Notes from Fall 2005 These
notes are only intended to indicate what was covered in class, and
occasionally to include some extra info. We'll see if and how they
evolve. Tu
12/05: (see 2004 Notes for more details on some of this) + Einstein field eqn with matter:
Newton's eqn sets the trace of the tidal tensor -phi,ii equal to -4\pi
\rho_mass. The trace of the relativistic tidal tensor for geodesics in
the frame of a 4-velocity u^a is -R_ab u^a u^b. Hence the field
equation should presumably take the form R_ab = 4\pi Y_ab, where Y_ab
is some tensor with the property that Y_ab u^a u^b reduces in the
Newtonian limit to the mass density. The obvious first guess is Y_ab =
T_ab, where T_ab is the energy momentum tensor. Indeed this was
Einstein's first guess, but it runs into an inconsistency with energy
conservation. The covariant divergence of T_ab vanishes, whereas the
contracted Bianchi identity implies that the covariant divergence of
R_ab is 1/2 R,a. Thus local energy conservation of matter is only
consistent with this proposed field eqn if the Ricci scalar is
constant. This is bad, since the trace of this field eqn implies R =
4\pi T, so the trace of the stress tensor would have to be constant,
which is too restrictive. For example in the Newtonian limit T is just
the negative of the mass density, which is surely not generally
constant in space and time! To fix this problem one might just say "ok,
just subtract 1/2 R g_ab from R_ab in the field equation",
producing R_ab - 1/2 R g_ab = 4\pi T_ab. This would have the
virtue that the equation is now consistent with energy
conservation---indeed it would IMPLY energy conservation---, but we
would have messed up the agreement with the Newtonian limit. To see
why, solve this last eqn for R by taking the trace of both sides,
yielding -R = 4\pi T (since the trace of g_ab is g^ab g_ab = 4).
Substituting back into the equation we reexpress the latter as R_ab
= 4\pi(T_ab - 1/2 Tg_ab). In the Newtonian limit T is just
-\rho_mass, so now the problem is evdent: if we contract the rhs with
u^a u^b we get in the Newtonian limit 4\pi(\rho_mass
-1/2\rho_mass) = 2\pi\rho_mass. So it's too small by a
factor of two. But this is easy to fix: just multiply the rhs by 2! So
we arrive at the Einstein eqn:
R_ab - 1/2 R g_ab = 8\pi
T_ab
The lhs is also called the Einstein
tensor G_ab := R_ab - 1/2 R g_ab. Eliminating R in favor
of T we can also write this instead as
R_ab = 4\pi (2 T_ab - T
g_ab)
What
happened is that there are two relativistic quantities that reduce to
the mass density in the Newtonian limit: the
energy density T_ab u^a u^b and minus the trace -T. The source of
gravitational attraction in the Einstein equation is a combination of
these two. Contracting twice with a 4-velocity u^a we find that the
source of attraction is 2 \rho + T, where \rho is the energy density.
For example, consider the general class of perfect fluid stress tensors
T_ab = \rho v_a v_b + p(v_a v_b + g_ab), where \rho is the energy
density, p is the pressure, and v^a is the 4-velocity of the fluid. For
such stress tensors, the trace is T = -\rho + 3p, so the source of
attraction is \rho + 3p. In Newtonian experience p is always much less
than \rho, since \rho includes the rest energy. However for a radiation
fluid, or a vacuum energy, or some other exotic things, the source can
be very different from the energy density, and even can be negative...
# vacuum energy: T_ab = -\rho_v g_ab. For the divergence to be
zero \rho_v must be a constant. So this type of stress tensor is
proportional to the metric. It is equivalent to a cosmological constant
term in the field equation. It is locally Lorentz invariant, produces
the
same energy density in all reference frames, and also corresponds to a
perfect fluid with p = -\rho_v. So if the energy density is positive,
the pressure is negative. In this case the source of attraction is
-2\rho_v, i.e. it is repulsion if \rho_v > 0. This is how a positive
consmological constant or vacuum energy works to explain the current
acceleration of the universe.
# dust: T_ab = \rho u_a u_b. For the divergence to be zero, we showed
two things must hold: u^a is geodesic and (\rho u^a);a = 0. So the
Einstein eqn implies the eqn of motion of the dust particles! This is a
perfect fluid with zero pressure, so the source of attraction is just
the energy density in this case.
# general perfect fluid (see above). In this case the vanishing
of the divergence implies the Euler equation for the fluid.
+ I emphasized the analogy with Maxwell theory. There one also has
something like the Bianchi identity, and the field eqn implies charge
conservation. I recalled the fact that indeed Maxwell discovered the
need for the displacement current term by requiring consistency with
charge conservation. Th
12/01: (see 2004 Notes for more details on some of this) - Index
antisymmetrizers and symmetrizers and their properties
- Another
index symmetry of Riemann: R_[abcd] = 0, which together with the other
symmetries implies R_a[bcd]=0.
- Counting independent components of Riemann: in 4d, it's like a
symmetric 6x6 matrix, minus one component, hence 6x7/2 - 1 = 20. In 3d
it works out to be 6. In 2d it's 1.
- Einstein tidal tensor: pointed out it is orthogonal on both indices
to the 4-velocity of the geodesic, so it reduces to a spatial
linear operator. Setting its trace to zero is the statement that
Newton's law of gravity holds in the frame of that geodesic. Assuming
relativity holds so that this should hold in all frames, we infer that
the Ricci tensor Rab = Rmamb
vanishes. This is the vacuum Einstein equation. The Ricci tensor
is symmetric, so has the same number of independent components as the
metric, 10 in 4d. So the vanishing of Ricci does not imply vanishing of
Riemann. (In 3d, Ricci has 6 independent components, the same as
Riemann, so the vacuum field equation implies there is no curvature at
all, hence no gravity. This means that 3d Einstein gravity does not
reduce to 3d Newtonian gravity.
I talked about various aspects of the Einstein equation and contrasted
with Newton's field equation: 10 nonlinear coupled pdes involving both
space and time derivatives. The time derivatives enter in such a way
that initial data at one time determines the field at later times...up
to an arbitary time-dependent coordinate transformation. The vacuum
equation determines the Schwarzschild solution and the Kerr solution
for example. It also determines the form of the static field outside a
star (=Schwarzschild), and it admits wave solutions, like Maxwell's
equation. Something I forgot to say: the nonlinearity corresponds
physically to the fact that a gravitational field creates more gravity.
In particular, the superposition of two solutions is not a solution.
- See-saw identity on contracted indices.
- Geometric meaning of the Einstein eqn: a spherical ball of test
particles initially at rest wrt each other distorts to an
ellipsoid with the same volume,
up through order t^2 where t is the time since they are released.
Another formulation of this: a beam of light rays with initially
circular cross section and initially not distorting, distorts to a beam
with an elliptical cross section with the same area, up through
quadratic order in the affine paramter along the null geodesic. So
spacetime acts like an astigmatic lens.
- What to replace the Newtonian mass density by? Energy density is a
component of energy-momentum 4-vector density. This is described by a
two-index tensor. We first looked at the relativistic description of a
charge density: it is the "time component" of a 4-current density, i.e.
- jau a, where ua is the 4-velocity of the
pbserver. The space components are the charge current density.
Charge conservation is imposed by the continuity equation ja;a
= 0. To describe energy-momentum 4-vector we add an index. The meaning
of the various components is as described in Hartle, Chapter 22.
Tu
11/29:
- Inverse
metric = metric with indices raised by inverse metric
- Curvature, geodesic deviation
equation: see 11/18 in 2004
Notes.
- symmetry properties of the Riemann tensor indices: see textbook and 11/23 in 2004 Notes.
- Lie bracket of two vector
fields: [X,Y]^n = X^m Y^n,m - Y^m X^n,m is a tensor. If the
partial derivatives are replaced by covariant derivatives, the
connection terms cancel because of symmetry of the Christoffel symbols.
This is also called the Lie derivative
of Y with respect to X, and also the commutator
of the two vector fields. Thinking of a vector field as a differential
operator on functions, the commutator can be defined as the commutator,
i.e. [X,Y](f) = X(Y(f))-Y(X(f)). The commutator can be seen to be
another vector field by checking the Liebniz identity: the composition
XY does not satisfy the Liebniz rule: XY(fg) = X((Yf)g + fYg) = (XYf)g
+ (Yf)(Xg) + (Xf)(Yg) + f (XYg). However if we subtract the same as
this
with X and Y reversed, the middle terms cancel, and the Leibniz rule is
satisfied.
- Newtonian gravity in terms of the tidal tensor: the infinitesimal
separation vector connecting two free-fall trajectories has
acceleration equal to the tidal tensor contracted with the separation
vector: S^i,00 = - phi,ij S^j (upstairs/downstairs indices equivalent
in Cartesian coordinates in Euclidean space). Newton's vacuum field
equation states that the tidal tensor is traceless. See 11/23 in 2004 Notes.
Tu
11/22:
-
Kronecker delta as an invariant (1,1) tensor.
- Invariants formed by contracting contra- and co-variant index pairs.
- Vectors as differential operators:
sometimes the vector Vaisidentified with the
differential operator (Va ∂/∂xa), which is nice since
it is a coordinate invariant definition. If the vector is a coodrdinte
basis vector, eg (0,0,0,1) in some coordinate system, this differential
operator is the corresponding partial derivative, ∂/∂x3.
An
example would be the time translation Killing vector in Schwarzschild
spacetime, ∂/∂t,
or the horizon generator Killing vector in Kerr, ∂/∂t + Omega_H ∂/∂phi.
- Metric tensor: covariant
tensor of type (0,2). Inverse metric: contravariant tensor of type
(2,0), denoted gab. (Exercise:
Show that the inverse metric is a tensor.) Contraction with the metric
or inverse metric converts a contravariant into a covariant index or
vice versa.
- Derivatives: Start
with "acceleration" of a curve: the second derivative of coordinates
with respect to parameter is not a vector, since the transformation
rule is spoiled by a term involving the derivative of the Jacobian. So
the condition of vanishing second derivative is not a covariant
condition. So how did we characterize geodesics before anyway? We set
to zero the variation of the action \int L ds = 0, where L = 1/2 g_ab
dx^a/ds dx^b/ds. The resulting Euler-Lagrange equation E_a = 0 sets a
covariant vector to zero. How do we know it's a covariant vector? Two
ways: 1) transform it and check (I encourage you to do this, but didn't
assign it since no one got it right last year), or 2) argue from the
fact that the action is a scalar, so its variation is a scalar. This
variation has the form \int E_a (delta
x)^a ds, and (delta x)^a is
an arbitrary vector field along the curve, so E_a must be a covector.
So what's the relation to "acceleration"? I rewrote the E_a = 0 as g_ab
A^b = 0, where A^b = (xddot^b + Gamma^b_mn xdot^m xdot^n), with
Gamma^b_mn the Christoffel symbols
(see textbook), is the covariant acceleration. The metric
is invertible, so the geodesic eqn is equivalent to the vanishing of
the covariant acceleration. Note that in a local
inertial coordinate system at a point the Christoffel symbols vanish,
so the covariant acceleration is simply equal to the second derivative
of the curve coordinates with respect to the path parameter.
I then showed explicitly that similarly the ordinary partial derivative
of a vector field does not yield a tensor field. The covariant
acceleration idea can be exported to define a covariant derivative of
vector and other tensor fields. See Chapter 20 of Hartle for the
details. I'll give a slick proof next time that these definitions yield
tensors. Note that again, in a locial inertail coordinate system at a
point the covariant derivative reduces to the ordinary partial
deriviative. In fact it can be DEFINED by that property, together
with the fact that it transforms as a tensor. But to compute with
it one often needs the formula with the Christoffel symbols in a
particular coordinate system.
I forgot to mention something crucial: the
covariant derivative of the metric tensor is zero! This easily
seen by evaluating it in a local inertial coordinate system at a point:
all terms vanish. For an exercise, write out all the terms in an
arbitrary coordinate system, and show that they combine to give zero.
- parallel transport - see
section (20.4) of the textbook. Gave the example that the tangent to a
geodesic is parallel transported along the geodesic, i.e. it is
covariantly constant. Some tensor facts:
1. Vanishing contractions
(a) If
PaVa= 0 for all Va,
then Pa= 0, and the same with the roles of Va and Pa interchanged.
(b) If QabVaWb= 0 for all Va
and Wa, then Qab= 0.
(c)
If TabVa
Vb = 0 for all Va,
then the symmetric part T(ab) = (Tab + Tba)/2
must vanish. (Use Va = Xa, Ya, and Xa + Ya to reduce this to
2(c).)
(d) The contraction of a
symmetric index pair with an
antisymmetric index pair always vanishes. 2. Inferring tensors from contractions
If PaVais a
scalar for all vectors Va, then Pais a covector.
Similarly,
if it is a scalar for all covectors Pa, then Vais a vector.
(Set the difference
between PaVa
and its coordinate
transform equal to zero, and use the result of 2(a).) Most generally,
if the contraction of T with S on some set of indices is a tensor for
all tensors S, then T is a tensor. 3. Nested symmetrizers Parentheses
around a clump of indices stands for symmetrization, i.e. sum over all permutations
divided by the number of permutations. Similarly square brackets stand
for anti-symmetrization, i.e. sum over all permutations with + sign
for even and - sign for odd permutations, divided by the number of
permutations. For example, B[abc] = (1/3!)(Babc + Bbca + Bcab - Bbac - Bcba - Bacb). Nested index
symmetrizers or antisymmetrizers can be simplified as((ab)c) = (abc) and [[ab]c] =
[abc]. Th
11/18:
-
Nucleosynthesis (see Box 19.1)
- Baryogenesis (see 11/16 notes for a reference)
- Covariant and Contravariant Vectors, Tensors
+ Coordinates: four arbitrary functions x^m labeling events in
spacetime. Other coordinates y^m(x^a) related by "smooth" functions.
How smooth? Differentiable enough times to write Einstein's equation, I
suppose. (This "differentiable structure" on spacetime is not a
metrical notion but it seems to me it has physical content. See the Spacetime
Primer for a discussion of this issue.)
+ Curve, tangent vector, tangent space:
x^m(s), s and arbitrary smooth path parameter. Consider a small
displacement between s and s+Ds on the curve. The difference x^m(s+Ds)
- x^m(s) is meaningless, since a coordinate change can make it have any
value whatsoever. If we divide by Ds and take the limit as Ds --> 0
we get the derivative dx^m/ds, which can still take any value
whatsoever, but the change under a coordinate transformation is more
tame: dy^m/ds = (∂y^m/∂x^a) (dx^a/ds), all derivatives being evaluated
at one point x^m(s). Since this is a linear
change, the operation of adding two such tangent vectors or multiplying one
by a scalar commutes with coordinate change, hence is meaningful. This
means there is a vector space associated with each point of spacetime,
called the tangent space.
+ Another way to generate a vector in flat spacetime is by taking the
gradient of a function ∂S/∂x^m. But this is really not a vector. Why
not? It transforms under coordinate change as ∂S/∂y^m = (∂x^a/∂y^m) (∂S/∂x^m), which looks the
same but upon closer inspection is different: here the new coordinate
is in the denominator whereas in the vector transformation rule the new
coordinate was in the numerator. The difference is whether it is the
Jacobian or inverse Jacobian that enters. The distinction is crucial.
Consider the function F(x^m(s)). Being a function of the path parameter
only, it is manifestly coordinate invariant, so the 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, so the
contraction is invariant!
+ A set
of quantities that transforms like the tangent vectors, by the
Jacobian, is called a contravariant
vector, or just vector.
A set that transforms like the gradient, by the inverse Jacobian, is
called a covariant vector,
or covector or dual vector or one-form or 1-form. To distinguish these we use
the index convention that vectors have upstairs (superscript) indices
and covectors have downstairs (subscript) indices. This convention
derives from the original convention that coordinates have a
superscript index. The tangent vectors inherit this superscript, while
the gradient ∂S/∂x^m
has the
coordinate in the denominator, so the index is downstairs! The
summation convention is only applied to upstairs-downstairs index
pairs, since only such sums are coordinate-invariant. The operation of
summing over the values of such an index pair is called contraction.
+ Tensors: Outer products of
vectors V^m W^n transform linearly as well, with a Jacobian factor for
each index. This is called a contravariant tensor of rank 2. This
generalizes in the obvious way to tensors of mixed type (r,s):
T^(m1...mr)_(n1...ns). Tu
11/16:
- How to
measure a(t): One way is to observe flux-redshift relation f_L(z) for
objects of known intrinsic luminosity L. Another is to observe angular
diameter-redshift relation D\phi_s(z) for objects of
known intrinsic
size s. We went through the derivation of these relations, showing how
they depend on the parameters of the cosmological model. (See textbook,
chapter 18.)
- Note on terminology: thought Hartle does not use these terms, one
often hears about "luminosity distance"
d_lum and "angular diameter distance"
d_ang. Their definitions, in terms of the above quantities, is via
f_L(z) = L/4\pi (d_lum)^2 and D\phi_s(z) = s/d_ang. Going back to the
derivations theref, we see that d_lum = d_eff (1+z) and d_ang =
d_eff/(1+z), so d_lum = d_ang (1+z)^2. A paper concerned with
observational confirmation of this relation between these two distance
measures: The distance
duality relation from X-ray and SZ observations of clusters,
Jean-Philippe Uzan, Nabila Aghanim, Yannick Mellier (http://arxiv.org/abs/astro-ph/0405620).
- Explained connection between "dimness" of the Ia supernovae and
acceleration of the expansion: If the expansion is accelerating, then
it used to be expanding not as fast, so a given amount of expansion
would take longer, so light from a given redshift would spend more time
traveling to us, so would have originated farther away, so would appear
dimmer.
- Discussed nature of CMB spectrum of angular correlations and its role
in observing flatness of space. A general review by Wayne Hu of CMB Temperature and
Polarization Anisotropy Fundamentals
(http://arxiv.org/abs/astro-ph/0210696) looks good and has lots of
references. See also the "CMB "trackback"
(http://arxiv.org/tb-display/astro-ph/0210696).
- Three independent arguments for dark energy: 1) need extra energy
density to have flatness (critical density), 2) need Universe older
than oldest stars, and 3) need negative pressure to produce
acceleration of scale factor implied by Type Ia supernovae
flux-redshift relation. There may be other arguments...
- next: inflation & nucleosynthesis Tu
11/08:
- scaled
Friedman equation and dynamics of scale factor in the general case.
- Friedman equation, dynamics of scale factor. Types of energy density:
Pressureless dust ~ 1/a^3, radiation ~ 1/a^4, and vacuum ~ constant.
Curvature term can also be interpreted as an effective energy density ~
1/a^2.
- Time dependence of scale factor in spatially flat case: dust ~ t^2/3,
radiation ~ t^1/2, vacuum ~ exp(Ht).
- Age of flat FRW matter dominated universe t_0 = 2/3H_0 ~ 9 Gy: too
short for oldest stars which are ~12 Gy.
- "Size" of observable universe: again for flat FRW matter dominated:
Dx = \int dt/a(t) = 3t_0/a_0. Projected to the time slice today t_0,
this corresponds to the distance 3t_0. Tu
11/01:
- Penrose
diagrams of collapsing black hole spacetime, and
collapse/evaporation black hole spacetime. Discussion of the topology
of the spatial slices in the maximally extended Schwarzschild
spacetime: S^2 X R, where the S^2 is the sphere of spherical symmetry,
and the R is the radial direction. Contrasted with R^3 for simple
collapse to a black hole. Discussion of the fact that no single
spacelike slice can cover the region to the future of the final
singularity as well as the interior of the black hole. Expressed my
opinion that in quantum gravity it seems most likely that the interior,
and the future of the singularity, is described by a sector of the
Hilbert space that is topologically disconnected from the outside. It
could be that our universe emerged from a black hole singularity.
Mentioned Lee Smolin's idea of natural selection of universes: if the
constants of nature undergo small changes at the birth of each baby
universe, then universes with constants that produce the most
black holes would tend to be more common in the ensemble of
universe branches. Here is a description by Lee,
a critique by Lenny
Susskind, and a discussion
of this idea from a broad set of perspectives from The Reality Club
forum at Edge.
- Cosmology: Overview of
distance scales in the universe and the evidence for homogeneity and
isotropy. Replacement of ten functions of four vairables by one
function of one variable in the general homogeneous isotropic
spacetime. The possibiities for spatial slices: R^3, S^3, H^3
(explained H^3 as the hyperboloid at constant timelike distance from a
point in Minkowski spacetime). The only freedom int he metric is the
spatial scale factor a(t).
Newtonian gravity admits (sort of) a static homogeneous isotropic
universe with constant mass density: any deviation would break the
symmetry. But in GR the scale factor can change without breaking
the symmetry. TO prevent this, Einstein in 1916 introduced the
cosmological constant term into his equations, which produced
anti-gravity that kept the universe from collapsing. He found this
implied that space has a constant positive curvature, hence is S^3, not
R^3, which made him very happy from the point of view of Mach's
principle. Friedman
in 1922 published time-dependent solutions, with expanding or
contracting universe. Einstein published a note (in the same issue of
the journal, I think) saying Friedman made a computational error, and
that no such solutions exist! Hubble's law, and its special
relativistic interpretation. Redshift from the point of view of GR. I
derived this in a simpler way than did Hartle: translational symmetry
implies that the coordinate wavelength is conserved as a wave
propagates, therefore the proper wavelength scales with the scale
factor. Hartle instead considered two light rays or wavefronts emitted
from the same position at two different times and received somewhere
else at two later times. If the wave travels at the speed of
light, then ds^2 = 0, so dx = dt/a(t). Since the integral of the
coordinate increment dx is the same for both rays, one infers a
relation between the emission and reception time intervals, which
reproduces the wavelength redshift when a(t) is nearly constant over
the emission interval. This implies that not just frequency of
radiation but the lapse of time during any process is "redshifted".
Finally, I showed how Hubble's law is recovered. The separation between
two comoving observers at one cosmic time is d(t) = a(t) (x2-x1). (The
time rate of change of this "global distance" can be arbitrarily large,
even larger than the speed of light, even though each observer world
line is timlike, i.e. travels locally at less than the speed of light.)
The rate of change yields d-dot(t) = (adot(t)/a(t)) d(t), which can be
identified with Hubble's law if the separation d(t) is sufficiently
small that a(t) changes little during the light propagation time from
x1 to x2. That is, d is the distance, d-dot is the velocity,and H_0 =
adot(t_0)/a(t_0) is the present value of Hubble's constant. Th
10/27:
- Kruskal coordinates
- Penrose diagrams Tu
10/25:
- Class
taught by Prof. Chris Reynolds of UMD Astronomy Department, on
astrophysical evidence for black holes, observational signatures of
accretion disks, and observational methodologies. You can read about
his research and view web pages covering the material he spoke to us
about at his web site, http://www.astro.umd.edu/~chris/ Th
10/20:
- Ingoing
and outgoing Eddington-Finkelstein coordinates.
- Example of flat spacetime in hyperbolic polar coordinates (a.k.a.
Rindler coordinates): ds^2 = -x^2 dt^2 + dx^2. Showed how to transform
this to ingoing and outgoing EF coordinates, as well as the usual
Minkowski coordinates. See 10/21 notes from 2004. This revealed that
the horizon at x=0 corresponds to the lines T = X AND T = -X in
Minkowski spacetime with ds^2 = -dT^2 + dX^2. I claim this shows how
the puzzle of HW#5, problem 15-10 (de Sitter space) is resolved: the
r=2M surface in the advanced EF coordinates is a different surface from
r=2M in retarded EF coordinates. I quickly drew a picture of the
similar coordinate diagram in the Schwarzschild case. Tu
10/18:
- Angular
velocity of the horizon or black hole: The combination of Killing
vectors (xi + Omega_H eta) is null on the horizon. Turns out a
fixed Omega_H
works at all latitudes, so the horizon is not
differentially rotating. Since it is a constant, this combination is a
Killing vector, which on the horizon is tangent to the horizon
generators. The angular coordinate of the horizon generators varies
with
Killing time as dphi/dt = Omega_H.
- Planck units: length, time
and mass constructed from hbar, G, and c. These are "natural" units. We
expect quantum gravity to become important at or before reaching
distances as small as the Planck length, 10^-33 cm. The Planck time is
10^-43 s and the Planck mass is 10^-5 g = 10^19 GeV.
- Hawking radiation: Zeldovich
argued that the Penrose process can occur spontaneously in the vacuum,
with a positive-negative energy vacuum fluctuation pair going
"on-shell". But to substatiate this with a calculation, one needs to
understand the initial vacuum state. The idea that this is the ground
state is problematic, since there is no ground state of the Hamiltonian
generating the usual time translations inthe Kerr background: there is
an unbounded spectrum of negative energy states in the ergoregion. To
deal with this Hawking considered the scenario where the black hole
forms from collapse of a spinning body, so the initial state would be
understood to be the usual vacuum plus the collapsing body. As a
warm-up he considered the non-rotating case, and found to his surprise
that even for a non-rotating black hole there is a continuous outgoing
stream of radiation. How
can that be? The non-rotating black hole has an ergosphere too, but it
is only inside the horizon. But that just means that the vacuum
fluctuation must sit just at the horizon, so the negative energy
partner can go on-shell inside the horizon while the positive energy
one escapes to infinity. Hawking doubted his result, but was convinced
by the fact that the spectrum is perfectly thermal, at the temperature
T_H \hbar \kappa/2\pi. This identifies the constant factor in the black
hole entropy as 1/4, S_BH = A/4L_P^2. (By the way this is a humongous
entropy for a macroscopic black hole, far more than the entropy of the
star whose collapse might have produced the black hole.) The thermality
is not surprising: what else could it be? There is no information other
than the mass of the black hole. Thermal states maximize entropy. QED.
We worked out the black hole lifetime under Hawking evaporation. I
considered the black hole as an object with surface area equal to
horizon area and temperature T_H, and used the Stefan's law luminosity
~ T^4 A. In Planck units the lifetime is proportional to M^3.
Since the mass of the sun is 10^33 g and the Planck mass is 10^-5 g
this yields ~ (M/M_sun)^3 x 10^114 Planck times. Since the Planck time
is 10^-43 s this is (M/M_sun)^3 x 10^71
sec. Since the age of the universe is 10^17 s this is 10^54 times the
age of the universe for a solar mass
black hole, and for a black hole of mass 10^-15 g it is the age of the
universe. Note the typical wavelength of the thermal radiation is of
order \hbar c/T, which for the Hawking temperature is of order the
inverse surface gravity, which for a non-rotating black hole is of
order the Schwarzschild radius. For a solar mass black hole this is of
order a kilometer, so the radiation has very low temperature, in fact
of order 10^-7 K. A primordial 10^15 g black hole that might be just
expiring today would have started with a temperature 10^18 times
larger,
i.e. 10^11 K or 10 MeV (I think a more careful estimate yields a
temperature closer to 100 MeV.)
- The Black hole information "paradox":
the Hawking radiation carries away entropy, due to the fact that it is
thermal. Together with the partners inside the black hole the
state is pure, but outside one has access only to half of the entangled
pair. This preserves the second law as the black hole evaporates, but
it means that if a black hole forms and evaporates completely, the
process is not described by a unitary quantum evolution, since that
would necessarily conserve the total entropy. I argued that there is no
paradox or contradiction with the principles of quantum mechanics, but
that one must allow for a sector of the quantum state space to split
off from the rest of the universe inside the black hole at the
singularity, a "baby universe".
This view is a minority opinion, for three reasons: 1) people are
either too afraid or too unimaginative to allow for this new branch of
the universe to be born inside, and 2) people convinced themselves
(wrongly, i maintain) that there is a problem with rampant information
loss and violation of energy momentum conservation if virtual baby
universes can be born, and 3) string theory has identified systems
related to black holes but not black holes, for which a process
related to the Hawking effect is unitary without the need to introduce
any baby universes. Regarding 3) that would take a lot of discussion to
get into, but my view is that the relationship is being misinterpreted.
Th
10/13:
- Began
with some transparencies. One showed collapse of a star with a magnetic
dipole moment to a black hole. The em field radiates away,
illustrating the no-hair property of black holes. In this case, no
magnetic dipole moment. I was asked about the charged Kerr black hole.
Indeed it has a magnetic dipole moment, but that is determined by its
charge and spin, not by another free parameter. By the way, Ted Newman
with his relativity class found the Kerr-Newman charged generalization
of the Kerr solution. The other transparencies were from Roger
Penrose's famous article, "Gravitational
Collapse: the Role of General Relativity", at the inaugural meeting
of the European Physical Society in Florence, July 1969 Rivista del
Nuovo Cimento, 1, 252 (1969). (The link is to a copy I scanned. You
need to choose View, then Rotate Clockwise in your Acrobat Reader
menu.) There is a picture of a naked singularity (!) and a picure
of a rotating scaffolding he imagined being set up to tap the negative
energy states in the ergosphere. -
Coordinate r-theta diagrams of shape of ergosurface and
the circular (r=r+) horizon inside. See Hartle Figure 15.5 for an
example. The two meet at the poles. These don't have anything to do
with the intrinsic or extrinsic geometry, but they're kinda fund to
draw anyway. For a < Sqrt[3]/2 the ergosurface is convex at the
poles, for a > Sqrt[3]/2 it is concave, and for a=1 it becomes
conical.
- Explained Penrose process in terms of local 4-momentum conservation
at the break-up point (see eq. (15.30). Discussed how 4-momentum must
be oriented relative to Killing vector in order to have negative
Killing energy. Explained that physical states must have future
pointing timelike 4-momenta, in order to have positive local energy in
all reference frames, which is a requirement for stability. Note
carefully the distinction between Killing energy and local energy.
- Showed that negative energy particle must carry negative angular
momentum, thus slowing down the black hole spin. Looked at the point
where the particle crosses the horizon to determine the maximal
efficiency of energy extraction. On the horizon both Killing fields are
spacelike, but there is a future lightlike linear combination with a
constant coefficient Omega_H. This constant is called the angular velocity of the black hole.
Showed that dE is greater than or equal to Omaga_H dL, both dE and dL
being negative and Omega_H positive. The most energy per angular
momentum is thus when dE = Omega_H dL, or in terms of the black hole
mass M and angular momentum J, dM - Omega_H dJ = 0.
- "1st law" of black hole mechanics: dM - Omega_H dJ = (kappa/8\pi G)
dA, where kappa is the surface gravity
and A is the horizon area. The surface gravity can be defined in terms
of the derivative of the Killing field at the horizon and for a
nonrotating bh can be simply defined physically as the force per unit
mass exerted on a rope at infinity to hold a particle at rest just
outside the horizon.
- 2nd law of black hole mechanics, a.k.a. Hawking's area theorem: the horizona
area cannot decrease.
- Black hole entropy: These
are strikingly like thermodynamics, and together with information
theoretic considerations of missing information regarding how a black
hole formed led Jacob Bekenstein to propose that bh's really have an
entropy proportional to A. But entropy is dimensionless, Bekenstein
estimated the missing information and arrived at the conclusion that
the entropy is a number of order unity times Area/L_P^2, where L_P =
hbar G/c^3 is the Planck length, 10^-33 cm. Putting c=1 for
convenience, multiply and divide by Planck's constant in the first law
to get
(kappa/8\pi G) dA = (hbar kappa/8\pi)
d(A/L_P^2), thus identifying the temperature as proportional to hbar
kappa. It is suitable that quantum mechanics is required to get a
finite entropy. Bekenstein proposed that the "generalized second law" should hold:
ordinary entropy outside plus the black hole entropy cannot decrease,
and he noted that it did not seem to be true for a black hole in a heat
bath colder than the bh "pseudo-temperature". He came up with a bogus
way to wiggle out of this contradiction. He originally took seriously
the idea of bh temperature, but Brandon Carter talked him out of it,
since it was patent nonsense for a bh to have a temperature. But he
should have seen it the other way: the (generalized) second law DEMANDS
that a bh has a real temperature, so it will radiate into a colder
bath. His work was visionary and deeply conceived, but he failed to
take that last step. Had he taken it, it would now be called Bekenstein
radiation! Instead, a couple of years later Hawking discovered it quite
by accident...
Tu 10/11:
- orbits of Kerr: discussed the
distinction between orbits in the equatorial plane and general orbits.
The latter actually have a third conserved quantity, the Carter
constant, which is quadratic in momentum and related to the
existence of what is called a "Killing tensor". This is not related to
a globally conserved quantity, so its evolution cannot be tracked by
measuring flux at infinity. This is a difficulty for computing the
evolution of orbits under the effects of gravitational radiation
reaction. The rest of what I said is in the textbook, except how the
ISCO can be determined. The effective potential depends on r,e
(energy per unit mass), and l (angular momentum per unit mass). To
determine these three one can use the three equations,
V'(r)=0 (circular orbit), V''(r)=0 (stable and unstable
orbits---positive and negative curvature V''(r) stationary
points---coincide) , and V(r) = (e^2 -1)/2 (definition of effective
potential).
- event horizon at r=r+, first explained as in book: surface posesses
at each point one lightlike direction and two spacelike ones orthogonal
to it. Then explained general definition of a null surface, and how to recognize
one: it has a null normal vector. Consider a surface of constant F for
some function F. The normal vector n^a at each point of the surface
satisfies g_ab n^a v^b = 0 for all tangent vectors v^b tangent to the
surface at that point. Also the gradient of the function F satisfies
F,b v^b = 0 for all v^b tangent to the surface. Therefore g_ab n^a = A
F,b for some finite nonzero constant of proportionality A. That's how
the normal and the gradient of F are related. Then the squared norm of
the normal is g_ab n^a n^b = A n^b F,b. Applying this to the surface of
constant r, F=r, we get from the second equation that the squared norm
of n^a is A n^r. from the b=r component of the first equation we have that g_ar n^a = g_rr
n^r = A, so n^r = A/g_rr, (since there are no off-diagonal metric
components g_ar in the Kerr metric in Boyer-Lindquist
coordinates). The squared norm of n^a is thus A^2/g_rr. This goes
to zero when g_rr blows up, i.e. at r=r+ and r=r-, so those surfaces
are null surfaces.
- ergosphere: region where the
time translation Killing vector is in fact spacelike. Here there are
negative energy states available. A neutron star could have an
ergosphere, but as it turns out it would be unstable. The Kerr black
hole, by contrast, is stable. Penrose process explained, as in
textbook. Th
10/06:
-
Discussed some of the observational evidence for astrophysical black
hole horizons, as presented in the review article "Black Holes in
Astrophysics" by Ramesh Narayan, http://arxiv.org/abs/gr-qc/0506078.
In particular, to explain extraordinary dimness of supermassive black
hole despite high accretion rates one seems to need the absence of a
surface. Another set of observations relate to the different behavior
of neutron star (NS) and black hole (BH) candidates (distinguished by
mass):
(i) x-ray novae: BH much dimmer than NS
in the quiescent
state; also NS spectra have a thermal component (presumably from
surface) in addition to a power law (from disk), while BH have only the
latter.
(ii) In variable state time series of NS has
higher top frequency around 1kHz,
interpreted as orbital frequency of the viscous boundary layer where
the accretion disk
meets the NS, whereas the BH disk plunges after the ISCO.
(iii) Type-I x-ray bursts: nuclear explosion of
accreted surface layer. Thousands of examples for NS and not one for BH.
- rotating black holes, Kerr spacetime:
Sections 15.1,2. I also talked a bit about the structure of the ring
singularity, the maximal analytic extension of the spacetime with a
passage through the ring, the closed timelike curves on the other side,
and the "Cauchy horizon" beyond which an observer can "see", i.e. be
causally influenced by, the singularity. Mentioned how it is believed
that this Cauchy horizon is unstable and would be covered up by a
Schwarzschild-like singularity in nature. Also elaborated on cosmic
censorship, and its various meanings, and the fact that very special,
symmetrical initial conditions have been shown (using numerical
computation) to violate it. The current statement is that no open set
in the space of initial data produce a naked singularity. BTW, it would
be great if nature is not so modest, since by seeing a singularity we
could learn something about the breakdown of our present theory of
spacetime, probably involving quantum gravity...
Tu 10/04:
- Redshift
from accretion disk. See section 11.2 of Hartle.
- Types of black holes See Ch. 12 Intro and Ch. 13 Intro & Sec. 1,2.
Th
9/29:
- Timelike
geodesics of Schwarzschild. If angular
momentum L<Sqrt[12]M
there is no circular
orbit. If L>Sqrt[12]M there is a stable and an unstable circular
orbit. The innermost unstable one is at r=3M which is reached as L goes
to infinity. The innermost stable circular orbit (ISCO) is reached when L=Sqrt[12]M,
at r=6M. Energy per unit mass on this orbit is Sqrt[8/9]=0.94. Thus 6%
of rest energy is released as gravitational radiation if a mass spirals
into this orbit in an accretion process. The particle still has the
same rest mass, but it has lost this much "Killing energy".
- Neutron star: is the ISCO outside the surface? Usually M_NS = 1.4
M_sun = 1.4 x (3km/2) = 2.1 km, so ISCO = 12.6km, while and r_NS
= 10km,
so it seems the ISCO is barely outside the neutron star.
- Precession of the perihelion of
Mercury: see notes from Tu 9/27 for how to get the shape of the
orbit, i.e. the angle as a function of radius. This gives the angle
elapsed in one orbit. Mercury has an anomalous precession: 43 seconds
of arc per century. This is about ten times smaller than the Newtonian
perturbations from the other planets. (Turns out the solar oblateness
makes for a smaller effect.) Leverrier proposed to explain this with a
new planet Vulcan inside Mercury's orbit. Einstein trembled for two
days after finding his theory gave the correct precession. In order of
magnitude, we can guess that the precession angle as a fraction of 360
degrees will be of the order of the ratio of the relativistic term
GML^2/r^3 to the Newtonian term GM/r, i.e. it will be of order L^2/r^2
= v^2 = (v/c)^2. The maximum speed of Mercury in its orbit is of order
2 x10^-4 c, the orbital period of Mercury is 88 days or about 1/4 year,
so in one year it makes 4 orbits, or 4x360x60x60 = 5,184,000 seconds of
arc. Multiply this by (v/c)^2 = 4 x 10^-8 to get 0.2'' per year
or 20'' per century.
Note: I said the solar oblateness
created a "dipole" moment of the suns gravitational potential, but this
is not correct since
there is no reflection asymmetry about the equatorial plane. Rather it is a quadrupole moment.
Each multipole order brings another factor of 1/r in the potential, so
this contributes at order 1/r^3, like the GR term.
- Lightlike orbits: for any nonzero L there is an unstable circular
orbit at L=3M, the limit of the timelike unstable circular orbits. The
bending of light is found by the same method as the precession,
integrating d\phi/dr. Another effect is the Shapiro time delay, which I
described in class. It is also explained in some detail in the textbook.
- Gravitational redshift. I
explained this in a very simple way on 9/22. Today I went over the way
Hartle presents it (p. 189-191), which is much more complicated but
which has the virtue of being more flexible and conceptually general.
We'll apply it next time to the redshift of photons from an accretion
disk. The method is this: let k be the photon 4-wavevector, and u the
observer 4-velocity. Then the observed frequency is w = -k.u. If the
observer is static, i.e. u is parallel to the time translation Killing
vector X, then u = X/|X|, so w = -k.X/|X|. But k.X is conserved along a
geodesic, since k is proportional (with fixed proportionality constant)
to
the 4-velocity. So the ratio of frequency measured by two static
observers is w_2/w_1 = |X_1|/|X_2|. Finally, the norm of the Killing
vector |X| = Sqrt[-X.X] = Sqrt[-g_tt]. Tu
9/27:
- Schwarzschild geometry:
1/ the unique vacuum spherically symmetric spacetime
(Birkhoff's theorem)
2/ indep. of t-coordinate: "time" translation
symmetry (which becomes space translation inside horizon)
3/ r is defined intrinsic to spheres of symmetry, eg
"area coordinate" (A/4\pi)^1/2, not distance to "center"
4/ M is total energy/c^2 of spacetime, and is the
Newtonian attractive mass as viewed from far away.
5/ the spacetime is asymptotically flat (as r ->
infinity)
6/ r_s = 2GM/c^2 = 3km (M/M_sun) is the
"Schwarzschild radius", location of the horizon and a coordinate
singularity in Schwarzschild coordinates.
- The relation between the Eddington-Finkelstein and Schwarzschild
coordinates is v = t + r + 2M ln(r/2M -1). In Eddington-Finkelstein
coordinates there is no coordinate singularity at r=2M. As you've seen,
it is the event horizon.
- Newtonian treatment of orbits in a central potential V(r): Energy and
angular momentum are conserved: 1/2 mv^2 + V(r) = E = constant, and m
r^2 d\phi/dt = L = constant. Use the latter to solve for d\phi/dt and
substitute in the energy conservation law, when then takes the form 1/2
m (dr/dt)^2 + V_eff(r) = E, where the effective
potential is V_eff(r) = V(r) + L^2/2mr^2. The second term
is the centrifugal barrier.
This reduces the problem of orbits to a one dimensional problem. To
reconstruct the shape of the orbit, use d\phi/dr = (d\phi/dt)/(dr/dt)
and express this as a function of r, L, and E using the energy and
angular momentum conservation laws.
- GR treatment of orbits. Again use energy and angular momentum
conservation. Need one more conserved quantity since now we have an
extra coordinate for the path: the time coordinate as a function of
proper time t(\tau). This last conserved quantity is the norm of the
4-velocity: u.u = -1. Use the energy and angular momentum expressions
to eliminate dt/d\tau and d\phi/d\tau from u.u to arrive at a reduced
conservation law for just the radial motion. This is done at the
beginning of section 9.3 of Hartle. Th
9/22:
-
Non-affine parameterized geodesic equation.
- Conserved quanties: momentum conjugate to an ignorable coordinate is
conserved. For geodesics this means that if the metric components are
independent of some coordinate, the correpsponding momentum is
conserved.
- Killing vectors: to each symmetry coordinate q, we get a vector field
\xi^m_q whose components are zero \delta ^m_q. Illustrated with
translation and rotation Killing vectors ont he Euclidean plane. (See
example 8.6 in the book.) In the case of the Eddington-Finkelstein line
element, menetioned that the v-translation symmetry is timelike out
side the horizon, but lightlike ON the horizon, and spacelike inside.
Then since the conserved quantity associated with a spacelike
translation symmetry is momentum, and since momentum is not bounded
below, black holes have negative "energy" states inside. This makes no
CLASSICAL instability, since nothing can escape the black hole. But the
QUANTUM VACUUM is unstable because of this, producing what is called
the Hawking effect.
- Gravitational redshift effect:
explained much more simply/directly than in Hartle: outgoing light ray
follows some lightlike curve. Displace this whole thing uniformly in
the EF v-coordinate, and you get an equivalent such light ray,
since the metric is v-translation invariant. So the emission and
reception of the two light rays at radii r1 and r2 respectively are
separated by the same Dv, hence the ratio of the proper time
separations is tau1/tau2 =\sqrt{g_vv(r1)/g_vv(r2)}. This is also the
ratio of the frequencies w2/w1. If r2 is at infinity where g_vv=-1,
this is just \sqrt{-g_vv(r1}}. As r1 approaches the horizon, this goes
to zero. That's why the horizon is sometimes called an infinite
redshift surface. Mentioned the the role of redshift effect in the GPS.
This is discussed by Hartle in section 6.4.
- Energy extraction: argued that can extract 100% of rest energy as
work at infinity and no more, by lowering an object into a black hole.
In Newtonian gravity, if you had point masses you could extract an
infininte energy by lowering them together. But in GR a point mass
would make a black hole, and the horizon cuts off the lowering process
just when you have extracted the rest energy and no more. Tu 9/20:
- Geodesic
eqn is actually four coupled, non-linear 2nd order ODEs. Given initial
position and 4-velocity in spacetime, a unique solution is determined.
Physically this just means that the free-fall trajectory is determined
by the initial position and velocity.
- time/light/space-llike character of a geodesic is preserved along the
geodesic. Proof: we evaluated d(gmndxm/dl dxn/dl)/dl in lics at a point and
showed that it is zero, so if it starts out negative, it stays
negative, etc.
- Newtonian limit: (v << c, gmn = etamn + hmn , hmn
<< 1): Using
the
parametrization-invariant action \int L1 dl, and the coordinate t as
the parameter, the Lagrangian is approximately given by L = 1 -
½ v2 + ½ htt
- htivi
.The
constant 1 affects nothing. The next two terms would give the same
equations of motion as the
Newtonian LagrangianLNewton
= ½ mv2 - m Phi if the metric component gtt is related to the
Newtonian potential Phi by htt = -2 Phi = 2 GM/r
, so thatgtt
= -(1 -
2GM/r). The last term is smaller, and is analogous to the
velocity-dependent coupling of a charge to the electromagnetic vector
potential. This produces a gravito-magnetic force.
- Relation between Newtonian and Einsteinian gravity theories,
form of Einstein equation. Newton's field eqn is nabla^2 Phi
= 4 pi G rho. In Einstein's theory, nabla^2 must be replaced by
something including time derivatives, Phi is just a component of the
metric, the mass density rho is replaced by energy density, which is a
component of the energy-momentum tensor. Thus all forms of
energy gravitate. Finally, even gravitaitonal energy gravitates, which
means the field equation must be non-linear (no superposition
principle). In fact, this nonlinearity is forced by the requirement
that the equations hold true in any coordinate system, that is, by the
symmetry of general covariance.
- The Einstein equation renders the metric/inertial structure of
spacetime an equal player in the dynamics, not fixed a priori. The weak
field solutions to the Einstein equation in include gravitational waves
that propagate at the speed of light.
- Reviewed the Eddingon-Finkelstein diagram and some interpretation.
Discussed horizon and trapped surfaces and the infinite stretch at the
singularity. Noted that this is but one coordinate system for the
unique vacuum spherically symmetric solution, called the Schwarzschild
solution. The uniqueness is called "Birkhoff's theorem". Th
9/15:
-
Coordinate invariant condition for a geodesic in a Euclidean signature
space: minimizes arc length between sufficiently nearby points (the
nearby condition required to exclude geodesics that go "the long way
around" like on a great circle on the earth. In a Minkowski (a..k.a.
Lorentz) signature space the inertial motion/geodesic takes the maximum proper time between
sufficiently nearby points. In either signature the length integral is
unchanged to first order when the curve is varied, i.e. it is stationary. This is a coordinate
invariant condition.
- To develop a formula for the geodesic condition we wrote the integral
as \integral L1 dl,
where
L1 = (-gmndxm/dl dxn/dl
)1/2
.
The stationarity condition is the Euler-Lagrange equations for L1
(d/dl)(L-1gan dxn/dl) -
½ L-1 gmn,a dxm/dl dxn/dl = 0.
If the curve parameter
l is chosen to be the proper time, then L1=1, so this simplifies
to
(d/dl)(gan
dxn /dl) - ½ gmn,a dxm/dl dxn/dl = 0.
This is called the geodesic equation for affinely parameterized
geodesics. In a l.i.c. at a point this geodesic eqn. reduces to d2xm
/dl2 = 0, which is how we
previously defined a geodesic.what we had before.
- This
geodesic equation also follows directly from the variational condition
with Lagrangian
L = ½ gmndxm/dl dxn/dl,
without the square root. This action is not reparametrizationinvariant,
which explains why the resulting geodesic eqn is for a special class of
paramters, i.e. affine
parameters. Any two affine parameters are linearly related. In this
form, the variational principle makes sense also for null geodesics,
for which L=0. This tells us that null geodesics have a preferred
one-parameter family of parameters, i.e. the affine ones, unlike
timelike curves which have a uniquely defined parameter, even if not a
geodesic. One can interpret the ratio of affine parameters in terms of
the limit of ratios of proper times along a timelike geodesic that
approaches the null one. This is nice, but does not explain why it is
only null geodesics that have well-defined affine parameters, rather
than all null curves...
- A student asked whether non-geodesic null
curves ever arise 'physically'. I'm looking for an example with light
rays... Tu
9/13:
-
equivalence principle: no experiment can distinguish a uniform
gavitational field from an accelerating reference frame. Einstein
inferred from this that light is bent in a gravitational field;
Einstein changes what we mean by an inertial reference frame.
- gravitational tidal forces
are what's real; described by derivative of gravitational field, i.e.
second derivative of Newtonain potential, -\phi,ij. This object can be
called the "tidal tensor".
- line
element on a curved space (eg sphere), dl^2 = g_ij dx^i dx^j, Einstein
summation convention, g_ij is the metric
tensor.
- local Cartesian coordinates at a point p: g_ij(p) = delta_ij, and
g_ij,k(p) = 0, where ,k means partial derivative wrt x^k. Spherical
polar coordinates on sphere are locally Cartesian on the equator but
not anywhere else. Also gave example of sterographic coordinates for
sphere, dl^2 = 4(dx^2 + dy^2)/(1+x^2 + y^2)^2, which are locally
inertial at x=y=0 only.
- spacetime version: let indices become greek, written here still as
latin: ds^2 = g_mn dx^m dx^m. To have Lorentz (should be MInkowski, but
people call it Lorentz) signature, ds^2=0 should define a light cone,
which is same as saying g_mn can be diagonalized to form (-1,1,1,1),
which is same as saying it has one negative and three positive
eigenvalues. Example: ds^2 = dv^2 + dv dx + dx^2 + dy^2 has Minkowski
signature.
- In a Euclidean signature space, a geodesic
is a curve with vanishing coordinate acceleration wrt arc length
when evaluated at any point p in a locally Cartesian coordinate
system at p. we argued by this reasoning the equator of the sphere is a
geodesic, but a line of latitude other than the equator need not be
(and in fact is not), despite the fact that its coordinate acceleration
vanishes!
- In a spacetime, a free-fall or inertial motion is a geodesic, with
the same defintion as above, only replacing the arc length by proper
time. Th
9/08:
-
relativistic Doppler effect; wave 4-vector k, frequency as seen by
observer with 4-velocity u is w = -k.u
- Newtonian gravity:
distinction between inertial, active gravitational, and passive
gravitational mass.
- gravity as a fictitious force, "true" gravity as the variation
gravitational acceleration, i.e. the mismatching of the local inertial
frames at different points.
- geometrical analogy with locally flat approximations to patches of a
sphere.
- free-fall as straight line motion, variation of gravity as deviation
of straight lines from remaining parallel: curvature. Tu
9/06:
- Change of
HW1 due date: now due Tuesday, 9/13.
- Massless particles
- massive particles have future timelike 4-momentum vectors, massless
ones have future lightlike 4-momentum
- Energy-momentum conservation; this is a 4-vector relation, expressing
both energy and 3-momentum conservation.
- computational technique: take the relativistic dot product with
itself of each side of the conservation equation to obtain a SCALAR
equation that combines energy and momentum conservation. These dot
products are INVARIANT under Lorentz transformation, so can be
evaluated using the components of the 4-momenta in any fixed reference
frame, chosen for convenience.
-three examples:
- impossibility of photon decay
to electron-positron pair: the sum of two future timelike 4-vectors
cannot be lightlike.
- Compton scattering of photon by
electron at rest
- Ultra-high-energy cosmic rays
(UHECR): threshold energy for protons to collide with cosmic microwave
background (CMB) photon to produce a pion. The answer
is around 3 x 10^20 eV, corresponding to a gamma of order 10^11!
- GZK cutoff on cosmic ray energy spectrum brought about by the above
process; observational ambiguity; the Auger
detector (partially built, still under construction)
should resolve this in a
year or so. See for example The Curious Adventure of
the Ultrahigh Energy Cosmic Rays by F.W. Stecker for a discussion
of the GZK cutoff and the possibility that it is missing.
Th
9/01:
- Introduction to the class.
- Newtonian compared with (flat) Einsteinian spacetime structure:
Newton: absolute time function t,
Euclidean spatial metric dl^2 = dx^2 + dy^2 + dz^2 on t=constant
surfaces
Einstein: absolute interval ds^2 =
-dt^2 + dx^2
+ dy^2 + dz^2,
defines proper time of timelike intervals and proper length of
spacelike intervals
- time dilation & twin effect
- 4-velocity u, u^2 = -1, u = gamma*(1, v), v^i = dx^i/dt and gamma =
(1-v^2)^(-1/2)
- 4-momentum p = mu, p^2 = -m^2, p = (E,p^i), E^2 = p^2 + m^2, E = m +
1/2 mv^2 + ...
Physics 675