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Gravitational Waves
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:Author: bnewbold@mit.edu

.. note:: Most of this content is based on a 2002 Caltech course taught by
    Kip Thorn [PH237]_

Raw Info
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Rank 4 Riemann tensors, will cover different gages.
Waves are double integrals of curvature tensor...
 


Gravitons as Quantum Particles
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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 propogation.

Two polarizations: cross and plus, corresponding to spin of particles aligning wiht or against propagation?  (Ref: eugene vickner? reviews of modern physics)

Waves' multipole order $\geq$ spin of quantum = 2 for graviton ((??))

Waves don't propogate like E, because mass monopoles don't oscillate like charges.

:latex:`$ h \req \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 qudrupole moment dominates
Fourth term: angular momentum can't oscillate
Fifth term: current quadrupole

Energy
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Quick calculation: for a source with mass M, size L, period P, the quadupole moment $M_2 \req M L^2$, h \req 1/c^2 (newtonian potential energy) ????

h on the order of $10^{-22}$

Propogation
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When wavelength much less than curvature of universe (background), then gravitational waves propagate like light waves: undergo red shifts, gravitational lensing, inflationary redshift, etc. 

Spectrum
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High Frequency: Above 1 Hz, LIGO (10 Hz to 1kHz), resonant bars
    Small black holes (2 to 1k suns), neutron stars, supernovae

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
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$\Delta L = h L \lreq 4 \times 10^{-16} \text{cm}$

LIGO (10 Hz to 1kHz)
    Also GEO, VIRGO, TAMA (?), AIGO

LISA (10^-4 Hz to 0.1 Hz)

Resonant Bars
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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
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.. [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.