 Rotating Binary Blackhole (Kerr) Solution Approximation

+ 2 like - 0 dislike
76 views

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.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:$\varnothing\hbar$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.