# Black Hole-Black Hole Evolution

## Introduction

Here we have a binary black hole system (BHBH) in a quasi-equilibrium circular orbit. This is puncture BHBH initial data, evolved with the BSSN formulation. The primary black hole has a spin that is at a 45 degree angle with respect to its angular momentum axis, and the secondary black hole has a spin that is at 225 degrees with with respect to this axis. $S_1$ and $S_2$ are defined as $S_1/{M_1}^2 = a_1/M_1$ and $S_2/{M_2}^2 = a_2/M_2$. The mass of the primary black hole is $M_1 = 36 M_{\odot}$ and the mass of the secondary black hole is $M_2 = 29 M_{\odot}$, which gives a mass ratio of $M_{1}/M_{2} = 1.24$. We define M to be the ADM mass at the initial time which is $M = M_{ADM}(0) = 65 M_{\odot}$. Similarly, we define $J = J_{ADM}(0)$, where $J/M^{2} = 0.949$.

 Fig. 1-1: Initial Configuration Fig. 1-2: Initial Masses Fig. 1-3: Initial Spins

## Black Hole-Black Hole Evolution

The initial and final spins are indicated by the green arrows, and are normalized to their dimensionless values. The binary makes approximately six orbits prior to merging at $t \approx 335 \; ms$. As the binary merges, we see the formation of a common horizon, which oscillates until settling down at $t \approx 345 \; ms$. The simulation continues until $t = 416 \; ms$ to demonstrate the stability of the resulting Kerr black hole. Notice that for this case, we draw a gray plane along the initial orbital plane, which we use to emphasize the precession of the orbit as the black holes spiral in towards merger. Also, after merger, the resulting black hole has a drift velocity, which we calculated to be $v_{drift} = 26 \; km/s$. The arrow at the end of the visualization indicates the direction of the final angular momentum, and is scaled to the dimensionless angular momentum, $J_{BH}/{M_{BH}}^2$.

 Fig. 1-2: Binary Black Holes at time t = 0 ms Fig. 1-2: Binary Black Holes at time t = 329 ms Fig. 1-2: Binary Black Holes at time t = 416 ms
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## Final Black Hole Parameters

Listed in the table below is the dimensionless spin of the black hole at the end of our simulation. Also shown are the radiated energy and angular momentum from gravitational wave emission. Here, M is the initial binary ADM mass whereas MBH is the final ADM mass of the black hole remnant.
 Quantity Dimensionless Value $M_{BH}/M$ 0.95 $J_{BH}/M^2$ 0.69 $\Delta E_{GW}/M$ 0.040 $\Delta J_{GW}/M^2$ 0.337 $\delta E \equiv (M-M_{BH}- \Delta E_{GW})/M$ $4.37 \times 10^{-4}$ $\delta J \equiv (J-J_{BH}- \Delta J_{GW})/J$ $-2.16 \times 10^{-2}$

Our simulation maintains fair conservation of energy and momentum, since $\delta E$ and $\delta J$ are on the order of $10^{-4}$ and $10^{-2}$, respectively.