of Physics, University of Maryland, Fall 2006, Prof. T. Jacobson
(due Friday Dec. 15, 5pm, at Room 4115)
+ Chapter 21,
+ Chapter 23,
7-9 (curvature and limits to flattening the coordinate
system) Note that in a space of any dimension the number of second
of the coordinates is equal to the number of first derivatives of the
metric, since both have one symmetric index pair
and another independent index. Therefore the first derivatives of the
metric can always be set to zero at a given point.
Consider the problem as written to be part (c). Let parts (a) and (b)
be the same problem in 2 and 3 spacetime dimensions respectively.
Add a further part (d): Argue that the symmetries of the Riemann tensor
R_abmn = - R_bamn = - R_abnm = R_mnab imply that
in 1,2,3, and 4 dimensions R_abcd has 0,1,6, and 21 independent
components. Explain why the additional symmetry
R_[abmn] = 0 is empty in less than four dimensions, and constitutes
just one condition in four dimensions, thus reducing the
count of 21 independent components to 20.
(linearized plane wave
in two gauges)
13S-1 Synchronous or
Gaussian normal coordinates: For any spacetime metric, one can
always find coordinates (t, x^i)
such that the line element takes the form ds^2 = - dt^2 + g_ij
dx^idx^j (i, j = 1, 2, 3), although the coordinates
will in general be singular beyond some region. To construct such a
coordinate system, start with an arbitrary
3-dimensional spacelike surface S, labeled with coordinates x^i. At
each point of S fire the geodesic orthogonal
to S and label the points on this geodesic by the x^i at the
launch point together with the proper time along the
geodesic. By construction then we have g_tt = -1 everwhere, and on S we
have g_it = 0. Show that the geodesic equation
for the orthogonal geodesics implies g_it,t = 0 everywhere. This shows
the line element takes the above form
everywhere, i.e. for all t (until the geodesics cross, where a single
point is labeled by two different sets of coordinates).
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
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 +
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
(a) Show that the linearlized Einstein equation in Lorentz gauge
(Hartle's (23.6)) implies the equivalent equation
box h_ab =
-16\pi(T_ab - 1/2 T\eta_ab). Please note that it is assumed in deriving
(23.6) that one is perturbing arount a
flat metric expressed in
Minkowski coordinates, so that the background metric is diag(-1,1,1,1).
In particular the
space coordinates are Cartesian.
(b) Use part (a) to solve for the static perturbation h_ab with a
string source. (Hint: it
reduces to Poisson's
equation in 2d with a delta function source term.) Express your answer
in terms of \mu.
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
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,
produce two images, separated by this angle. So this deficit is a
(e) Using the energy density and diameter of the string mentioned
above, estimate the value of the deficit angle.
(I suggest you work in
(due Thursday Dec. 7)
+ Chapter 21, Sections
2-4 (Section 1 was in hw10)
+ Chapter 22
+ The Meaning of Einstein's
Equation (http://arxiv.org/abs/gr-qc/0103044) (This
is not required, just highly recommended.)
derivative of the metric vanishes)
21-5 (static weak field limit of curvature)
tensor of a gas)
12S-1 Compute the Riemann tensor, Ricci tensor, Ricci scalar, and
tensor by hand (it's good for the soul to do this at least once in your
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
(c) Under what conditions on a(t) does the curvature
vanish? Is your answer
precisely the same for cases (a) and (b)?
(d) Are there any
spatially flat, 4d RW solutions to the vacuum Einstein
equation besides flat spacetime?
12S-2 Consider the field equation R_ab - 1/4 R g_ab = 0. The
trace (contraction with g^ab) of the left hand side vanishes
so this does not imply R = 0, and it is really only 9 independent
(a) Show using the contracted Bianchi identity (22.50) that this
equation does however imply that R is a constant. Thus this equation is
equivalent to the vacuum Einstein equation with an undetermined
cosmological constant term.
(b) Now consider the field equation R_ab - 1/4 R g_ab = 8\pi(T_ab -
1/4 T g_ab), where T_ab is the matter stress tensor and T is its trace.
Show using the Bianchi indentity together with the local conservation
of stress energy (22.40) that this equation implies the Einstein
equation (22.51) with an additional undetermined cosmological constant
equations: the electromagnetic 4-vector potential Aa is
covariant vector. The field strength is defined by Fab = Ab,a
As we showed in class this transforms as a covariant tensor.
(a) Show that F[ab,c] = 0, and argue that this
represents four independent conditions. The bracket means the totally
i.e. sum over all permutations of abc with + sign
for even and - sign for odd permutations, and divide by the number of
(Simplifying Tip: Don't
expand out the antisymmetrizer. Use (i) Fab = 2A[b,a], (ii) [[ab]c] = [abc], and (iii) the
fact that mixed partials commute.)
(b) Choose coordinates x0,xi, and define the
electric field by Ei = F0i and the magnetic field
by Bi = -1/2 \epsilonijk Fjk, where \epsilonijk is
the alternating symbol and
\epsilon123=1. Express the content of the identity F[ab,c] = 0 in terms of the electric and
To which of the Maxwell equations does this identity
Tip: Express the identity F[ab,c] = 0 as epsilonabcd Fab,c= 0,
where epsilonabcd is the alternating symbol and
\epsilon0123 = 1. Then consider two cases: d=0 and d=i.)
(c) The rest of the Maxwell equations depend upon the spacetime metric.
These take the form Fab;a
= jb, where the semicolon denotes
covariant derivative, the indices are raised by contraction with
the inverse Minkowski metric, and where jb is the current
time component is the charge density and whose space component
is the 3-current density. Show that this reduces to the remaining
equations in 3-vector form in the case of flat spacetime with Minkowski
(d) (i) Show that the Maxwell equation in part (d) implies
jb;b = 0, and (ii) explain why this expresses
conservation in flat spacetime.
(It also does in curved spacetime but
understanding that is slightly more involved.) (Simplifying
TIp for part (i): Either crank it out in
local inertial coordinates, or use the relation shown in class between
commutator of covariant
derivatives and the Riemann tensor,
the symmetry of the Ricci tensor, and the fact that contraction of a
symmetric index pair with an
antisymmetric pair is always zero.)
(due Thursday Nov. 30)
Chapter 14, read
and Box 14.1; skim Sections 4,5,6
+ Reading from last week, if you didn't
that the Kronecker delta with one contravariant and one covariant index
(and equal in any coordinate system to 1 when the indices match and 0
when they differ)
one sense this is obvious since upon
with a vector it give the vector
just the identity transformation on vectors, which
independent linear operation.
show that the components of the Kronecker delta in two
coordinate systems are related by the
transformation rule for a tensor of type 1-1.
the Christoffel symbols by hand for the Schwarzschild metric.
(Check Appendix B to see if you've got it right.)
(Simplifying Tip: You can just
crank it out from the definition (20.53) of the Christoffel symbols,
but often a more efficient method is to write the Lagrangian we've been
using (not the square root)
given coordinate system and then write out the Euler-Lagrange
equations. Comparing with (8.42)
or Problem 1 of HW\#3
you can then just "read off" the nonzero components of the Christoffel
(Killing's equation) (This characterizes Killing vector fields in a
20-20 (Killing vectors on the Euclidean plane) (Note you are
assume the metric of the Euclidean
For part (c), the linear combination has constant
coefficients so is obviously a Killing vector.)
11S-3 (freely falling
gyroscopes) As explained in section 14.2 and in Example 20.10, the spin
of a freely falling gyroscope is parallel
transported, i.e. its covariant derivative along the free-fall geodesic
(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).
that the magnitude of the spin remains constant. (Hint: The
covariant derivative of
the metric is zero.)
11S-4 Consider a gyro orbiting in a Schwarzschild metric. Adopt
Schwarzschild coordinates and assume
the orbit lies in the equatorial plane theta=pi/2.
(a) Show that if
spatial projection of the spin 4-vector is initially perpendicular to
the plane of the orbit,
then its r, phi, and t components vanish everywhere along the orbit.
the differential equation for the theta
component, and show that if the orbit is
circular the theta component is constant.
(c) For non-circular orbits show
explicitly using your result for (b) that the magnitude of the spin
vector is constant, despite the
fact that the theta component is not
constant. (Hint: Evaluate the
of the squared magnitude.)
(d) Show that if the spatial projection of the spin 4-vector is initially in the
plane of the orbit, it remains in the plane of the orbit.
(due Tuesday Nov. 21)
Spacetime Primer (figures here),
by T. Jacobson:
Ch. 2, Sections 1-3
Hartle: Ch. 20 (skip p. 424-5);
Ch. 21, Section 1
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
redshift for any FRW model with finite horizon size today. (b) Find the
redshift beyond which the angular size
increases in a flat,
FRW model. (c) Given that their angular size is larger, do more distant
Explain your answer!
[Hint: For part (a),
the angle can be evaluated just using the intrinsic size of the object,
the distance from
our comoving world line to the object at the emission time, and the
the spatial geometry of the emission time slice.
Think about the implications of the fact that the scale factor becomes
smaller and smaller at early times.
For a different
point of view, think about what happens to the area of a spatial cross
section of our past light cone as the
recedes into the past toward the big bang.]
19-7 (Number counts of galaxies) Instead of doing the
written, find a formula for N_gal(z) that applies for any
expressing your answer in terms of the function \chi(z) defined in eqn
(19.9). Assume that no galaxies are born or die.
size of horizon at last scattering
Show that the angular radius of a causal patch on the surface of last
scattering (SLS) is about 4 degrees, assuming a
FRW model all the way back to the big bang. More explicitly, show that
the horizon of a point
on the SLS subtends an angle of 2(a_ls/a_0)^1/2 radians, or about 4 degrees as viewed today.
[Guidance: The angle
viewed today is the same as the angle subtended
from our co-moving world line at the time t_ls of
last scattering, so
do the calculation all on the t_ls surface. Referring to Fig. 19.3 (p.
407) of the textbook, this is the angle
subtended by 2\chi_c (transverse to the line of sight) viewed at a
distance of \chi_ls (these
are coordinate distances, but the
angle is their ratio which is the
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
universe inflates for a time t with Hubble constant Hi (as a
result of a vacuum
energy density 3Hi^2/8\pi G) the
increases by a factor of
exp(Hi t). Assume
that such inflation occurs, and ends abruptly at "reheating" (t_rh),
at which time
this vacuum energy density is instantly converted to an equal thermal
radiation energy density.
Assume that after
radiation remains thermal and simply reshifts to lower temperature as
the scale factor
(a simplification, but it captures the leading order physics). The time
from t_rh to the last scattering time t_ls is much shorter
than the time from last scattering to the present, and most of that
time is well-described by a
matter-dominated flat FRW
Hence for the purpose of evaluating the
present horizon size it is a good approximation to just consider two
phases: the exponential
expansion, followed by the flat matter dominated FRW phase. Under these assumptions,
Show that the number of e-foldings N = Hi t required to solve the
horizon problem is N ~ ln[(2Hi/H0)(arh/a0)].
a 1 compared to an exp(Hi t) ).
that N ~ ln(EiT0/H0), up to
an additive constant of order unity (using Planck units).
Here Ei is the energy scale of inflation, defined
by setting the
energy density equal to (Ei)^4.
N from (b) as a function of Ei (in Planck units).
[Guidance: The horizon
problem is solved if all events visible to us on last
scattering surface have at least one event
common past. (Strictly speaking, more than this is needed to
for the homogeneity of the CMB.) Explain why
this condition of nonempty common past amounts to the statement that
the conformal time (defined in (18.44)) from the
beginning to the end of inflation is approximately equal to the
conformal time from the end of inflation until today.
It is ok to neglect the time between reheating and last
evaluate the logarithm in (b) divide each
quantity by the corresponding Planck unit, to make it dimensionless.
divide the energy Ei by the Planck energy ~10^19 GeV,
and divide H0
by the inverse Planck time ~ 10^-43 s.
This results in pure numbers, so
it must be the correct dimensionless argument of the logarithm we would
kept track of the powers of G, c, hbar and k_B.]
(due Tuesday Nov. 14):
matter dominated FRW models) (Note:
In part (d) the "circumference" can be taken at constant theta and phi.)
18-16 (deceleration parameter) (Hint: Express q_0 in terms of the
second time derivative of the scale factor at t_0, and use the
derivative of the scaled
Friedman equation (18.77) to express this in terms of the cosmological
18-19 (de Sitter space) Add parts (b) and (c): treat also
and open cases (b) k=0 and (c) k = -1.
[Comment: These are
all different coordinate patches for the same spacetime! This is a
peculiarity of constant energy density.
The k=-1,0,+1 cases correspond to slicings of
the de Sitter hyperboloid by timelike, null, and spacelike slices
Only the k=+1 case covers the entire hyperboloid within the coordinate
patch. For a discussion
of de Sitter spacetime and
seven different coordinate systems thereon
see Les Houches Lectures
de Sitter Space.]
9S-1 (Milne universe) While we're at it, consider the case
(a) Show that this is compatible with k = 0 or -1 but not +1, and find
the solutions for the spacetime metric in these cases.
(b) Both of these correspond to flat
spacetime. Explain this for the k=-1 case, with a rough 2+1 dimensional
spacetime diagram sketch
showing surfaces of constant FRW time and lines of constant space
(c) Use your sketch from part (b) to explain why the the scale factor
goes to zero at some time t*, i.e. why the proper length of any fixed
spatial coordinate curve goes to zero at t*.
(d) Purely Optional: Find the
transformation relating the non-Minkowski to the Minkowski coordinates.
18-24 (Einstein static universe)
9S-2 Consider some stuff satisfying the simple "equation of
state" p = w
(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
(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
time, tearing apart everything including nuclei and nucleons (the "Big Rip").
(due Tuesday Nov. 7)
Ch. 18, sections 18.1-18.6
17-5 (homogeneity scale of the universe
from 2dF Galaxy Redshift Survey)
18-3 (particle motion
in expanding universe) (Show that what Hartle says is true only if a(t)
increases more rapidly than the square root of t.)
18-5 (cosmological redshift of CMB)
18-6 (cosmological redshift of timescales)
18-8 (cosmological redshift via momentum conservation)
(spatial curvature lens)
(due Tuesday Oct. 31)
1) Hartle: Ch. 16, all.
1-9 (or more if you like) of Listening
to the Universe with Gravitational Wave Astronomy
by Scott Hughes.
3) Optional: You might like reading Interferometric
gravitational wave detection: Accomplishing the impossible.
(http://www.iop.org/EJ/abstract/0264-9381/17/12/315/, accessible from
campus computers), by Peter Saulson
16-7 (only two
gravitational wave polarizations)
7S-1 Write the Lagrangian for test particle motion in the metric
(16.2b) and derive the result (16.8) from
the corresponding Euler-Lagrange equations.
Assume as does Hartle that the particles are freely falling
zero initial velocity.
7S-2 Referring to the
article by Scott
Hughes, (a) Derive equation (4) using dimensional analysis.
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
the electric field produced by a radiating charge dipole; (c) Verify
7S-3 (gravitational wave energy flux) LIGO expects to detect
gravitational waves at frequencies of ~ 200 Hz
that cause a dimensionless strain of deltaL/L ~ 10-21.
(a) What is the flux of energy in such waves incident on Earth in ergs
per square centimeter per second?
(b) What is the ratio of the gravitational wave energy flux in part (a)
to (i) the solar flux 1466 W/m2, (ii) the energy
sound waves at the threshold of human detection, 10-11 erg/cm2-s,
and (iii) the
50,000 Watt WAMU radio
transmitter at a
distance of 20km (assuming that spherical wavefronts
(c) At what distance from the sun is the solar energy flux equal to the
gravitational wave energy flux in part (a)?
Give your answer in light-hours.
7S-4 Derive the analog of Kruskal coordinates for the de Sitter
spacetime (cf. problem 15-10, hw#5) following the
method explained in my Phys 675 lecture notes of 10/19/04 (posted at
the course web page). Give the coordinate
transformation from the Schwarzschild -like coordinates as well as
the metric components in the new coordinates.
(due Thursday Oct. 19)
Ch. 12: 12.2,3 (The rest of Ch. 12 was assigned last week.)
Ch. 13: 13.3 (The rest of Ch. 13 was assigned
Note: There are a lot of problems here, but
they are mostly elementary computations.
I assign those for physical
insight and practice using Planck units. The only ones
requiring some sustained thought are probably 12-15 and 6S-1.
light rays in a spacetime diagram projected to two dimensions)
12-13(a) (feet first into a black hole) Skip part (b), do part (a)
an Eddington-Finkelstein and a Kruskal diagram of the situation.
12-15 (escape from near a black hole) [Hint:
Assume the rocket ejects the
fuel all at one instant, and further assume that the ejecta has the
possible energy it can have for a given momentum, i.e. it has a null
(AGN lifetime estimate) [Estimate an upper limit to the lifetime,
none of the rotational energy goes into the black hole
itself, which is to say that the
irreducible mass, or what is the same
the area, remains constant. See problem 6S-4
for some data, and do the calculation using Planck units if you like.
The Planck unit of
power is c^5/G ~ 10^59 erg/s.]
(pair production distance estimate near a rotating black hole)
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
Determine the maximum radiated energy (as a fraction of M) allowed by
Hawking's area theorem. What is the result for the extremal case J=M^2,
and the non-spinning case J=0?
6S-2 (surface gravity) Consider a mass m at radius r suspended from a
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
equal to the change of the Killing energy of the mass. (a) Using this
set-up, find the
force per unit mass exerted at infinity to slowly
lift---or just hold---the mass at radius r.
(b) What is the surface
gravity, i.e. the limit of this force per unit mass as r approaches
the horizon? Give your
answer as a function of the black hole mass M.
6S-3 (thermodynamics for nonspinning black holes) Show that for a small
change of mass
of a Schwarzschild black hole, dM = (kappa/8\piG) dA,
where kappa is the surface gravity,
A is the horizon area, and c=1.
6S-4 (BH thermo and practice with Planck units)
(a) Estimate the entropy of the sun and compare it to that of a solar
mass black hole.
(b) If a single proton is dropped into a solar mass black hole, roughly
how much does its
entropy go up and by roughly many years does its Hawking lifetime
Keep only orders of magnitude
(powers of ten) as you calculate in these problems, dropping all
coefficients of order unity. Do the computation using Planck
units (G=c=\hbar=1). As always,
set Boltzmann's constant to unity as well. The idea is to learn to do
this sort of calculation
without looking up any numbers. You should commit to memory the
following, in which I keep
only the power of 10:
L_P = 10^-33 cm
t_P = 10^-43 s
M_p = 10^-5 g
E_p = 10^19 GeV
Treat the sun as a ball of thermal radiation at the temperature 10^7 K,
which is close to the
temperature at the core. The entropy density of thermal radiation is (I
(4 \pi^2/45) (T/\hbar c)^3, i.e. ~T^3 in Planck units. Note that 1 eV =
11,600 K ~ 10^4 K,
and the radius of the sun is about 700,000 km ~ 10^6 km. For part (b)
the variation of the entropy and lifetime to first order in the mass
variation. The mass of the
sun is ~ 10^33 g, the mass of a proton is ~ 1GeV/c^2, and one year is
\pi x 10^7 s (to within
half a percent!).
(due Thursday Oct. 12)
Ch. 12: Intro, 12.1
Ch. 13: Intro, 13.1,2
Ch. 15: all
15-3 (EF coordinates for Kerr) (If
you want to check your result you can look it up somewhere,
e.g. MTW, or in http://arxiv.org/abs/gr-qc/9910099.)
(a) The Boyer-Lindquist angle accumulated by an infalling zero angular
momentum, unit energy
particle is discussed in Example 15.1. Show that the accumulated angle
goes to infinity as the
particle crosses the horizon.
(b) This is bizarre, and suggests that the BL angle is a bad coordinate
at the horizon.
Show that the accumulated EF angle psi defined in prob. 15-3 is finite.
15-6 (The surface r = r+
in Kerr is a null surface.)
(de Sitter horizon) [Hartle doesn't tell you but this is the line
element for the de Sitter spacetime.
Solve this problem in the
(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
(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.
diagram corresponding to the opposite sign choice for the dv dr term
and h(r). This
= R is also a "past horizon" for an observer at r=0. I.e. no
signal issuing from the observer can
ever cross beyond r=R. Hunh? How can r=R be both a future and a past
horizon??! (We'll answer this later.)
(a) What is the ratio J/M2 for a uniform sphere of mass M,
radius R, and angular velocity Omega
in Newtonian physics? Write your answer in terms of R, the tangential
velocity at the surface,
Schwarzschild radius of the mass M.
(b) Apply your result to the earth and the sun to obtain a crude
estimate for J/M^2 for these bodies.
(c) If the sun were to collapse without shedding any mass or angular
momentum, would it form a
black hole or a naked singularity?
(due Tuesday Oct. 3)
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
9-10 (velocity of
orbit wrt local static observer) [Suggestions: You might do this with
the help of the angular
velocity found in problem 4S-1. Alternatively,
you might use the total energy to find
the energy measured
by the static observer, and from that find the
velocity measured by that observer.]
9-18 (Nordstrom theory) [As the book requests, do this by
finding the effective potential for null geodesics,
then find d\phi/dr
and show that it is independent of the parameter M. Note that the
result from last week's
homework problem 2 implies the same result by a simpler argument.]
12-6 (orbit of closest approach) [What the book
says about crossing 3M makes no sense. Interpret it as
close to 3M for a long time. Show that the limiting case is where the
probe takes an infinite
amount of time to spiral in ever closer to
r=3M, and this occurs when angular momentum and energy go
to infinity, with
impact parameter 271/2 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
4S-2: In class we showed using the weak field, slow motion limit
of the geodesic equation that the
metric perturbation h_tt is identified with minus twice the Newtonian
potential, -2 Phi. Show that the same
conclusion can be reached using the gravitational redshift of frequency
for a photon.
To make the connection with Newtonian gravity, note that if a particle with mass m moves in
between points separated by a gravitational potential change DPhi the
fractional change in the energy
in a static frame is approximately (-m
DPhi)/(mc^2)=-DPhi/c^2, since to leading order
the energy is mc^2. Thus DE/E=-DPhi/c^2, independent of the particle
mass, so the same should apply
to photons. Einstein's relation E=hf between photon frequency and
energy implies DE/E = Df/f. Use the
gravitational redshift to express Df/f in terms of h_tt, and hence to
relate h_tt to Phi.
(due Tuesday Sept. 26)
Hartle: Ch. 8, first three pages; Ch. 9 (as far as you can get)
(linked to a pdf file)
(due Tuesday Sept. 19)
Hartle: Ch. 6 & Secs. 7.1,2,3,4
time and orbits) Change
part (d) to the following: The elapsed time for the fixed particle in
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
B starting from A. Show that the total elapsed time
for this particle is longer than for the fixed particle in part (a).
Use the Newtonian approximation (6.26) that holds
for slow motion (compared to c) and weak gravitational fields. (Hints: (i) Make use of Newtonian
(ii) even with the Newtonian approximation there remains a subtle step
to argue that the total proper time is longer.)
Extra credit: Find a way to show
that the proper time is longer in part (d) without making the Newtonian
7-12 (warp drive proper time)
2S-1. (a) Show that the line element ds2 = dv2
+ dv dz + d x2 + dy2 has Minkowski signature by
showing that the
metric tensor has one negative and three positive
eigenvalues. (b) Since all the metric components are constants
this must be quivalent to the line element for flat spacetime. Show
this directly here by replacing v and z by new
t(v,z) and w(v,z) such that the line element takes the standard
form. Since the Minkowski
line element is invariant under Lorentz
transformations, t and w will be determined only up to a Lorentz
(and translation). You should just
make a simple choice that does the job.
2S-2. The Eddington-Finkelstein
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
Find a coordinate transformation that brings it to the standard
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
this surface are spacelike, except ones at fixed theta and phi, which
are lightlike. The surface contains no
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 <
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,
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?
(due Tuesday Sept. 12)
Organizational notes (page xxii)
Chapters 1 (Gravitational Physics) and 5 (Special Relativistic
Other chapters as needed for you.
Appendices A (Units) and D (Pedagogical Strategy)
2-7 (a coordinate transformation)
5-1 (4-vectors and dot product)
5-3 (free particle world line)
5-13 (pion photoproduction)
5-14 (energy of highest energy cosmic rays)
S-1 (relativistic beaming)
S-2 (null vectors and null planes) a) Show that a null vector is
Minkowski-orthogonal in four spacetime dimensions to a three
dimensional "hyperplane" (a subspace of one dimension less than the
whole space) spanned by itself and two independent spacelike vectors,
and is not orthogonal to any timelike vector or any other null vector.
The hyperplane orthogonal to a null vector is called a "null
hyperplane", and is tangent to the light cone since it contains one and
only one null direction. b) Sketch a light cone and a tangent
null hyperplane in 2+1 spacetime dimensions. Also sketch a spacelike
hyperplane (orthogonal to a timelike vector) and a timelike hyperplane
(orthogonal to a spacelike vector).