean space. A 3-hyperboloid can be constructed as the surface - w^2 + x^2 + y^2 + z^2 = - R^2. This can be interpreted as a surface in flat 4d Minkowski space, consisting of the points to the future at proper time R from an origin.)

Once one of these possibilities is chosen, the only thing that can differ from layer to layer is the scale of the spatial metric. By homogeneity the worldlines orthogonal to these homogeneous and isotropic layers are all equivalent, and we can use the proper time t along these worldines as a coordinate on spacetime. The scale factor for the spatial metric can then depend upon t, a(t). The line element takes the general form ds^2 = -dt^2 + a^2(t) dl^2, where dl^2 is the metric on a fixed R^3, S^3, or H^3.

We began with the spatially flat case, and studied the redshift due to the expansion of the universe. In addition to Hartle's approach, we deduced the effect in another way, noting that the coordinate wavelength of electromagnetic radiation is conserved as the wave propagates, hence the physical wavelength changes in proportion to a(t). The relation between redshift and scale factor is then z = (a_0/a_e) - 1. Taylor expanding about the present time and neglecting all higher order terms yields z = (adot/a)_0 (d/c), thus identifying the Hubble constant with the present value of adot/a.

Th 10/28:

+ 3-sphere and hyperbolic spaces

+ Friedman equation: motivation from Newtonian gravity in pressure-free case

+ matter, radiation, and vacuum energy: scaling with a(t), redshifting of blackbody spectrum. The occupation number for a photon mode stays the same (since the expansion of the universe is extremely slow compared to the period of oscillation of the photons, i.e. the expansion is 'adiabatic') as the frequency redshifts, so the blackbody spectrum persists, but at a redshifted temperature T/a. Since the energy density is proportional to T^4, it scales as 1/a^4. You can also think of this as occuring since each photon energy redshifts as 1/a and the number density goes like 1/a^3.

+ scale factor dynamics: matter dominated: a(t) ~ t^2/3 ; radiation dominated a(t) ~ t^1/2; vacuum dominated a(t) ~ e^Ht (de Sitter space)

+ critical density

Tu 11/02:

+ Fractions of critical density: defined critical density, and Omega_m,r,v which I write here as O_m,r,v.

+ Dynamics of the scale factor: We expressed the Friedman eqn in the form 1/2 (dA/dT)^2 + U_eff(A) = 1/2 O_c, where: A := a/a_0, t = t H_0 = t/t_H, Ueff = -1/2(O_v A^2 + O_m A^-1 + O_r A^-2), and O_c = - k/(a_0 H_0)^2. At the present time t_0 we have A=1 and dA/dT=1, hence as an identity, O_m + O_r + O_v + O_c=1 . For the presently favored parameters , O_v = 0.7, O_m = 0.3, O_r = 8 x 10^-5, the plot for the effective potential in Figure 18.9 is not correct. In fact, at A=1 we are already on the downward slope, i.e. the scale factor is accelerating today. You can see a correct plot here. For an open universe, k = -1, O_c > 0, so we always have expansion forever. For a closed universe k = +1, O_c < 0, it is possible that O_c/2 is less than the maximum of U_eff. In that case there is a turning point in the evolution, either a maximum of expansion after which the universe recollapses, or a minimum of contraction (no big bang), from which the present universe has emerged.

+ Age of universe: Derived expression for the age in terms of the presently measured cosmological parameters. The oldest stars are about 12 Gyr old, while a matter dominated FRW model is 9 Gyr old. Adding a vacuum energy term can stretch this to be more than 12 Gyr.

+ Measuring a(t) to determine cosmological parameters : What we see from far away and long ago is affected by both the expansion of the universe and the possible curvature of space. We derived the redshift-magnitude relation and redshift-angular size relations as explained in the textbook, and discussed the application of the former to the Type Ia supernova observations. The redshift-magnitude relation is f(z)/L= 1/[(1+z)^2 4\pi d_eff(z)^2], where f(z) is the measured flux today of a source with intrinsic luminosity L, emitting at a time corresponding to a redshift z, and d_eff(z) = d_eff(z; H_0, O_m, O_r, O_v) is a function of both z and the cosmological parameters, defined by the condition that 4\pi d_eff(z)^2 is the area today of a spherical wavefront of light emitted at a redshift z. Measuring f(z) thus in principle allows us to compare with the above formula, and if a FRW model with matter, radiation, and vacuum energy is valid, to determine O_m,r,v. The redshift-angular size relation is Dphi(z)/Ds = (1+z)/d_eff, where Dphi(z) is the angle subtended today by a source of instrinsic size Ds at a redshift z, and d_eff is the same function as defined above.

Th 11/04:

+ Thermal history of the universe:

Cosmic Microwave Background (CMB): free since "last scattering" a.k.a. "recombination" of electrons and protons in to neutral hydrogen at T ~ 0.3 eV ~ 3000K, t ~ 400,000 y. This termperature determined by the ionization rate and the photon/baryon ratio ~ 2 x 10^9. The CMB is almost perfectly uniform blackbody radiation. It has a dipole anisotropy of order 10^-3 caused by the Doppler effect due to the motion of the earth relative to the CMB rest frame. The structure we see today in the universe arose from clumping of "seed" anisotropies that must have existed at the time of last scattering. These were first observed by the COBE (Cosmic Background Explorer) sattelite in 1992, and have since been measured with ever greater precision in balloon borne and South Pole expperiments, as well as by the WMAP (Wilkinson Microwave Anisotropy Probe) satellite. An upcoming important future experiment is called Planck, dedicated to measuring in particular polarization features of the CMB. COBE and WMAP were used to put lower limits on the size of any nontrivial topology of space (a recent Physics Reports article extensively discusses the topology of space and its measurability). An excellent web site with information at all levels about the CMB is maintained by Wayne Hu at Unviersity of Chicago.

Neutron-proton freezout and Nucleosynthesis: In the early universe neutrons and protons readily interconverted, but these reactions froze out around T ~ 1 MeV. This temperature is determined by the photon/baryon ratio, the reaction rates, and the expansion rate of the universe H=adot/a. The expansion rate is determined by the Friedman equation H^2 = (#)T^4, where # is a numerical factor that depends on the number of massless species of particles in equilibrium. The n/p ratio at freezeout is determined by the Boltzmann factor to be n/p ~ exp(-(m_n-m_p)/T) ~ 1/6. This freezeout temperature is still high enough to break apart nuclei under these conditions, so the universe expanded after this for a while with the free neutrons decaying, until a temperature of T ~ 0.1 MeV was reached around t ~ 3min, at which time light stable nuclei formed. By this time some of the neutrons had decayed to protons, so the ratio was ~ 1/7. Essentially all the neutrons were "cooked" into nuclei, almost all 4He. To make one helium nucleus takes two neutrons and two protons. For each neutron there were seven protons, so for each 4He nucleus there are 12 extra protons. Hence the 4He makes up about 25% of the mass in baryons. An accurate calculation yields 24%. In addition, of order one part in 10^5 is deuterium and also 3He. One part in 10^10 is 7Li (the rest of the nuclei were formed in stars). This prediction is fairly well confirmed, providing strong evidence that the hot expanding universe model is accurate at least that far back in time. The prediction depends on the photon/baryon ratio, or in other words on the baryon density at that time, which corresponds today to Omega_baryon ~ 0.04. The account of nucleosynthesis, together with the observed abundances, provides (I think) the most accurate measurement of this ratio. It also limits the number of massless neutrino species in equilibrium to between three and four. This agrees with the collider measurements from the lifetime of the Z-boson

Tu 11/09:

+ flatness and horizon problems

+ inflation

Th 11/11:

+ angular size of horizon at last scattering

+ inflaton field

+ origin of primordial fluctuations, angular power spectrum

+ A nice reference: Inflationary Cosmology: Theory and Phenomenology, by Andrew R Liddle

+ cosmological constant problems: why so small (not Planck scale), why so close to matter density today (coincidence problem)

+ Christoffel symbols a.k.a. Levi-Civita connection a.k.a. connection components

+ vector nature of covariant acceleration, reason the coordinate acceleration is not a vector

+ covariant derivative; A nice reference: English translation excerpted from Einstein's article, "Die Grundlage der allgemeinen Relativitatstheorie", Annalen der Physik, 49, 1916.

Tu 11/16:

+ covariant derivative of an arbitrary tensor: I like to define it by the following two properties: (i) In l.i.c. at a point p it is simply the ordinary partial derivative at p, and (ii) it is a tensor. The first property defines it in a l.i.c. at any point, and the second property determines it at that point in any other coordinate system, via the tensor transformation rule. This definition immediately implies that the covariant derivative of the metric tensor is zero. A notation sometimes used for partial derivatives is comma with a subscript (which I've already used), and for covariant derivatives a semi-colon with a subscript. Thus the covariant derivative of a vector field V^a is V^a_;m = V^a_,m + G^a_,mnVⁿ, where G is the Christoffel symbols. I determined from this the formula for the covariant derivative of any contraviariant tensor by building up tensors by outer product of vectors, and I determined the formula for covariant derivative of a covariant vector by contracting with an arbitrary contravariant tensor. This revealed that for a covariant index the Christoffel symbols enter with a minus sign instead of a plus sign (see page 436 of the textbook). For a general tensor, there is one Christoffel symbol term for each index.

+ The geodesic equation says that the covariant derivative of the tangent along the curve vanishes.

+ gyro equation: a freely falling gyro's spin 4-vector has vanishing covariant derivative along the geodesic. This follows from local conservation of angular momentum and the equivalence principle. One says the spin is parallel transported along the curve. More generally, any tensor field with vanishing covariant derivative along a curve is said to be parallel transported along the curve. In particular, the geodesic equation states that the tangent to a geodesic is parallel propagated along the geodesic.

+ Geodetic precession of an orbiting gyro follows from the parallel transport rule. See sections 14.2 and 14.3, and Example 20.10 of the textbook. We raised an interesting question about angular momentum conservation and geodetic precession: while the parallel transport rule for the gyro spin 4-vector expresses local angular momentum conservation, the total angular momentum does not appear to be conserved, since the spin vector rotates direction after a complete orbit. There must be another, compensating change in the angular momentum of the system. I conjectured that it is probably a shift of the orbit, like a spin-orbit interaction. In fact, a spinning gyro does not follow a geodesic, but rather satisfies an equation, called the Papapetrou equation, that involves the curvature tensor. Perhaps gravitational radiation is also inextricably involved. I'll try to find out.

Th 11/18:

+ Geodetic precession, cont'd:
de Sitter applied this in 1916 (!) to the lunar orbit about the earth, as a gyro freely falling in orbit around the sun. It precesses with an angle of 0.02''/y, which has been confirmed to ~2% accuracy by lunar laser ranging (as of the late 1980's). It also applies to the Hulse-Taylor pulsar, the spinning neutron star being a gyro in orbit around its companion, but cannot be quantitatively observed there due to the complicated pulsar beam shape. Gravity probe B will attempt to measure the geodetic precession of 6.6''/y, as well as the Lens-Thirring precession (dragging of inertial frames due to spin of the earth) of 0.04''/y.

+ Computational method for Chirstoffel symbols:
a super-convenient trick-of-the-trade: rather than chasing indices, the easiest way to compute the Christoffel symbols is usually just to read them off of the geodesic equation, as obtained from the Euler-Lagrange equations for the lagrangian L = ½ g_mn dx^m/dl dxⁿ/dl. I illustrated this in class for the metric of a plane gravitational wave. This way one quickly obtains the non-zero components, and sees which components are zero.

+ Tidal effects, geodesic deviation equation, and the curvature tensor:
I discussed the ideas covered in sections 20.1,2 of the textbook. There is a derivation of the geodesic deivation equation at the book website. I gave a different derivation in class, based upon the following construction. Let x^m(s,t) describe an affinely parameterized geodesic with affine parameter t, for each value of s, hence it describes a one-parameter family of geodesics. The tangent vectors to the geodescis are T^m = x^m,_t, while the separating vectors between neighboring geodesics are S^m = x^m,_s. The relative covariant acceleration is the second covariant derivative of S^m along a geodesic. Because of the tedium of putting in sub and superscripts using my html editor I will write out this calculation without indices, which will be a bit obscure, but you can fill in the details. Let D stand for covariant derivative, and dot (.) stand for contraction, so T.D is the covariant derivative contracted with T, i.e. in the direction of the tangent to the geodesics. Then we have an identity, S.DT = T.DS, which is most easily established by evaluating it in local inertial coordinates at a point, in which case it just reads x^m,_ts= x^m,_st. Then, for the second covariant derivative of S we calculate:

(T.D T.D) S = T.D (S.DT) (use the identity above)
= (T.DS). DT + T.S.DD T (product rule)
= (S.DT). DT + T.S.DD T (use the identity again)
= S.D(T.DT) - S.T.DDT + T.S.DD T (integrate by parts in the first term)
= - S.T.DDT + T.S.DD T (use geodesic equation for T)
= T.S.(DD-DD)T (rename indices in the first term)

where in the last line the index order on the second DD pair is opposite to that on the first. In a flat spacetime, D is the ordinary partial derivative, so the DD-DD = 0, so there is no relative acceleration of neighboring geodesics. In a curved spacetime, the covariant derivatives don't commute. Actually we checked that they do commute when acting on a scalar function, but they don't commute when acting on a vector. In fact, cranking out the terms we saw that

(D_aD_b - D_bD_a)T^m = R^m_nab T^m,

where R^m_nab is the Riemann tensor, as defined in Hartle's (21.20). It is a tensor since the lhs is a tensor for all T. The geodesic deviation equation thus reads

(T.D T.D) S^m = (R^m_nab TⁿT^a) S^b.

This may appear to differ by a minus sign from Hartle's (21.19), but note that his second u is contracted on the last index of the Riemann tensor, whereas my second T is contracted on the third index. Since the Riemann tensor is antisymetric in these two indices, the two expressions are actually identical. This equation tells us that the second covariant derivative of the separation vector S is linear in S itself, infact given by the action on S of a matrix given by the Riemann tensor twice contracted with the tangent to the geodesic.

matrix notation, electromagnetic and Yang-Mills (non-abelian gauge theory) analogy:
I explained how to think of the Christoffel symbols G^m_an as a matrix G_a with one explicit index, the mn pair being suppressed in matrix notation. Then the formula for Riemann looks like D_aG_b - D_bG_a + [G_a,G_a]. This is more memorable. It is also reminiscent of the expression for electromagnetic field strength F in terms of the antisymmetrized derivative of the vector potential A. In this analogy, the potential plays the role of the Christoffel symbols, and the field strength is a measure of the failure of the gauge-covariant derivatives d+A to commute for different values of the (suppressed) coordinate index. In that sense, the electromagnetic field strength is a "curvature". In Yang-Mills gauge theory, which describes the weak and strong interactions, A is replaced by a matrix, and the analogy with geometry is even more appararent, as there is also a commutator term [A,A] in the field strength.

Tu 11/23:

+ derived the algebraic identities (symmetries) satisfied by the Riemann tensor by first finding a nice expression for the tensor at a point in local inertial coordinates at that point: R_mnab = 2 d_[m g_n][a,b] (like in section 21.3, only nicer notation using antisymmetrizers).

+ counted the number of independent components (20) and explained that is equal to the number of second partials of the metric (100) minus the number of free numbers in the third partial derivatives of a coordinate change (80). That is, the curvature characterizes the obstruction to setting all the 2nd partials to zero at a point.

+ explained the Newtonian tidal equation, analog of the geodesic deviation equation. (as in section 21.1)

+ expressed Newton's field equation for the gravitational potential in terms of the vanishing of the trace of the tidal tensor. (strangely, Hartle does not quite point this out.)

+ explained how to carry this over to GR, thus "deriving" Einstein's vacuum field equation: vanishing of the Ricci tensor.

+ mentioned the biography of Einstein, "Subtle is the Lord", by Abraham Pais, in which Einstein's path to working out GR is detailed, including the twists and turns, in a fascinating way.

+ contrasted Einstein's and Newton's equations: Newton's is a single second order, time-independent, linear pde for the gravitational potential. Einstein's is ten coupled, second order, time-dependent, non-linear pdes for the metric components. Once the coordinate freedom and initial value constraints are fixed these equations are wave-like.

+ discussed the initial value formulation of Einstein's equation (evolution in time from initial data). pointed out that since coordinate changes provide four free functions of space and time, which affect the form the metric takes, one cannot expect the field equation to determine uniquely the evolution. However, the Ricci tensor and metric both have ten independent components, so it seems as if the field equations are enough in number to determine the evolution of everything. the resolution of this paradox is that not all ten of the field equations involve second order time derivatives. in fact, four of them do not! these are initial value constraint equations. I explained how this point was very confusing to Einstein initially, and at one point led him to conclude, erroneously, that a physical theory could not be described by generally covariant equations!

+ explained why the vanishing of the trace of the tidal tensor is equivalent to the statement that a ball of dust particles initially at rest relative to each other, when freely falling will distort into an ellipsoid in a volume preserving manner (up though the terms of order t^2), and this fully characterizes the content of the vacuum Einstein equation.

Tu 11/30:

+ uniqueness of Einstein equation, psychological impact on Einsten

+ linearized (weak field) limit: basically the material is in Hartle's section 21.5. Things I discussed that are not in Hartle:

* writing the linearized Riemann tensor as 2 d_[m h_n][a,b]

* linearized Riemann tensor is invariant under linearized coordinate transformations.

* analogy with electromagnetism: field strength F_mn = 2 d_[m A_n], invariance under gauge transformations since mixed partial derivatives commute. In the case of gravity, the linearized coordinate transformation is like an electromagnetic gauge transformation with an extra index.

* "Lorentz gauge" should really be "Lorenz" gauge, referring to a different guy, not just a different spelling. However, this gauge was not widely adopted until Lorentz used it! The electromagnetic version of this can be accessed by solving a wave equation with source for the gauge parameter. Ditto for gravity

Th 12/02:

+ Showed explicitly how the Lorentz gauge condition V_a = h_ab^,b - 1/2 h,a = 0 can be accessed by making a gauge transformation with gauge parameter satisfying box \xi_a = -V_a.

+ The residual gauge freedom is given by those parameters satisfying box \xi_a = 0. I asserted without proof (though it is easy to see) that using this freedom one can set the time-space compnents and the trace of h_ab to zero: h_0i=0 and h = 0 in vacuum. (The fact that we can use this residual freedom to impose four more conditions means that the gauge freedom can be used twice to impose conditions on the field. My grad school prof. Claudio Teitelboim used to say "the gauge always hits twice".) Together with the Lorentz gauge condition this implies that h_00 is time-independent. The field equation in vacuum box h_ab = 0 then implies that h_00 satisfies Laplace's equation. If there are now sources, so this is satisfied everywhere in space, it implies that h_00 = 0.

+ Electromagnetic waves: Maxwell theory is quite analogous to linearized gravity. The field strength tensor is invariant under gauge transformations A'_m = A_m + f,m. Using the Lorentz gauge A_m^,m = 0 Maxwell's equation takes the form box A_m = 0. Using the residual gauge freedom (the gauge hits twice) one can set A_0 = 0. This is sometimes called radiation gauge. Together with this the Lorentz gauge implies A_i^,i = 0, i.e. the spatial divergence of the spatial vector potential vanishes. Plane wave solutions to the field equation have the form A_m = a_m exp(ikx), where a_m is a constant polarization vector, and kx = k_m x^m. The field equation is satisfied if k_m is a null vector: k_m k^m = 0. The Lorentz gauge condition implies k^m a_m = 0, Radiation gauge implies a_0 = 0. Thus k^i a_i = 0, so the polarization vector is spatial and orthogonal to the propagation direction, i.e. transverse. For each wavevector k_i there are two linearly independent wave solutions corresponding to the two polarizations orthogonal to the wave vector. The frequency k_0 is determined by the condition that k_m be null.

+ Gravitational waves: Very similar to electromagnetic waves. The plane waves have the form h_mn = a_mn exp(ikx), where a_mn is the polarization tensor. The field equation implies k_m is null, and the gauge conditions imply that the only nonzero components of a_mn are the spatial ones transverse to the propagation direction. For example, a wave in the z-direction has polarization components a_xx, a_xy=a_yx, and a_yy. The tracefree gauge condition h=0 impliues that a_xx = - a_yy. Thus although we begin with ten independent components of the metric, there are only two linearly independent polarizations for each wave vector, + (a_xx = - a_yy) and x (a_xy = a_yx). We thus recovered the wave metrics written earlier in the semester. This is called the transverse-traceless gauge, or TT gauge.

+ Inclusion of matter: introduced the question...

Tu 12/07:

+ Newton's field equation: Trace of tidal tensor = -\phi,^i_i = - nabla^2 phi = - 4\pi G\rho_mass. In relativity, the trace of the tidal tensor in the frame of a 4-velocity u^a is -R_ab u^a u^b, but by what should the mass density \rho_mass be replaced? In special relativity, mass is not conserved, but is rather just one form of energy, so replace by the energy density (divided by the squared speed of light). What kind of object is energy density from a transformation point of view? If it is to be equated to R_ab u^a u^b it must be of the form T_ab u^a u^b for some tensor T_ab, called the energy-momentum tensor , or stress-energy tensor, or just stress tensor for short. But why should energy density take this form?

+ Under a Lorentz boost the energy transforms since it is a projection of the energy-momentum 4-vector, and a density transforms since the volume undergoes a Lorentz contraction. Let's take these one at a time, first the density. If we have a scalar quantity like charge, a density of charge is the time component of a charge current 4-vector - j_a u^a. In a basis adapted to the frame u^a, the time component of j^a is the charge density, and the space components are the charge current density. Now if we want to describe the density of a vector quantity like 4-momentum rather than a scalar like charge, we must add an index to the current, thus something of the form -T_ab u^a describes the energy-momentum 4-vector density. Finally, if we want to specify just the energy density we must contract with u^b, obtaining T_ab u^a u^b.

+ It turns out that the energy-momentum tensor is always symmetric (if suitably defined). Physical interpretations of all its components are explained in Hartle's Chapter 22. In brief, in a given Lorentz frame. T_00 is energy density, T_0i is momentum density, T_i0 is energy flux (which is equal to momentum density; my index conventions might be the reverse of Hartle's), and T_ij are the stress components, the diagonal ones being pressures. The trace T = T_a^a is a scalar. If the pressures are small compared to the energy density, as they are for nonrelativistic matter, T is approximately equal to -T_00. Thus just from the Newtonian limit we cannot tell whether to use -T or T_ab u^a u^b as the energy density in guessing the relativistic replacement for the mass density in Newton's field equation.

+ Examples of stress tensors:

> pressureless dust: T_ab = rho u_a u_b, where u^a is dust 4-velocity and rho is rest mass density
> prefect fluid: T_ab = rho u_a u_b + p(g_ab + u_a u_b), where p is pressure and it multiplies the spatial metric
> vacuum: T_ab = - rho_v g_ab, where rho_v is the (constant) energy density in all frames!
(This stress tensor is locally Lorentz invariant, since the metric is.)
> Electromagnetic field: T_ab = (1/4\pi)[F_an F_b^n - 1/4 g_ab F_mn F^mn]
> In general one finds the stress tensor by varying the matter field action with respect to the metric.

+ Gravitational field equation: set the trace of the tidal tensor equal to something that matches minus the mass density:

R_ab u^a u^b = 4\pi [x (-T) + (1-x)T_ab u^a u^b]

= 4\pi [xT g_ab + (1-x)T_ab] u^a u^b

If true for all frames, this implies the tensor equation

R_ab = 4\pi [x T g_ab + (1-x)T_ab]

Now how are we to fix x? Einstein's first guess was x=0. I don't think he thought about the trace. But he quickly realized that this choice is inconsistent with energy conservation!

+ Local energy-momentum conservation for the matter is expressed by the condition T_ab^;b=0. In local inertial coordinates at a point this is just the statement that the stress tensor is divergenceless. Why should that express the conservation? Just think about charge conservation which is expressed by the continuity equation rho,t + j^i_,i = 0. In 4-vector notion this reads j^a_,a=0. Energy-momentum conservation is the same, with an extra index.

+ Suppose with Einstein's original proposal that we choose x=0. Then energy conservation together with the field equation implies R_ab^;b=0. There is an identity, true for all metrics, which states that (R_ab - 1/2 R g_ab)^;b = 0. This is called the contracted Bianchi identity. In view of this identity, the field equation with x=0 would imply that R_;b=0, i.e. the Ricci scalar is a constant everywhere. This is far too restrictive. It would imply the trace of the stress tensor is constant everywhere, which in the Newtonian limit would permit only constant mass densities! The opposite choice, x=1, leads to the same conclusion. To avoid this jam we must choose x so that the contracted Bianchi identity and energy-momentum conservation are compatible. This amounts to saying that the field equation should set R_ab -1/2 R g_ab proportional to T_ab. To find the x that does this, let's first eliminate T in favor of R. Tracing the last equation we find

R = 4\pi [4xT + (1-x)T] = 4\pi G(1+3x) T,

hence

R_ab - [x/(1+3x)]R g_ab = 4\pi (1-x) T_ab.

Therefore x must be chosen to that x/(1+3x)=1/2, i.e. x=-1. This yields the Einstein equation,

G_ab:= R_ab - 1/2 R g_ab = 8\pi T_ab,

G_ab is called the Einstein tensor. The Einstein equation implies energy-momentum conservation for the matter!

+ There is a striking analogy between this and Maxwell's discovery of the displacement current. Since div.curl B = 0 for any vector field B (due to equality of mixed partials), Ampere's law, curl B = j, implies that the current j is divergenceless. This is fine in static situations, but is certainly not generally true. In general, charge conservation implies div j = - rho,t. Maxwell realized he had to modify the field equation to be compatible with this charge conservation. Since Gauss' law implies div E = rho, the continuity equation can be expressed as div j = -div E,t, i.e. div(j + E,t) = 0. This would not only be consistent, but would be implied by the field equaion if one replaced Ampere's law with curl B = j + E,t. The term E,t is Maxwell's displacement current. With this term, Maxwell's equations imply the existence of propagating wave solutions in empty space! An incredibly profound discovery, based on a simple point of theoretical consistency.

+ The analogy to the situation with Einstein's equation and energy-momentum conservation is quite close formally as well, when expressed in covariant terms. The half of the Maxwell equations not identically implied by the definition F_mn = A_n,m - A_m,n are F^mn_,m = j^n. Charge conservation is expressed by the continuity equation j^n_,n=0, which is implied by the the Maxwell equation, since F^mn_,mn vanishes identically, due to the equality of mixed partials and the antisymmetry of F^mn.

Th 12/09:

+ I scanned Ken's notes from class. Today's class begins on page 2. Here I give a summary as well.

+ We can write the Einstein eqn as R_ab = 8\pi(T_ab -1/2 T g_ab). The scalar R_ab u^a u^b is minus the trace of the tidal tensor in the frame u^a. This scalar measures the gravitational attraction of the source, which in the Newtonian limit is equal to 4\pi x(mass density). Thus in Einstein's theory, the source of attraction is 4\pi(2T_ab u^a u^b + T). That is, the mass density is replaced by twice the energy density minus the trace of the energy-momentum tensor. For a perfect fluid this is rho + 3p, the energy density plus thrice the pressure.

+ Pressureless dust thus attracts with a source equal to the energy density. For the electromagnetic fields, the Maxwell stress tensor is tracefree (as a consequence of conformal invariance, by the way). Thus the source is twice the energy density. (For a thermal fluid of radiation as in cosmology, the equation of state is p=rho/3.) (By the way, I think one can infer from this that when a photon passes by a gravitating mass, it is in effect deflected with twice the "acceleration" as a slowly moving particle would be, which explains why the angle of deflection of light by the sun in general relativity is twice what Einstein first calculated based on the Newtonian acceleration. Vacuum energy has a stress tensor T_ab = -rho_v g_ab, which corresponds to the equation of state p = -rho, so the source of attraction is -2 rho, minus twice the energy density! That's why a positive vacuum energy density produces cosmic acceleration: it is a negative source of attraction.

+ Previous class I cited the contracted Bianchi identity G_ab^;b = (R_ab - 1/2 R g_ab)^;b = 0. To prove this one first establishes the Bianchi identity, R_mn[ab;c]=0, where the semicolon stands for covariant derivative. A simple way to establish this is to adopt a local inertial coordiante system at a point, in which case the components of the expression just written are 1/2 d_[m g_n][[a,b]c] (see lecture from 11/3). The abc antisymmetrizer takes care of the ab antisymmetrizer, so we can rewrite the last as 1/2 d_[m g_n][a,bc]. The antisymmetrization makes this vanish since mixed partials commute. This establlishes the Bianchi identity. Contraction over ma and nb then produces the contracted Bianchi identity. But to see this, one must expand out the antisymmetrizer in the Bianchi indentity. Doing this in principle generates six terms, with a coefficient 1/3! and the index arrangments and signs (abc + bca + cab - bac -cba - acb). However since the Riemann tensor is already antisymmetric in the last two indices, the last three terms are equal to the first three terms, hence we have just a coefficient 1/3 and three terms, (abc + bca + cab).

+ The contracted Bianchi identity explains why four of the equations are not evolution equations (see lecture of 11/3): The identity can be written out as G^ab_;b = G^a0_,0 + G^ai_,i + G-Gamma terms = 0. Therefore G^a0 cannot contain any second derivatives wrt x^0, otherwise the term G^a0_,0 would contain third derivatives with respect to x^0 and no other term would contain these, so the indentity could not hold for all metrics! This means that G^a0 = 8\pi T^a0 expresses four constraints on the initial values of the metric and its time derivative (and the matter source), rather than evolution equations. If these constraints are satisfied initially, then the rest of the Einstein equations guarantee that they are automatically preserved in time. The reason? In vacuum, it is just the contracted Bianchi identity itself. As you see here, that identity relates the time deriative G^a0_,0 to other components of the Einstein tensor that vanish at the initial time. In the presence of matter, one must also invoke the fact that the matter stress tensor is conserved, which is true if the matter satisfies its own equations of motion. Then we have also T^ab_;b = 0, and hence (G^ab - 8\pi T^ab)_;b = 0. Then the same argument shows that G^a0 - 8\pi T^a0 = 0 is automatically preserved in time.

+ There is a perfect analogy with electromagnetism. The field strength F_mn = 2 d_[m A_n] is gauge invariant, so the evolution of the vector potential cannot be entirely determined. In fact there is a free gauge function of time. Thus one of the four Maxwell equations F^ab_,b = j^a must not be a true evolution equation. In fact, F^0b_,b = F^0i_,i contains no second time derivatives, since F^00 = 0.
This is just the Gauss law equation div E = 4\pi \rho, which is an initial value constraint equation, not an evolution equation. This follows from a kind of "Bianchi identity", that F^ab_,ab = 0, which follows from the symmetry of mixed partials. This identity implies charge conservation j^a_,a = 0. The Gauss law constraint is preserved in time as a consequence of the other Maxwell equations.

+ Like the Gauss law, which sets the Laplacian of the scalar potential equal to a source term, the Einstein initial value constraint equations are elliptic pde's, but they are more complicated since non-linear and coupled with many variables. Thus, to even set up initial data for future evolution by numerical methods there is already a nontrivial problem to solve, which in general itself requires numerical methods!

+ Action principle for gravity: the Einstein equation can be derived from a very simple action principle, the Einstein-Hilbert action. For pure gravity, the action functional is S[g] = \integral R (-det g)^{1/2} d^4x. The determinant of the metric is needed to obtain the proper volume element for integration (consider a diagonal metric to convince yourself that this is plausible). The Ricci scalar is the only scalar that can be formed from the metric and up to two derivatives (other than just a constant from the trace of the metric, which corresponds to a cosmological constant in the action). The requirement that the action be stationary under infinitesimal variations of the metric implies the vacuum Einstein equation. How? The metric appears in three places. Lets vary the inverse metric at x. Since R=g^ab R_ab that plucks out a factor of R_ab at x. Varying the metric in the determinant can be shown to contribute -1/2 g_ab R at x. The last part to consider is the variation of the metric in R_ab. This is complicated, since it appears non-linearly, and differentiated. But the Lord is merciful, and this contribution turns out to be a total derivative, hence integrates away to nothing. This produces the Einstein tensor R_ab -1/2 g_ab R. In vacuum that must vanish, yielding the Einstein equation. Matter is described by its own action S[matter, g], which depends upon both the matter fields and the metric. Varying the matter action with respect to the metric produces the energy momentum tensor for the matter. With the appropriate coefficients, the vanishing of the full variation with respect to the metric yields the Einstein equation with the matter source term.