6.6 Tests of general relativity

Gravitational wave measurements of black holes automatically test general relativity in its strong-field regime. Observations of the mergers of comparable-mass black holes will be rich in details of their strong-field interactions. If measurements can determine the masses and spins of the initial black holes, as well as the eccentricity and orientation of their inspiral orbit, then one would hope to compare the actual observed waveform with the output of a numerical simulation of the same system. If measurements can also determine the final mass and spin (say from the ringdown radiation) then one can test the Hawking area theorem (the final area must exceed the sum of the areas of the initial holes) and the Penrose cosmic censorship conjecture (the final black hole should have 2 J ∕M < 1).

Observations of stellar mass black holes inspiraling into SMBHs, the extreme mass ratio inspirals (EMRIs), have an even greater potential for testing general relativity. The stellar mass black hole spends thousands of precessing (both of periastron and the orbital plane) orbits along highly-eccentric trajectories and slowly inspirals into the larger black hole. The emitted gravitational radiation literally carries the signature of the spacetime geometry around the central object. So fitting the orbit to theoretical templates could reveal small deviations of this geometry from that of Kerr. For example, if we know (from fitting the waveform) the mass and spin of the central black hole, then all its higher multipole moments are determined. If we can measure some of these and they deviate from Kerr, then that would indicate that either the central object is not a black hole or that general relativity needs to be corrected [17870].

6.6.1 Testing the post-Newtonian approximation

Current tests of general relativity rely on experiments in the solar system (using the sun’s gravitational field) and observations of binary pulsars. In dimensionless units, the gravitational potential on the surface of the sun is about one part in a million and even in a binary pulsar the potential that each neutron star experiences due to its companion is no more than one part in ten thousand. These are mildly relativistic fields, with the corresponding escape velocity being as large as a thousandth and a hundredth that of light, respectively.

Thus, gravitational fields in the solar system or in a binary pulsar are still weak by comparison to the largest possible values. Indeed, close to the event horizon of a black hole, gravitational fields can get as strong as they can ever get, with the dimensionless potential being of order unity and the escape velocity equal to that of the speed of light. Although general relativity has been found to be consistent with experiments in the solar system and observations of binary pulsars, phenomena close to the event horizons of black holes would be a great challenge to the theory. It would be very exciting to test Einstein’s gravity under such circumstances.

The large SNR that is expected from SMBH binaries makes it possible to test Einstein’s theory under extreme conditions of gravity [47Jump To The Next Citation Point48Jump To The Next Citation Point]. To see how one might test the post-Newtonian structure of Einstein’s theory, let us consider the waveform from a binary in the frequency domain. Since an inspiral wave’s frequency changes rather slowly (adiabatic evolution) it is possible to apply a stationary phase approximation to compute the Fourier transform H (f) of the waveform given in Equation (124View Equation):

[ ] H (f) = 𝒜f − 7∕6 exp iΨ (f) + iπ- , (127 ) 4
with the Fourier amplitude 𝒜 and phase Ψ (f) given by
∘ --- ∑ 𝒜 = --𝒞--- 5-νM 5∕6, Ψ (f) = 2πf tC + ΦC + -3--- αk (πM f)(k−5)∕3 . (128 ) D π2∕3 24 128ν k
Here ν is the symmetric mass ratio defined before (see Equation 31View Equation), 𝒞 is a function of the various angles, as in Equation (124View Equation), and tC and ΦC are the fiducial epoch of merger and the phase of the signal at that epoch, respectively. The coefficients in the PN expansion of the Fourier phase are given by
α0 = 1, α1 = 0, α2 = 3715-+ 55ν, α3 = − 16π, 756 9 ( ) 15293365 27145 3085 2 38645 65 [ ( 3∕2 )] α4 = -508032-- + -504--ν + -72--ν , α5 = π -756--− 9-ν 1 + ln 6 πM f , ( ) 11583231236531-- 640- 2 6848- 15737765635-- 2255- 2 α6 = 4694215680 − 3 π − 21 γ + − 3048192 + 12 π ν 76055 127825 6848 + ------ν2 − -------ν3 − -----ln(64 πM f ), 1(728 1296 63 ) 77096675- 378515- 74045- 2 α7 = π 254016 + 1512 ν − 756 ν . (129 )
These are the PN coefficients in Einstein’s theory; in an alternative theory of gravity they will be different. In Einstein’s theory the coefficients depend only on the two mass parameters, the total mass M and symmetric mass ratio ν. One of the tests we will discuss below concerns the consistency of the various coefficients. Note, in particular, that in Einstein’s gravity the 0.5 PN term is absent, i.e., the coefficient of the term v is zero. Even with the very first observations of inspiral events, it should be possible to test if this is really so.

Figure 16View Image shows one such test that is possible with SMBH binaries [47Jump To The Next Citation Point48Jump To The Next Citation Point]. The observation of these systems in LISA makes it possible to measure the parameters associated with different physical effects. For example, the rate at which a signal chirps (i.e., the rate at which its frequency changes) depends on the binary’s chirpmass. Given the chirpmass, the length of the signal depends on the system’s symmetric mass ratio (the ratio of reduced mass to total mass). Another example would be the scattering of gravitational waves off the curved spacetime geometry of the binary, producing the tail effect in the emitted signal, which is determined principally by the system’s total mass [8786Jump To The Next Citation Point]. Similarly, spin-orbit interaction, spin-spin coupling, etc. depend on other combinations of the masses.

The binary will be seen with a high SNR, which means that we can measure the mass parameters associated with many of these physical effects. If each parameter is known precisely, we can draw a curve corresponding to it in the space of masses. However, our observations are inevitably subject to statistical (and possibly systematic) errors. Therefore, each parameter corresponds to a region in the parameter space (shown in Figure 16View Image for the statistical errors only). If Einstein’s theory of gravitation is correct, the regions corresponding to the different parameters must all have at least one common region, a region that contains the true parameters of the binary. This is because Einstein’s theory, or an alternative, has to be used to project the observed data onto the space of masses. If the region corresponding to one or more of these parameters does not overlap with the common region of the rest of the parameters, then Einstein’s theory, or its alternative, is in trouble.

In Brans–Dicke theory the system is expected to emit dipole radiation and the PN series would begin an order v−2 earlier than in Einstein’s theory. In the notation introduced above we would have an α−2 term, which has the form [75Jump To The Next Citation Point76]

2 α −2 = − -5𝒮---. (130 ) 84ωBD
Here 𝒮 is the the difference in the scalar charges of the two bodies and ω BD is the Brans–Dicke parameter. Although this term is formally two orders lower than the lowest-order quadrupole term of Einstein’s gravity (i.e., it is −2 𝒪(v ) order smaller), numerically its effect will be far smaller than the quadrupole term because of the rather large bound on ωBD ≫ 1. Nevertheless, its importance lies in the fact that there is now a new parameter on which the phase depends. Berti, Buonanno and Will conclude that LISA observations of massive black-hole binaries will enable scientists to set limits on 4 5 ωBD ∼ 10 –10.

A massive graviton theory would also alter the phase. The dominant effect is at 1 PN order and would change the coefficient α2 to

128ν π2DM α2 → α2 − ------2-------, (131 ) 3 λg(1 + z)
where ν is the symmetric mass ratio. This term alters the time of arrival of waves of different frequencies, causing a dispersion, and a corresponding modulation, in the wave’s phase, depending on the Compton wavelength λg and the distance D to the binary. Hence, by tracking the phase of the inspiral waves, one can bound the graviton’s mass. Will [393] finds that one can bound the mass to 1.7 × 1013 km using ground-based detectors and 1.7 × 1017 km using space-based detectors, as also confirmed by more recent and exhaustive calculations [75Jump To The Next Citation Point]. These limits might improve if one takes into account the modulation of the waveform due to spin-orbit and spin-spin coupling, but the latter authors [75] looked at spinning, but nonprecessing, systems only.
View Image

Figure 16: By fitting the Fourier transform of an observed signal to a post-Newtonian expansion, one can measure the various post-Newtonian coefficients ψ (m ,m ),k = 0,2,3, 4,6,7 k 1 2 and coefficients of log-terms ψ5l(m1,m2 ) and ψ6l(m1, m2 ). In Einstein’s theory, all the coefficients depend only on the two masses of the component black holes. By treating them as independent parameters one affords a test of the post-Newtonian theory. Given a measured value of a coefficient, one can draw a curve in the m1m2 plane. If Einstein’s theory is correct, then the different curves must all intersect at one point within the allowed errors. These plots show what might be possible with LISA’s observation of the merger of a binary consisting of a pair of 6 10 M ⊙ black holes. In the right-hand plot all known post-Newtonian parameters are treated as independent, while in the left-hand plot only three parameters ψ0,ψ2 and one of the remainingpost-Newtonian parameter are treated as independent and the procedure is repeated for each of the remaining parameters. The large SNR in LISA for SMBH binaries makes it possible to test various post-Newtonian effects, such as the tails of gravitational waves, tails of tails, the presence of log-terms, etc., associated with these parameters. Left-hand figure adapted from [48], right-hand figure reprinted with permission from [47].  External LinkThe American Physical Society.

6.6.2 Uniqueness of Kerr geometry

In Section 3 we pointed out that LISA should be able to see many hundreds of signals emitted by compact objects – black holes, neutron stars, even white dwarfs – orbiting around and being captured by massive black holes in the centers of galaxies. But for LISA to reach its full potential, we must model the orbits and their emitted radiation accurately. Since the wave trains may be many hundreds or thousands of cycles long in the LISA band, the challenge of constructing template waveforms that remain accurate to within about one radian over the whole evolution is significant.

The range of mass ratios is also wide. LISA’s central black holes might have masses between 103 and 107M ⊙. Inspiraling neutron stars and white dwarfs might have masses between 0.5 and 2M ⊙. Inspiraling stellar-population black holes might be in the range of 7 –50M ⊙, while intermediate-mass black holes formed by the first generation of stars (Population III stars) might have masses around 300M ⊙ or even 1000M ⊙. So the mass ratios might be anything in the range 10–7 to 1.

The techniques that must be used to compute these signals depend on the mass ratio. Ratios near one are treated by post-Newtonian methods until the objects are so close that only numerical relativity can follow their subsequent evolution. For ratios below 10–4 (a dividing line that is rather very uncertain and that depends on the bandwidth being used to observe the system, i.e., on how long the approximation must be valid for), systems are treated by fully-relativistic perturbation theory, expanding in the mass ratio. Intermediate mass ratios have not been studied in much detail yet; they can probably be treated by post-Newtonian methods up to a certain point, but it is not yet clear whether their final stages can be computed accurately by either numerical relativity or perturbation theory.

Post-Newtonian methods have been extensively discussed above. The basics of perturbation theory underlying this problem are treated in two Living Reviews [294322]. Once one has sufficiently good waveform templates, there remain the challenge of finding weak signals in LISA’s noise. This depends on a number of factors, including the rates of sources. A recent study by a number of specialists [170] has concluded that the event rate is high enough and the detection methods robust enough for us to be very optimistic that LISA will detect hundreds of these sources. In fact, the opposite problem might materialize: LISA might find it has a confusion problem for the detection of these sources, as for the galactic binaries. Recent estimates of the magnitude of the problem [68] suggest that LISA’s noise may at worst be raised effectively by a factor of two, but in return one gets a large number of sources of this kind.

6.6.3 Quantum gravity

It seems inevitable that general relativity’s description of nature will one day yield to a quantum-based description, involving uncertainties in geometry and probabilities in the outcome of gravitational observations. This is one of the most active areas of research in fundamental physics today, and there are many speculations about how quantum effects might show up in gravitational wave observations.

The simplest idea might be to try to find evidence for “gravitons” directly in gravitational waves, by analogy with the way that astronomers count individual photons from astronomical sources. But this seems doomed to failure. The waves that we can observe have very low frequency, so the energy of each graviton is extremely small. And the total energy flux of the waves is, as we have seen, enormous. So the number of gravitons in a detectable gravitational wave is far more than the number of photons in the light from a distant quasar.

Quantum gravity might involve new gravity-like fields, whose effects might be seen indirectly in the inspiral signals of black holes or neutron stars, as we have noted above. String theory might lead to the production of cosmic strings, which might be observed through their gravitational wave emission [143Jump To The Next Citation Point]. If our universe is just a 4-dimensional subspace of a large-scale 10 or 11-dimensional space, then dynamics in the larger space might produce gravitational effects in our space, and in particular gravitational waves [306].

It might be possible to observe the quantum indeterminacy of geometry directly using gravitational wave detectors, if Hogan’s principle of holographic indeterminacy is valid [195]. Hogan speculates that quantum geometry might be manifested by an uncertainty in the position of a beam splitter, and that this could be the explanation for an unexpectedly large amount of noise at low frequencies in the GEO600 detector. In this connection it is interesting to construct, from fundamental constants alone, a quantity with the dimensions of amplitude spectral noise density (Sh)1∕2. This has units of s1∕2, so one can define the “Planck noise power” SPl = tPl = (G ℏ∕c5)1∕2. Then the amplitude noise is 1∕2 5 1∕4 − 22 −1∕2 S Pl = (G ℏ∕c ) = 2.3 × 10 Hz. This is comparable to or larger than the instrumental noise in current interferometric gravitational wave detectors, as shown in Figure 5View Image. This in itself does not mean that Planckian noise will show up in gravitational wave detectors, but Hogan argues that the particular design of GEO600 might indeed make it subject to this noise more strongly than other large interferometers.

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