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7 Going Beyond Einstein

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Figure 18: The spectral (redshift) and timing capabilities required for an observatory to probe different strengths of gravitational fields. Phenomena that occur in the vicinities of neutron stars and stellar-mass black holes experience large redshift and occur over sub-millisecond timescales.

Testing general relativity in the strong-field regime with neutron stars and black holes will require advanced observatories that will be able to resolve various phenomena in the characteristic energy and time scales in which they occur. The two parameters used to quantify the strength of a gravitational field in Section 3.1 are also useful in discussing the specifications required for such future observatories.

The potential and the curvature in a gravitational field are directly related, respectively, to the characteristic energy and time scales that need to be resolved in order for an observation to be able to probe a particular region of the parameter space. The potential ε directly gives the gravitational redshift z according to

z = 1 − (1 − 2ε)−1∕2 , (30 )
the measurement of which is the goal of spectroscopic observations; for weak gravitational fields z ≃ ε. At the same time, the curvature ξ is directly related to the dynamical timescale τ in the same region of a gravitational field by
2π- −1∕2 τ = c ξ . (31 )
As shown in Figure 18View Image, only observatories with excellent spectroscopic and millisecond timing capabilities will be able to resolve phenomena that occur in the strongest gravitational fields found in astrophysics, i.e., those in the vicinities of neutron stars and stellar-mass black holes.
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Figure 19: The parameter space that will be probed by an experiment based on gravitational wave detection with LIGO and LISA, for an assumed source at a distance of 1 Mpc.

One of the most promising avenues of testing strong-field general relativity is via the detection of the gravitational waves emitted during the coalescence of compact objects. In the simple case in which two compact objects of mass M are orbiting each other in circular orbits with separation a, slowly approaching because of the emission of gravitational waves, the characteristic period P of the gravitational wave is half of the orbital period and, therefore, is related to the spacetime curvature by

π P = -ξ− 1∕2 . (32 ) c
At the same time, the strain h detected by an observatory on Earth for a gravitational wave emitted by such a source placed at a distance D, is [58]
( ) ( ) GM--- GM--- h = ac2 Dc2 . (33 )
Given the distance to the source and the measurement of strain, the curvature of the gravitational field probed is
5 ( )−2 ( )− 2 ξ = -ε---= 10−3ε5 --h--- --D---- cm− 2 . (34 ) h2D2 10− 23 1 Mpc
The sensitivity of each gravitational wave detector depends strongly on the period of the wave. Using Equations (32View Equation) and (34View Equation), the sensitivity curve of a detector can be converted into a region of the parameter space that can be probed, given the distance to the source. This is shown in Figure 19View Image for the advanced LIGO and LISA for an assumed source distance of 1 Mpc. Gravitational waves detected by LISA will probe the same curvatures as current tests of general relativity, but significantly larger potentials. On the other hand, gravitational waves detected by the advanced LIGO have the potential to directly probe directly the strongest gravitational fields found around astrophysical objects.

In the near future, a number of observatories will exploit new techniques and open new horizons in gravitational physics by exploring the strong-field region of the parameter space shown in Figure 18View Image. Observations with the Square Kilometre Array [151] may lead to the discovery of the most optimal binary systems for strong-field gravity tests with pulsar timing, in which a pulsar is orbiting a black hole [78]. High-energy observations of black holes and neutron stars with IXO [74] and XEUS [185] will detect highly-redshifted atomic lines and measure their rapid variability properties. Finally, gravitational wave observatories, either from the ground (such as LIGO [88], GEO600 [62], TAMA300 [163] and VIRGO [175]) or from space (such as LISA [108]) will directly detect, for the first time, one of the most remarkable predictions of general relativity, the generation of gravitational waves from orbiting compact objects and black-hole ringing.


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