Physics 675

Introduction to

Relativity, Gravitation and Cosmology

Homework Assignments

HW13 (due ?):

reading:
Chapter 22

problems:

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 magnetic fields.)

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).

HW12 (due Tuesday Nov. 30):

reading:
+ finish reading everything that has been assigned up to now
+ The Meaning of Einstein's Equation (http://arxiv.org/abs/gr-qc/0103044)

problems:

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)

HW11 (due Thursday Nov. 18):

reading:
Chapter 14, Sections 2,3
Chapter 21, Sections 2-5
Einstein excerpt on covariant differentiation

problems:

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 is zero.)

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.)

HW10 (due Thursday Nov. 11):

reading:
Chapter 19, Section 2
Chapter 21, Section 1

problems:

19-4 (approximate redshift-magnitude relation)
(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.]

19-6 (Standard 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 not?

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).

HW9 (due Thursday Nov. 4):

reading:
Chapter 18, Sections 6,7
Chaper 19, Section 1

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 email.

1. 18-11 (closed, matter dominated FRW models)
2. 18-14 (spatial curvature lens)
3. 18-16 (deceleration parameter) Hint: Differentiate the Friedman equation.
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 3.
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").

HW8 (due Thursday Oct. 27):

reading:
Chapter 18, Sections 1-5

problems:
1. 16-7 (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)

HW7 (due Thursday Oct. 20): This week the homework is mainly reading.

reading:
1) 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 Universe Observed

problems:
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 (5).
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?

HW6 (due Thursday Oct. 13)
reading: No new reading. Catch up if you need to.
problems:

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 A_a is a co-variant vector. The field strength is defined by F_ab = A_b,_a - A_a,_b. (a) Show that F_ab transforms as a covariant tensor, although A_a,_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 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? (d) The rest of the Maxwell equations depend upon the spacetime metric. Let's consider just flat spacetime here. Then these take the form F^ab,_a = j^b, where the indices are raised by contraction with the 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 how this reduces to the remaining Maxwell equations in 3-vector form. (e) Show that the equation in part (d) implies j^b,_b = 0, and explain why this expresses charge conservation.

3. Show that the inverse metric g^ab (defined by g^abg_bc = (\delta)^a_c) transforms as a contravariant tensor.

HW5 (due Thursday Oct. 7)

reading:
Hartle: read Ch. 13 & 15; skim Ch. 14

problems:
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: 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)

HW4 (due Thursday Sept. 30)

reading:
Hartle: Ch. 11, skim 11.1, read 11.2,3; Ch. 12, skim 12.1,2 (already covered in class), read 12.3,4

problems:
9-10 (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 g_ab and A²(x)g_ab 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.

HW3 (due Thursday Sept. 23)

reading:
Hartle: Ch. 9

problems:
1. The geodesic equation
(d/dl)(g_an dxⁿ/dl) - ½ g_mn,_a dx^m/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 element.

HW2 (due Thursday Sept. 16)

reading:
Hartle: Ch. 6; Secs. 7.1,2,3,4; Sec. 8.1, first three pages
Spacetime Primer (see the course syllabus): Secs. 2.1,2,3 (and 2.4,5 if you like); Ch. 3

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. Calculate the total elapsed time for this particle. Use the approximations described at the end of problem 6-13.
7-11 (warp drive speed)
7-12 (warp drive proper time)
7-20 (embedding diagram of spatial slice of Schwarzschild black hole)
S1. Show that if T_abV^a V^b=0 for all V^a, then the symmetric part T_(ab)= (T_ab + T_ba)/2 must vanish.
S2. 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 this field.

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
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 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?

HW1 (due Thursday Sept. 9)

reading:
Organizational notes (page xxii)
Chapters 1 and 5
Appendices A and D
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)
5-17 (relativistic beaming)[See problem with better notion in Errata for Printings 1-3 (pdf) at the book companion website.]