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-=======================
-Gravitational Waves
-=======================
-
-.. warning:: This is a rough work in progress!! Likely to be factual errors, poor grammar, etc.
-
-.. note:: Most of this content is based on a 2002 Caltech course taught by
-    Kip Thorn [PH237]_
-
-Raw Info
------------------
-Rank 4 Riemann tensors, will cover different gauge.
-Waves are double integrals of curvature tensor...
- 
-
-
-Gravitons as Quantum Particles
----------------------------------
-Invariance angles: (Spin of quantum particle) = :latex:`$2 pi$` / (invariance angle)
-
-Graviton has :latex:`$\pi$` invariance angle, so it is spin 2; photons have unique :latex:`$\arrow{E}$` vector, so invariance angle is :latex:`$2\pi$`, spin 1
-
-Also describes spin by the group of Lorentz transformations which effect propagation.
-
-Two polarizations: cross and plus, corresponding to spin of particles aligning with or against propagation?  (Ref: Eugene Vickner? reviews of modern physics)
-
-Waves' multipole order :latex:`$\geq$` spin of quantum = 2 for graviton ((??))
-
-Waves don't propagate like E, because mass monopoles don't oscillate like charges.
-
-:latex:`$ h \approx \frac{G}{c^2} \frac{M_0}{r} + \frac{G}{c^3} \frac{M'_1}{r} + \frac{G}{c^4} \frac{M''_2}{r} + \frac{G}{c^4} \frac{S'_1}{r} + \frac{G}{c^5} \frac{S''_1}{r}$` 
-First term: mass can't oscillate
-Second term: momentum can't oscillate
-Third term: mass quadrupole moment dominates
-Fourth term: angular momentum can't oscillate
-Fifth term: current quadrupole
-
-Energy
-----------------
-
-Quick calculation: for a source with mass M, size L, period P, the quadrupole
-moment :m:`$M_2 \approx M L^2$`, :m:`$h \approx 1/c^2` (Newtonian potential
-energy) ????
-
-h on the order of :m:`$10^{-22}$`
-
-Propagation
------------------
-
-When wavelength much less than curvature of universe (background), then gravitational waves propagate like light waves: undergo red shifts, gravitational lensing, inflationary red shift, etc. 
-
-Sources
--------------
-
-Inspirals of bodies into super-massive black holes
-    Eg, white dwarfs, neutron stars, small black holes.
-    Super-massive black holes are expected near the centers of galaxies.
-    Low frequencies (LISA); waveforms could hold data about spacetime curvature
-    local to the black hole.
-    Waveforms could be very difficult to predict.
-
-Binary black hole mergers
-    Broadband signals depending on masses.
-
-Neutron Star/Black hole mergers
-    Stellar mass objects existing in the main bodies of galaxies.
-    Higher frequencies (LIGO and AdvLIGO).
-
-Neutron Star/Neutron Star mergers
-    Have actual examples in our galaxy of these events; but final inspiral rate
-    is so low that we have must listen in other galaxies. Merger waves will
-    probably be lost in higher frequency noise, so can't probe local 
-    gravitational curvature. 
-    May observe "tails" of waves: scattering off of high curvature around the 
-    binary.
-
-Pulsars (spinning neutron stars)
-    Known to exist in our galaxy.
-
-Spectrum
-----------------
-
-High Frequency: Above 1 Hz, LIGO (10 Hz to 1kHz), resonant bars
-    Small black holes (2 to 1k suns), neutron stars, supernovas
-
-Low frequency: 1Hz and lower, LISA (10^-4 Hz to 0.1 Hz), Doppler tracking of spacecraft
-    Massive black holes (300 to 30 million suns), binary stars
-
-Very Low Frequency: 10^-8 Hz, Pulsar timing (our clocks shifted by gwaves, average of distance pulsars are not over long periods)
-
-Extreme Low Frequency: 10^-16 Hz, Cosmic Microwave Background anisotropy
-
-Detectors
------------------
-
-:m:`$\Delta L = h L ~ \leq 4 \times 10^{-16} \text{cm}$`
-
-LIGO (10 Hz to 1kHz)
-    Also GEO, VIRGO, TAMA (?), AIGO
-
-LISA (10e-4 Hz to 0.1 Hz)
-
-Resonant Bars
-~~~~~~~~~~~~~~~
-First by Webber. 
-Currently in Louisiana State University (Allegro), University of West Australia (Niobe), CERN (Explorer), University of Padova (Auriga), and University of Rome (Nautilus)
-
-References
-----------------
-
-.. [PH237] `Gravitational Waves`:title: (aka ph237), a course taught by Kip Thorne at Caltech in 2002. See http://elmer.tapir.caltech.edu/ph237/ for notes and lecture videos.