First law of binary black hole mechanics in general relativity and post-Newtonian theory

First laws of black hole mechanics, or thermodynamics, come in a variety of different forms. In this paper, from a purely post-Newtonian (PN) analysis, we obtain a first law for binary systems of point masses moving along an exactly circular orbit. Our calculation is valid through 3PN order and includes, in addition, the contributions of logarithmic terms at 4PN and 5PN orders. This first law of binary point-particle mechanics is then derived from first principles in general relativity, and analogies are drawn with the single and binary black hole cases. Some consequences of the first law are explored for PN spacetimes. As one such consequence, a simple relation between the PN binding energy of the binary system and Detweiler’s redshift observable is established. Through it, we are able to determine with high precision the numerical values of some previously unknown high-order PN coefficients in the circular-orbit binding energy. Finally, we propose new gauge-invariant notions for the energy and angular momentum of a particle in a binary system.

Figure 1

A gravitationally bound isolated matter source is formed at $T=T_0$, and starts emitting gravitational radiation. The ADM mass $M_{\mathrm{ADM}}$, as computed on that time slice, coincides with the binding energy $M(T_0)=m+E(T_0)$, which is itself equal to the Bondi mass $M_{\mathrm{B}}(U_0)$, as computed on the asymptotically null hypersurface $U=U_0$. At a later time $T=T_1$, the binding energy $M(T_1)=M_{\mathrm{B}}(U_1)$ has decreased. The difference $M_{\mathrm{ADM}}-M_{\mathrm{B}}(U_1)$ with respect to the constant ADM mass is equal to the energy taken away from the source by the gravitational waves emitted between $T=T_0$ and $T=T_1$, or equivalently between $U=U_0$ and $U=U_1$.

Figure 2

A nonradiative PN spacetime containing an ever-lasting gravitationally bound isolated matter source of constant size $\mathcal{R}$. The ADM mass $M_{\mathrm{ADM}}$ coincides with the binding energy $M(T)=m+E(T)$, which is itself equal to the Bondi mass $M_{\mathrm{B}}(U)$, at all times.

Figure 3

A spatial slice $\Sigma$ of a circular-orbit compact binary spacetime. The black hole is characterized by its horizon area $A$ and uniform surface gravity $\kappa$, while the point particle has a mass $m$ and redshift factor $z=1/u^t$, and is such that its four-velocity reads $u^{\alpha }=u^t{K}^{\alpha }$.