Department of Physics, University of Maryland, Fall 2007, Prof. T. Jacobson

Physics 675

Introduction to

Relativity, Gravitation and Cosmology

Homework Assignments

HW13 (due Tuesday Dec. 11)

reading:
Hartle: Ch. 22

problems:

22-5 (stress tensor of a gas)

13S-1 Consider the energy-momentum tensor T^ab = rho u^a u^b + p(u^a u^b + g^ab) for a perfect fluid,
where rho is the rest energy (=mass) density, p is the pressure, and u^a is the 4-velocity.

(a) For the case with p = 0 (pressureless "dust"), show that the zero divergence condition T^ab;b = 0
implies that (i) the rest energy of the dust is conserved just as electric charge is conserved,
i.e. (rho u^a);a = 0, and (ii) the fluid worldlines are geodesics. (Tips: Consider the components of
T^ab;b=0 along u^a and perpendicular to u^a. Also, it is convenient to keep the combination rho u^a
(the rest mass current density) together as one entity.)

(b) Next consider the case where p is not zero, and show that (i) the "first law of thermodynamics" holds,
i.e. (rho u^a);a = - p u^a;a and (ii) the acceleration of the fluid worldlines is governed by the relativistic
Euler equation, (rho +p) u^a;b u^b = - (g^ab + u^a u^b) p,b. In words: The 4-acceleration of a fluid element
is minus the gradient of the pressure, projected orthogonal to the 4-velocity, divided by the sum of the
energy density and pressure.

Notes: Part (b-i) corresponds to the 1st law since u^a;a is the fractional rate of change (dV/dt)/V of an
infinitesimal 3-volume V of fluid. Part (b-ii) shows that the role of inertial mass of the fluid is played by
(rho+p)/c^2. This is strange but true. It means for example that for a star in hydrostatic equilibrium,
the pressure gradient required to hold up a given fluid element is greater than it would be if the inertial
mass were only due to energy density

13S-2 Maxwell's equations & charge conservation

Show that Maxwell's equations in curved spacetime imply charge conservation, j^b_;b = 0.

(Tip: Either crank it out in local inertial coordinates, being careful that non-zero partial derivatives of connection components
enter, which you must show vanish when contracted with F^ab, or more elegantly use the relation between commutator
of covariant derivatives and the Riemann tensor, the symmetry of the Ricci tensor, and the fact that contraction
of a symmetric index pair with an antisymmetric pair is always zero.)

Background: The electromagnetic 4-vector potential A_a is a covariant vector. The field strength is defined by
the antisymmetric co-variant tensor F_ab = A_b,_a - A_a,_b. (The electric field measured by an observer with 4-velocity
u^a corresponds to F_ab u^b and the magnetic field corresponds to -1/2 epsilon^abcd F_bc u_d, where epsilon^abcd
is totally antisymmetric and epsilon^abcd epsilon_abcd = -4!) It follows that F_[ab,c] = 0, which represents four independent
conditions (a,b,c must have distinct values, so the independent components can be labelled by the missing value).
These correspond to divB=0 and Faraday's law of induction curl E = - dB/dt. These equations are independent of the
spacetime metric. The rest of Maxwell's equations are F^ab_;a = 4pi j^b, where the semicolon denotes covariant derivative
(which of course depends upon the metric), the indices are raised by contraction with the inverse metric, and j^b is the
charge current density. This is four equations, which correspond to Gauss' law div E = 4pi rho and the Ampere-Maxwell
equation curl B = 4pi j + dE/dt.

13S-3 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 the trace of this stress tensor vanishes identically. (This property results from the conformal invariance of
electromagnetism.) (b) 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.

Background: Up to an overall constant this agrees with the formulae in terms of electric and magnetic fields in Hartle's
problem 22-6. Maxwell's equations with no sources (vanishing charge and current density) imply the conservation of this
stress tensor (i.e. vanishing covariant divergence). With charges they imply that the divergence of the stress tensor is
equal (up to a coefficient) to - F_ab j^b, which is minus the rate at which the field does work on the charges per unit volume.

13S-4 Light rays in curved spacetime

We've been saying all along this semester that light rays travel on null geodesics. Now let's deduce this from Maxwell's
equations. While we're at it, let's show that the polarization vector is parallel transported along the light rays, and that
the squared amplitude (~ photon number density) varies inversely as the cross sectional area of the beam.

(a) The field strength F_mn is unchanged if the gradient of a scalar is added to the vector potential A_m. This is called a
gauge transformation of A_m and has no physical effect. Using the gauge freedom one can arrange for the covariant Lorentz
gauge condition D^m A_m = 0 to hold (where D_m is the covariant derivative). Show that Maxwell's vacuum equation
D^m F_mn = 0 in Lorentz gauge takes the form D^m D_m A_n = R_mn A^m.

(b) Assume A_m has the form A_m = A e_m e^iS, where A and e_m are both slowly varying compared to the phase S.
A is the amplitude and e_m is a unit "polarization" vector. Assume also that the spacetime geometry is also slowly
varying compared to the phase (i.e., we are looking at "short wavelength radiation".)
(i) Subsitute this into Maxwell's equation, and drop all terms that do not contain at least one derivative of S.
(ii) Taking real and imaginary parts show that the gradient of the phase k_m = S,m is a null vector. According to last week's
homework, this means that the surfaces of constant phase are null surfaces, indicating that the electromagnetic waves
travel at the speed of light. Also k_m is tangent to affinely parametrized null geodesics. These are the "light rays".
(iii) Show that k^m D_m e_n = 0, i.e. the polarization is parallel transported along the light rays.
(iv) Show that D_m(A^2 k^m) = 0. (Squared amplitude varies inversely with beam area.)

(Tip: For parts (iii) and (iv) take the components perpendicular and parallel to e_m respectively.)

HW12 (due Tuesday Dec. 4)

reading:
The Meaning of Einstein's Equation (http://arxiv.org/abs/gr-qc/0103044)

problems:
20-19 (Null generators of null surfaces are geodesics) The problem as
written is incorrect unless you assume also that the surfaces f(x^a)=C are
null for every C in a neighborhood of C=0. Add this assumption and make
this part (a). Add a part (b):
Assuming now only that the C=0 surface is null, show that the vector l_a = f,a
satisfies the non-affinely parametrized geodesic equation.

(Note: the result of this problem shows that the null generators of null surfaces
are always null geodesics. In fact, null geodesics can be fully characterized in this
way, which is really neat since it does not refer to derivatives of the metric,
Christoffel symbols, etc. It answers a question we posed long ago at the
beginning of the semester: how can you distinguish the null curves that are
geodesics from those that are not? The answer: the null geodesics are
tangent to (generate) null surfaces. Since null surfaces are identical for metrics
related by a conformal factor, this also shows why null geodesic curves are
the same for conformally related metrics. Moreover, since the boundary of the
future of any point is a null surface, this also means that null geodesics
"surf the causal structure" of spacetime. That is, departing from a point p,
the null geodsics run along the boundary of the future of p, at least initially.
They can later leave the boundary due to lensing.)

12S-1
(a) Compute the Christoffel symbols and Riemann tensor BY HAND
(it's good for the soul to do this at least once in your life) for the line element
ds² = -dt² + a²(t)q_ij dx^i dx^j, where q_ij is a t-independent n-dimensional metric
on the space labeled by coordinates x^i, i = 1,,,,n.
Express your result in terms of a and adot=da/dt, and the Christoffel symbols and
curvature tensor for the metric q_ij. (You should find there are G^t_ij, G^k_tj = G^k_jt,
and G^k_ij components of the Christoffel symbol G, and all others vanish, and that
the nonzero curvature components will have either no t indices or two t indices.)

(b) Specialize your result to the case n = 1, and characterize all the cases in which the
1+1-dimensional curvature vanishes. Explain how the time-dependent case is flat, by
explaining how the (t,x) coordinates sit in the (1+1) Minkowski spacetime.

(c) In the case n > 1, characterize all the cases in which the n+1 dimensional curvature vanishes.
Explain how the time-dependent case is flat, by explaining how the (t,x^i) coordinates sit in the
(n+1) Minkowski spacetime.

(d) (i) Now assume n = 3 and compute the Ricci tensor, Ricci scalar, and Einstein tensor. We will
use these to write the Friedman equations for homogeneous isotropic cosmological spacetimes.
(ii) Characterize all the cases in which the vacuum Einstein equation is satisfied. Are there any
besides flat spacetime?

12S-2 Consider the field equation R_ab - 1/4 R g_ab = 0. The trace (contraction with g^ab) of the
left hand side vanishes identically, so this does not imply R = 0, and it is really only 9 independent equations.

(a) Show using the contracted Bianchi identity (22.50) that this equation does however imply that
R is a constant. Thus this equation is equivalent to the vacuum Einstein equation with an
undetermined cosmological constant term.

(b) Now consider the field equation R_ab - 1/4 R g_ab = 8\pi(T_ab - 1/4 T g_ab), where T_ab is the
matter stress tensor and T is its trace. Show using the Bianchi indentity together with the local conservation
of stress energy (22.40) that this equation implies the Einstein equation (22.51) with an additional
undetermined cosmological constant term.

HW11 (due *Thursday* Nov. 29)

reading:
Hartle: Ch. 21

problems:
20-17 (Covariant derivative of the metric vanishes)

20-18 (Killing's equation) (This characterizes Killing vector fields in a coordinate independent manner.)

20-20 (Killing vectors on the Euclidean plane) (Assume the metric of the Euclidean plane.)

11S-1 (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 is zero.)

11S-2 Consider a gyro orbiting in the "equatorial" plane of a static, axially and reflection symmetric metric.
That is, the gyro orbits in the z=0 plane of a spacetime with metric ds^2 = A dt^2 + B dr^2 + C dphi^2 + D dz^2,
where A,B,C,D depend only on r and z, and are even functions of z. For example, orbits in the equatorial plane of
the Schwarzschild metric satisfy these conditions.

(a) Show that on the z=0 plane, the only nonzero Christoffel symbols G^a_bc with a lower index z are G^r_zz, G^z_rz, and G^z_zr.

(b) Show that if initially the only nonzero component of the spin 4-vector S^a is S^z, then
(i) S^z remains the only nonzero component of S^a for all times, and (ii) S^z is constant only if the orbit is circular.
(iii) Show explicitly by computing the relevant Christoffel symbols that when S^z is not constant, S^2 = g_ab S^a S^b
is neverthelss constant (in agreement with the general result of 11-S1(b).)

(c) Show that if S^z is initially zero, it remains zero. (In this case, the spin undergoes geodetic precession in the orbital plane.)

HW10 (due Tuesday Nov. 20)

reading:
Hartle: Ch. 20 (skip p. 424-5);
Chapter 14, read Sections 1,2,3 and Box 14.1; skim Sections 4,5,6

problems:
19-5 (d_eff and H(z)) (Just a simple change of variables, but good to know.)

19-6 (Standard rulers) The general expression for the angular size was derived in the textbook for a general FRW model.
Hence answer only the other two questions.

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). Assume that galaxies are born
at the big bang and live forever without coalescing (not valid!).

10S-1 Angular size of horizon at last scattering

(a) Compute the maximum angular separation of two points on the surface of last scattering (SLS) that are "in causal contact",
i.e. whose past light cones intersect before the big bang. Assume a flat, matter-dominated FRW model all the way back to the
big bang. Give both the exact result and the leading order approximation gelecting higher order terms in 1/z_ls. (b) Next do the
calculation assuming a flat, matter plus radiation plus vacuum energy model, with Omega_m = 0.3 and Omega_r = 8 E-5 and
Omega_v = 1 - Omega_m - Omega_r. Then you probably cannot do the integral exactly, but you can argue that the O_v contribution
can be neglected (why?), and then do the integral exactly. (c) Compare the results for (a) and (b) and explain why they are fairly
close to each other, which one is larger, and why.

[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 you may 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 2\chi_c (but transverse to the line of sight) 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 may neglect
the difference between \chi_ls and \chi_horiz. Why?]

HW9 (due *Thursday* Nov. 8)

reading: Hartle, Ch. 17 & 18

problems:
17-5 (homogeneity scale of the universe from 2dF Galaxy Redshift Survey)

18-3 (particle motion in expanding universe) (Show that what Hartle says is true only if a(t) increases more rapidly than the square root of t.)

18-11 (closed, matter dominated FRW models) (Note: In part (d) the "circumference" can be taken at constant theta and phi.)

18-14 (spatial curvature lens)

18-16 (deceleration parameter) (Hint: Express q_0 in terms of the second time derivative of the scale factor at t_0, and use the
derivative of the scaled Friedman equation (18.77) to express this in terms of the cosmological parameters (Omega_r,m,v).)

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! This is a peculiarity of constant energy density.
The k=-1,0,+1 cases correspond to slicings of the de Sitter hyperboloid by timelike, null, and spacelike slices respectively.
Only the k=+1 case covers the entire hyperboloid within the coordinate patch. For a discussion of de Sitter spacetime and
seven different coordinate systems thereon see Les Houches Lectures on de Sitter Space.]

18-24 (Einstein static universe)

9S-2 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 show that rho varies as the -3(1+w) power of a.
(b) What values of w correspond to matter, radiation, vacuum, and curvature terms in the Friedman equation? (Think of the curvature term as due
to a fluid with energy density that varies as 1/a^2.)
(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").

HW8 (due Tuesday Oct. 30)

reading:
1) Hartle: Ch. 16
2) Pages 1-9 (or more if you like) of Listening to the Universe with Gravitational Wave Astronomy
(arxiv.org/abs/astro-ph/0210481), by Scott Hughes.
3) Optional: You might like reading Interferometric gravitational wave detection: Accomplishing the impossible.
(http://www.iop.org/EJ/abstract/0264-9381/17/12/315/, accessible from campus computers), by Peter Saulson

problems:
16-7 (only two gravitational wave polarizations) Add part (b): Show that under a rotation by pi the original
polarization returns to itself. This reflects the fact that gravitational waves carry a spin-2 representation
of the rotation group.

8S-1 Write the Lagrangian for test particle motion in the metric (16.2b) and derive the result (16.8) from
the corresponding Euler-Lagrange equations. Assume as does Hartle that the particles are freely falling
with zero initial velocity.

8S-2 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
quadrupole moment Q. (b) Using similar reasoning derive a corresponding formula for the amplitude
of the electric field produced by a radiating charge dipole; (c) Verify equation (5).

8S-3 (gravitational wave energy flux) LIGO expects to detect gravitational waves at frequencies of ~ 200 Hz
that cause a dimensionless strain of deltaL/L ~ 10^-21.
(a) What is the flux of energy in such waves incident on Earth in Watts per square meter?
(b) What is the ratio of the gravitational wave energy flux in part (a) to (i) the solar flux 1466 W/m², (ii) the energy
flux of sound waves from a vacuum cleaner, 10^-4 W/m², and (iii) the 50,000 Watt WAMU radio
transmitter at a distance of 20km (assuming that spherical wavefronts are emitted).

HW7 (due Tuesday Oct. 23)

reading:
Hartle:
Ch. 15, if you haven't finished it.

problems:

15-16 (AGN lifetime estimate) [Estimate an upper limit to the lifetime, assuming that
none of the rotational energy goes into the black hole itself, which is to say that the
irreducible mass, or equivalently the area, remains constant.]

15-18 (pair production distance estimate near a rotating black hole)

7S-1 Suppose two Kerr black holes, each of mass M and angular momentum J,
collide and coalesce and radiate away energy and angular momentum in
gravitational waves and eventually settle down to a final single black hole.
Determine the upper bound on radiated energy (as a fraction of M) allowed by
Hawking's area theorem. What is the result for the extremal case J=M^2,
and the non-spinning case J=0?

7S-2 (surface gravity of Kerr) In problem 5S-1 you computed the surface gravity
of Schwarzschild as the force exerted at infinity to suspend a unit mass at the horizon.
Use the same method for Kerr, suspending the mass along the axis of rotation at the
pole, and show that you obtain (15.37). (It is a general theorem that the surface gravity,
properly defined, is constant over the horizon.) [Hint: You may wind up with a
result that doesn't look as if it is equal to (15.37), but in fact they are equal. This took
me a while to see, even though it's just elementary algebra! But the algebra is simpler if
you write g_tt as g_tt = -(Delta - a^2 sin^2(theta))/rho^2, and use the fact that Delta = 0 at the
horizon, and sin(theta)=0 on the axis.]

7S-3 (Hawking temperature and analytic continuation) As you may have learned elsewhere,
a thermal state at temperature T in quantum mechanics has the feature that the correlation between
observables separated by a given time is periodic if that time is translated by the imaginary amount
ihbar/T. There is a slick way to use this to obtain the Hawking temperature.
Let's do this in a few steps, and learn about the Unruh acceleration temperature along the way.
(There are a lot of words in the following, but all the computations are totally trivial. If you are doing
something complicated, then you are not doing it right!)

a) Consider Minkowski spacetime in Rindler coordinates: ds^2 = - L^2 dw^2 + dL^2. (i) Replace
w by an imaginary coordinate ib, and find the period of b required for the resulting Euclidean
signature space to be smooth at L = 0 (i.e. no conical singularity, so it is just the Euclidean plane).
[Hint: L = 0 is a coordinate singularity, since ds²=0 even when dw is nonzero. In order for
L= 0 to be a regular point in a Euclidean space you must interpret it as the origin of polar coordinates,
with L identified as the radial coordinate and w identified as the angular coordinate.]
(ii) Assuming this can be interpreted as the thermal periodicity mentioned above, what is
the corresponding temperature? (Since w is a dimensionless, hyperbolic angle coordinate,
your temperature will have dimensions of hbar rather than energy.) (In fact, the Minkowksi
vacuum state of quantum fields, restricted to the Rindler wedge, is indeed "thermal" at this temperature.)

b) The proper time along the hyperbola L = L_0 in Rindler coordinates is L_0 dw. When w is imaginary,
this hyperbola becomes a circle. What is the circumference of this circle, given what you found in part a),
and what is the corresponding proper temperature? (This is the physical temperature that an observer
following that hyperbola would perceive when the quantum field is in the Minkowski vacuum state.)

c) (i) Find the acceleration a of the hyperbolic worldline in terms of L_0. (ii) Use this to show that the
temperature you found in part b) is equal to the famous Unruh acceleration temperature, hbar a/2\pi.
[Hint: The worldline of the hyperbola in Minkowski coordinates is (t,x) = L_0 (sinh w, cosh w, 0, 0).
Find the 4-acceleration by differentiating twice with respect to the proper time along the worldline,
and compute the magnitude of the acceleration 4-vector.]

d) Consider the Schwarzschild black hole line element, using proper spatial distance L from the
horizon rather than the usual area radius coordinate: ds^2 = - N^2(L) dt^2 + dL^2 + r^2(L) dO^2.
We don't need to find N(L) explicitly. At the horizon we have N(0) = 0. Expanding N(L) in powers of L
around the horizon, the lowest order term is N(L) = kappa L. (i) Show that this kappa agrees with the
definition of surface gravity from problem 5S-1 or 7S-2. (ii) Replace t by an imaginary time coordinate
ib, and find the period of b required for the resulting Euclidean Schwarzschild space to be
smooth at L = 0 (no conical singularity). (iii) Show that the temperature corresponding to this period
is the Hawking temperature T_H = hbar kappa/2\pi.

7S-4 (BH thermo and practice with Planck units)
(a) Estimate the entropy of the sun and compare it to that of a solar mass black hole.
(b) If a single proton is dropped into a solar mass black hole, roughly how much does its
entropy go up and by roughly many years does its Hawking lifetime increase?

Keep only orders of magnitude (powers of ten) as you calculate in these problems, dropping all
coefficients of order unity. Do the computation using Planck units (G=c=\hbar=1). As always,
set Boltzmann's constant to unity as well. The idea is to learn to do this sort of calculation
without looking up any numbers. You should commit to memory the following, in which I keep
only the power of 10:
L_P = 10^-33 cm
t_P = 10^-43 s
M_p = 10^-5 g
E_p = 10^19 GeV
Treat the sun as a ball of thermal radiation at the temperature 10^7 K, which is close to the
temperature at the core. The entropy density of thermal radiation is (I think)
(4 \pi^2/45) (T/\hbar c)^3, i.e. ~T^3 in Planck units. Note that 1 eV = 11,600 K ~ 10^4 K,
and the radius of the sun is about 700,000 km ~ 10^6 km. For part (b) just consider
the variation of the entropy and lifetime to first order in the mass variation. The mass of the
sun is ~ 10^33 g, the mass of a proton is ~ 1GeV/c^2, and one year is \pi x 10^7 s (to within
half a percent!).

HW6 (due Tuesday Oct. 16)

reading:
Hartle: Ch. 15
(we'll do Ch. 14 later)

problems:

15-3 (EF coordinates for Kerr) Treat the problem in the book as part (a).
(b) Show that the lines of constant v, theta and psi are null geodesics with
nonzero angular momentum (except for those coming in along the axis of
rotation which have zero angular momentum), on which r is an affine parameter. (c) Show that,
unlike EF for Schwarzschild, the surfaces of constant v surface are timelike,
rather than null. (You can check your result e.g. in MTW (Gravitation, by Misner,
Thorne and Wheeler), or in http://arxiv.org/abs/gr-qc/9910099.)

5S-1
(a) The Boyer-Lindquist angle accumulated by an infalling zero angular momentum, unit energy
particle is discussed in Example 15.1. Show that the accumulated angle goes to infinity as the
particle crosses the horizon. [Hint: It is convenient to write Delta as (r - r_-)(r - r₊), and then to expand about r₊.
If you use eqn (15.23), note that the a^2/r^2 term in the radical should have the opposite sign.]

(b) This is bizarre, and suggests that the BL angle is a bad coordinate at the horizon.
Show that the accumulated EF angle psi defined in prob. 15-3 is finite.

15-6 (The surface r = r₊ in Kerr is a null surface.) Treat the problem in the book as part (a).
Since the Boyer-Lindquist coordinates t and phi are singular at the horizon, one might and should worry about
whether the null generator (15.10) is really a finite vector. To alleviate any such concern, do this problem
in BOTH BL coordinates and EF coordinates. (b) Prove the same result a different way: show that the
three-dimensional horizon surface has at least two independent spacelike tangent vectors at each point,
and the determinant of the metric restricted to the surface is zero in EF coordinates, which are
known to be regular there. (Had we shown that that determinant vanishes in BL coordinates, we wouldn't be
sure without further analysis whether this was a result of the coordinates being bad on this surface, or
because it is truly a null surface.)

5S-2
(a) (i) What is the dimensionless ratio J/M² for a uniform sphere of mass M, radius R, and angular velocity
Omega in Newtonian physics? (We are using geometrical units, G=c=1.)
Write your answer in terms of R, the tangential velocity at the surface, and the Schwarzschild radius of the mass M.
(ii) Apply your result to the earth and the sun to obtain a crude estimate for J/M^2 for these bodies.
(iii) If the sun could collapse without shedding any mass or angular momentum, is your calculation precise
enough to detemrine if it would it form a black hole or a naked singularity?

(b) (i) Find J/M^2 for a Newtonian binary system of two bodies with mass m in a circular orbit of radius R,
treating M=2m as the total mass, i.e. neglecting binding energy and kinetic energy.
(ii) Find the conditions for which the binary satifies J > M^2, under the above assumptions.

HW5 (due Tuesday Oct. 9)

reading:
Hartle:
Ch. 12
Ch. 13

problems:

12-10 (non-radial light rays in a spacetime diagram projected to two dimensions)

12-13(a) (feet first into a black hole) Skip part (b). Answer all questions in part (a)
and sketching BOTH an Eddington-Finkelstein and a Kruskal diagram of the situation.
Show worldlines of the head and feet, and several light rays that leave the feet at
different times.

12-15 (escape from near a black hole) [Hints: 1) Assume the rocket ejects the
fuel all at one instant. 2) Assume that the ejecta has the minimum
possible energy it can have for a given momentum, i.e. it has a null 4-momentum
vector. 3) Use local energy-momentum conservation. 4) Use the conservation of
Killing energy for the escaping rocket. 5) There are many ways to handle
the algebra, but I liked doing it using invariants (rather than 4-momentum
components), since that kept things simple.]

15-10 (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:

(a) 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),
as described in section 12.1. 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 (taking v to increase toward the future).

(b) Make an EF diagram like Fig. 12.2, showing the constant v and constant R lines, 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" for an observer at r = 0. I.e. no signal from beyond
r = R can ever reach the observer.

(c) Make another diagram corresponding to the opposite sign choice for the dv dr term and h(r).
This illustrates how r = R is also a "past horizon" for an observer at r=0. I.e. no signal issuing
from the observer can ever cross beyond r=R.
Hunh? How can r = R be both a future and a past horizon??!

5S-1 (surface gravity) Consider a mass m at radius r suspended from a non-stretchable
string running from r to infinity above a nonrotating black hole (Schwarzschild spacetime).
If the string is lifted a proper distance dl at infinity, the work dW done by the lifter at infinity
is equal to the change of the Killing energy of the mass. (a) Using this set-up, find the
force per unit mass exerted at infinity to slowly lift---or just hold---the mass at radius r.
(b) What is the surface gravity, i.e. the limit of this force per unit mass as r approaches
the horizon? Give your answer as a function of the black hole mass M. (It should agree
with the definition from hw3, problem 3b, though that definition looks rather different.)

5S-2 (thermodynamics for nonspinning black holes) Show that for a small change of mass
of a Schwarzschild black hole, dM = (kappa/8\piG) dA, where kappa is the surface gravity,
A is the horizon area, and c=1.

HW4 (due Tuesday Oct. 2)

reading:
Hartle:
Ch. 9, rest of the chapter
Ch. 10
Ch. 11, skim 11.1 (at least), read 11.2,3

problems:
9-10 (velocity of orbit wrt local static observer) [Suggestions: You might do this with the help of the angular
velocity found in problem 4S-1. Alternatively, you might use the total energy to find the energy measured
by the static observer, and from that find the velocity measured by that observer by solving for v from gamma.]

9-18 (Nordstrom theory) [As the book requests, do this by finding the effective potential for null geodesics,
then find d\phi/dr and show that it is independent of the parameter M. Note that the result from last week's
homework problem 2 implies the same result by a simpler argument.]

4S-1:
(a) The textbook derives the expression (9.46) relating the angular velocity to the radius of a circular orbit
in the Schwarzschild metric using the condition that the radius is a minimum of the effective potential.
Derive it instead using just the r-component of the geodesic equation.

(b) Show that the energy and angular momentum per unit mass on a circular orbit of Schwarzschild are
given by e = (1-2M/r)/(1-3M/r)^1/2 and l= (rM)^1/2/(1-3M/r)^1/2. [I suggest you employ units with M=1,
and restore the factors of M at the end of your calculation using dimensional analysis.]

4S-2:
(a) Orbit of closest approach: Find the closest visit one could make to a nonrotating black hole and return
to infinity without expending any rocket fuel during the trip, i.e. falling freely. Show that the limiting case is
where the probe takes an infinite amount of time to spiral in ever closer to r=3M, and this occurs on an orbit
for which angular momentum and energy are infinite, with impact parameter 27^1/2 M. How should you modify
this case so as to return to large r?

(b) Budget tourist orbit: The orbit of closest approach is only for the rich, since to get onto this orbit
requires infinite energy! Find the limiting closest approach and return trajectory that you can make for free,
i.e. starting at rest at spatial infinity. Explain how to arrange the initial conditions at infinity to be on this orbit.
(Be careful if you decide to take this trip: just a tiny error in your initial impact parameter will send you into
the bottomless pit in the potential, unless you have enough rocket fuel to escape...)

HW3 (due Tuesday Sept. 25)

reading:
Hartle: Sections 8.2,3; Sections 9.1,2,3 (only part of 9.3 this week)

problems: (linked to a pdf file)

HW2 (due Tuesday Sept. 18)

reading:
Hartle: Ch. 7 (The Description of Curved Spacetime) Sections1,2,3,4,5;
Chapter 8 (Geodesics) Section 8.1

problems:
6-14 (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. Show that the total elapsed time
for this particle is longer than for the fixed particle in part (b). Use the post-Newtonian approximation (6.26) that holds
for slow motion (compared to c) and weak gravitational fields. (Hints: (i) Make use of Newtonian energy conservation;
(ii) even with this approximation there remains a subtle step to argue that the total proper time is longer. Actually,
the exact relativistic treatment will turn out to be simpler...)

7-11 (warp drive speed)

7-12 (warp drive proper time)

2S-1. (a) Show that the line element ds² = dv² + dv dz + d x² + dy² has Minkowski signature by showing that the
metric tensor has one negative and three positive eigenvalues. (b) Since all the metric components are constants
this must be quivalent to the line element for flat spacetime. Show this directly here by finding new
coordinates t(v,z) and w(v,z) for which the line element takes the standard Minkowski form,
ds² = -dt² + d w² + d x² + dy². Since the Minkowski line element is invariant under Lorentz transformations,
t and w will be determined only up to a Lorentz transformation (and translation). You should just find a simple
choice that does the job.

2S-2. 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 gravitating mass as measured at infinity.

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
within this surface are spacelike, except ones at fixed theta and phi, which are lightlike. The surface contains no
timelike displacements.

e) Suppose an outgoing light flash is emitted from the spherical surface at r = r₀and 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 with |dr/d(tau)| greater than or equal to (2M/r - 1)^1/2, where tau 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?

HW1 (due Tuesday Sept. 11)

reading:
Organizational notes (page xxii)
Chapters 1 (Gravitational Physics), 5 (Special Relativistic Mechanics), and 6 (Gravity as Geometry)
(Chapters 2,3,4 as needed for you.)
Appendices A (Units) and D (Pedagogical Strategy)
Textbook companion website (http://wps.aw.com/aw_hartle_gravity_1/0,6533,512494-,00.html)

problems:
2-7 (a coordinate transformation)
5-1 (4-vectors and dot product)
5-3 (free particle world line)
5-13 (pion photoproduction)
S-1 (relativistic beaming)
S-2 (null vectors and null planes) a) Show that a null vector is Minkowski-orthogonal in four spacetime dimensions to a three dimensional "hyperplane" (a subspace of one dimension less than the whole space) spanned by itself and two independent spacelike vectors, and is not orthogonal to any timelike vector or any other null vector.
(You can show this using vector components in a conveniently chosen coordinate system.) The hyperplane orthogonal to a null vector is called a "null hyperplane", and is tangent to the light cone since it contains one and only one null direction. b) Sketch a light cone in 2+1 spacetime dimensions, and on that sketch show three planes that pass through the
vertex of the light cone: (i) a null hyperplane, (ii) a spacelike hyperplane (orthogonal to a timelike vector), and (iii) a timelike hyperplane (orthogonal to a spacelike vector).