Gravitational Radiation

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Gravitational Radiation
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General Relativity
Riemann Tensor
Einstein Tensor
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lhs curvature term rhs matter term, with the
energy-momentum tensor
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• weak field leaves local spacetime flat
• h is the perturbation off Minkowski
• 'nearly flat' regime allows a solution for field
  equations

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In the Weak Field
where we define

• substitute the new metric
• keeping everthing to first order in h
• define the trace referse

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Linearized Equations
intrducing the Lorentz gauge
gives,
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• assuming the solution and evaluating the
  derivative gives the wave vector condition
• wave propagates according to the above wave
  vector
• travels at the speed of light

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using the Lorentz gauge for this soltion
still have gauge freedoms left

• introduce a fixed four velocity U
• pick a Lorentz frame where wave is traveling in
  z-direction
• transverse wave is purely spatial and
  perpendicular to propagation direction
• tracless

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• two independent degrees of freedom
• this is what we expect for graviataional radiation

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Must consider affect on free particles from
rest U(1,0,0,0)

• substituting for the metric terms gives 0
• this is a coordinate-dependent value
• this does not tell the whole story

we should consider a coordinate-free number...
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Proper Distance

• proper distance depends on metric
• changes with time
• spacetime, definition of distance, changes

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test ring of paricles
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Resonant Detectors
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Detector Response Equation
must examine free particles in TT coordinates
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Detector Sensitivity
To make a sensitive detector one should pick a
target frequency The typical wave solutoin is,
where
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Energy of Detector
it is assumed that the detector starts from rest
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Resonance
resonance occurs when the frequencies of wave and
detector match or when,
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• mathematical considerations lead to an inexact
  solution
• we assume the time-dependent part of
  energy-momentum tensor

is small relative to the wavelength
one can assume that the space where
conservation of energy momentum tensor
Gauss's theorem, and tensor virial theorem
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Inexact Solution and the Quadrupole moment
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because we are working with the linearized wave
equations we can evaluate the reduced quadrupole
moment one term at a time
first we examine the last term
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side note for waves in the y direction solution
is same for waves in the x direction
now we examine the middle term
combining the two gives
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The Orbit Equation (from FG Fcentrifugal )
Choice of Coordinate System (assumed circular)
Radiation (circularly polarized)
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• detectors oscillate therefore they get energy
• where does this energy come from?
• sources
• Binary systems
• detectors
• masses on a spring

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• assume a change in amplitude signifies a change
  in energy carried by wave
• get an expression for the energy as a function of
  amplitude
• use the expression along with the form of the
  incident wave to find the energy lost by a source

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ii. Formula for energy versus in terms of
amplitude
ii. formula for energy loss using reduced
quadrupole moment of binary system
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using the derived for energy carried away by a
source
What is this in terms of the Newtonian Energy of
the system
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taking the logarithm of E above, and
differentiating gives,
this now gives us a useful way to measure the
energy loss in terms of the period,
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10-13
this value is actually assumes a circular
orbit accounting for the eccentricity of the
orbit gives a value 12 time greater
Experimental Measurement
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Summary

• Einstein's equations describe how space time
  curves and gives radiation solutions
• We can detect waves with massive systems
• Can see how waves are generated
• binary systems
• detectors
• Energy is carried away from massive systems
  through gravitational radiation