# Pretorius (2005): Evolution of Binary Black-Hole Spacetimes **Frans Pretorius**, Theoretical Astrophysics, California Institute of Technology; Department of Physics, University of Alberta *Physical Review Letters*, 95, 121101 (2005). arXiv:gr-qc/0507014. The first successful numerical simulation of a binary black hole merger through inspiral, coalescence, and ringdown. This paper exists because GR cannot analytically solve even the two-body problem for comparable masses. ## Overview Pretorius describes "early success in the evolution of binary black hole spacetimes with a numerical code based on a generalization of harmonic coordinates." The simulation evolves two equal mass, non-spinning black holes through approximately one orbit, plunge, merger, and ringdown, producing a single Kerr black hole with spin parameter $a \approx 0.70$. The paper required decades of computational development to achieve. It solved what was at the time one of the most pressing unsolved problems in general relativity. ## The opening statement (p. 1) ![[Pretorius2005_p1.png]] > "One of the more pressing, unsolved problems in general relativity today is to understand the structure of spacetime describing the evolution and merger of binary black hole systems." > [!critical] > In 2005, 90 years after the field equations were written, GR still could not describe what happens when two black holes merge. The two-body problem in GR was unsolved. This is not a three-body problem. This is two bodies. Newton solved the two-body problem with Kepler orbits in the 17th century. GR could not solve it for comparable masses after nearly a century of effort. ## Why numerical relativity was needed > "during the last several orbits, plunge, and early stages of the ring down, it is thought a numerical solution of the full problem will be needed to provide an accurate waveform." Perturbative methods (like the EIH post-Newtonian expansion from [[1938_Einstein_Infeld_Hoffmann_Equations_of_Motion]]) work during the early inspiral when the bodies are far apart and moving slowly. But as the bodies approach merger: - Velocities approach $c$ - Fields become strong (not weak) - The post-Newtonian expansion breaks down - Only direct numerical integration of the full Einstein field equations works > [!note] > This is the same pattern as the Newtonian three-body problem, but it happens already at two bodies. The field equations provide the instantaneous state. To find out what happens next, you integrate step by step on a computer. There is no formula $\mathbf{r}(t) = f(t, \text{initial conditions})$ that gives you the answer. ## The numerical method The simulation uses: 1. **Generalized harmonic coordinates** with constraint damping 2. **Adaptive mesh refinement** (up to 10 levels during inspiral and ringdown) 3. **Dynamical excision** that tracks the black holes through the grid 4. **Numerical dissipation** to control high frequency instabilities 5. **Scalar field collapse** to prepare initial data (Lorentz boosted scalar field profiles) The field equations are discretized in the form: $g^{\delta\gamma}g_{\alpha\beta,\gamma\delta} + g^{\gamma\delta}_{,\beta}g_{\alpha\delta,\gamma} + g^{\gamma\delta}_{,\alpha}g_{\beta\delta,\gamma} + 2H_{(\alpha,\beta)} - 2H_\delta\Gamma^\delta_{\alpha\beta} + 2\Gamma^\delta_{\beta\gamma}\Gamma^\delta_\alpha = -8\pi(2T_{\alpha\beta} - g_{\alpha\beta}T) - \kappa(n_\alpha C_\beta + n_\beta C_\alpha - g_{\alpha\beta}n^\gamma C_\gamma)$ > [!note] > This is not a textbook equation being "solved." This is a system of coupled nonlinear partial differential equations being discretized on a spatial grid and stepped forward in time. The computer computes the metric at each grid point at each timestep from the metric at the previous timestep. This is numerical integration. It is kinematics on a grid. ## Results - Two equal mass black holes ($M_0$ each) with initial proper separation $\sim 16.6 M_0$ - Orbital eccentricity $\sim 0$ to $0.2$ - Merge within approximately one orbit - Final black hole: mass $M_f \approx 1.9 M_0$, spin $a \approx 0.70$ - Roughly 5% of the initial rest mass radiated as gravitational waves - Runtime: "a few days for the lowest resolutions attempted, to several months at the higher resolutions" on a Xeon Linux cluster > [!critical] > The simulation took months of supercomputer time to evolve two bodies through a single orbit. This is the state of the art for the GR two-body problem. Compare with Kepler's solution for Newtonian two-body: an ellipse, computable by hand. GR does not improve on Newton for multi-body problems. It makes them vastly harder. ## The energy estimate problem The energy radiated as gravitational waves is estimated using the Newman-Penrose scalar $\Psi_4$: $\frac{dE}{dt} = \frac{R^2}{4\pi}\int p \, d\Omega, \quad p = \int_0^t \Psi_4 \, dt \cdot \int_0^t \bar{\Psi}_4 \, dt$ > [!warning] > The energy calculation is "quite susceptible to numerical error, as we are summing a positive definite quantity over all time." The gravitational wave energy depends on coordinate-dependent extraction at finite radius. This connects directly to Infeld and Plebanski's (1960) finding that radiation depends on "the choice of arbitrary harmonic functions." See [[1960_Infeld_Plebanski_Motion_and_Relativity]]. ## Relevance to the three-body null hypothesis This paper demonstrates that: 1. GR cannot solve the two-body problem analytically for comparable masses 2. The only method that works is numerical integration (kinematic computation on a grid) 3. Decades of effort were required to get even this far 4. The three-body problem in GR is incomparably harder, since it inherits all of this plus Poincare's chaos If the two-body problem requires months of supercomputer time for a single orbit, the three-body problem is not merely unsolved, it is practically intractable at the strong-field level. ## Cross-references - [[1938_Einstein_Infeld_Hoffmann_Equations_of_Motion]] -- The perturbative approach that breaks down where this paper begins - [[1960_Infeld_Plebanski_Motion_and_Relativity]] -- The radiation ambiguity that the energy estimate in this paper inherits - [[1890_Poincare_Three_Body_Problem]] -- If adding one body to Newton makes the problem unsolvable, adding one body to numerical relativity makes it worse - [[Three_Body_Null]] -- The null hypothesis note this source supports ## Citation Pretorius, F. (2005). "Evolution of Binary Black-Hole Spacetimes." *Physical Review Letters*, 95, 121101. arXiv:gr-qc/0507014.