Relativity, Gravitation and
1. 21-26 (plane wave in
a different gauge)
2. The Lorentz gauge is the linearized version of a gauge that can be used
in the fully nonlinear setting called harmonic gauge or de Donder gauge.
This can be defined as a coordinate choice for which each of the four coordinate
functions satisfies the covariant wave equation. Let's call the coordinates
x^(m), where I put the index m in parentheses to emphasize that it is NOT
a vector index. The harmonic gauge condition is box x^(m) = x^(m)_;ab g^ab
= 0, for all four values of m. This gauge can always be accessed: we need
only choose the coordinates to be four solutions of the wave equation. (a)
Show that harmonic gauge is equivalent to the condition Gamma^m_ab g^ab =
0. (b) Show that in the linearized limit this becomes the Lorentz gauge condition.
3. Gravitomagnetism: Show that the geodesic equation in a weak gravitational
field with only time-space components h_0i takes the form of the Lorentz
force on a charge in a magnetic field with unit charge/mass ratio, du/dt
= u x B. Here u = dx/ds is the spatial part of
the 4-velocity (with s proper time), t is coordinate time, and the gravitomagnetic
field B is constructed from h_0i just as the magnetic field is constructed
from the electromagnetic potential A_i. (The other half of this story is
how energy currents---i.e. 0i components of the stress tensor---give rise
to 0i components of the metric, in analogy to how charge currents produce
4. 22-4 (symmetry of stress tensor)
5. 22-5 (stress tensor of a gas)
6. Stress tensor of electromagnetic field: The Maxwell stress tensor for
the electromagnetic field takes the form
T_ab = F_an F_b^n - 1/4 g_ab F_mn F^mn.
(a) Show that up to an overall constant this agrees with the formulae in
terms of electric and magnetic fields in Hartle's problem 22-6. (b) Show
that the trace of this stress tensor vanishes. (This property results from
the conformal invariance of electromagnetism.) (c) Argue that blackbody
distribution of electromagnetic radiation at rest in a given frame has a
stress tensor of the form diag(rho, p, p, p), with p = 1/3 rho. (d) Show
that the Maxwell equations in covariant form (see HW6, problem 2) with no
sources (vanishing charge and current density) imply the conservation of
the stress tensor (i.e. vanishing covariant divergence).
(due Tuesday Nov. 30):
+ finish reading everything that
has been assigned up to now
+ The Meaning of Einstein's Equation
1. Compute the Christoffel symbols by hand for the Schwarzschild metric.
(Check Appendix B to see if you've got it right.)
2. Consider a gyro orbiting in a Schwarzschild metric. (a) Assuming the
spin 4-vector is initially perpendicular to the plane of the orbit, determine
how the spin vector changes along the orbit if the orbit is (i) circular,
and (ii) elliptical. In case (ii), show that your result is consistent with
constancy of the magnitude of the spin vector. (b) Assuming the spin 4-vector
is initially in the plane of the orbit, show that it remains in the plane
of the orbit.
3. Compute the Riemann tensor, Ricci tensor, Ricci scalar, and Einstein
tensor by hand for (a) the 2d line element ds2 = -dt2 + a2(t)dx2, and (b) the 4d, flat RW line element ds2 = -dt2 + a2(t)(dx2 + dy2 + dz2). (You can check the 4d case in Appendix
B.) (c) Under what conditions on a(t) does the curvature vanish? Is your answer
precisely the same for cases (a) and (b)? (d) Can the Ricci tensor
vanish without the Riemann tensor vanishing for either of these metrics?
4. 21-5 (Static weak field limit of curvature)
5. 21-18 (Birkhoff's theorem)
(due Thursday Nov. 18):
Chapter 21, Sections 2-5
Einstein excerpt on covariant differentiation
1. Angular size of horizon at last scattering
Show that the angular radius of a causal patch on the surface of last
scattering (SLS) is about 2 degrees, assuming a matter-dominated flat FRW
model all the way back to the big bang. More explicitly, show that the horizon
of a point on the SLS subtends an angle of 2 degrees as viewed today. (Guidance:
The angle viewed today is the same as the angle subtended from our co-moving
world line at the time t_ls of last scattering, so do the calculation all
on the t_ls surface. Referring to Fig. 19.3 (p. 407) of the textbook, this
is the angle subtended by \chi_c viewed at a distance of \chi_ls (these
are coordinate distances, but the angle is their ratio which is the same
as the ratio of the corresponding physical distances). For this purpose
you can neglect the difference between \chi_ls and \chi_horiz.)
2. Number of e-foldings required to solve the horizon problem
If the universe inflates for a time t with Hubble constant H (vacuum
energy density 3H^2/8\pi G) the scale factor increases by a factor of exp(Ht).
If the increase in the horizon size during inflation is greater than the
present horizon size at the end of inflation, then the horizon problem is
solved in the sense that all events visible to us have at least one point
in their common past. (Strictly speaking, more than this is needed to account
for the homogeneity of the CMB.)
(a) Show that the number of e-foldings (Ht) required to solve the horizon
problem is N=ln[(2H/H_0)(a_rh/a_0)], assuming the inflationary period ends
abruptly at "reheating" (rh), and assuming a matter-dominated flat FRW
model from reheating to the present. For the purpose of evaluating the
present horizon size it is a good approximation to treat the reheating time
as if it were the big bang.
(b) The energy scale of inflation E_infl is defined by setting the energy
density equal to (E_infl)^4, where we have set hbar=c=1. Assume this vacuum
energy density is instantly converted to an equal thermal radiation energy
density, and assume that after reheating the radiation remains in
equilibrium and reshifts to lower temperature as the scale factor grows.
Show that under these assumptions we have N ~ ln(E_infl T_0/H_0), up to
constants of order unity (using Planck units G=c=hbar=1).
(c) The argument of the logarithm doesn't look dimensionless, but remember
we are using Planck units. To evaluate this we should just divide each
quantity by the corresponding Planck unit, to make it dimensionless. Eg.
divide the energy E_infl by the Planck energy ~10^19 GeV, and divide H_0
by the inverse Planck time ~ 10^-43 s. This results in pure numbers, so
it must be the correct dimensionless result we would have obtained had we
kept track of the powers of G, c, and habr. Using this method and these
numbers, evaluate N assuming the energy scale of inflation is 10^15 GeV.
(I obtain N ~ 58 using this rough approach. Since N depends logarithmically
the various assumptions it is not all that sensitive to them.)
3. (freely falling gyroscopes)
As explained in section 14.2 and in Example 20.10, the spin 4-vector s
of a freely falling gyroscope is parallel transported, i.e. its covariant
derivative along the free-fall geodesic is zero. (a) Show that s
remains orthogonal to the geodesic tangent u if it is initially orthogonal
(i.e. s is purely spatial in the free-fall frame). (b) Show that the magnitude
of the spin remains constant. (Hint: The covariant derivative of the metric
4. 20-18 (Killing's equation) (This covariant equation characterizes
Killing vector fields in a coordinate independent manner.)
5. 20-20 (Killing vectors on the Euclidean plane) (Note the book forgets
to mention that you are to assume the metric of the Euclidean plane. For
part (c), the linear combination must have constant coefficients, otherwise
it would not be a Killing vector.)
(due Thursday Nov. 11):
Chapter 21, Section 1
19-4 (approximate redshift-magnitude
(a) As in the textbook problem, find the constant c_1 and show that
it is equal to (- 1 - O_v + O_r + (.5)O_m). However, don't sketch the curves.
Add part (b): c_1 is of order unity, suggesting that the approximation
is not so good at z = 1. For example, in the flat, matter plus vacuum case,
the constant is -1.55, hence the approximation 1 + c_1 z is negative for z beyond 1/1.55, whereas
the exact result is always positive. To pursue the question further, make
a plot showing four curves on the same graph: in each of the two cases
mentioned in the problem, plot both the exact function (f(z)/L)*(4\pi (z/H0)2) and the approximation 1 + c_1 z. (They should be tangent at z=0.) Is
the approximation good for z=0.6, corresponding to the upper end of the
clump of supernova data points in Fig. 19.2? [I pose this problem both
for what you may learn about cosmology, as well as to gain useful experience
in approximating a complicated expression, as well as numerical evaluation
and plotting. Use whatever math program you like---Mathematica, Maple,
Matlab, ... If you don't already have access to a computer with one of
these, you can use a University computer in one of the labs on campus.
To find one visit http://www.oit.umd.edu/projects/wheretogo/searchSW.cfm,
which lists the labs where each software tool is available. If you can't
print for some reason, save a pdf or html version of your output and email
it to me. I used Mathematica. I defined a function I(z) equal to the chi-integral
times a_0 H_0, and included that in the function being plotted. It worked.]
rulers) The general expression for the angular size was derived in the
textbook for a general FRW model. Add part (a): Explain clearly in plain
English (not math) why the angular size increases with large enough
redshift for any FRW model with finite horizon size today. (b) Find the
redshift beyond which the angular size increases in a flat, matter-dominated
FRW model. (c) Give a complete answer, also in plain English, to the final
question: do more distant objects therefore appear brighter? Why or why
19-7 (Number counts of galaxies) Instead of doing the problem as
written, find a formula for N_gal(z) that applies for any FRW model, expressing
your answer in terms of the function \chi(z) defined in eqn (19.9).
(due Thursday Nov. 4):
Chaper 19, Section
problems: Note: Although there are a lot of problems
here, they are mostly very quick and simple. If you get stuck, don't
spend a lot of time struggling. Ask someone or send me a question by
(closed, matter dominated FRW models)
2. 18-14 (spatial curvature lens)
3. 18-16 (deceleration parameter) Hint: Differentiate the Friedman
4. 18-19 (de Sitter space) Add parts (b) and (c): treat also the
flat and open cases (b) k=0 and (c) k = -1. [Comment: these are
all different coordinate patches for the same spacetime! The k=+1 case
covers the entire spacetime, but the others do not. For a discussion
of de Sitter spacetime and seven different coordinate systems thereon
see Les Houches Lectures on
de Sitter Space.]
5. (Milne universe) While we're at it, consider the case of vanishing
energy density. (a) With which values of k is this compatible, and what
are the solutions in these cases? (b) All of these correspond to flat spacetime.
Explain how that can be true and sketch a diagram illustrating it. (c)
Optional: Find the explicit coordinate transformation relating the non-Minkowski
to the Minkowski coordinates.
6. 18-24 (Einstein static universe)
7. 18-28 (big bang singularity theorem) Hint: See hint for problem
8. 18-29 (negative vacuum energy)
9. Consider some stuff satisfying the simple "equation of state"
p = w rho. (a) Assuming this stuff doesn't interact with anything else,
use the first law of thermodynamics to determine how rho varies with
the scale factor a. (b) What values of w correspond to matter, radiation,
vacuum, and curvature terms in the Friedman equation? (c) The case w <
-1 has been called "phantom energy". Show that if there is any of this
nasty stuff the universe will blow up to infinite scale factor in a finite
time, tearing apart everything including nuclei and nucleons (the "Big Rip").
(due Thursday Oct. 27):
(only two gravitational wave polarizations)
2. 16-13 (g-wave energy flux) Compare with the energy flux of
sound waves at the threshold of human detection, 10-11 erg/cm2-s,
and to the flux from the 50,000 Watt WAMU radio transmitter at a distance
of 20km (assuming unrealistically that spherical wavefronts are emitted).
3. 16-14 (g-wave energy flux, again)
4. 18-3 (particle motion in expanding universe)
5. 18-5 (cosmological redshift of CMB)
6. 18-6 (cosmological redshift of timescales)
7. 18-8 (cosmological redshift via momentum conservation)
(due Thursday Oct. 20): This week the homework is mainly reading.
Pages 1-9 of Listening
to the Universe with Gravitational Wave Astronomy (http://arxiv.org/abs/astro-ph/0210481),
by Scott Hughes.
2) Chapter 17, The
1. Referring to the article by Scott Hughes,
(a) Derive equation (4) using dimensional analysis. That is, assume
h is proportional to GQ/r on the general physical grounds discussed in
the article, and deduce the missing power of the speed of light c and
the number of time derivatives of the qudrupole moment Q. (b) Verify equation
2. 17-5 (Homogeneity
scale of the universe from 2dF Galaxy Redshift Survey.)
3. 17-8 (Main sequence color index distance determination.)
4. 17-9 (Cepheid variable distance determination.)
5. Hawking's area theorem: That the black hole
horizon area cannot decrease in the Penrose process was discussed in
class and in the textbook. In fact Hawking showed that in complete generality---in
arbitrary processes including far from stationary conditions---that the
horizon area cannot decrease, provided that (1) matter energy is positive
in a suitable sense and (2) there are no naked singularities, i.e. cosmic
censorship holds. (The horizon is defined generally as the boundary of
the region that can communicate causally with distant observers.) Thus for
example if two Kerr black holes collide and coalesce and radiate away energy
and angular momentum in gravitational waves and eventually settle down to
a final Kerr black hole, the area of the final black hole horizon must be
greater than or equal to the sum of the areas of the two initial horizons.
Determine the maximum energy radiated allowed by Hawking's area theorem when two Kerr black holes, each of mass M
and angular momentum J, coalesce. What does your result give in the
non-spinning case J=0 and the extremal case J=M^2?
(due Thursday Oct. 13)
reading: No new reading. Catch up if you need
1. Euclidean section and Hawking temperature of Schwarzschild
spacetime: Thermal averages at temperature T have a property of
periodicity in imaginary time with period \hbar/T. This can be used
to "derive" the Hawking temperature as follows. Replace the Schwarzschild
time coordinate t by iw, where w is real. This yields a Euclidean
signature metric, called the "Euclidean section" of the Schwarzschild
geometry. For each set of spherical angles, the resulting geometry
is that of a curved 2-d surface parameterized by r and w, with w-translation
interpreted as a rotational symmetry. (a) Show that there is a conical
singularity at r = 2M unless the coordinate w is identified with
a period of 2\pi/\kappa, where \kappa = 1/4M is the surface gravity of
the black hole. This corresponds to a temperature \hbar \kappa/2\pi, the
Hawking temperature. (b) Roughly sketch an embedding diagram of the r-w
space in this case.
[Hints: You can compute the proper circumference and proper
radius of a circle of constant r and demand that their ratio is 2\pi
in the limit that r approaches 2M. It will be sufficient to expand the
to lowest order in (r - 2M) in a neighborhood of r = 2M. Alternatively, you can look for a coordinate
transformation into standard polar coordinates, again expanding around
r = 2M, and read off what is the angle whose period must be 2\pi.]
2. Maxwell's equations: the electromagnetic vector
potential Aa is a co-variant vector. The field strength
is defined by Fab = Ab,a - Aa,b. (a) Show that Fab transforms as a covariant tensor, although
Aa,b by itself does not. (b) Show that F[ab,c] = 0, and argue that this represents
four independent conditions. The bracket means the totally antisymmetrized
part, i.e. sum over all permutations of abc with + sign for even
and - sign for odd permutations, and divide by the number of permutations
3!. (c) Choose coordinates x0,xi, and define the
electric field by Ei = F0i and the magnetic field
by Bi = 1/2 \epsilonijk Fjk, where \epsilonijk is the alternating symbol and \epsilon123=1.
Express the content of the identity F[ab,c] = 0 in terms of the electric and magnetic
fields. To which of the Maxwell equations does this identity correspond?
(d) The rest of the Maxwell equations depend upon the spacetime metric.
Let's consider just flat spacetime here. Then these take the form Fab,a
= jb, where the indices are raised by contraction with
the Minkowski metric, and where jb is the current density
whose time component is the charge density and whose space component
is the 3-current density. Show how this reduces to the remaining Maxwell
equations in 3-vector form. (e) Show that the equation in part (d) implies
jb,b = 0, and explain why this expresses charge
that the inverse metric gab (defined by gabgbc = (\delta)ac)
transforms as a contravariant tensor.
(due Thursday Oct. 7)
Hartle: read Ch. 13 & 15; skim Ch. 14
(de Sitter horizon) [Hartle doesn't tell you but this is the line
element for the de Sitter spacetime. Solve this problem in the following
way: Transform to the Eddington-Finkelstein form ds2 = -(1 - r2/R2)
dv2 - 2 dv dr + r2((d theta)2
+ sin2 theta (d phi)2) . The transformation is
similar to what worked for the Schwarzschild line element: v = t + h(r).
Find the function h(r), being careful to note that I've chosen the sign
of the dv dr term to be negative. Because of this choice, the constant
v surfaces describe outgoing
rather than ingoing light rays. Make an EF diagram like we did for the black
hole, showing the constant v and constant R surfaces, and then add some incoming
radial light rays, paying particular attention to how they behave
near r=R. This illustrates how r = R is a "future horizon". Make another
diagram, using the opposite sign choice for h(r). This illustrates how r
= R is also a "past horizon". How can it be both a future and a past horizon??!
15-11 (circular photon orbits in extremal Kerr spacetime)
15-13 (circumference of Kerr ISCO's)
15-16 (AGN lifetime estimate) [Estimate the maximum lifetime,
i.e. assuming that none of the rotational energy goes into the black
hole itself, which is to say that the irreducible mass, or what is
the same the area, remains constant.]
15-18 (pair production estimate near a rotating black hole)
(due Thursday Sept. 30)
Hartle: Ch. 11, skim 11.1, read 11.2,3; Ch. 12, skim
12.1,2 (already covered in class), read 12.3,4
(velocity of orbit wrt local static observer) (Suggestion: First find
the energy measured by the static observer.)
9-11 (decay of unstable orbit)
12-6 (orbit of closest approach) [Note what the book
says about crossing 3M makes no sense. Interpret it as just coming
close to 3M for a long time.]
S1: Conformal invariance of null geodesics: Two metrics
related by an overall scalar multiple function are said to be "conformally
related", or related by a "Weyl rescaling" or "Weyl transformation".
The light cones of two such metrics gab and A2(x)gab
are obviously the same, and hence so are the null curves. Show that
in fact the null geodesic curves are also the same, but that the affine
parameters are not the same.
(due Thursday Sept. 23)
Hartle: Ch. 9
The geodesic equation
(d/dl)(gan dxn/dl) - ½
gmn,a dxm/dl dxn/dl = 0
was derived in class from the condition that a scalar
action functional be stationary w.r.t. curve variations. This condition
is coordinate independent, hence if the geodesic equation holds in
one coordinate system it must hold in all coordinate systems. Verify
this explicitly by showing that the complete left hand side transforms
as a covariant vector under coordinate transformations (although the
two terms by themselves do not).
2. Consider radial light rays in the Edddington-Finkelstein
(EF) line element (see hw2).
(a) Show that the radial coordinate r is an affine
parameter along both ingoing and outgoing null geodesics (light rays),
except for the outgoing one that sits on the horizon.
(b) The null geodesics on the horizon are called "horizon
generators". Show that the ("advanced time") coordinate v is related
to the affine parameter on the horizon generators by d2v/dl2 = - k (dv/dl)2, where l is an affine
parameter and k = 1/4M is the "surface gravity" of the black hole. This means
that v is not an affine parameter along the horizon generators.
(c) Show that exp(kv) = al + b for constants a and b
along the horizon generators. This implies that as v goes to negative
infinity l covers only a finite range. This means that the EF coordinate
patch does not cover the whole spacetime. We'll see later what's missing.
Whatever it is, is it not relevant in a situation where the black hole
formed at some finite time in the past from gravitational collapse, since
the spacetime inside the collapsing stuff is not described by the EF line
Thursday Sept. 16)
Hartle: Ch. 6; Secs. 7.1,2,3,4; Sec. 8.1, first three
Spacetime Primer (see the course syllabus): Secs. 2.1,2,3
(and 2.4,5 if you like); Ch. 3
(proper time and orbits) Change part (d) to the following:
The elapsed time for the fixed particle in part (b) is longer than
for the orbiting particle in part (a). Since the longest time
must be on a free-fall path, there must be another free-fall path connecting
A and B that has longer time. This is the path where the particle
goes up and comes down, starting with just the right velocity to reach
B starting from A. Calculate the total elapsed time for this particle.
Use the approximations described at the end of problem 6-13.
(warp drive speed)
7-12 (warp drive proper time)
7-20 (embedding diagram of spatial slice
of Schwarzschild black hole)
S1. Show that if TabVa Vb
=0 for all Va, then the symmetric part T(ab)
= (Tab + Tba)/2 must vanish.
The Eddington-Finkelstein line element
ds2 = -(1 - 2M/r) dv2 +
2 dv dr + r2((d theta)2 + sin2
theta (d phi)2) (EF)
is one way to present the (unique) spherically symmetric,
vacuum solution to Einstein's equation.
It is written above in geometrical units, with G = c
= 1. In general units M would be replaced by GM/c2.
The parameter M is the mass of the source that produces
a) Show that (EF) defines a Lorentzian metric for all
r > 0.
b) If M = 0, the line element (EF) corresponds to flat
spacetime (Minkowski space).
Find a coordinate transformation that brings it to the
standard Minkowski form.
c) Show that a line of constant r,theta,phi is timelike
for r > 2M, lightlike for r = 2M, and spacelike for r < 2M.
d) The three-dimensional surface r = 2M is the event
horizon of a black hole. Show that all displacements
on this surface are spacelike, except ones at constant
angles, which are lightlike. The surface contains no
e) Suppose an outgoing light flash is emitted from the
spherical surface at r = r0 and v = 0.
Show that the
area of the wavefront grows with v for r0
> 2M, stays constant for r0 = 2M, and decreases for
r0 < 2M.
f) Show that any particle (not necessarily in free fall)
inside the horizon must decrease its radial coordinate
at a rate given by |dr/ds| >/= (2M/r - 1)1/2,
where s is the proper time along the particle world line.
g) Show that the maximum proper time before reaching
the singularity at r = 0 for any observer inside the
horizon is \pi M. How long is this for a solar mass
black hole? For a 108 solar mass black hole?
(due Thursday Sept. 9)
Organizational notes (page xxii)
Chapters 1 and 5
Appendices A and D
companion website (http://wps.aw.com/aw_hartle_gravity_1/0,6533,512494-,00.html)
2-7 (a coordinate transformation)
5-1 (4-vectors and dot product)
5-3 (free particle world line)
5-13 (pion photoproduction)
5-14 (energy of highest energy cosmic rays)
5-17 (relativistic beaming)[See problem with better notion in Errata for Printings 1-3 (pdf) at the book companion