Strong field point particle limit

3.1 Strong field point particle limit

In a relativistic compact binary system in an inspiralling phase, each star is well approximated by a point particle with a few low order multipole moments. This is because in the inspiralling phase, the stellar size divided by the orbital separation is much smaller than unity.

If we wish to apply the post-Newtonian approximation to the inspiraling phase of binary neutron stars, the strong internal gravity must be taken into account. The usual post-Newtonian approximation explicitly assumes the weakness of the gravitational field everywhere including inside the material source. It is argued by applying the strong equivalence principle that the external gravitational field which governs the orbital motion of the binary system is independent of the internal structure of the components up to tidal interaction. Thus it is expected that the results obtained under the assumption of weak gravity also apply for the case of a neutron star binary. Experimental evidence for the strong equivalence principle is obtained only for systems with weak gravity [159 , 160 , 161 , 164 ], but at present no experiment is available in a case with strong internal gravity.

In the theoretical aspect, the theory of extended objects in general relativity [69 ] is still in a preliminary stage for an application to realistic systems. The matched asymptotic expansion technique has been used to treat a system with strong gravity in certain situations [47 , 65 , 66 , 104 , 105 ]. Another way to handle strong internal gravity is by the use of a Dirac delta distribution type source with a fixed mass [73 ]. However, this makes the Einstein equations mathematically meaningless because of their nonlinearity. Physically, there is no such source in general relativity because of the existence of black holes. Before a body shrinks to a point, it forms a black hole whose size is fixed by its mass. For this reason, it has been claimed that no point particle exists in general relativity.

This conclusion is not correct, however. We can shrink the body keeping the compactness $(M ∕R )$ , i.e. the strength of the internal field fixed. Namely we should scale the mass $M$ just like the radius $R$ . This can be fitted nicely into the concept of the regular asymptotic Newtonian sequence defined in Section 2 because there the mass also scales along the sequence of solutions. In fact, if we take the masses of two stars as $M$ , and the separation between two stars as $L$ , then $Φ ∼ M ∕L$ . Thus the mass $M$ scales as $ε2$ if we fix the separation⁴ . In the above we have assumed that the density scales as $ε2$ to guarantee this scaling for the mass while keeping the size of the body fixed. Now we shrink the size as $2 ε$ to keep the compactness of each component. Then the density should scale as $−4 ε$ . We shall call such a scheme the strong field point particle limit since the limit keeps the strength of internal gravity⁵ . The above consideration suggests the following initial data to define a regular asymptotic Newtonian sequence which describes a nearly Newtonian system with strong internal gravity [81 ]. The initial data are two uniformly rotating fluids with compact spatial support whose stress-energy tensor and size scales as $−4 ε$ and $2 ε$ , respectively. We also assume that each of these fluid configurations would be a stationary equilibrium solution of the Einstein equations if the other were absent. This is necessary for the suppression of irrelevant internal motions of each star. Any remaining motions are tidal effects caused by the other body, which will be of order $ε6$ smaller than the internal self-force. These data allow us to use the Newtonian time $τ = εt$ as a natural time coordinate everywhere including the interior region of the stars.

As for multipole moments, the scaling $R = 𝒪 (ε2)$ enables us to incorporate the multipole expansion of the stars into the post-Newtonian approximation. In inspiralling compact binaries the tidally and rotationally induced quadrupole moments are too small to affect the binary orbital motion. The time to coalesce is too short for the binary to be tidally locked and corotate [15 , 44 , 110 ]. The phase shift due to the quadrupole-orbit couplings may be negligible in the LIGO bandwidth [44 ]. However, to detect gravitational waves from inspiralling binaries, we have to have highly accurate prior knowledge about the binary orbital motion, say, 4 PN equations of motion or so. The effect of quadrupole moments on the orbital motion can be of about the same order as that of spin-spin interactions [132 ], while the spin-spin interactions appear at about 2.5 PN order for slowly rotating stars. Also, at the late inspiralling phase, an effect of extendedness of the stars on the motion will be important. Thus it is important to take the multipole moments of the stars into account in a way that is suitable for compact stars when we derive the equations of motion for an inspiralling compact binary.