Physics:Kerr-Schild perturbations
Kerr-Schild perturbations are a special type of perturbation to a spacetime metric which only appear linearly in the Einstein field equations which describe General Relativity. They were found by Roy Kerr and Alfred Schild in 1965.[1]
Form
A generalised Kerr-Schild perturbation has the form [math]\displaystyle{ h_{ab}=V l_a l_b }[/math], where [math]\displaystyle{ V }[/math] is a scalar and [math]\displaystyle{ l_a }[/math] is a null vector with respect to the background spacetime.[2] It can be shown that any perturbation of this form will only appear quadratically in the Einstein equations, and only linearly if the condition [math]\displaystyle{ l^a \nabla_a l_b =\phi l_b }[/math], where [math]\displaystyle{ \phi }[/math] is a scalar, is imposed. This condition is equivalent to requiring that the orbits of [math]\displaystyle{ l^a }[/math] are geodesics.[2]
Applications
While the form of the perturbation may appear very restrictive, there are several black hole metrics which can be written in Kerr-Schild form, such as Schwarzschild (stationary black hole), Kerr (rotating), Reissner–Nordström (charged) and Kerr-Newman (both charged and rotating).[2][3]
References
- ↑ Kerr, R. P.; Schild, A. (2009). "Republication of: A new class of vacuum solutions of the Einstein field equations". General Relativity and Gravitation 41 (10): 2485–2499. doi:10.1007/s10714-009-0857-z.
- ↑ 2.0 2.1 2.2 Phys. Rev. D 94, 084009 (2016) - Generating exact solutions to Einstein's equation using linearized approximations. doi:10.1103/PhysRevD.94.084009.
- ↑ Balasin, Herbert; Nachbagauer, Herbert (1994). "Distributional energy--momentum tensor of the Kerr--Newman spacetime family". Classical and Quantum Gravity 11 (6): 1453–1461. doi:10.1088/0264-9381/11/6/010.
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