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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

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Gravitational radiation energy

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


  • 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!


  • 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
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 cont1
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
Radial In-fall into a Schwarzschild Black Hole Using Curved Background

  • Davis, Ruffini, Press and Price [1971] 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
Relativistic In-fall to Schwarzschild Background

  • Smarr [1977] 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
Infall Along the Z-Axis of a Kerr Black Hole Background

  • Nakamura and Sasaki [1982] 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 [2006] 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
GW From spinning particle into spinning BH Background

  • Mino, Shibata, Takahiro [1995] 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