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) = 2 pi / (invariance angle)

Graviton has \pi invariance angle, so it is spin 2; photons have unique \vec{E} vector, so invariance angle is 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 \geq spin of quantum = 2 for graviton ((??))

Waves don’t propagate like E, because mass monopoles don’t oscillate like charges.

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_2 \approx M L^2, $h 1/c^2 (Newtonian potential energy) ????

h on the order of 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

\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 (aka ph237), a course taught by Kip Thorne at Caltech in 2002. See http://elmer.tapir.caltech.edu/ph237/ for notes and lecture videos.