Department of Physics, University of Maryland, Fall 2006, Prof. T. Jacobson

Physics 675

Introduction to

Relativity, Gravitation and Cosmology

Homework Assignments

HW13 (due Friday Dec. 15, 5pm, at Room 4115)

reading:
+ Chapter 21, Section 5
+ Chapter 23, Section 1

problems:

7-9 (curvature and limits to flattening the coordinate system) Note that in a space of any dimension the number of second partials
of the coordinates is equal to the number of first derivatives of the metric, since both have one symmetric index pair
and another independent index. Therefore the first derivatives of the metric can always be set to zero at a given point.
Consider the problem as written to be part (c). Let parts (a) and (b) be the same problem in 2 and 3 spacetime dimensions respectively.
Add a further part (d): Argue that the symmetries of the Riemann tensor R_abmn = - R_bamn = - R_abnm = R_mnab imply that
in 1,2,3, and 4 dimensions R_abcd has 0,1,6, and 21 independent components. Explain why the additional symmetry
R_[abmn] = 0 is empty in less than four dimensions, and constitutes just one condition in four dimensions, thus reducing the
count of 21 independent components to 20.

21-26 (linearized plane wave in two gauges)

13S-1 Synchronous or Gaussian normal coordinates: For any spacetime metric, one can always find coordinates (t, x^i)
such that the line element takes the form ds^2 = - dt^2 + g_ij dx^idx^j (i, j = 1, 2, 3), although the coordinates
will in general be singular beyond some region. To construct such a coordinate system, start with an arbitrary
3-dimensional spacelike surface S, labeled with coordinates x^i. At each point of S fire the geodesic orthogonal
to S and label the points on this geodesic by the x^i at the launch point together with the proper time along the
geodesic. By construction then we have g_tt = -1 everwhere, and on S we have g_it = 0. Show that the geodesic equation
for the orthogonal geodesics implies g_it,t = 0 everywhere. This shows that the line element takes the above form
everywhere, i.e. for all t (until the geodesics cross, where a single point is labeled by two different sets of coordinates).

13S-2 Cosmic String: Unified theories of elementary particle physics sometime admit configurations of the fields that are
cylindrically symmetric, time-independent, and invariant under Lorentz boosts in the direction along the symmetry axis.
In a cylindrical coordinate system (t,z,r,\phi) for which the line element has the form

ds^2 = A(r)(-dt^2 + dz^2) + B(r) dr^2 + C(r) d\phi^2,

the energy-momentum tensor for the matter that makes up the string takes the form

T_ab = diag(w(r), -w(r), T_rr, T_\phi\phi).

Note that the pressure along the axis is the negative of the energy density, so this is a highly non-Newtonian source.
The energy density w(r) vanishes outside some radius r=R, where R might be around (10^16 GeV)^-1 ~ 10^-30 cm in
a grand unified theory with "symmetry breaking scale" 10^16 GeV, and the central density might be around (10^16 GeV)^4
(in units with \hbar=c=1). In effect, the core of the string consists of false vacuum, with an energy-momentum tensor that
looks like an anisotropic form of vacuum energy. In the physical case of interest, it is a good approximation to neglect the
transverse pressures T_rr and T_\phi\phi, and to treat w(r) as a two-dimensional Dirac delta function. Then the
energy-momentum tensor takes the form

T_ab = \mu \delta^2(r) diag(1,-1,0,0),

where \mu is the energy per unit length of the string. Use this form in the problem.

(a) Show that the linearlized Einstein equation in Lorentz gauge (Hartle's (23.6)) implies the equivalent equation
box h_ab = -16\pi(T_ab - 1/2 T\eta_ab). Please note that it is assumed in deriving (23.6) that one is perturbing arount a
flat metric expressed in Minkowski coordinates, so that the background metric is diag(-1,1,1,1). In particular the
space coordinates are Cartesian.

(b) Use part (a) to solve for the static perturbation h_ab with a cosmic string source. (Hint: it reduces to Poisson's
equation in 2d with a delta function source term.) Express your answer in terms of \mu.
Choose the integration constants so that the perturbation vanishes at r=R. Your solution will blow up at both large and small r,
which means that it cannot be trusted in those limits, since it was derived under the assumption that it is small.
Find expressions for the value of r where the perturbation components are +1 and -1.

(c) Show that the linearized Riemann tensor vanishes off the string, so despite the presence of the string the spacetime
is locally flat. (It turns out that the exact Riemann tensor also vanishes off the string.)

(d) Even though the spacetime is locally flat, it is not globally flat, since there is a conical deficit. To understand this,
compute the rate of change of circumference of a circle at fixed t,z, and r, with respect to the proper radial distance
(which is not just dr). Show that to linear order in the perturbation your result is what you'd get on a cone with deficit
angle 8\pi\mu. (Another way to see this is to find a coordinate transformation that makes the metric the flat space
form in cylindrical coordinates, but with an angle whose range is 2\pi minus this deficit angle.)
If a source of light lies directly far behind a cosmic string oriented perpendicular to the line of sight,
it will produce two images, separated by this angle. So this deficit is a physical effect!

(e) Using the energy density and diameter of the string mentioned above, estimate the value of the deficit angle.
(I suggest you work in Planck units.)

HW12 (due Thursday Dec. 7)

reading:
+ Chapter 21, Sections 2-4 (Section 1 was in hw10)
+ Chapter 22
+ The Meaning of Einstein's Equation (http://arxiv.org/abs/gr-qc/0103044) (This is not required, just highly recommended.)

problems:

20-17 (Covariant derivative of the metric vanishes)

21-5 (static weak field limit of curvature)

22-5 (stress tensor of a gas)

12S-1 Compute the Riemann tensor, Ricci tensor, Ricci scalar, and Einstein tensor by hand (it's good for the soul to do this at least once in your life)
for (a) the 2d line element ds² = -dt² + a²(t)dx², and (b) the 4d, spatially 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) Are there any spatially flat, 4d RW solutions to the vacuum Einstein equation 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.

12S-3 Maxwell's equations: the electromagnetic 4-vector potential A_a is a covariant vector. The field strength is defined by F_ab = A_b,_a - A_a,_b.
As we showed in class this transforms as a covariant tensor.

(a) 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!.
(Simplifying Tip: Don't expand out the antisymmetrizer. Use (i) F_ab = 2A_[b,_a], (ii) [[ab]c] = [abc], and (iii) the fact that mixed partials commute.)

(b) Choose coordinates x⁰,xⁱ, and define the electric field by E_i = F_0i and the magnetic field by Bⁱ = -1/2 \epsilon^ijk F_jk, where \epsilon^ijk 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? (Simplifying Tip: Express the identity F_[ab,c] = 0 as epsilon^abcd F_ab,c= 0,
where epsilon^abcd is the alternating symbol and \epsilon⁰¹²³= 1. Then consider two cases: d=0 and d=i.)

(c) The rest of the Maxwell equations depend upon the spacetime metric. These take the form F^ab_;a = j^b, where the semicolon denotes
covariant derivative, the indices are raised by contraction with the inverse Minkowski metric, and where j^b is the current density whose
time component is the charge density and whose space component is the 3-current density. Show that this reduces to the remaining
Maxwell equations in 3-vector form in the case of flat spacetime with Minkowski coordinates.

(d) (i) Show that the Maxwell equation in part (d) implies j^b_;b = 0, and (ii) explain why this expresses charge conservation in flat spacetime.
(It also does in curved spacetime but understanding that is slightly more involved.) (Simplifying TIp for part (i): Either crank it out in
local inertial coordinates, or use the relation shown in class 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.)

HW11 (due Thursday Nov. 30)

reading:
Chapter 14, read Sections 1,2,3 and Box 14.1; skim Sections 4,5,6
+ Reading from last week, if you didn't finish it.

problems:

11S-1 Show that the Kronecker delta with one contravariant and one covariant index
(and equal in any coordinate system to 1 when the indices match and 0 when they differ)
is a tensor. In one sense this is obvious since upon contraction with a vector it give the vector
back, i.e. it is just the identity transformation on vectors, which is a manifestly coordinate
independent linear operation. However, show that the components of the Kronecker delta in two
different coordinate systems are related by the tensor transformation rule for a tensor of type 1-1.

11S-2 Compute the Christoffel symbols by hand for the Schwarzschild metric.
(Check Appendix B to see if you've got it right.)
(Simplifying Tip: You can just crank it out from the definition (20.53) of the Christoffel symbols,
but often a more efficient method is to write the Lagrangian we've been using (not the square root)
in the given coordinate system and then write out the Euler-Lagrange equations. Comparing with (8.42)
or Problem 1 of HW\#3 you can then just "read off" the nonzero components of the Christoffel symbols.)

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

20-20 (Killing vectors on the Euclidean plane) (Note you are to assume the metric of the Euclidean plane.
For part (c), the linear combination has constant coefficients so is obviously a Killing vector.)

11S-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 is zero.)

11S-4 Consider a gyro orbiting in a Schwarzschild metric. Adopt Schwarzschild coordinates and assume
the orbit lies in the equatorial plane theta=pi/2.
(a) Show that if the spatial projection of the spin 4-vector is initially perpendicular to the plane of the orbit,
then its r, phi, and t components vanish everywhere along the orbit.
(b) Write the differential equation for the theta component, and show that if the orbit is circular the theta component is constant.
(c) For non-circular orbits show explicitly using your result for (b) that the magnitude of the spin vector is constant, despite the
fact that the theta component is not constant. (Hint: Evaluate the time derivative of the squared magnitude.)
(d) Show that if the spatial projection of the spin 4-vector is initially in the plane of the orbit, it remains in the plane of the orbit.

HW10 (due Tuesday Nov. 21)

reading:
A Spacetime Primer (figures here), by T. Jacobson:
Ch. 2, Sections 1-3
Hartle: Ch. 20 (skip p. 424-5); Ch. 21, Section 1

problems:

19-6 (Standard rulers) The general expression for the angular size was derived in the textbook for a general FRW model.
Change the problem as follows: (a) Explain clearly in plain English (no equations) 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) Given that their angular size is larger, do more distant objects therefore
appear brighter? Explain your answer!

[Hint: For part (a), remember that the angle can be evaluated just using the intrinsic size of the object, the distance from
our comoving world line to the object at the emission time, and the the spatial geometry of the emission time slice.
Think about the implications of the fact that the scale factor becomes smaller and smaller at early times. For a different
point of view, think about what happens to the area of a spatial cross section of our past light cone as the time slice
recedes into the past toward the big bang.]

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 no galaxies are born or die.

10S-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 4 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(a_ls/a_0)^1/2 radians, or about 4 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 2\chi_c (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 can neglect
the difference between \chi_ls and \chi_horiz.]

10S-2 Number of e-foldings required to solve the horizon problem

If the universe inflates for a time t with Hubble constant H_i (as a result of a vacuum energy density 3H_i^2/8\pi G) the
scale factor increases by a factor of exp(H_i t). Assume that such inflation occurs, and ends abruptly at "reheating" (t_rh),
at which time this vacuum energy density is instantly converted to an equal thermal radiation energy density.
Assume that after reheating the radiation remains thermal and simply reshifts to lower temperature as the scale factor grows
(a simplification, but it captures the leading order physics). The time from t_rh to the last scattering time t_ls is much shorter
than the time from last scattering to the present, and most of that time is well-described by a matter-dominated flat FRW model.
Hence for the purpose of evaluating the present horizon size it is a good approximation to just consider two phases: the exponential
expansion, followed by the flat matter dominated FRW phase. Under these assumptions,

(a) Show that the number of e-foldings N = H_i t required to solve the horizon problem is N ~ ln[(2H_i/H₀)(a_rh/a₀)].
(neglecting a 1 compared to an exp(H_i t) ).

(b) Show that N ~ ln(E_iT₀/H₀), up to an additive constant of order unity (using Planck units).
Here E_i is the energy scale of inflation, defined by setting the energy density equal to (E_i)^4.

(c) Evaluate N from (b) as a function of E_i (in Planck units).

[Guidance: The horizon problem is solved if all events visible to us on last scattering surface have at least one event in their
common past. (Strictly speaking, more than this is needed to account for the homogeneity of the CMB.) Explain why
this condition of nonempty common past amounts to the statement that the conformal time (defined in (18.44)) from the
beginning to the end of inflation is approximately equal to the conformal time from the end of inflation until today.
It is ok to neglect the time between reheating and last scattering.

To evaluate the logarithm in (b) divide each quantity by the corresponding Planck unit, to make it dimensionless.
Eg. divide the energy E_i by the Planck energy ~10^19 GeV, and divide H₀ by the inverse Planck time ~ 10^-43 s.
This results in pure numbers, so it must be the correct dimensionless argument of the logarithm we would have
obtained had we kept track of the powers of G, c, hbar and k_B.]

HW9 (due Tuesday Nov. 14):

reading:
Chapter 18.7
Chaper 19

problems:

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

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.)

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.]

9S-1 (Milne universe) While we're at it, consider the case of vanishing energy density.
(a) Show that this is compatible with k = 0 or -1 but not +1, and find the solutions for the spacetime metric in these cases.
(b) Both of these correspond to flat spacetime. Explain this for the k=-1 case, with a rough 2+1 dimensional spacetime diagram sketch
showing surfaces of constant FRW time and lines of constant space coordinates.
(c) Use your sketch from part (b) to explain why the the scale factor goes to zero at some time t*, i.e. why the proper length of any fixed
spatial coordinate curve goes to zero at t*.
(d) Purely Optional: Find the explicit coordinate transformation relating the non-Minkowski to the Minkowski coordinates.

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 Nov. 7)

reading: Hartle:
Ch. 17
Ch. 18, sections 18.1-18.6

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.)
18-5 (cosmological redshift of CMB)
18-6 (cosmological redshift of timescales)
18-8 (cosmological redshift via momentum conservation)
18-14 (spatial curvature lens)

HW7 (due Tuesday Oct. 31)

reading:
1) Hartle: Ch. 16, all.
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)

7S-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.

7S-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).

7S-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 ergs per square centimeter per second?
(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 at the threshold of human detection, 10^-11 erg/cm²-s, and (iii) the 50,000 Watt WAMU radio
transmitter at a distance of 20km (assuming that spherical wavefronts are emitted).
(c) At what distance from the sun is the solar energy flux equal to the gravitational wave energy flux in part (a)?
Give your answer in light-hours.

7S-4 Derive the analog of Kruskal coordinates for the de Sitter spacetime (cf. problem 15-10, hw#5) following the
method explained in my Phys 675 lecture notes of 10/19/04 (posted at the course web page). Give the coordinate
transformation from the Schwarzschild -like coordinates as well as the metric components in the new coordinates.

HW6 (due Thursday Oct. 19)

reading:
Hartle:
Ch. 12: 12.2,3 (The rest of Ch. 12 was assigned last week.)
Ch. 13: 13.3 (The rest of Ch. 13 was assigned last week.)

problems:

Note: There are a lot of problems here, but they are mostly elementary computations.
I assign those for physical insight and practice using Planck units. The only ones
requiring some sustained thought are probably 12-15 and 6S-1.

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), do part (a) sketching BOTH
an Eddington-Finkelstein and a Kruskal diagram of the situation.

12-15 (escape from near a black hole) [Hint: Assume the rocket ejects the
fuel all at one instant, and further 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.]

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 what is the same the area, remains constant. See problem 6S-4
for some data, and do the calculation using Planck units if you like. The Planck unit of
power is c^5/G ~ 10^59 erg/s.]

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

6S-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 maximum 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?

6S-2 (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 dr 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.

6S-3 (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.

6S-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!).

HW5 (due Thursday Oct. 12)

reading:
Hartle:
Ch. 12: Intro, 12.1
Ch. 13: Intro, 13.1,2
Ch. 15: all

problems:

15-3 (EF coordinates for Kerr) (If you want to check your result you can look it up somewhere,
e.g. MTW, 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.

(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.)

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??! (We'll answer this later.)

5S-2
(a) What is the ratio J/M² for a uniform sphere of mass M, radius R, and angular velocity Omega
in Newtonian physics? Write your answer in terms of R, the tangential velocity at the surface,
and the Schwarzschild radius of the mass M.

(b) Apply your result to the earth and the sun to obtain a crude estimate for J/M^2 for these bodies.

(c) If the sun were to collapse without shedding any mass or angular momentum, would it form a
black hole or a naked singularity?

HW4 (due Tuesday Oct. 3)

reading:
Hartle:
Ch. 9 (if you have not yet read it)
Ch. 10, skim (at least)
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.]

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.]

12-6 (orbit of closest approach) [What the book says about crossing 3M makes no sense. Interpret it as
just coming close to 3M for a long time. 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 when angular momentum and energy go
to infinity, with impact parameter 27^1/2 M. How should you modify this case so as to return to large r?]

4S-1: Derive the expression (9.46) relating the angular velocity to the radius of a circular orbit using just
the r-component of the geodesic equation.

4S-2: In class we showed using the weak field, slow motion limit of the geodesic equation that the
metric perturbation h_tt is identified with minus twice the Newtonian potential, -2 Phi. Show that the same
conclusion can be reached using the gravitational redshift of frequency for a photon.
To make the connection with Newtonian gravity, note that if a particle with mass m moves in free-fall
between points separated by a gravitational potential change DPhi the fractional change in the energy
DE/E measured in a static frame is approximately (-m DPhi)/(mc^2)=-DPhi/c^2, since to leading order
the energy is mc^2. Thus DE/E=-DPhi/c^2, independent of the particle mass, so the same should apply
to photons. Einstein's relation E=hf between photon frequency and energy implies DE/E = Df/f. Use the
gravitational redshift to express Df/f in terms of h_tt, and hence to relate h_tt to Phi.

HW3 (due Tuesday Sept. 26)

reading:
Hartle: Ch. 8, first three pages; Ch. 9 (as far as you can get)

problems: (linked to a pdf file)

HW2 (due Tuesday Sept. 19)

reading:
Hartle: Ch. 6 & Secs. 7.1,2,3,4

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 (a). Use the 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 the Newtonian approximation there remains a subtle step to argue that the total proper time is longer.)
Extra credit: Find a way to show that the proper time is longer in part (d) without making the Newtonian approximation.

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 replacing v and z by new
coordinates t(v,z) and w(v,z) such that the line element takes the standard Minkowski form. 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 make 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. 12)

reading:
Organizational notes (page xxii)
Chapters 1 (Gravitational Physics) and 5 (Special Relativistic Mechanics)
Other chapters 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)
5-14 (energy of highest energy cosmic rays)
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. 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 and a tangent null hyperplane in 2+1 spacetime dimensions. Also sketch a spacelike hyperplane (orthogonal to a timelike vector) and a timelike hyperplane (orthogonal to a spacelike vector).