Scalings on the initial hypersurface

3.3 Scalings on the initial hypersurface

Following [79

, 82 , 138

], we use the initial value formulation to solve the Einstein equations. As initial data for the matter variables and the gravitational field, we take a set of nearly stationary solutions of the exact Einstein equations representing two widely separated fluid balls, each of which rotates uniformly. We assume that these solutions are parametrized by $ε$ and that the matter and field variables have the following scalings.

The density scales as $−4 ε$ (in the $i (t,x )$ coordinates), implied by the scalings of $m$ and $R$ . The scaling of the density suggests that the natural dynamical time (free fall time) $η$ inside the star may be $η = ε− 2t$ . Then if we cannot assume almost stationary condition on the stars, it is difficult to use the post-Newtonian approximation [80 , 81 ]. In practice, however, our formalism is still applicable to pulsating stars if the effect of pulsation is not important in the orbital motion.

The velocity of stellar rotation is assumed to be $𝒪 (ε)$ . In other words, we assume that the star rotates slowly and is pressure supported⁶ . By this assumption, the spin-orbit coupling force appears at 2 PN order rather than the usual 1.5 PN order. The slowly spinning motion assumption is not crucial: In fact, it is straightforward to incorporate a rapidly spinning compact body into our formalism.

From these initial data we have the following scalings of the star $A$ ’s stress-energy tensor components $T μAν$ in the body zone coordinates: $T τAτ= 𝒪 (ε− 2)$ , $TAτi= 𝒪 (ε−4)$ , $ij TA-= 𝒪(ε−8)$ . Here the underlined indices mean that for any tensor $Ai$ , $Ai-= ε−2Ai$ . In [94 ] , we have transformed $T μν N$ , the components of the stress-energy tensor of the matter in the near zone coordinates, to $T μν A$ using the transformation from the near zone coordinates to the (generalized) Fermi normal coordinates at 1 PN order [13 ]. It is difficult, however, to construct the (generalized) Fermi normal coordinates at an high post-Newtonian order. Therefore we shall not use it. We simply assume that for $T μν N$ (or rather $Λ μν N$ , the source term of the relaxed Einstein equations; see Equation (63 )),

T&#x03C4;&#x03C4; = &#x1D4AA; &#x0028;&#x03B5;&#x2212;2&#x0029;, &#x0028;45 &#x0029; N&#x03C4;i TN- = &#x1D4AA; &#x0028;&#x03B5;&#x2212;4&#x0029;, &#x0028;46 &#x0029; ij &#x2212;8 TN = &#x1D4AA; &#x0028;&#x03B5; &#x0029;, &#x0028;47 &#x0029;

as their leading scalings.

As for the field variables on the initial hypersurface, we simply assume that the field is of 2.5 PN order except for the field determined by the constraint equations. Note that the radiation reaction effect to the stars first appears at the 2.5 PN order. Futamase showed that even if one takes the field of order 1 PN, initial value of the field does not affect the subsequent motion of the system up to 2.5 PN order [80 ]. Thus, we expect that the initial value of the field does not affect the orbital motion of the system up to 3 PN order, though a detailed calculation has not been done yet.

It is worth noticing that the initial value formulation has some advantages. First, by using the initial value formulation one can avoid the famous runaway solution problem in a radiation reaction problem. Second, one can construct an initial condition on some spacelike hypersurface rather than at past null infinity. Putting an initial condition for the field in the past null infinity requires a prior knowledge about the spacetime, which is obtained through the time evolution of the field from the initial condition. The initial value formulation can give in a sense a realistic initial condition. In our universe there may be no past null infinity because of the big bang.

An interesting initial condition is the statistical initial condition [138 ]. Here the binary system is in the background gravitational radiation bath for which we know only its statistical properties. For example, the phase of the radiation is assumed to be random and irrelevant to the motion of the binary. The origins of the radiation are cosmological, or related to the evolution of the system before the initial hypersurface. Then we can evaluate the expected time evolution of the binary system by letting the system evolve from a set of possible initial conditions and taking a statistical ensemble average over the initial conditions.