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=======================
Gravitational Waves
=======================
:Author: bnewbold@mit.edu
.. 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 gages.
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 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
----------------
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
-----------------
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
----------------
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
-----------------
$\Delta L = h L \lreq 4 \times 10^{-16} \text{cm}$
LIGO (10 Hz to 1kHz)
Also GEO, VIRGO, TAMA (?), AIGO
LISA (10^-4 Hz to 0.1 Hz)
Resonant Bars
~~~~~~~~~~~~~~~
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.
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