In general relativity, Regge–Wheeler–Zerilli equations are a pair of equations that describes gravitational perturbations of a Schwarzschild black hole, named after Tullio Regge, John Archibald Wheeler and Frank J. Zerilli. The perturbations of a Schwarzchild metric is classified into two types, namely, axial and polar perturbations, a terminology introduced by Subrahmanyan Chandrasekhar. Axial perturbations induce frame dragging by imparting rotations to the black hole and change sign when azimuthal direction is reversed, whereas polar perturbations do not impart rotations and do not change sign under the reversal of azimuthal direction. The equation for axial perturbations is called Regge–Wheeler equation and the equation governing polar perturbations is called Zerilli equation.
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| - In general relativity, Regge–Wheeler–Zerilli equations are a pair of equations that describes gravitational perturbations of a Schwarzschild black hole, named after Tullio Regge, John Archibald Wheeler and Frank J. Zerilli. The perturbations of a Schwarzchild metric is classified into two types, namely, axial and polar perturbations, a terminology introduced by Subrahmanyan Chandrasekhar. Axial perturbations induce frame dragging by imparting rotations to the black hole and change sign when azimuthal direction is reversed, whereas polar perturbations do not impart rotations and do not change sign under the reversal of azimuthal direction. The equation for axial perturbations is called Regge–Wheeler equation and the equation governing polar perturbations is called Zerilli equation. (en)
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| - In general relativity, Regge–Wheeler–Zerilli equations are a pair of equations that describes gravitational perturbations of a Schwarzschild black hole, named after Tullio Regge, John Archibald Wheeler and Frank J. Zerilli. The perturbations of a Schwarzchild metric is classified into two types, namely, axial and polar perturbations, a terminology introduced by Subrahmanyan Chandrasekhar. Axial perturbations induce frame dragging by imparting rotations to the black hole and change sign when azimuthal direction is reversed, whereas polar perturbations do not impart rotations and do not change sign under the reversal of azimuthal direction. The equation for axial perturbations is called Regge–Wheeler equation and the equation governing polar perturbations is called Zerilli equation. The equations take the same form as the one-dimensional Schrödinger equation. The equations read as where characterizes the polar perturbations and the axial perturbations. Here is the tortoise coordinate (we set ), belongs to the Schwarzschild coordinates , is the Schwarzschild radius and representing the time-dependence of the perturbations appearing in the form . The Regge–Wheeler potential and Zerilli potential are respectively given by where and characterizes the eigenmode for the coordinate. For gravitational perturbations, the modes are irrelevant because they do not evolve with time. Physically gravitational perturbations with (monopole) mode represents a change in the black hole mass, whereas the (dipole) mode corresponds to a shift in the location and value of the black hole's angular momentum. The shape of above potentials are exhibited in the figure. Remember that in the tortoise coordinate, denotes event horizon and is equivalent to i.e., to distances far away from the back hole. The potentials are short ranged as they decay faster than ; as , we have and as , we have Consequently, the asymptotic behaviour of the solutions for is (en)
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