Gravitational Radiation Energy From Radial In-fall Into Schwarzschild and Kerr Geometries Project for Physics 879, Professor A. Buonanno, University of Maryland, 15 May 2006 M. A. Chesney
Motivation • Radial in-fall simplifies calculations tremendously • Total energy output and waveforms calculable by multiple methods • Results of various methods and authors easy to compare • Intense source of gravitational radiation • May be first detected by LIGO • Reaches limits of mass-energy to GW conversion • Look at size and shape of craters on the moon!
Newton • No GW, action at a distance, instantaneous propagation • But, as derived in class combine with Linearized Gravity gives fairly good result! Solving the differential equation, setting Rs = 2 G M / c2, and z[tmax] = R
Linearized Gravity Given the energy output and the Mass Quadrupole Moment We integrate from -infinity to tmax and get Setting R = Rs the prediction is 0.019 m c2 (m / M)
Mathematica code, cont. The above result is about twice the value predicted when the background curved spacetime is taken into account. See Animation for a hint as to why***
Radial In-fall into a Schwarzschild Black Hole Using Curved Background • Davis, Ruffini, Press and Price  were among the first to accurately describe the radial in-fall problem in terms of the wave form of the gravitational radiation and the total energy of the outgoing waves. • Employed Fourier transformations of metric perturbations crafted by Regge and Wheeler. • Interprets the gravity wave generation as the result of vibrations or quasinormal modes in the horizon caused by the impacting mass. • Paper did not detail the methods for solution or give the explicit amplitude solutions for any of the modes but they did give an overall result of • Etotal = 0.0104 m c2 (m / M) . • Listed methods as direct integration of Zerilli wave equation with a numerical search technique so that incoming waves only at Rs and outgoing waves only at infinity are found. Also Greens function technique. • Next slide shows equations to be solved.
Relativistic In-fall to Schwarzschild • Smarr  predicted a zero-frequency limit (ZFL) dE/dw, w -> 0 that is easily calculable for masses with a non-zero velocity at infinity • Able to make an order of magnitude estimate of energy for each polarization and the angular dependence. • Cardoso confirms ZFL of Smarr, and estimate of energy using Regge-Wheeler-Sasaki formalism • (dE/dw) w -> 0 = 0.4244m2 c2g2 • Etotal = 0.26m c2g2 (m / M) • Perturbation calculation gives excellent agreement with Smarr’s ZFL method as v -> 1 at infinity. • (dE/dw) w -> 0 = 4/(3 p) m2 c2g2 = 0.4244m2 c2g2 • Etotal = 0.2m c2g2 (m / M) • Next few slides detail Smarr’s ZFL method.
Infall Along the Z-Axis of a Kerr Black Hole • Nakamura and Sasaki  have taken the analysis a step further and found the gravitational radiation produced by the in-fall of a non-rotating test mass along the z-axis of a spinning black hole. • Using a finite difference method they found the total energy radiated • 0.0170m c2 (m / M) when a = 0.99. • Cardoso and Lemos  used the above Sasaki-Nakamura formalism to find the gravitational radiation when the impacting object is highly relativistic at infinity. • They found that for a collision along the symmetry axis the total energy is 0.31m c2g2 (m / M) for a = 0.999 M. • The equatorial limit is 0.69m c2g2 (m / M) for a = 0.999 M, v -> 1 • Represents the most efficient gravitational wave generator in the universe • See animation for hint why Equatorial > Radial GW generator****
GW From spinning particle into spinning BH • Mino, Shibata, Takahiro  examined z-axis, 0 in-fall velocity case • 0.0106m c2(m / M) , Parallel a = 0.99 M and m • 0.0298m c2(m / M) , Antiparallel a = 0.99 M and m • The parallel spin case is nearly same as spin zero case of Davis in 1971 • 0.0104 m c2 (m / M) • Intermediate to result when no spin falls into spin = 0.99 M • 0.0170m c2 (m / M) • Two factors at work here • Spin-spin interaction described by Papapetrou-Dixon equations • Energy-momentum tensor of the spinning particle • Magnificent treatment, combining Papapetrou-Dixon equations with Teukolski formalism, using the Sasaki-Nakamura formulation