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chapter on symmetries
static and stationary
Killing vectors and conserved quantities
Birkhoff's theorem ?
No hair theorem
Kerr metric is infamous for being nasty to derive from scratch.
Expand it in powers of the angular momentum and try to use the series technique I used for Schw. metric
possible hw
verify that big bang singularity in Friedmann solutions is a real singularity
(text claims that R^a_a diverges and sing occurs at finite proper time in past)
understand Rindler metrics better
grav waves
Should discuss LIGO, etc.
Should discuss the possible misconception that LIGO can't work because the photons get stretched:
ch. 5:
Show that when a particle is dropped from rest at infinity and falls into a black hole, a shell observer (in the sense
of the Taylor-Wheeler book) at r observes it to have velocity (2M/r)^1/2 as it goes by. To make this a good hw problem,
I should solve it myself, and make sure I understand all about the shell obvserver.
in section on geodesic eqn, do an example by 2 techniques: (1) geodesic eqn; (2) path ABC, varying B;
I use technique 2 later in book, should probably use 1
I think my problems with toc are related to use of mytocloft instead of the real tocloft.
ch 4., hw:compton
could use a figure is goofy, had discriminator set too low, and also has a funky "backscatter" peak in it that I don't
understand myself
ch. 4 does OOM est of deflection of light; I have more recent notes at
with some nice qualitative stuff, like Einsteinian versus Newtonian factor of 2 (cf. Newtonian black hole), 2=1+gamma, where gamma is PPN param,
time delay versus angular deflection, waves versus particles; should also fiddle with to get time delay; I'm not totally clear on
how time delay should be defined
could do Schild's ladder in section on parallel transport
I introduce the frequency four-vector. Rindler has some nice applications to Doppler shifts, etc.
could do Faraday tensor, E&M, etc.; hw: show that force in frame 1 is propto uv, to lowest order (a la SN)
in discussion of four-potential, would be nice to have an example where a charged particle moves through the solenoid's internal region, show that least-action makes sense
in section "The tensor transformation laws", could discuss partial derivative operators as basis for tangent plane; then back-link from frequency four-vector
I need to understand the deeper reasons for this remark in Carroll: "Notice that, unlike the partial derivative, it makes sense to raise an index on the covariant
derivative, due to metric compatibility." (I just repeated this fact in the book at "With the partial derivative..."
tests of GR:
...long review article by Will on tests of gen rel \url{}
... a whole bunch of good tests of GR: \url{}
...exptl test of gravitational deflection of light waves: Shapiro, S. S.; Davis, J. L.; Lebach, D. E.; Gregory, J. S. (2004), "Measurement of the solar gravitational deflection of radio waves using geodetic very-long-baseline interferometry data, 1999", Phys. Rev. Lett. 92: 121101, doi:10.1103/PhysRevLett.92.1211011979 ;
...Testing General Relativity with Pulsar Timing, \url{}
...more good tests, Penrose, p. 470
...tests of GR:
... more tests of GR:
..."The geodetic effect provides us with a sixth test of general relativity (after the three classical tests plus Shapiro delay and the binary pulsar),"
...Shapiro test: see Penrose, p. 467
...\url{} ... I should read this, then give url in footnote if it seems good; skeptical take on complementarity.
...information: qm says conserved; naked singularity would product information out of nothing; black hole appears to absorb information hole thermodynamics... ; cool paper: A Larger Estimate of the Entropy of the Universe, it possible to do the analog of the Sch. metric for cylindrical or Cartesian symmetry? sec:more-energy-mom-tensor, should show why this is all true; note that if g is non-diagonal, T with lower indices is non-diagonal, but $T\indices{^\mu_\nu}$ is still diagonal;
the reason the spatial components are all equal is because we assume a perfect fluid
interesting remark on connecting Riemannian curvature to Gaussian curvature:
WP has nice map of globe, interprets impossibility of global orthonormal coords as
consequence of theorema egregium; I have to think about this more; I've referred to this fact before; put a forward ref to this argument
cosmology applications
nice example of Ricci versus Weyl curvature: Linder, p. 56, clumpy universe distances
also surface brightness fluctuations, Linder p. 54, and survey, ; but Linder doesn't explicitly say
anything about curvature, and neither does the paper; the survey only goes out to .01 redshift; can you do this with deeper fields,
so that the fluctations count Poisson statistics of galaxies?
it would be cute to know the actual average numerical value of, say, $R_{xy}$ in our universe
...hw: verify $U^aV_a=U_aV^a$
...I never explicitly introduced contraction
...use a 1-d circle as an example of R normal coords
...pp-waves = plane-fronted with parallel rays; are somewhat more general than plane waves;
turns out exact pp-waves can be added linearly (which is surprising) in the sense of adding the $H$ function;
...TA has book by griffiths on colliding grav waves; can always transform into a coordinate system where waves
propagate either in same dir or opposite dirs;
...colliding gravitational waves --
...overview of wave solutions \url{}
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