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;
energymomentum 4vector
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 4wavevector of the photon
and u is the 4velocity 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 4velocity to the
observer frame,
rather than transforming the observed 4wavevector to the source frame
(as did Hartle). It
seemed simpler to me, since the source 4velocity 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 4wavevector, 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 noninertial 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 noninertial 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 nonuniform 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 nonuniformity is negligible.
Curvature Analogy:
freefall <=> straight line in space
relative acceleration of freefall trajectories <=> curvature of
space
Illustrated with great circles on a sphere.
Description of curvature: line element with nonconstant 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
77).
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 79). 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
10080=20 nonzero 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.
Twodimensional 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 nonzero 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 4veclocity 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 upperlower 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
contravariant tensor indices.
 B^{m} a scalar for
all vectors B^{m}
implies
A_{m }is a covector, and
vice versa.
 d^{2}x^{m}/dl^{2} is not a vector
 freefall = inertial = unaccelerated = geodesic motion is
characterized by
d^{2}x^{m}
/dl^{2} = 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 dx^{n}/dl
)^{1/2
} . The stationarity
condition is the EulerLagrange equations
(which
we rederived).
Th 9/16:
 For the Lagrangian for the proper time, the EulerLagrange
equations become
(d/dl)(L^{1}g_{an} dx^{n}/dl) 
½ L^{1 }g_{mn},_{a} dx^{m}/dl dx^{n}/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}
dx^{n} /dl)  ½ g_{mn},_{a} dx^{m}/dl dx^{n}/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 d^{2}x^{m}
/dl^{2} = 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 d^{2}theta /dl^{2}
= 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
covector, ensuring
that this
is true.
 This
geodesic equation also follows directly from the variational condition
with Lagrangian
L = ½ g_{mn}
dx^{m}/dl dx^{n}/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
parametrizationinvariant form, and the coordinate t as the
parameter, the Lagrangian is approximately
given by L = 1  ½ v^{2} + ½ h_{tt}
.The
constant 1 affects nothing. This gives the same equations of
motion as
the Newtonian Lagrangian L^{Newton}
= ½ mv^{2}  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 EddingtonFinkelstein (EF)
and Schwarzschild line elements:
v = t + r + 2M ln(r2M)
 EF spacetime diagram: interpreting the coordinates, light cones, and
light rays (see Fig. 12.2).
Horizon at r=2M, curvature singularity at r = 0.
 vtranslation 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 momentumlike 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
 tcoordinate pileup 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 Lagrangiana symmetry
coordinateis conserved. For free particles, this means that if
the metric components do not depend on a coordinate w, the
corresponding momentum g_{wb}u^{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 4velocity 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} dx^{n}/dl
, so that the
geodesic equation can be written (d/dl)(k_{a})  ½ g_{mn},_{a} dx^{m}/dl dx^{n}/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 Killingfrequency,
i.e.
frequency wrt the ignorable tcoordinate, then the radiation field will
have the same tfrequency, since the metric is invariant under
ttranslation. 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 wavefrontsi.e. constant phase surfacesof
electromagnetic radiation are null hypersurfacesi.e.
3surfaces 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}
v^{n} = k_{n} v^{n} = S,_{m} v^{n} = 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 4velocity 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
(10^610^9 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 "nohair 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 BoyerLindquist 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 nonorthogonal 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 nonzero. 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 rcoordinate 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
timetranslation 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 timetranslation 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 nonembeddabilty).
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,
timetranslation 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 informationtheoretic
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 nonrotating
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:
EddingtonFinkelstein diagram from homework 5,
GibbonsHawking 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 EddingtonFinkelstein coordinates, and they
seem to differ. In fact, they cover different overlappping patches of
the extended spacetime.
+ maximal analytic extension of Schwarzschild metric: KruskalSzekeres
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(VU), 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:
ds^{2} = (11/r) dt^{2} + (11/r)^{ 1} dr^{2}
= (11/r) [dt^{2}  dr_{ *}^{2}]
r_{*}
= r + ln(r  1), "tortoise coordinate"
= r^{1}(r1) du dv
u = t  r_{*},
v = t + r_{*}
= r^{1}e^{r}
(e^{ u/2} du) (e^{v/2} dv)
r  1 = e^{r} e^{r*} = e^{r}
e^{v/2}e^{u/2}
= r^{1}e^{r}
dU dV
U = 2e^{u/2}, V =
2e^{v/2}
The relation between t,r and U,V:
4(1r)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 UV 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 UV 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 3geometry of this (or any other) constant
t slice has a minimal 2sphere of radius 1 at the center, with
2spheres
of increasing radii out to both sides. This is the "EinsteinRosen
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:
ds^{ 2} = x^{2}dt^{2} + dx^{2}
= x^{2} [dt^{2}  dx_{*}^{
2}] x_{*}
= ln x
= x^{2} du dv
u = t  x_{*}
, v = t + x_{*}
= (e^{u} du) (e^{v} dv)
x = e^{x}_{ *}
= e^{v/2}e^{u/2}
=  dU dV
U
= e^{u}, V = e^{v}
= dT^{2} + dX^{2}
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 nonspherical 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)/(1v/c)} =
1 + v/c + O(v^2/c^2), hence z is approximately v/c. Hubble's
law can thus be reexpressed 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 3space, a 3sphere,
and a 3hyperboloid.
(A 3sphere can be constructed as the surface w^2 + x^2 + y^2 + z^2 =
R^2 in flat 4d Euclidean space. A 3hyperboloid 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:
+ 3sphere and hyperbolic spaces
+ Friedman equation: motivation from Newtonian gravity in
pressurefree 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
redshiftmagnitude relation and redshiftangular size relations as
explained in the textbook, and discussed the application of the former
to the Type Ia supernova observations. The redshiftmagnitude
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 redshiftangular
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.
Neutronproton 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_nm_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 Zboson
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. LeviCivita 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
semicolon with a subscript. Thus the covariant derivative of a vector
field V^{a} is V^{a}_{;m} = V^{a}_{,m}
+
G^{a}_{,mn}V^{n},
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 4vector 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 4vector 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 spinorbit
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
HulseTaylor 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
LensThirring precession (dragging of inertial frames due to spin of
the earth) of 0.04''/y.
+ Computational method for Chirstoffel symbols:
a superconvenient trickofthetrade: 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 EulerLagrange
equations for the lagrangian L = ½ g_{mn}
dx^{m}/dl dx^{n}/dl. I illustrated this in
class for the metric of a plane gravitational wave. This way one
quickly obtains the nonzero 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 oneparameter 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.(DDDD)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 DDDD = 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_{a}D_{b}  D_{b}D_{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} T^{n}T^{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 YangMills (nonabelian 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_{a}G_{b}  D_{b}G_{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
gaugecovariant
derivatives d+A to commute for different values of the (suppressed)
coordinate
index. In that sense, the electromagnetic field strength is a
"curvature".
In YangMills 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, timeindependent, linear pde for the gravitational
potential. Einstein's is ten coupled, second order,
timedependent, nonlinear pdes for the metric components. Once
the coordinate freedom and initial value constraints are fixed these
equations are wavelike.
+
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 timespace
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 timeindependent.
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 zdirection 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 transversetraceless 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 4velocity 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 energymomentum
tensor , or stressenergy 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 energymomentum 4vector, 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 4vector  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 4momentum 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 energymomentum 4vector 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 energymomentum 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 4velocity 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) + (1x)T_ab u^a u^b]
= 4\pi [xT g_ab + (1x)T_ab] u^a u^b
If true for all frames, this implies the tensor equation
R_ab = 4\pi [x T g_ab + (1x)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
energymomentum 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 4vector notion this
reads j^a_,a=0. Energymomentum 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
energymomentum 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 + (1x)T] = 4\pi G(1+3x) T,
hence
R_ab  [x/(1+3x)]R g_ab = 4\pi (1x) 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 energymomentum 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
energymomentum 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
energymomentum 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 + GGamma 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 nonlinear 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 EinsteinHilbert 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 nonlinearly, 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.