Pre-Phase A Report - Lisa - Nasa

12 Chapter 1 Scientific Objectives
The most difficult part of the 2-body problem is the case of two objects of comparable
mass in a highly relativistic interaction, such as when two black holes merge. This can only
be studied using large-scale numerical simulations. One of the NSF’s Grand Challenge
projects for supercomputing is a collaboration among 7 university groups in the USA to
solve the problem of inspiralling and merging black holes. Within 10 years good solutions
could be available.
Mathematically, the field equations can be formulated in terms of a set of 10 fields that
are components of a symmetric 4 × 4 matrix {hαβ, α = 0...3, β = 0...3}. These
represent geometrically the deviation of the metric tensor from that of special relativity,
the Minkowski metric. In suitable coordinates the Einstein field equations can be written
∇ 2 − 1
c 2
∂2 ∂t2
hαβ = G
(source), (1.1)
c4 where “(source)” represents the various energy densities and stresses that can create the
field, as well as the non-linear terms in hαβ that represent an effective energy density and
stress for the gravitational field. This should be compared with Newton’s field equation,
∇ 2 Φ=4πGρ , (1.2)
where ρ is the mass density, or the energy density divided by c 2 .Sinceρ is dimensionally
(source)/c 2 , we see that the potentials hαβ are generalisations of Φ/c 2 , which is dimensionless.
This correspondence between the relativistic h and Newton’s Φ will help us to
understand the physics of gravitational waves in the next section.
Comparing Equation 1.1 with Equation 1.2 also shows how the Newtonian limit fits into
relativity. If velocities inside the source are small compared with c, then we can neglect
the time-derivatives in Equation 1.1; moreover, pressures and momentum densities will
be small compared to energy densities. Similarly, if h is small compared to 1 (recall that
it is dimensionless), then the nonlinear terms in “(source)” will be negligible. If these two
conditions hold, then the Einstein equations reduce simply to Newton’s equation in and
near the source.
However, Equation 1.1 is a wave equation, and time-dependent solutions will always
have a wavelike character far enough away, even for a nearly Newtonian source. The
transition point is where the spatial gradients in the equation no longer dominate the
time-derivatives. For a field falling off basically as 1/r and that has an oscillation frequency
of ω, the transition occurs near r ∼ c/ω = λ/2π, whereλis the wavelength of
the gravitational wave. Inside this transition is the “near zone”, and the field is basically
Newtonian. Outside is the “wave zone”, where the time-dependent part of the gravitational
acceleration (∇Φ) is given by Φ/λ rather than Φ/r. Time-dependent gravitational
effects therefore fall off only as 1/r, not the Newtonian 1/r2 .
1.1.2 The nature of gravitational waves in general relativity
Tidal accelerations. We remarked above that the observable effects of gravity lie in
the tidal forces. A gravitational wave detector would not respond to the acceleration
produced by the wave (as given by ∇Φ), since the whole detector would fall freely in this
3-3-1999 9:33 Corrected version 2.08