# Rotating Binary Blackhole (Kerr) Solution Approximation

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Since the Kerr spacetime metric $$ds^2=-c^2d\tau^2=-\left(1-\frac{r_sr}{\Sigma}\right)c^2dt^2+\frac{\Sigma}{\Delta}dr^2+\Sigma d\theta^2\\+\left(r^2+a^2+\frac{r_sra^2}{\Sigma}sin^2\theta\right)sin^2\theta d\phi^2-\frac{2r_sra sin^2\theta}{\Sigma}cdtd\phi$$ is the axially symmetric solution of the Einstein field equations for a mass $M$ of angular momentum $J$, for $a=\frac{J}{Mc},$ $\Sigma=r^2+a^2cos^2\theta,$ and $\Delta=r^2-r_sr+a^2,$ in local Boyer–Lindquist coordinates $(x^0,x^1,x^2,x^3)=(ct,r,\theta,\phi),$ how does one numerically approximate the metric tensor that corresponds to the spacetime $(M,\mathcal{O},\mathcal{A},g)$ of a system of interacting, rotating binary blackholes with respective angular momenta of $J_1$ and $J_2$? Furthermore, what is the action associated with such a system?

Attempt: I thought that the localized Lagrangian may be given (invoking a $O(2)$ gauge theory) as $\mathcal{L}_{loc}=\frac{1}{2}(\nabla_{\mu}\Phi)^T\nabla^{\mu}\Phi-\frac{1}{2}m^Tm\Phi^T\Phi$ where $\nabla_{\mu} =\partial_{\mu}+igA_{\mu}$ is the metric-induced Levi-Civita connection (with $g$ the field coupling constant and $A(\bf x)$ the guage field), $\Phi:=(Tr[G^{\mu \nu}g_{\mu \nu}]_1,Tr[G^{\mu \nu}g_{\mu \nu}]_2)^T$ is the vector of fields for $\phi_i=Tr[G^{\mu \nu}g_{\mu \nu}]_i$ the gravitational scalar field associated with the $i$-th blackhole body, and $m=(M_1,M_2)^T$ is the vector of masses of the respective blackholes. (Unfortunately, this assumes that the metric describing the entire Lorentzian spacetime can be obtained linearly from individual Schwarzchild metrics of the binary in question, which is, of course, not true.) The Lagrangian thereby has local $O(2)$ guage group-invariance, preserved under the transformation $\Phi\mapsto\Phi'=G\Phi$ for $G\in O(2)$ a function of spacetime (i.e. $G:=G(\bf x)$), since the covariant derivative transforms identically as $\nabla_{\mu}\Phi\mapsto (\nabla_{\mu}\Phi)'=G(\nabla_{\mu}\Phi).$

I referred to Sean Carroll's Spacetime and Geometry; however, the section on binary pulsars does not go into much detail. Please note that I am new to this site, so constructive criticism is very much appreciated.

edited Feb 12, 2018

Large question perhaps too much general ? 1st The Kerr spacetime: A brief introduction ; note §6 ; then   Metric for two equal Kerr black holes  and for an attempt to generalize to multiple Kerr bh. Its useful references too, like Measuring multipole moments of Weyl metrics by means of gyroscopes . You may also find answers in spin netwoks studies. It is always about special cases with additional initial conditions and often, speculative assumptions.

Your references are very much appreciated!

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