4.4 Equations of motion and 4 Strong Gravity and Gravitational 4.2 Motion and gravitational radiation

4.3 Einstein's equations in ``relaxed'' form 

The Einstein equations tex2html_wrap_inline5011 are elegant and deceptively simple, showing geometry (in the form of the Einstein tensor tex2html_wrap_inline5013, which is a function of spacetime curvature) being generated by matter (in the form of the material stress-energy tensor tex2html_wrap_inline4449). However, this is not the most useful form for actual calculations. For post-Newtonian calculations, a far more useful form is the so-called ``relaxed'' Einstein equations:


where tex2html_wrap_inline5017 is the flat-spacetime wave operator, tex2html_wrap_inline5019 is a ``gravitational tensor potential'' related to the deviation of the spacetime metric from its Minkowski form by the formula tex2html_wrap_inline5021, g is the determinant of tex2html_wrap_inline5025, and a particular coordinate system has been specified by the de Donder or harmonic gauge condition tex2html_wrap_inline5027 (summation on repeated indices is assumed). This form of Einstein's equations bears a striking similarity to Maxwell's equations for the vector potential tex2html_wrap_inline5029 in Lorentz gauge: tex2html_wrap_inline5031, tex2html_wrap_inline5033 . There is a key difference, however: The source on the right hand side of Eq. (45Popup Equation) is given by the ``effective'' stress-energy pseudotensor


where tex2html_wrap_inline5035 is the non-linear ``field'' contribution given by terms quadratic (and higher) in tex2html_wrap_inline5019 and its derivatives (see [94], Eqs. (20.20, 20.21) for formulae). In general relativity, the gravitational field itself generates gravity, a reflection of the nonlinearity of Einstein's equations, and in contrast to the linearity of Maxwell's equations.

Equation (45Popup Equation) is exact, and depends only on the assumption that spacetime can be covered by harmonic coordinates. It is called ``relaxed'' because it can be solved formally as a functional of source variables without specifying the motion of the source, in the form


where the integration is over the past flat-spacetime null cone tex2html_wrap_inline5039 of the field point tex2html_wrap_inline5041 . The motion of the source is then determined either by the equation tex2html_wrap_inline5043 (which follows from the harmonic gauge condition), or from the usual covariant equation of motion tex2html_wrap_inline5045, where the subscript tex2html_wrap_inline5047 denotes a covariant divergence. This formal solution can then be iterated in a slow motion (v <1) weak-field (tex2html_wrap_inline5051) approximation. One begins by substituting tex2html_wrap_inline5053 into the source tex2html_wrap_inline5055 in Eq. (47Popup Equation), and solving for the first iterate tex2html_wrap_inline5057, and then repeating the procedure sufficiently many times to achieve a solution of the desired accuracy. For example, to obtain the 1PN equations of motion, two iterations are needed (i.e. tex2html_wrap_inline5059 must be calculated); likewise, to obtain the leading gravitational waveform for a binary system, two iterations are needed.

At the same time, just as in electromagnetism, the formal integral (47Popup Equation) must be handled differently, depending on whether the field point is in the far zone or the near zone. For field points in the far zone or radiation zone, tex2html_wrap_inline5061 (tex2html_wrap_inline5063 is the gravitational wavelength tex2html_wrap_inline5065), the field can be expanded in inverse powers of tex2html_wrap_inline5067 in a multipole expansion, evaluated at the ``retarded time'' t - R . The leading term in 1/ R is the gravitational waveform. For field points in the near zone or induction zone, tex2html_wrap_inline5073, the field is expanded in powers of tex2html_wrap_inline5075 about the local time t, yielding instantaneous potentials that go into the equations of motion.

However, because the source tex2html_wrap_inline5055 contains tex2html_wrap_inline5019 itself, it is not confined to a compact region, but extends over all spacetime. As a result, there is a danger that the integrals involved in the various expansions will diverge or be ill-defined. This consequence of the non-linearity of Einstein's equations has bedeviled the subject of gravitational radiation for decades. Numerous approaches have been developed to try to handle this difficulty. The ``post-Minkowskian'' method of Blanchet, Damour and Iyer [19, 20, 21, 45, 22, 15] solves Einstein's equations by two different techniques, one in the near zone and one in the far zone, and uses the method of singular asymptotic matching to join the solutions in an overlap region. The method provides a natural ``regularization'' technique to control potentially divergent integrals. The ``Direct Integration of the Relaxed Einstein Equations'' (DIRE) approach of Will, Wiseman and Pati [152Jump To The Next Citation Point In The Article, 105] retains Eq. (47Popup Equation) as the global solution, but splits the integration into one over the near zone and another over the far zone, and uses different integration variables to carry out the explicit integrals over the two zones. In the DIRE method, all integrals are finite and convergent.

These methods assume from the outset that gravity is sufficiently weak that tex2html_wrap_inline5051 and harmonic coordinates exists everywhere, including inside the bodies. Thus, in order to apply the results to cases where the bodies may be neutron stars or black holes, one relies upon the strong equivalence principle to argue that, if tidal forces are ignored, and equations are expressed in terms of masses and spins, one can simply extrapolate the results unchanged to the situation where the bodies are ultrarelativistic. While no general proof of this exists, it has been shown to be valid in specific circumstances, such as at 2PN order in the equations of motion, and for black holes moving in a Newtonian background field [39Jump To The Next Citation Point In The Article].

Methods such as these have resolved most of the issues that led to criticism of the foundations of gravitational radiation theory during the 1970s.

4.4 Equations of motion and 4 Strong Gravity and Gravitational 4.2 Motion and gravitational radiation

image The Confrontation between General Relativity and Experiment
Clifford M. Will
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de