Physics 675

Introduction to

Relativity, Gravitation and Cosmology

Fall 2005

Homework Assignments

HW13 (due Thursday Dec. 15, 5pm, at room 4115)

reading:
+ Chapter 23, Section 1
+ Chapter 24, Sections 2,3

problems:
1. 22-5 (stress tensor of a gas)

2. 24-6 (incompressible stars) Let's change the problem a bit. Do the following parts: (a) Find the pressure as a function of radius for a star of a given density and surface radius R. (b) Show that there is a maximum possible surface radius R_max, that is related to the maximum mass by R_max = 2.25 M_max (note it's a bit outside the Schwarzschild radius of M_max). (c) Evaluate R_max for the density of (i) water (1 g/cm^3) and (ii) nuclear density (~ 10^15 g/cm^3).

3. 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 perturbation h_ab with a cosmic string source. 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 G=\hbar=c=1.)

HW12 (due Tuesday Dec. 6)

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

problems:

1. 21-5 (static weak field limit of curvature)

2. 21-26 (linearized plane wave in two gauges)

3. 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.

4. 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, 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 vacuum solutions to the Einstein equation besides flat spacetime?

5. 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.

(a) Show that F_ab transforms as a covariant tensor, although A_a,_b by itself does not. (Simplifying Tip: Just write out A'_a,_b and show that the term that spoils the tensor transformation rule is symmetric in the pair ab.)

(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!. (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.)

(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? (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.)

(d) The rest of the Maxwell equations depend upon the spacetime metric. These take the form F^ab_;a = j^b, where 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.

(e) (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: Use the relation between commutator of covariant derivatives and the Riemann tensor, and the fact that contraction of a symmetric index pair with an antisymmetric pair is always zero.)

HW11 (due Tuesday Nov. 29)

reading:
Chapter 14, read Sections 1,2,3 and Box 14.1; skim Sections 4,5,6
Chapter 20. I recommend ignoring everything that talks about bases and dual bases at this stage (which is a significant part of the chapter). Also, I don't like blurring the distinction between vectors and dual vectors as Hartle does with the help of the metric and its inverse.

problems:

1. A function F(x) on spacetime induces a function F(s)=F(x(s)) on a curve with path paramter s whose derivative dF/ds is coordinate invariant. Using the chain rule we have dF/ds = (∂F/∂x^m)(dx^m/ds). Each factor changes under a coordinate transformation, but they change inversely to each other. Show explicitly that the contraction is invariant.

2. Compute the Christoffel symbols by hand for the Schwarzschild metric. (Check Appendix B to see if you've got it right.)

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. 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.

5. 20-18 (Killing's equation) (This covariant equation characterizes Killing vector fields in a coordinate independent manner.)

6. 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. 17):

reading:
A Spacetime Primer (figures here), by T. Jacobson:
Ch. 2, Sections 1-3 (and 4-5 if you like); Ch. 3, Sections 1-2 and Problem 7
Hartle, Chapter 21, Section 1 (do not read Chapter 20)

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) Do more distant objects therefore appear brighter? Explain your answer!

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 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 (a_ls/a_0)^1/2 radians or about 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.)

10S-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 on the various assumptions it is not all that sensitive to them.)

HW9 (due Thursday Nov. 10):

reading:
Chapter 18.6-7,
Chaper 19, all

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 rescaled Friedman equation (18.77).
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. 18-29 (negative vacuum energy)
8. 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 Thursday Nov. 3)

reading: Hartle:
Ch. 17, skim (at least)
Ch. 18, sections 18.1-18.5

problems:
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)

HW7 (due Thursday Oct. 27)

reading:
1) Hartle: Ch. 16, all

2) Pages 1-9 of Listening to the Universe with Gravitational Wave Astronomy (arxiv.org/abs/astro-ph/0210481), by Scott Hughes.

problems:
16-7 (only two gravitational wave polarizations)
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).
16-14 (g-wave energy flux, again)
7S-1 Derive the result (16.8) using the Lagrangian method. 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 qudrupole 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).

HW6 (due Thursday Oct. 20)

reading:
Hartle:
Ch. 12, the rest (sections 12.1-4)
Ch. 13, the rest (section 13.3)
Ch. 14, Intro, sections 14.1 & 14.4, and Box 14.1

problems:
13-7 (distance to a primordial black hole explosion)
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.]
5S-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 energy radiated allowed by Hawking's area theorem. What does your result give in the non-spinning case J=0 and the extremal case J=M^2?
5S-2 Suppose a particle of unit energy and zero angular momentum falls into a Kerr black hole. What is the total advance of the particle's azimuthal angle up to the point it crosses the event horizon?
5S-3 (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 as r approaches the horizon? Give your answer as a function of the black hole mass M.
5S-4 (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.

HW5 (due Thursday Oct. 13)

reading:
Hartle:
Ch. 12, just Intro. and Box 12.1 now
Ch. 13, just Intro and sections 1 & 2 now
Ch. 15, all

problems:
12-10 (non-radial light rays in a spacetime diagram projected to two dimensions)
12-11 (negative mass Schwarzschild spacetime) Let the book part be (a). Add parts: (b) Are massive particles attracted or repelled by the object? Justify your answer by use of the geodesic equation or effective potential. (c) If a radial photon emitted from radius r with frequency w* as measured by a static observer, what is the frequency of the photon measured by a static observer at infinity? What happens in the limit that r approaches 0?
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: 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. 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??!
15-18 (pair production estimate near a rotating black hole)

HW4 (due Tuesday Oct. 4)

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) (a) As the book requests, first 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. (b) Next, use the result from last week's homework problem 2 to find the same result in a simpler way.

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 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.

HW3 (due Tuesday Sept. 27)

reading:
Hartle: Ch. 8, first three pages; Ch. 9

problems: (linked to a pdf file)

HW2 (due Tuesday Sept. 20)

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. 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)
2S-1. Projective coordinates on a unit 2-sphere: If the sphere sits tangent to the x-y plane at the south pole, a straight line from the north pole punctures the sphere at some point and then goes on to hit the plane at some (x,y). Using this (x,y) to label the puncture point, show that the line element on the sphere takes the form ds² = (d x² + dy²) /(1 + (x² + y²)/4)².
2S-2. In class we showed 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. Show it 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-3. 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/d(tau)| >/= (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 Thursday Sept. 8)

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)
1-S (relativistic beaming)