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.

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.

- 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 R

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. - j

- 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.

- Invariants formed by contracting contra- and co-variant index pairs.

- Vectors as differential operators: sometimes the vector V

- Metric tensor: covariant tensor of type (0,2). Inverse metric: contravariant tensor of type (2,0), denoted g

- 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.

(b) If Q

(c) If T

(d) The contraction of a symmetric index pair with an antisymmetric index pair always vanishes.

If P

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

- 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).

- 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 Mlmenstdt

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

- 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

- dE = -pdV implies d(rho)/dt = -3(\rho + p) (da/dt)/a.

- second order Friedman eqn: d^2a/dt^2 = -(4\pi/3)(\rho + 3p)a.

- Age of Universe as a function of cosmological parameters.

- 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.

- 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.

- Penrose diagrams

- 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.

- 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.

"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.

- 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...

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.

(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...

- Types of black holes See Ch. 12 Intro and Ch. 13 Intro & Sec. 1,2.

- 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].

- 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.

- 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.

- time/light/space-llike character of a geodesic is preserved along the geodesic. Proof: we evaluated d(g

- Newtonian limit: (v << c, g

- 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".

- To develop a formula for the geodesic condition we wrote the integral as \integral L1 dl, where

L1 = (-g

The stationarity condition is the Euler-Lagrange equations for L1

(d/dl)(L

If the curve parameter l is chosen to be the proper time, then L1=1, so this simplifies to

(d/dl)(g

This is called the geodesic equation for affinely parameterized geodesics. In a l.i.c. at a point this geodesic eqn. reduces to d

- This geodesic equation also follows directly from the variational condition with Lagrangian

L = ½ g

without the square root. This action is not reparametrization

- A student asked whether non-geodesic null curves ever arise 'physically'. I'm looking for an example with light rays...

- 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.

- 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.

- 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.

- 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 + ...