Phys675  Introduction to Relativity, Gravitation and Cosmology
University of Maryland, College Park
Fall 2014, Professor: Ted Jacobson
Homework
Problems from Hartle, Gravity

HW0 - due at beginning of class, Tuesday 9/09/14

HW14 - due at 6pm, Monday 12/15/14

S14-1 (Number of e-foldings required to solve the horizon problem)

(a) The horizon problem is "solved" if the past light cones of any two points on the surface of last scattering (ls) overlap. If the universe inflates for a time t with Hubble constant H the scale factor increases by a factor of exp(Ht).  Show that the number of e-foldings (Ht) required to solve the horizon problem is approximately N=ln[(2H/H_0)(a_rh/a_0)]. Assume the inflationary period ends abruptly at "reheating" (rh). For the purpose of this calculation, the period of radiation domination between rh and ls does not make much of a contribution, so to simplify the calculation just assume a matter-dominated, flat FRW model from reheating to the present.

(b) The energy scale of inflation E_inf is defined by setting the energy density equal to (E_inf)^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 redshifts to lower temperature as the scale factor grows. Show that under these assumptions (which are roughly justified) we have N ~ ln(E_inf T_0/H_0), up to an additive constant 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_inf by the Planck energy ~10^19 GeV, and divide H_0 by the inverse Planck time ~ 5 10^-44 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 in terms of the ratio (E_inf/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.)

S14-2 Exercise 2, Baumann Lectures (m^2 phi^2 inflation)

S14-3 Exercise 3, Baumann Lectures (coherent scalar field oscillations) [The overbar indicates average over an oscillation.]

S14-4 (a) Assumimg the form (119) for the metric perturbations, derive the second order action for tensor (graviton) fluctuations (208, Baumann). (b) Derive the "canonically normalized" action (212) from (210), and the relation (213) that holds in de Sitter spacetime.  [You may simplify the calculation as follows: In part (a), set all the perturbations to zero except h_ij before evaluating the action. (Cross-terms between different types of perturbations in the SVT decomposition will not survive.) Also, neglect terms in which neither of the two h's has a derivative acting on it. (Coordinate invariance of the action implies that these must vanish.)  Hint: Pay close attention to the consequences of h_ij being transverse and traceless.]

HW13 - due at beginning of class, Tuesday 12/09/14

TASI Lectures on Inflation - Daniel Baumann
week of Dec.1 - Chapter I; Chapter II, Sections 3-5
week of Dec.8 - Chapter II, Section 6; Chapter VII

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. (Answer: z = 5/4, I think...)

13S-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 would be "in causal contact", i.e. whose past light cones intersect before the big bang, in a flat, matter-dominated FRW model. Give both the exact result and the leading order approximation neglecting 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 integrals exactly. Instead,  argue that the Omega_v  contribution can be neglected from one of the integrals, and then do that integral exactly. Evaluate the other integral numerically. (c) Compare the results for (a) and (b) and explain why they are fairly close to each other.

[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 of the angle 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?]

BONUS PROBLEM : Hartle gives in Example 18.5 on p. 388 a Newtonian interpretation of the Friedman equation in the case of zero pressure. Conversely, his eqn. (18.66) could be derived from just the Newtonian argument, except for one aspect: the number k in the Friedman equation of GR is necessarily 0,+1, or -1, whereas in the Newtonian derivation it is related to the "total mechanical energy" of the matter, and could have any value whatsoever. Find a simple physical argument explaining the GR restriction on the values of k. (If you solve this problem let's write a paper on it!)

HW12 - due at beginning of class, Tuesday 12/02/14

week of Nov. 24

19. Which Universe and Why?

problems
:

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

18-14 (spatial curvature lens)

18-24 (Einstein static universe) [Tip: It may be helpful to refer to Problem 12S-1.]

12S-1 Show that the Friedman equation and the fluid local energy conservation equation imply 3 a,tt/a = - 4πG(rho + 3p). [This is a combination of the Einstein evolution equation and the Friedman equation. It is also the equation for the trace of the tidal tensor for the isotropic geodesics, i.e. the left hand side is equal to R_ab u^a u^b = R_tt.]

12S-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.)
(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").

12S-3 (Milne, de Sitter, and Anti-de Sitter à la Friedman) In this problem let's survey all the solutions to the Friedman equation with a constant energy density (equivalently, no energy density but a cosmological constant) and k = 0, 1, or -1.   For notational simplicity, choose units with 8πG/3=1.
(a) For vanishing energy density, (i) show that there are solutions only with k = 0 or -1, (ii) find the solutions, and (iii) argue that they are both locally Minkowski spacetime, and explain what portion of Minkowski spacetime they cover.
(b) For constant negative energy density, (i) show that there is a solution only for k = -1, (ii) find the solution. [It coincides with a patch of Anti-de Sitter space.]
(c) For constant positive energy density, (i) show that there are solutions for k = 0, 1, and -1, (ii) find the solutions. [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. The k = 0 case covers half, and the k = -1 case covers a subspace of the k=0 case. For a discussion of de Sitter spacetime and seven different coordinate systems thereon see Les Houches Lectures on de Sitter Space.]

HW11 - due at beginning of class, Thursday 11/20/14

week of Nov. 17

17. The Universe Observed
18. Cosmological Models

problems
:

S11-1 (charge conservation in curved spacetime) Show that the curved spacetime Maxwell equation F^ab;b = j^a implies the charge current has vanishing covariant divergence, j^a;a=0. [In flat spacetime we have j^a,a = F^ab,ab = 0, because of the antisymmetry of F^ab and the symmetry of mixed partial derivatives, but in curved spacetime covariant derivatives don't commute, so the proof is not as simple.] Prove it two ways: (a) using the fact that the commutator of covariant derivatives involves the Riemann tensor (see Oct. 28 lecture notes), and (b) using the expression for the covariant divergence of vectors and antisymmetric tensors from hwS8-4.

S11-2 (Maxwell's equations in covariant Lorentz gauge) In problem S8-6, you used the fact that in covariant Lorentz gauge D^m A_m = 0 (where A_m is the vector potential and D_m is the covariant derivative) Maxwell's vacuum equation D^m F_mn = 0 takes the form  D^m D_m A_n = R_mn A^m, where R_mn is the Ricci tensor. Derive this equation.

S11-3 (conformal invariance of Maxwell's equations) The action for the Maxwell field is (-1/4)∫ d^4x √-g g^ab g^cd F_ac F_bd. (a) Find the stress-energy tensor T_ab = (-2/√-g) dS/dg^ab. (b) Evaluate the trace of T_ab and show that it is zero. (c) (i) Show that the Maxwell action is invariant under a "Weyl rescaling" of the metric g_ab -> W g_ab, and (ii) use this to show directly [i.e. without appeal to the result of part (a)] that the trace of T_ab is zero. (d) Argue that a blackbody distribution of electromagnetic radiation at rest in a given frame in flat spacetime has a stress-energy tensor of the perfect fluid form with pressure equal to 1/3 the energy density. (e) Use the result of (c) to argue that a solution to Maxwell's equations on a spacetime with metric g_ab is also a solution on a spacetime with metric W g_ab.

S11-4 (invariant volume element) In class we saw that for diagonal metrics that dV = d^4x √-g defines the invariant volume of the region defined by the dx's. To show that this is true for any metric, note that any metric can always be made equal to the Minkowski metric at a point, so we need only check that dV is coordinate invariant. To show this, (i) derive the transformation law for √-g under a coordinate transformation, and (ii) recall the transformation law for a multiple integral under change of coordinates.

S11-5 (conservation of scalar field stress-energy tensor) Consider a scalar field with Lagrangian density  √-g [g^ab ½ f,a f,b + V(f)].  (a) Find the field equation for f by requiring that the action be stationary under variations of f. (b) Find the stress-energy tensor. (c) Show that the covariant divergence of the stress-energy tensor is zero when the f satisfies its field equation.

S11-6 (homogeneous cosmological scalar field)  Consider the scalar field of S11-5 in a spacetime with line element -dt^2 + a(t)^2 dx^i dx^i. (a) Find the stress-energy tensor in configurations with f(t,x^i) = f(t), i.e. independent of x^i. (b) Show that this stress-energy tensor has the form of a perfect fluid, and identify the 4-velocity of the fluid, the energy density, and the pressure.

HW10 - due at beginning of class, Thursday 11/13/14

week of Nov. 10

23. Gravitational Wave Emission: Sections 1-3.

problems
:

22.5 (stress-energy tensor of a gas)

22.11 (no local gravitational energy exists) (Hint: This problem should take you 30 seconds to solve.)

S10.1 (accessing TT gauge) Consider plane wave solutions [cf. (106) in the notes, or Hartle (21.64)] to the linearized vacuum Einstein equation in Lorentz gauge. (a) Show that the residual gauge freedom allows a gauge function of the form xi_a = w_a exp(ik.x), where k_a is the wave 4-vector of the original solution, and w_a is any constant covector. (b) Show that w_a can be chosen so as to impose the supplementary gauge conditions h_0i = 0 = h [(105), or (21.66)] on the gauge-transformed solution.

S10.2 (Linearized geodesic deviation in TT gauge) (a) Show using the linearized Riemann tensor in TT gauge that the linearized tidal tensor is given by 1/2 h_ij,00 [(108) in the notes]. (b) Show by evaluating the linearized covariant acceleration" of the connecting vector [cf. the left hand side of (77) in the notes, or (21.19)] of the geodesic congruence in problem S9.1,  that it is equal to 1/2 h_ij,00 for the (TT gauge) metric in that problem, contracted with the connecting vector. (Hint: You need to compute some components of the linearized Christoffel symbol for this metric.)

S10.3 (symmetries and conserved energy-momentum currents) Suppose a spacetime has a Killing vector X^a, and on that spacetime there is some matter with a locally conserved stress-energy tensor T^ab. Show that the contraction J^a = T^ab X_b defines a conserved current. (Hint: This problem should take you 30 seconds to solve.)

S10.4 (the tracefree part of Einstein's equation)  Suppose that R_ab k^a k^b = 0 for all null vectors k^a. As shown in HW8, then R_ab = f g_ab for some function f. That is, R_ab is "pure trace", i.e., its tracefree part vanishes. (a) Show that f = R/2 + C for some constant C. Thus this equation is equivalent to the vacuum Einstein equation with an undetermined cosmological constant term. (b) Now suppose (R_ab - 8πG T_ab) k^a k^b = 0 for all null vectors k^a, where T_ab is the matter stress tensor. Show that this implies the Einstein equation with an undetermined cosmological constant term.

S10.5 (Maximally symmetric Einstein spaces) A maximally symmetric space is one with the maximum number of independent Killing vectors, i.e. the maximum dimension of the symmetry group. For example, in two dimensions, with Euclidean signature, the plane is maximally symmetric. It has two independent translational and one rotational symmetry about every point. The other maximally symmetric two-dimensional spaces are the sphere (whose symmetry group is the 3d rotation group) and the hyperbolic plane (whose symmetry group is the 3d Lorentz group SO(2,1) in 2+1 dimensions). These symmetry groups can be inferred by viewing the sphere as a surface of points at constant distance from the origin in 3d flat Euclidean space, and viewing the hyperbolic plane as a surface of points at constant proper time from the origin in 2+1d Minkowski spacetime. In general a maximally symmetric n-dimensional space has an n + n(n-1)/2 = n(n+1)/2 dimensional symmetry group. All points in a maximally symmetric space are equivalent, in that nothing in the geometry distinguishes one point from another. (Such spaces are called "homogeneous".)

(a) Argue that if a space is maximally symmetric, then its Riemann curvature tensor must be constructed entirely out of the metric and some constant with dimensions of inverse length^2, and write a formula for the Riemann tensor in terms of the metric and this constant. (b) Using your result from part (a), compute the Ricci tensor. (c) Show that the space satisfies the vacuum Einstein equation with a possibly nonzero cosomological constant. (d) If the cosmological constant is positive (negative), show that there is a constant repulsive (attractive) tidal force that is isotropic and has the same value for every congruence of timelike geodesics. (e) Consider the hyperboloid  X^2 + Y^2 + Z^2 + W^2 = T^2 + R^2 in 5d Minkowski spacetime, where R is a constant and (X,Y,Z,W,T) are Minkowski coordinates. Argue that this is a maximally symmetric 4d Lorentzian space, with symmetry group SO(4,1), and an Einstein space with positive cosmological constant. [It is called de Sitter (dS) space.] (f) Consider the "surface"  X^2 + Y^2 + Z^2 = W^2 + T^2 - R^2 in 5d spacetime with a metric diag(1,1,1,-1,-1). Argue that this is a maximally symmetric 4d Lorentzian space, with symmetry group SO(3,2), and an Einstein space with negative cosmological constant. [It is called anti-de Sitter (AdS) space. As defined here it has closed timelike curves. We can pass to the covering space" to unwrap these curves.] (g) Argue that all timelike geodesics that leave a point in AdS reconverge at another point, after the same lapse of proper time. What time is that? (Hint: Think about the analogy with the Euclidean 2-sphere.) [Note that in the limit in which the geodesic reaches the speed of light it reaches all the way to infinity and back, arriving at the same point as the finite, timelike geodesics!]

HW9 - due at beginning of class, Thursday 11/06/14

week of Nov. 3

22. The Source of Curvature
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.
Optional: Interferometric gravitational wave detection: Accomplishing the impossible by Peter Saulson

problems
:

21.18 (Birkhoff's theorem) [Note: Ignore the nonsensical "for some redefined r".]

21.19 (static weak field solution) [Clarification for part (a): For this result to hold the source must be not only "time-independent" but also "static". You may just assume that this means that the metric perturbation is time-reversal invariant.]

S9-1 (particles in plane wave metric)  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.

S9-2 Referring to the article by Scott Hughes (see above), (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).

S9-3 (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).

S9-4 rope hanging above a black hole

HW8 - due at beginning of class, Thursday 10/30/14

week of Oct. 27

21. Curvature and the Einstein Equation

problems
:

S8-1 Show that the Kronecker delta is a tensor.

S8-2 Prove the quotient rule for a vector-covector contraction. (If w_a V_a is a scalar for all vectors V^a, then w_a is a covector, and vice versa.)

S8-3 (a) Show that if a symmetric tensor T_ab satisfies T_ab V^a V^b = 0 for all vectors V^a, then T^ab = 0. (b) Show that the same conclusion follows even if V^a is restricted to be timelike. (c) Show that if V^a is restricted to be null then it follows that T_ab = f g_ab for some function f.

S8-4 In hw7 you showed for diagonal metrics that the trace of the Christoffel symbol is given by (1/√g)(√g),a  where g is the determinant of the metric. This actually holds for any Riemannian metric (which has positive determinant). If the determinant is negative then the result is (1/√-g)(√-g),a . Use this to show the following (where semicolon index represents covariant derivative):

(i) (J^a);a = (1/√-g)(√-g J^a),a  where J^a is any vector field
(ii) (F^ab);a = (1/√-g)(√-g F^ab),a where F^ab is any antisymmetric contravariant tensor field
(iii) g^ab f;ab = (1/√-g)(√-g g^ab f,a),b where f is any scalar field (this is the covariant d'Alembertian wave operator)

S8-5 (curvature of cosmological metric)
(a) Compute the Christoffel symbols and Riemann tensor BY HAND for the line element ds2 = -dt2 + a2(t) dx^i dx^i, with i=1,...,n. Express your result in terms of a and adot=da/dt.
(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) Show that for n > 1, the n+1 dimensional curvature  vanishes only if a(t) is constant.
(d) (i) Now assume n = 3 and compute the Ricci tensor, Ricci scalar, and Einstein tensor. (ii)  Show that among these metrics only flat spacetime satisfies the vacuum Einstein equation.

S8-6 (Light propagation 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.  The field strength F_mn = A_n,m - A_m,n 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).  In Lorentz gauge Maxwell's vacuum equation D^m F_mn = 0 takes the form  D^m D_m A_n = R_mn A^m, where R_mn is the Ricci tensor. 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 varies slowly compared to the phase (i.e., we are looking at "short wavelength radiation"), so that it makes sense to assume a complex solution of this form exists.

(i)  Substitute A_m into the Lorentz gauge condition and into Maxwell's equation in Lorentz gauge, and drop all terms that do not contain at least one derivative of S. Use the result to establish:
(ii) k_m := S,m is a null vector.
(iii) k^m e_m = 0.
(iv) k^m D_m e_n = 0.
(v) D_m(A^2 k^m) = 0.

By a result of last week's homework, (ii) means that the surfaces of constant phase are null surfaces,  so the phase velocity of electromagnetic waves is the speed of light. That homework also showed that k_m is tangent to affinely parametrized null geodesics. These are the "light rays".  (iii) states that the polarization is orthogonal to the light ray direction, and (iv) means that the polarization is parallel transported along the light rays. (v) implies that squared amplitude varies inversely with beam area (an easy way to see this is to integrate the divergence and use the divergence theorem; in this class we haven't yet covered integration in curved spacetime). (Tip: For parts (iv) and (v) take the components  perpendicular and parallel to e_m respectively.)

HW7 - due at beginning of class, Thursday 10/23/14

week of Oct. 20

14. A Little Rotation: Sections 1,2 carefully; 3-6 lightly
20. A Little More Math: All, except skip p. 424-425 "Working with Bases and Dual Bases",  and if it's not clear to you don't worry about the four pages of 20.4 discussing the derivation of the expression (20.54) for the covariant derivative. I will derive this in a simpler way.

problems
:

14-1 (freely falling gyroscopes) (Note: See example 20.10, which explains that (14.3) is equivalent to (20.73). )

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

20-19 (Null generators of null surfaces are geodesics)  The problem as written is incorrect unless you assume also that the surfaces f = 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. 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.)

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

20-21 (trace of Christoffel symbol)

S7-1 Consider the energy-momentum tensor (22.39, Hartle) for a perfect fluid (the symbols are defined on the previous page).

(a)  For the case with p = 0 (pressureless "dust"), show that the zero divergence condition (22.40) implies that (i) (rho u^a);a = 0 [the semicolon index is a subscript, and means covariant derivative: u^a;a = 𝛁aua], and (ii) the fluid worldlines are geodesics. (Tips: Consider the components of (22.40) 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.) (Note: (i) states that the rest energy of the dust is conserved; see (22.9), or wait for this to be explained in class.

(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.

HW6 - due at beginning of class, Thursday 10/16/14

week of Oct. 13

15. Rotating Black Holes (if you haven't yet finished)
Recommended: take a look at The Kerr Metric - review paper by Saul Teukolsky

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 supermassive rotating black hole)

S6-1 Suppose two Kerr black holes, each of mass M and angular momentum J, start far away from each other, then 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 the area theorem. What is the result for the extremal case J=M^2, and the non-spinning case J=0?

S6-2 (surface gravity of Kerr) In problem S5-3 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 gtt as
gtt = -(∆ - a2 sin2(theta))/rho2, and use the fact that ∆ = 0 at the horizon, and sin(theta)=0 on the axis.]

S6-3 (horizon geometry) Show that any 2d slice of the Schwarzschild horizon (a 3d null surface) has the same intrinsic geometry. (
The result holds also for the Kerr horizon, and in fact for any null surface generated by a null Killing vector.) [Hint: Show this using the spacetime metric written in coordinates that are regular at the horizon.]

S6-4 (embedding diagram for Schwarzschild time slice) The embedding diagram shown in Box 12.4 of Hartle shows the equatorial geometry of a Schwarzschild time slice. It can be described as the surface z = z(r) in 3d Euclidean space, where r is the radial coordinate on the xy plane. Find the function
z = z(r).

HW5 - due at beginning of class, Thursday 10/09/14

week of Oct. 6

13. Astrophysical Black Holes
15. Rotating Black Holes

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) Consider the given problem as part (a). Add: (b) Evaluate your result for R=3M, and (c) If instead the ship is suspended from a tether anchored at infinity, how much energy must be supplied to the winch to lift the ship back up to infinity? (This energy could have been stored when the ship was lowered.)

[Hints: 1) Assume the ship 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 ship. 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.

S5-1 (Rindler space) Consider a two-dimensional spacetime with line element ds2 = - x
2 dt2 + dx2. (a) Show that this is just Minkowski space, for which (x,t) are "hyperbolic polar coordinates". Do this by defining coordinates X(x,t)  and T(x,t) such that ds2 = -dT2 + dX2. (b) In the (x,t) coordinates the vector ∂/∂t  is manifestly a Killing vector. Find the expression for this vector in the (X,T) coordinates.

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the following two are due Tuesday Oct. 14
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S5-2 (Painlevé-Gullstrand coordinates) The line element for the unique spherically symmetric, vacuum solution to the Einstein equation can be written as

ds
2 = - dT2 + (dr + v dT)2 + r22,        v = (2M/r)1/2

in units with c = G =1. Note that the 3-geometry of a surface of constant T is simply flat Euclidean space! (a) Which value of r corresponds to the event horizon? Give a clear and precise explanation of your answer, using the properties of the metric extracted directly from the above expression (i.e. without reference to some other coordinate system, for example). (b) Find the coordinate transformation relating these coordinates to the usual Schwarzschild coordinates (t,r,\theta,\phi). (c) Show that the radial curves with dr = -v dT are timelike, and that T is the proper time along these curves. (d) Show that (i) the curves in part (c) are geodesics and are asymptotically at rest at infinity, and that (ii) they are orthogonal (in the sense of the spacetime metric) to the surfaces of constant T. (e) Draw a spacetime diagram of the (r,T) plane showing lines of constant r as vertical and lines of constant T as horizontal. On your diagram, indicate (i) one of the radial curves discussed in part (c), (ii) the light cone at various values of r, and (iii) a curve of constant Schwarzschild time t.

S5-3 (surface gravity) Consider a test mass m at radius r suspended from a non-stretchable string running from r to infinity above a nonrotating black hole (Schwarzschild spacetime) of mass M. (a) What is the Killing energy of the mass? (b)
Find the force per unit mass that must be exerted at infinity to slowly lift---or to just hold---the mass at radius r. (c) Evaluate the limit of this force per unit mass as r approaches the horizon. (This limit is the surface gravity. It should agree with the definition from hw4, problem S4-2(b), though that definition looks rather different.)

HW4 - due at beginning of class, Thursday 10/02/14

week of Sept. 30
11. Relativistic Gravity in Action (focus on 11.2; skim 11.1 & 11.3)
12. Gravitational Collapse and Black Holes

problems
:

9-18 (Nordstrom theory) [As the book requests, do this by finding the effective potential for null geodesics,  then find dϕ/dr and show that it is independent of the parameter M. The result from problem S4-1 implies the same result by a simpler argument.]

S4-1 (Conformal invariance of null geodesics)

S4-2 (Affine parameter on black hole horizon generators)

S4-3
(Christoffel symbol)

HW3 - due at beginning of class, Thursday 9/25/14

week of Sept. 22
9 (The Geometry Outside a Spherical Star) Sect. 3,4
10 (Solar System Tests of General Relativity) (N.B. Some of the tests described are, of course, not the best current measurements.)

problems
:

8-12  (The Hyperbolic Plane) (Hint: I would use the form of the geodesic equation derived in class (and written in the notes). The x-component equation is simple because the Lagrangian has x-translation symmetry. Instead of the y-component equation, use the fact that the tangent vector is a unit vector when the curve is parametrized by arc length.

9-10 (velocity of orbit wrt local static observer) [Suggestions: You might do this with the help of the angular velocity found in problem S3-1. 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.]

S3-1:
(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've set G=c=1. I suggest you employ units with also M=1, and restore the factors of M at the end of your calculation using dimensional analysis.]

S3-2:
(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 271/2 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...)

S3-3 (precession of perihelion of Mercury due to GR effects)
(a) Consider a perturbation of any stable circular orbit in the Schwarzschild metric, and compute the radial oscillation frequency w_r in terms of the radius r.  (Eliminate the explicit dependence on the angular momentum l by solving for l in terms of r). (b) The precession rate of the orbit in radians is w_p = w_phi - w_r, where w_phi is the angular velocity. Find the leading order precession rate by expanding in 2GM/rc^2, assumed to be small. (c) Evaluate the precession rate for the perihelion of Mercury, expressing the answer in seconds of arc per century. For the radius use the semi-major axis of Mercury's orbit. [Tip: To simplify the algebra choose units with GM = c = 1. Once you have the result, restore the dimensionful quantities using dimensional analysis.] (Answer: w_p = 3r^(-5/2),  42"/century.)

HW2 - due at beginning of class, Thursday 9/18/14

week of Sept. 15
8 (Geodesics)
9 (The Geometry Outside a Spherical Star) Sect. 1,2

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; dtau/dt is smaller on part of the journey and greater on the rest, in such a way that the total proper time is longer.)

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 equivalent 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 (in spherical polar coordinates.)

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 = r0  and v = 0. Show that the area of the wavefront grows with v for r0 > 2M, stays constant for r0 = 2M, and decreases for r0 < 2M.

f) Show that any particle (not necessarily in free fall) inside the horizon must decrease its radial coordinate at a rate 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 at beginning of class, Thursday 9/11/14

week of Sept. 1
Organizational notes (page xxii)
Chapters 1 (Gravitational Physics), 5 (Special Relativistic Mechanics)
(Chapters 2,3,4 as needed for you.)
Appendices A (Units)  and D (Pedagogical Strategy)
take a look at the Textbook companion website (http://wps.aw.com/aw_hartle_gravity_1/0,6533,512494-,00.html)

week of Sept. 8
6 (Gravity as Geometry)
7 (The Description of Curved Spacetime) Sect. 1-5
8 (Geodesics) Sect. 1

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).