Notes

Extensive notes were written for Fall 2004 and Fall 2005. I will try to be much more brief this semester.

The main purpose of these notes as I start out is just to indicate what was covered in class,

and occasionally to include some extra info that was not in the previous notes or the textbook.

is not done with any binning.

+ A "spin-j object" (particle or field) is one transforming under the spin-j representation of the

rotation group. One aspect of this is that a 2\pi/j rotation brings the system back to its original

configuration. Examples: j=1, 2\pi rotation, eg photon polarization vector; j=1/2, 4\pi rotation,

electron spin; j=2, \pi rotation, gravitational wave polarization tensor (think of the pattern:

compression in one direction and expansion in the orthogonal direction).

+ Conservation laws and Killing vectors: now that we know 1) the covariant derivative,

and 2) the covariant form of Killing's equation X_(a;b)=0 for a Killing vector X^a,

we can derive conservation laws in a slick way. The derivative of the inner product u^b X_b

along a geodesic with tangent u^a is u^a (u^b X_b);a = u^a u^b;a X_b + u^a u^b X_b;a.

The first term vanishes by the geodesic equation and the second by Killing's equation since

contraction with u^a u^b projects out only the symmetric part X_(b;a). [Note that in the

first expression the covariant derivative could have been written as an ordinary partial

derivative since it is acting on a scalar, but we can just as well think of it as a covariant derivative

in anticipation of applying the product rule for covariant derivatives, which we want to do in

order to apply the geodesic and and Killing equations.

That was for a particle, for a field we have a stress energy tensor T^ab satisfying local

energy-momentum conservation T^ab;b=0. This is a very local form of conservation,

while to formulate a globally conserved quantity we need a conserved current,

i.e. a divergence free vector. If there is a Killing vector we can make a conserved current:

Define j^a = T^ab X_b, and look at the divergence of j^a:

j^a_;a = (T^ab X_b);a = T^ab;a X_b + T^ab X_a;b.

The first term vanishes if the stress tensor is divergence free, and the second vanishes

since the symmetric stress tensor projects out the symmetric part X_(b;a) which vanishes

by Killing's equation.

Killing tensor: I didn't get to the following in class, but the Carter constant for geodesic motion

in the Kerr geometry arises from the following generalization: suppose there is a tensor

X_ab satisying X_(ab;c)=0. Then if u^a is a geodesic tangent vector we have

u^a (u^b u^c X_bc);a = u^a u^b u^c X_bc;a + terms that vanish with the geodesic equation.

Since the tree u's are totally symmetric in abc this projects out X_(bc;a), which vanishes

for a Killing tensor. The Kerr metric admits a Killing tensor.

+ Bianchi identity: arises from the fact that mized partials commute. To prove it simply, use

local inertial coordinates at a point to show that R^a_b[cd;e]=0. Then contract this on ac and

be to deduce the contracted Bianchi indentity R^a_b;a = 1/2 R,b.

+ Friedman equations from the Einstein equation: G_tt = 8\pi T_tt yields the Friedman equation

involving only the first time derivative of the scale factor. The pure second derivative equation

a''/a = - (4\pi/3)(rho+3p), which we obtained previously by taking the derivative of the Friedman

equation and using the first law of thermo, dE + pdV = 0, is equivalent to

(G_tt + 3G_ss) = 8\pi (T_tt + 3T_ss).

+ The first law of thermo follows from the u_a component of T^ab;b=0 applied to the perfect fluid.

The spatial component of this equation is the relativistic Euler equation, (rho + p) u^a u_i;a = - p,i,

where i stands for a spatial component orthogonal to u. This means that the sum (rho+p) plays

the role of the inertial mass density in determining the acceleration in response to an external

force.

+ Let (t,x^i) be the coordinates. Then in general the tt and ti components of the Einstein equation

contain no second t-derivatives, so they comprise four initial value constraint equations. This

can be deduced direcctly from the Bianchi identity and the conservation of the stress tensor.

If these initial value constraints are satisfied at one time, then the Bianchi identity implies they are

satisfied at all times. So instead of 10 independent evolution equations the Einstein equations

yield only 10-4=6 evolution equations. This is just what the doctor ordered: evolution in a generally

covariant theory MUST be non-unique at the level of tensor components, since there are

4 arbitrary functions that can be chosen during the evolution, corresponding to the coordinate freedom.

This point tripped up Einstein, who at one stage concluded from it that a physical theory cannot be

generally covariant. But later he realized that physical observables lsuch as for example the existence

of a collision between two particle worldlines would be unaffected by the evolution ambiguity of the

tensor components araising from the coordinate freedom.

so transforms under Lorentz transformation. Also "density" refers to inverse volume, which undergoes

Lorentz contraction. So energy density transforms "twice". This is captured by the fact that it is the

"time-time" component of a symmetric tensor T_ab, the "energy-momentum tensor", also called the

"stress tensor", or "stress-energy tensor". The energy density in the rest frame of an observer with

4-velocity u^a is T_ab u^a u^b. This tensor is in fact determined completely by the energy density

for all such observers, or even for a suitable finite collection of such observers.

+ To better understand the stress tensor and the conservation law it is helpful to first look at charge density,

which is the "time component" of a 4-current density, i.e. - j_a u^a, where u^a is the 4-velocity of the observer.

The space components of j^a are the charge 3-current density. Charge conservation is imposed by the

continuity equation j^a_;a = 0. To describe the energy-momentum 4-vector 4-current density we add an index,

yielding what is almost always written as T^ab. The meaning of the various components is as described in

Hartle, Chapter 22. Local energy-momentum conservation is expressed by T^ab_;b=0.

+ Ambiguity: there is an ambiguity in the definition of T^ab in field theory, since one can sometimes find

identically divergence-less symmetric two index tensors that can be added to T^ab. (This can even yield

a T^ab that is not symmetric.) In GR the ambiguity is resolved by choosing the T^ab that results from

varying the matter action with respect to the metric g_ab.

+ Examples of energy-momentum tensors:

pressureless dust T^ab = rho u^a u^b, vacuum T^ab = -rho_v g^ab, perfect fluid

T^ab = rho u^a u^b + p(u^a u^b + g^ab). Notice that the vacuum stress tensor is proportional

to the metric, hence Lorentz invariant, so all observers see the same components, in particular

the same energy density rho = T_ab u^a u^b = rho_v.

+ Coming back to the source term in the Einstein equation, besides the energy density T_ab u^a u^b,

also the negative of the trace T = T_ab g^ab is approximately equal to the mass density in the Newtonian limit.

Thus for the field equation with sources introduced in the 12/05 lecture we presume the form

R_mn = 4pi Y_mn = 4pi [(1-x)T_mn + x Tg_mn]. The trace of this implies R = 4pi (3x+1)T. Taking the

covariant divergence of both sides, using the conservation of energy and the contracted Bianci identity

R^mn_;n = 1/2 R,n we discover that it is only consistent if x=-1. Thus the Einstein equation must be

R_mn = 4pi (2T_mn - Tg_mn), or equivalently G_mn = R_mn - 1/2 R g_mn = 8pi T_mn. Here G_mn is

the Einstein tensor, which is identically divergenceless by virtue of the contracted Bianchi indentity.

+ The trace of the tidal tensor is thus proportional, for geodesics with tangent u^m,

to (2T_mn - T gmn)u^m u^n = 2 rho + T. For a perfect fluid this yields rho + 3p, which we encountered

before as the source in the acceleration of the cosmic scale factor. Thus pressure contributes to the

source of gravity, not just energy density...

of the metric vanish. In 4d, 80 of the 100 2nd partials can be set to zero using the freedom to choose the

3rd partials of the old coordinates with respect to the new ones. The remaining 20 match in number

the independent components of the Riemann tensor.

+ Curvature and relative acceleration of neighboring geodesics: geodesic deviation equation.

For a proof, see the 2004 notes for 11/18.

+ Tidal tensor in Newtonian gravity and in GR: in GR, it is R^a_mnb T^m T^n for a geodesic with tangent T^m.

The vacuum field equation R_mn = 0 is equivalent to the statement that the trace of this tidal tensor vanishes for all T^m.

A geometrically, this means that the second derivative of the volume of an infinitesimal sphere of test particles that

start out mutually at rest vanishes initially. That is, the sphere distorts to an ellipsoid, but with the same volume.

Example: sphere falling toward earth center. Example: gravitational wave. (There is a version of this that works

for null geodesics: if the cross-section of a beam of lightrays is initially a circle and not changing, then the

second derivative of the area is zero, i.e. the circle will distort into an ellispe with the same area.

+ In the presence of sources, the trace of the Newtonian tidal tensor is -4pi rho_m, the mass density.

The relativistic generalization must replace rho_m by something of the form Y_mn T^m T^n which for Newtonian

sources agrees with the mass density. Thus it could be the energy density, but it turns out that there is

another contribution, required by energy conservation, which is negligible for Newtonian sources: the sum of the

principal pressures, or just 3p for an isotropic fluid.

+ Gyro spin propagation along an acelerated worldine: "Fermi-Walker transport", "Thomas precession".

The equation is (u.D)s = (a.s) u, where u is the worldline tangent, dot is index contraction (with a metric

in the a.s term), D is the covariant derivative operator, s is the spin vector, and a is the covariant acceleration.

Note that with this equation the covariant derivative of s.u along the curve is zero, as it should be since s.u=0

as a consequence of the definition of the spin 4-vector: (u.D)(s.u)=((u.D)s).u + s.(u.D)u = (a.s)u.u + s.a = 0.

+ Commutator of covariant derivatives vanishes on a scalar, but when acting on a tensor there is a factor

of the Riemann tensor contracted with each free index, with a + sign for contravariant and a - sign for

covariant indices. For example, (D

- conceptual definition: equal to partial derivative when evaluated at a point p in l.i.c. at p,

equal to whatever the tensor transformation rule gives in other coordinate systems.

- why the ordinary partial derivative of a tensor is not a tensor

- the anti-symmetrized partial deriv. of a co-vector is a tensor. (Example: the electromagnetic

field strength F_ab = A_b,a - A_a,b. The time-space components determine the electric field and

the space-space components determine the magnetic field.)

- general formula for covariant derivative in any coordinate system

+ Symmetry of tensor indices:

- Symmetric and antisymmetric parts of a two-index tensor, T_ab = T_(ab) + T_[ab], where

T_(ab) = 1/2(T_ab + T_ba) and T_[ab] = 1/2(T_ab - T_ba)

- Contraction of a symmetric tensor S^ab with an anti-symmetric tensor A_ab vanishes:

S^ab A_ab = S^ba A_ba = S^ab (-A_ab) = - S^ab A_ab, hence it is zero.

- Corollary: S^ab T_ab = S^ab T_(ab) for any tensor T_ab, i.e. only the symetric part of T_ab contributes.

+ Parallel transport:

- If the covariant directional derivative of a tensor along the tangent to a curve is zero, the tensor is said

to be "parallel transported" along the curve. This can be understood at each point as the statement that

in a l.i.c. system at that point the components of the tensor are constant to first order.

- The geodesic equation is equivalent to the statement that the tangent to the curve is parallel transported

along the curve.

- gyroscope equation: the spin 4-vector is orthogonal to the 4-velocity of the gyro center of mass curve. If the

curve is a geodesic, the spin 4-vector is parallel transported along the curve. A student asked about what happens

if the curve is not geodesic. I think there is an acceleration-induced precession. I'll check on it...

that it gives a way to define a vector in a coordinate-independent way, i.e. instead of saying a vector is

a 4-tuple of numbers that change in a specified way when the coordinate system is changed. A function

is coordinate independent (a "scalar"), as is the directional derivative of the function along a given vector.

So we can define vectors as directional derivative operators that are linear operations taking functions to

functions. To fully specify vectors in this way we must add one requirement: the product rule is satisfied:

v(fg)=v(f) g + f v(g), where v(f) is the vector acting on f. This description of a vector requires no coordinates.

- The gradient of a function is the prototype of a different type of object, a "covariant vector", which transforms

by the inverse Jacobian (rather than the Jacobian) under coordinate change. The original thing we called

vector is then more precisely called a "contravariant vector". (I don't know the historical reason for the co-

and contra- assignment of names...) For a shortened terminology, these are often just called vectors

and covectors. Covectors are also called dual vectors, or one-forms.

- Index placement convention: contravariant indices are superscripts ("upstairs"), co-variant indices are

subscripts ("downstairs").

- Contraction: Summing over a co- and contra- index pair yields a scalar (since the Jacobian and

inverse Jacobian cancel).

- raising and lowering indices with the inverse metric and the metric. If a contravariant index is lowered,

the resulting index is of covariant type, since the contraction is invariant and the remaining index on the

metric is covariant. Similarly for raising indices. Thus index raising and lowere converts covariant to

contravariant indices and vice versa. The notational convention is to keep the same letter for the object,

but it is important to know that objects are generally born with a definite placement of their indices,

and when those have been raised or lowered there is hidden metric dependence. Note the metric with

the indices rasied is equal to the inverse metric, so it is consistent to denote the inverse metric by g^ab,

i.e. g_ab with the indices raised.

- general tensors of "rank r-s" or "type r-s", meaning r contravariant and s covariant indices.

- Illustrated some of this with the electromagnetic vector potential (born with the index down), and field

strength (atisymmetrized derivative of the vector potential), and with the Riemann curvature tensor

(not yet defined in this class) R^a_bcd (born with one index up and three down), Ricci tensor

R_bd:= R^a_bad (contraction on an index pair), and Ricci scalar R:= g^ab R_ab ("trace" of the Ricci

tensor; think of it as a double contraction with the inverse metric). The vacuum Einstein equation is

R_ab=0. The Einstein equation with matter sources is G_ab = 8\pi G T_ab, where the G_ab is the

Einstein tensor, G_ab:= R_ab - 1/2 R g_ab, and T_ab is the stress-energy tensor.

- Tensor equations: the vacuum Einstein eqn is R_ab=0. Since R_ab is a tensor, and the tensor transformation

rule is linear, R_ab vanishes in one coordinate system if and only if it vanishes in any coordinate system.

This is how a tensor equation is coordinate-independent. The Einstein equation with matter sets one tensor

equal to another. they are equal in one coordinate system if and only if they are equal in all coordinate systems,

since they transform the same way under a change of coordinates.

- Coordinate velocity dx^a/dl is a vector, but acceleration d^2x^a/dl^2 is NOT, as we showed by making

a coordinate transformation. The spoiling term is the derivative of the Jacobian. The "covariant acceleration"

is d^2x^a/dl^2 + G^a_bc dx^b/dl dx^c/dl, where G^a_bc is the Christoffel symbols. The geodesic equation

is the statement that this vanishes, a coordinate independent condition. The first term is not a vector, and the second

term is not a vector either, since G^a_bc is not a tensor. (It is easy to prove G^a_bc is ot a tensor: it vanishes at

a point if the coordinates are chosen so that the partial derivatives of the metric all vanish at that point. If it were

a tensor it would thus have to vanish in all coordinate systems, but it does not.) The non-vectorness of the two

terms cancel, so the sum, the covariant acceleration, is a vector. Note that the covariant acceleration at a point

p agrees with the coordinate acceleration at p if the coordinates are locally inertial at p.

its relic abundance. The numbers suggest that an LSP expected for particle physics reasons to help

solve the hierarchy problem would survive with a relic density of order the dark matter density. So this

is taken seriously as a candidate...

- The question arose how low can the energy scale of inflation be. Apparently in the simplest

("one-field") models it should be somewhat near the Planck scale, but in more complicated models

of the fields generating inflation it could be much lower. One paper discussed for example inflation

at 10 TeV...observations by themselves apparently don't rule this out. I'll ask around and add a

comment if I find out otherwise.

- Clarified how to approach the homework problems on the angular size of a casual patch on the sky

and the number of e-foldings required to solve the horizon problem.

- the trans-Planckian problem: the vacuum fluctuations that gave rise to the structure we have today

(not to mention vavuum modes at smaller scales today) all arose from field modes with wavelength

shorter than the Planck length at the onset of inflation...but should we trust that physics? No, but we

don't need to: we just need to assume that somehow these modes showed up in their vacuum state

early on but when their wavelength was longer than Planckian.

- ...but we can be more ambitious: where did these modes come from?? if there is a cut-off at short distances

on the Hilbert space then dimensions of Hilberrt space must be literally created...which goes beyond standard

QM. I believe the story is not over: ultimately we should explain where fields and space come from...

- Vectors: defined tangent vector of curves, showed how it transforms linearly via the Jacobian of the

coordinate transformation, defined the tangent vector space at a point and discussed its local nature.

N.B. In measuring the decay time one should correct for the time dilation due to redshift...did they??

- Cosmic variance: The prediction for the CMB angular power spectrum is a quantum expectation

value. There is a variance, producing irreducible uncertainty. Expand the temperature fluctuation

as DT/T = a_lm Y_lm summed over lm. The variance in a_lm a*_lm is large (in fact relative to the

mean it is 2^1/2, as I explained in the next class. It can be shown that the C_l angular power spectrum

coeffcient is the average over m values of the 2l+1 quantities a_lm a*_lm. The relative variance in the

average is down by a factor of (2l+1)^(-1/2), so the relative variance of C_l is ~ l^(-1/2). The low

multipoles are therefore rather subject to "cosmic variance", quantum uncertaintly in the prediction

that cannot be reduced...except as I mentioned in the next class, what is done is to bin neighboring

values of l do get an average with lower variance. This presumably degrades the information in

the spectrum, but I couldn't find a discussion in the literature of how the bin size is chosen, which

presumably is to optimize the available information...

- Note Alpher and Herman in 1948, then in various combinations sometimes with Gamow made various

predictions from BBN that there should be a CMB with a temperature between 5K and 28K...

- Baryogenesis, leptogenesis, Sakharov conditions (baryon # violated, CP violation, out of equilibrium).

- monopole, flatness and horizon "problems". Critiqued these. Horizon problem is a problem if an isotropization

mechanism is required since it has insufficient time to operate. But if the universe came out of a zero volume

initial state, is there really any problem? Monopole problem is a problem if monopoles exist. Flatness problem

does seem like a real fine tuning problem to me. Regarding the monopole problem, it was asked in class

how the monopoles when created could necessarily overclose the universe. Any density of monopoles

would be balanced in theFriedman equation by a sufficiently fast expansion rate. After looking into it,

I found this clarification: one can only conclude the monopoles overclose if one puts in the additional

information that the monopoles today do not provide ALL the energy density and in particular there

is a lot of entropy in the CMB.

- Inflation introduced to solve these problems: the size of a causally connected patch is exponentially

expanded, and initial curvature is flattened out. But the real bonus is that inflation predicts a particular

initial state: vacuum! The vacuum fluctuations produce the tiny variations in the CMB which

reflect the origin of structure, and the decay of the vacuum energy ("reheating") accounts for the matter

and radiation in the universe. Quite neat! Wht doesn't the decay of the vacuum mess up the quantum

fluctuations? Beccause the modes relevant for the fluctuations are at a much longer length scale,

in fact a scale that became larger than the Hubble length during inflation. Only after inflation ended

did the presence of the fluctuations make themselves felt.

- Some references on cosmology:

TASI Lectures: Introduction to Cosmology

Mark Trodden, Sean M. Carroll

http://arxiv.org/abs/astro-ph/0401547

An Introduction to Cosmological Inflation

Andrew R. Liddle

http://arxiv.org/abs/astro-ph/9901124

Inflationary Cosmological Perturbations of Quantum-Mechanical Origin

Jerome Martin

http://arxiv.org/abs/hep-th/0406011

The Physics of Microwave Background Anisotropies

Wayne Hu, Naoshi Sugiyama, Joseph Silk

http://arxiv.org/abs/astro-ph/9604166

Baryogenesis

Johannes Mulmenstadt

http://www-cdf.lbl.gov/~jmuelmen/www/baryo-rep.html

redshift-angular size relation (useful when know instrinsic size, i.e. a "standard ruler").

The idea is this: we cannot directly measure a(t) or even z(t) since we have no direct access to t.

We can measure for example f(z;L), the flux of energy from a source of intrinsic luminosity

L at a redshift of z. This function is determined by the function a(t), and conversely determines a(t).

- CMB angular power spectrum. Predicted by inflation as a result of vacuum fluctuations. The

origin of the large scale structure we see today.

- The first acoustic peak as a "standard ruler": it is predicted using the expansion rate at last scattering

and the baryon density and I don't know what else. The value of the angular multipole at which the peak

occurs is dependent on the spatial curvature. Evidently this sensitivity is enough to strongly constrain

the curvature to be near zero. (By the way, I think this angular scale corresponds to the largest

structures today, the walls and filaments at the 100 Mpc scale.)

- Last scattering occurs at about 3000K, or 0.3 eV, well below the ionization energy 13.6 eV of Hydrogen.

This is determined by the Planck distribution of photons, the number of photons per baryon (2E9), the

ionization energy, interaction rate, and expansion rate (I think...)

- Primordial nucleosynthesis ("Big bang nucleosynthesis, BBN", first (?) proposed by Alepherr and Gamow):

I gave a bit more detail than the textbook.

The neutron/proton ratio at "freezout" is determined by the temperature and the n-p mass difference to be about

1/6. Some neutrons decay subsequently making this effectively 1/7. Most of the neutrons are cooked into

He-4 (nnpp), so out of 2n and 14p (ratio of 1/7) we get one He-4 and 12 H, so He-4 is about 25% by mass.

hidden factors of G or c). For massive particles energy density is of order the mass density times c^2,

whch is much larger than the pressure for planets or large stars or even for white dwarfs. But for neutron

stars, or thermal radiation, or vacuum energy, pressure is important. It turns out that in GR the source

of gravity is energy density PLUS 3 times the pressure (for an isotropic fluid). Newton didn't notice this

since his sources had little pressure.

- Energy conservation in the form dE = -pdV, together with V = a^3 V_coord and E = rho V implies that

for radiation p = 1/3 rho and for vacuum p = -rho. (Aside for those who have the necessary background:

The result for radiation is also a consequence of the fact that the stress energy tensor for electromagnetic

fields is traceless, which is a consequence of scale invariance.)

- Differentiating the Friedman equation yields d^2 a/dt^2 = -(4 pi/3)(rho + 3p)a. Thus acceleration of the

scale factor can only happen if p < -rho/3. This is satisfied by vacuum energy. Anything satisfying this might

be called "dark energy". Actually I think it's also important that dark energy does not "clump" in galaxies,

but is rather uniform. Observations require p much more negative than -rho/3 for some "dark energy",

but not necessarily p = -rho.

- Effective potential for the Friedman equation, in terms of scaled variables (cf. sections 18.4 and 18.7).

I looked specifically at the examples of (i) pure matter, showing closed, open and flat behaviors.

Closed goes from big bang to big crunch; (ii) pure vacuum energy, which is de Sitter spacetime,

and (iii) matter + vacuum energy. In case (ii) I asserted that all three cases k=-1,0,1 are the same

spacetime, sliced differently (k=-1 timelike, k=0 null, k=+1 spacelike). Only the k=+1 slicing is global

and it makes clear that the universe bounces at a minimal radius. In case (iii) I put in the values

Omega_m=0.3 and Omega_v=0.7 and noticed that the maximum of the potential happens around

a(t)/a_0=0.6=1/(1+z), corresponding to a redshift of z=0.67. So with these parameters today we are

already past the peak and sliding down the slope. (We must already be accelerating to explain the

supernovae luminosities and the age of the universe.)

- Age of the universe (see example 18.7).

- Mentioned weak lensing as a way of mapping dark matter, i.e. "seeing" the gravitational effect of the

matter hanging around in clusters of galaxies. A web page from CITA in Toronto introducing weak

lensing in an elementary way is http://www.cita.utoronto.ca/~hoekstra/lensing.html.

- The history of Hubble's original measurements and the subsequent evolution of measurements of

the value of H_0 as they changed by a factor of 7 is described at http://cfa-www.harvard.edu/~huchra/hubble/.

CMBR, primordial nucleosynthesis.

- matter, radiation, and vacuum energy densities, what fraction of current contents they seem to compose,

how they change with the scale factor.

- Friedman equation, a(t) for matter, radiation, and vacuum

- age of matter-dominated, flat universe; size of visible universe (and what that means).

area 4\pi r^2 and length \sqrt{2M/r -1} Dv for a section of EF advanced time lapse Dv. As r goes to zero this

length goes to infinity. If Dv = 1second and M = M_sun and r is such that the curvature radius is the Planck

length then the length is 1 million light years... The relevance to the information puzzle is this: there is another

exit door for the information, into a new region of spacetime. We don't know what happens on the other side

of the door, but that doesn't mean the door ain't there! One caveat: the black hole interior is unstable, and the

structure of the region near the singularity may be significantly different than the Schwarzschild metric.

- Cosmology: started this topic: Introduced the length scales in the universe, and a few tidbits of history:

Einstein's 1917 static S^3 universe with cosmological constant, Friedman's 1922 time-dependent solutions,

Slipher's 1914 -1922 redshift measurements up to z = 0.006 establishing the extragalactic nature of some nebulae

if interpreted as induced by Doppler recession persisting over a billion years (the then-current estimate of the age

of the earth), Hubble's 1924 use of Leavitt's Cepheid standard candles to directly establish the distance to distant nebulae,

and his 1929 announcement of Hubble's law, z = H_0 d. Then introduced homogeneous isotropic spacetime metrics

(FRLW form), and discussed the possibilities for the spatial geometry: flat, S^3, H^3, and topologically nontrivial

quotients thereof. and gave two derivations of the cosmological redhift relation 1+z = a(t_0)/a(t_e). Discussed the

meaning of relative velocity in such a spacetime, pointing out that the distance between two objects as measured

on a surface of constant t can increase faster than the speed of light. Many interesting points came up along the

way which I won't attempt to write out here. You can find some more details in the 2004 and 2005 class notes around

this date, and of course the textbook has a lot of material.

energy, i.e. freely falling observers at the horizon actually measure negative average energy density. This is

distinct from the KILLING energy states that we know exist INSIDE the horizon. Even in flat space,

quantum fields can have negative energy densities over a sufficiently small region, although the TOTAL

energy of any state is greater than or equal to zero. (These results go under the name of quantum energy.)

inequalities. A student asked me for insight into why there is a negative energy flux across the horizon

in the Hawking effect, aside from the fact that total energy is conserved so there must be such a flux. I

said I could offer no coherent insght. But I suspect that some intuition should be possible...

- Meaning of black hole entropy: Bekenstein originally suggested, and many since him have thought, that

black hole entropy may be a measure of the number of ways a black hole could have formed. But this count

is irrelevant to thermodynamics. I prefer a meaning that is relevant dynamically, and the entanglement

f quantum field fluctuations across the horizon is a prime candidate. Only problem: it is infinite...but perhaps

it is cut off at short distances. Long story.

- Analytic extensions of coordinates, Rindler space, Kruskal cordinates. See 2004 notes from 10/19/04 and

10/21/04.

Please see 2005 notes from 10/18/05 for details. Some other references:

Introduction to quantum fields in curved space-time and the Hawking effect and related topics: http://arxiv.org/abs/gr-qc/0308048.

Review of Horizon Entropy: http://arxiv.org/abs/gr-qc/0302099,

Debate on the meaning of black hole entropy http://arxiv.org/abs/hep-th/0501103.

Lecture notes on Black Hole Thermodynamics:

1) My Phys776 class web page from 2005, http://www.physics.umd.edu/grt/taj/776b/

2) other lecture notes: http://www.physics.umd.edu/grt/taj/776b/lectures.pdf

3) other lecture notes: http://arxiv.org/abs/hep-th/9510026

Derivation of Einstein Equation as equation of state: http://arxiv.org/abs/gr-qc/9504004

Extension of that argument: http://arxiv.org/abs/gr-qc/0602001

on evidence for astrophysical black holes, observational signatures of accretion disks, and observational

methodologies. The powerpoint slides for his presentatation are linked here. You can read about Chris'

research and view web pages covering the material he spoke to us about at his web site, http://www.astro.umd.edu/~chris/

http://www.physics.umd.edu/grt/taj/776b/

Includes some lecture notes, links to articles, homework assignments and student project reports.

- Penrose's article: "Gravitational Collapse: The Role of General Relativity"

Rivista del Nuovo Cimento, 1, 252 (1969); a scan of it is posted here: http://www.physics.umd.edu/grt/taj/776b/PenroseReview.pdf

It was reprinted in General Relativity and Gravitation in '02 as a 'golden oldie', http://www.springerlink.com/content/8b4lb6ge7jle4e4y/fulltext.pdf

(See note from 10/12 on how to access this article from an off-campus computer.)

I showed pictures from the article and talked about them. One question that came up is whether the little circles of light should be circles

or ellipses. This would depend on the coordinates plotted. Seems not to be true even for the non-rotating case (Schwarzschild)

in Eddington-Finkelstein type coordinates, or rather (v-r, r). Either Penrose was just sketching roughtly, or he used another

coordinate system...The pictures show the stationary limit surface and event horizon, and the ratating scaffolding erected to

extract rotational energy from the black hole. I also discussed a picture of the future light cone of a point, or rather the boundary

of the future of the point. No generators enter the boundary, but they can leave. Turning it upside down we get the boundary of the

past of a point, which is like an event horizon (except that for the event horizon the "point" becomes the asymptotic region "at infinity".

No generators can leave the horizon...provided the horizon is nonsingular. This is one of the key assumptions in Hawking's area theorem,

i.e. that the horizon is nonsingular. (The other key assumption is that matter has locally positive energy.) Actually a stronger theorem

assumes only that no singularities are visible from infinity---i.e. there are no naked singularities. The Cosmic Censor hypothesis

asserts that naked singularities will not arise from non-singular initial data. It is not strictly true, since spherically symmetric examples

violate it. But it may be true as a statement about "generic" initial data. Given the link between the area theorem and the second law

of thermodynamics, there seems to be a link between cosmic censorship and the second law...

- Discussed Bekenstein's analysis of the minimal area increase of a black hole, to derive the proportionality betwen area and entropy

from the perspective of entropy as missing information. Result is S_BH = \eta A/L_Pl^2 for some dimensionless coeffcient eta of order unity.

- Bekenstein's "effective temperature", T_eff = hbar \kappa/8\pi \eta.

- Violation of the generalized second law (GSL) in the regime of quantum fluctuations. He also

argued in the same article that the entropy should undergo quantum fluctuations, sometimes decreasing. If you asked him at the time

would a black hole be in equilibrium with thermal radiation at the temperature T_eff, he might have said yes, with just the right T_eff.

Then it would have been a small step to say that in vacuum the black hole would radiate at temperature T_eff. But alas, he did not. Instead

he said: "We emphasize that one should not regard T_eff as THE temperature of the black hole; such an identification can easily lead

to all sorts of paradoxes, and is thus not useful."

"Reversible and Irreversible Transformations in Black-Hole Physics," http://prola.aps.org/abstract/PRL/v25/i22/p1596_1

- Area of Kerr horizon

- Hawking's area theorem, "Gravitational Radiation from Colliding Black Holes," http://prola.aps.org/abstract/PRL/v26/i21/p1344_1

- Laws of black hole thermodynamics

- Bekenstein's BH entropy and generalized second law, "Black Holes and Entropy," http://prola.aps.org/abstract/PRD/v7/i8/p2333_1

Note: UMD has a subscription so you can access these linked PRD articles from a campus computer, or from elsewhere using your

library card number, at http://www.lib.umd.edu/. Click the Research Port.

depends on e as well as l. Discussed how to find ISCO, showed plots including one showing simultaneously

the radius, binding energy and orbital frequency of the ISCO for co- and counter rotating orbits. (This last

plot came from the revier article "Black Holes in Astrophysics" by Ramesh Narayan, http://arxiv.org/abs/gr-qc/0506078.)

(Caution: According to Chirs Reynolds some of the things in this review are not considered well-founded.)

- The non-equatorial orbits possess another, independent conserved quantity, the Carter Constant. This is

quadratic in the 4-velocity of the particle, and is not assoicated with a global conservation law.

Its evolution can therefore not be found even for adiabatically changing orbits by computing a flux integral

at spatial infinity. However, apparently it can be evolved in such orbits using an expression that has been

worked out for the radiation reaction force. A recent paper describing the state of the art is

"Gravitational radiation reaction and inspiral waveforms in the adiabatic limit"

by Scott A. Hughes, Steve Drasco, Eanna E. Flanagan, Joel Franklin, http://arxiv.org/abs/gr-qc/0504015.

- Stationary limit surface (surface of ergosphere) and horizon of Kerr. Inside stationary limit surface

a future timeike vector MUST have a positive phi component, i.e. it must co-rotate: the spinning black

hoel drags the inertial frames around with it. We found the range of allowed angular frequencies, for

the example of a particle with no radial or polar motion.

- Physical states must have future pointing causal (timelike or null) 4-momentum. If the Killing vector is

spacelike this allows negative Killing energy states, i.e. whent he 4-momentum has a positive component

along the Killing vector. The Penrose process exploits these states to extract rotational energy from a

black hole.

precesses ~ 4 degrees/yr; new double pulsar, 2.4h orbit (Earth-Moon distance!), precesses ~ 17 degrees/yr.

- Thorne's theoretical upper limit for black hole spin: J ~ 0.998 M^2. The limit arises from

the fact that counter-rotating radiation from disk is more readily absorbed than co-rotating

radiation. I'm not sure how robust the limit is under different assumptions about the

accretion disk emission of radiation.

- Redshift of photons emitted from orbiting atoms in an accretion disk and observed at infinity.

(Went through the details of the computation. See textbook for details.)

- Kerr metric: wrote it down, and starting discussing it's properties.

- As promised a list of

----------------------------------

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Physics Colloquium

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Astronomy Colloquium

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11/21/06

[Topic: Laser interferometry for Advanced Detector Gravitational Wave Detection]

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Joint Astronomy-Physics Colloquium

is a curve that remains on the boundary of the future of any point p on it, at least for some interval from p.

Since the geodesic equation involves derivatives if refers to the differentiable structure of spacetime.

The null geodesics are differentiable curves so encode something about the differentiable structure.

In fact, they encode everything about the differentiable structure, as long as the spacetime dimension

is three or greater. Since they are determined by the causal structure alone, that means that the

causal structure determines the differentiable structure. It also determines the toplogical structure,

and the metric structure up to a local conformal factor. For a bit more discussion of this and references

see my notes A Spacetime Primer (and the associated figures). The two papers I mentioned in class

are S.W. Hawking, A.R. King, and P.J. McCarthy, "A new topology for curved space-time which

incorporates the causal, differential, and conformal structures," J. Math Phys. 17, 174 (1976),

and D. Malament, "The class of continuous curves determines the topology of spacetime",

J. Math Phys. 18, 1399 (1977).

- White dwarfs and neutron stars.

- According to current models of the nuclear equation of state together with observations it is

believed that neutron stars have a radius between 10 and 12 km, and a mass between 1.25 and 2.1 solar

masses, with R decreasing as M increases. With one plausible equation of state neutron stars with mass

above 1.3 solar masses have an ISCO. That would be a Schwarzswchild radius of 1.3 x 3km = 3.9 km,

so 6M is 3 x 3.9km = 11.7km... This has observational implications, since an accretion disk that meets

the star will behave differently than one whose inner edge is outside the star.

-perihelion precession: Mercury had an unaccounted for 43''/century. Solar oblateness would produce

a quadrupole moment lending a 1/r^3 term to the potential, but apparently not large enough. However

I think it was not until the 60's that people became confident that the quadrupole moment of the sun was

not large enough to explain or at lelast make an important contribution to the anomaly. (Please let me

know if you know some details about this.) Leverrier proposed explaining the anomaly with a new planet,

Vulcan, in an orbit between Mercury and the sun. In GR, the effect comes from the l^2/r^3 term in the potential.

The relative size of this effect is (GMl^2/r^3)/(GM/r) = l^2/r^2 =~ v^2/c^2 =~ GM/rc^2 = R_S/r.

The 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, as an estimate.

- Einstein said somewhere he trembled for two days after getting this result from GR.

In class I said he had found it three years before the final form of GR, with a lot of help from Besso,

and never acknowledged Besso. I just found an article by Michel Janssen that goes into this in great depth:

"What Did Einstein Know and When Did He Know It? A Besso Memo Dated August 1913."

To appear in Jürgen Renn et al., The Genesis of General Relativity: Documents and Interpretation.

Vol. 1. Einstein’s Zurich Notebook. Dordrecht: Springer, forthcoming. Apparently they found

18'', not 43''. They never published the result. Besso was going to write a major paper on

it, addressing solar oblateness, alternative gravity theories, the mass of Venus, and a host of

other topics. It was too ambitious, and they were scooped by Lorentz' student Droste on the

18'' calculation. Einstein does not mention solar oblateness in his 1915 article.

- energy and angular momentum of circular orbits of Schwarzschild: Solve V'(r,l)=0 to find l(r)

and V(r,l)=(e^2-1)/2 to find e(r). The result is

e =
(1-2M/r)/(1-3M/r)^1/2
and l= (r^1/2)/(1-3M/r)^1/2

- Rotating black hole: Kerr
solution. Not the metric outside a rotating star. Effective potential
for equatorialorbits depends on e as well as l. To find e(r) and l(r) for circular orbits solve V'(r;e,l)=0 and V(r; e,l)=(e^2-1)/2.

To find the ISCO solve in addition V''(r;e,l)=0. Find that the co-rotating ISCO approaches the event horizon

for a maximally spinning bh...

other material in accretion disk, or to gravitational radiation.

- Discussed meaning of conserved Killing energy again, and the distinction between this and locally

measured energy.

- photon orbits: effective radial potential, impact parameter b, unstable circular photon orbit at r=3M,

b = (27)^1/2 M = 2.6 R_Schwarzschild

- absorption cross-section of black hole: pi times the square of the impact parameter of the circular photon

orbit = 27 pi M^2

- deflection of light, gravitational lensing

- Shapiro time delay

- GR: classical violation of the 2nd law?

- interpretation of conserved Killing energy in the Newtonian limit: kinetic + potential energy

- derivation of gravitational redshift using conservation of the dot product of 4-momentum & Killing vector

- effective radial potential for Schwarzschild orbits

- Killing vectors; examples of Schwarzschild metric and Euclidean metric on the plane

- Conserved inner product of gedesic 4-velocity with Killing vector

- Gravitational redshift, derived using the fact that the Schwarzschild time coordinate separation between

successive wavecrests of light is constant at fixed radius, so the ratio of proper times is the ratio of the

values of (-g

- Energy extraction: lower mass to a black hole horizon and extract its entire rest mass as useful work at infinity.

- Orbits: reviewed reduction to the radial motion for a central force problem in Newtonian mechanics.

The effective potential consists of the original potential plus the angular momentum barrier. For any

angular momentum except zero there is a minimum in the potential, corresponding to a stable circular

orbit. In GR there is another term that is attractive and eliminates the stable orbit when the angular momentum

is too small. Also the GR orbits are not closed, and the ellipse axis precesses.

that g

constant plus the non-relativistic Lagrangian 1/2 v^2 - Phi, where the Newtonian potential Phi is identified

with -1/2 h

for a charged particle with the velocity contracted with the vector potential.

- affine parameter: this is defined as a parameter for which the geodesic equation takes the standard form.

For a non-affine parameter the geodesic equation has an extra term, of the form Ag

a function of the path parameter s. Affine parameters are only defined on geodesics, not on any random curve.

You can get the affine parameter on a null geodesic by snuggling up to it with a timelike geodesic, and considering

the ratios of proper times of segments to approach ratios of affine parameters. Why doesn't this work for ANY

curves? That is why does it fail for non-geodesics? One student suggested to me after class that it is because

there is no unique way to snuggle otherwise, and one can get different results by snuggling with timelike curves

that have "shrinking wiggles".

- Conserved quantities & symmetries: if g

cordinate are a symmetry of the spacetime, and the corresponding conjugate momentum is conserved.

For example, if the metric is independent of t then g

- Must reduce to Minkowsi form in
some coordinate system at each point.**
**

- ds^{2} = 0
defines a three-dimensional double cone at
each point.

- g_{ab }has
1 negative and three positive eigenvalues at each point.

- det g < 0. To choose (-+++) over (+---) must specify that there is more than one positive eigenvalue.

- local inertial coordinates- ds

- g

- det g < 0. To choose (-+++) over (+---) must specify that there is more than one positive eigenvalue.

- free-fall = geodesic motion: local maximum of the proper time between any two nearby points.

- geodesic equation as Euler-Lagrange equation

- geodesic equation in local inertial coordinates at a point p: d

- example of orbit of earth: geodesic but not the global maximum of proper time, which is achieved by

the radial up & down motion.

- Basic idea of GR: the spacetime metric g

- Einstein's motivations:

1) Newtonian gravity
instantaneous, inconsistent with special relativity

2) Inertial structure "should" depend on distribution of matter.

More precisely, he believed in Mach's principle: inertia is fully determined by matter.

This turns out to be overstated, but the spirit of it is correct: inertia is dynamical, not fixed a priori.

3) Gravity is a pseudo-force , i.e. equivalence principle: all bodies fall with the same acceleration,

there is no local gravity in a freely falling frame.

- It took 10
years from 1905 to 1915 and serious mathematical help to figure it out,
with many missteps 2) Inertial structure "should" depend on distribution of matter.

More precisely, he believed in Mach's principle: inertia is fully determined by matter.

This turns out to be overstated, but the spirit of it is correct: inertia is dynamical, not fixed a priori.

3) Gravity is a pseudo-force , i.e. equivalence principle: all bodies fall with the same acceleration,

there is no local gravity in a freely falling frame.

along the way. A nice account appears in Subtle is the Lord, a scientific biography of Einstein by A. Pais.

The path was twisted and confusing however, so Pais likely did not get everything right. Historians of science

have been looking at it very closely however. Some names of the people involved:

John Stachel, Jurgen Renn, Michel Janssen.

- Gravity as a pseudo-force, appearing when a non-inertial frame is adopted...but the local inertial frames

don't fit together into one global one. The non-fitting is due to gradients in the gravitational acceleration.

The gradient is true gravity. In GR, this is described by a g

metric by a coordinate transformation.

- Examples of metrics: Schwarzschild, Eddington-Finkelstein, Painleve-Gullstrand, cosmological.

- Birkhoff's theorem: the Schwarzschild soln is the unique spherically symmetric vacuum solution, up to

coordinate transformations.

- Coordinate freedom: 4 functions. Metric has 10 functions. So not all is arbitrary.

Analogy with gauge transformations of electromagnetic vector potential.

- the mass shell

- forbidden processes, thresholds (e.g. GZK cutoff), Compton (and inverse-Compton) scattering,

Doppler effect, relativistic beaming.

- A recent paper claims to have observed the GZK cutoff!

- Review of special relativity.

- Key idea of relativity: time is arc length along a spacetime curve.

- Comparison of relativistic with Newtonian spacetime structure:

Newton: 1) absolute time function, 2) Euclidean spatial metric on constant time slices, and

3) a preferred set of worldlines at absolute rest. The last ingredient serves to define the

inertial frames, but it is more than needed, since all unaccelerated frames are identical in

Newtonian physics. It can be replaced by a specification of the unaccelerated motions,

i.e. the "inertial structure". The mathematical object suited to this is called an affine connection.

Einstein: the invariant
interval
takes over the job of all three of these, determines the causal,

temporal, spatial metric and inertial structures of spacetime.

temporal, spatial metric and inertial structures of spacetime.

- Proper time, 4-momentum, rest mass.