Calculation of radiated gravitational energy using the second-order Einstein tensor

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Abstract
In general relativity there is a well-defined prescription for defining a quantity that represents the radiated energy of an exact, asymptotically flat solution of Einstein’s equation. This quantity is called the Bondi energy flux. However, in linearized gravity off a stationary and asymptotically flat background, the second-order Einstein tensor has been used as a stress-energy tensor for the perturbed gravitational field, enabling one to calculate the energy radiated away in gravitational radiation. It is natural to ask how this method compares to the exact method for calculating the Bondi energy flux. In this paper, it is shown that if the metric perturbation satisfies certain falloff and gauge conditions, then the radiated energy calculated using the second-order Einstein tensor equals the second-order contribution to the Bondi energy flux associated with the perturbation. As an application, the second-order Einstein tensor is used to demonstrate gravitational superradiance from a Kerr black hole. Also, the Appendix contains a theorem that makes precise the notion that if ∇(aξb) and its derivative is ‘‘small,’’ then ξa is close to a Killing field.

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