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Computational Methods for the Self-Force in Black Hole Spacetimes

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Book cover Mass and Motion in General Relativity

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 162))

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Abstract

We survey the set of computational methods devised for implementing the MiSaTaQuWa formulation in practice, for orbits around Kerr black holes. We focus on the gravitational self-force (SF) and review in detail two of these methods: (i) the standard mode-sum method, in which the perturbation field is decomposed into multipole harmonics and the MiSaTaQuWa regularization is performed, effectively, mode by mode; and (ii) m-mode regularization, whereby one regularizes individual azimuthal modes of the full perturbation. The implementation of these strategies involves the numerical integration of the relevant perturbation equations, and we discuss several practical issues that arise and ways to deal with them. These issues include the choice of gauge, the numerical representation of the particle singularity, and the handling of high-frequency contributions near the particle in frequency-domain calculations. As an example, we show results from an actual computation of the gravitational SF for an eccentric geodesic orbit around a Schwarzschild black hole, using direct numerical integration of the Lorenz-gauge perturbation equations in the time domain.

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Notes

  1. 1.

    In the original MiSaTaQuWa formulation, the SF is not expressed directly in terms of the gradient of \(\bar{{h}}_{\alpha \beta }^{\mathrm{tail}}\), but rather as a worldline integral over the gradient of the relevant retarded Green’s function (cf. Eq. 1.9.6 of Poisson [97]). The commutation of the derivative operator and the worldline integral produces local terms at x, which, however, vanish (in the vacuum case which concerns us here) upon contraction with k αβγδ at the limit x → z.

  2. 2.

    The quantity \({k}^{\alpha \beta \gamma \delta }\bar{{h}}_{\beta \gamma ;\delta }\) is the linear perturbation in the connection coefficients, δΓ γβ α(h μν), projected orthogonally to Γ; that is, \({k}^{\alpha \beta \gamma \delta }\bar{{h}}_{\beta \gamma ;\delta } = ({g}^{\beta \gamma } + {u}^{\beta }{u}^{\gamma })\delta {\Gamma }_{\beta \gamma }^{\alpha }\).

  3. 3.

    Here we ignore the vectorial nature of F α and F S α and treat each of their Boyer–Lindquist components as a scalar function. We do this for a mere mathematical convenience. See [62] for a more sophisticated, covariant treatment.

  4. 4.

    Note that the form of F αl and F S αl will depend on the specific off-worldline extension chosen for the tensor k. However, this ambiguity disappears upon taking the limit x → z, and the final SF is of course insensitive to the choice of extension.

  5. 5.

    It is easy to convince oneself that the two-sided limits δr → 0 ±  in Eq. 35 are equal in magnitude and different in sign: Notice ε0( − δr, − x, y) = ε0(δr, x, y) (for δt = 0), which, together with the fact that the integration domain is symmetric in x, implies that the integral in Eq. 35 is an odd function of δr.

  6. 6.

    The gauge conditions (1) do not fully specify the gauge: There is a residual gauge freedom within the family of Lorenz gauges, h αβ → h αβ + ξα; β + ξβ; α, with any ξμ satisfying □ ξμ = 0. It is easy to verify that both Eqs. 1 and 43 remain invariant under such gauge transformations.

  7. 7.

    To simplify the appearance of Eq. 45 we have used here a slightly different normalization than that of [15] for the functions \(\bar{{h}}^{(i)lm}\).

  8. 8.

    A full separation of variables in Kerr is possible within Teukolsky’s formalism, which, alas, brings about the metric reconstruction and gauge-related difficulties discussed in previous chapters. A full separation of the metric perturbation equations themselves, in Kerr, has not been formulated yet, to the best of our knowledge.

  9. 9.

    This can be shown by integrating S αβ res over a small 3-ball containing the particle (at a given time), and then inspecting the limit as the radius of the ball tends to zero [11].

  10. 10.

    It should not come as a surprise that at x = z the sum over m-modes F punc α, m − F S α, m (which are all zero) fails to recover the original function F punc α − F S α (which is discontinuous and hence indefinite). Recall that the formal Fourier sum at a step discontinuity (where the function itself is indefinite) is in fact convergent: it yields the two-side average value of the function at the discontinuity. Technically, this peculiarity of the formal Fourier expansion is due simply to the noninterchangeability of the sum and limit at the point of discontinuity.

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Acknowledgements

We acknowledge support from PPARC/STFC through grant number PP/D001110/1.

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  1. School of Mathematics, University of Southampton, Southampton, SO17 1BJ, United Kingdom

    Leor Barack

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  1. Institut d'Astrophysique de Paris, boulevard Arago 98bis, Paris, 75014, France

    Luc Blanchet

  2. Observatoire de la Cote d'Azur, Nice, France

    Alessandro Spallicci

  3. Dept. Physics, University of Florida, Gainesville, 32611-8440, Florida, USA

    Bernard Whiting

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Barack, L. (2009). Computational Methods for the Self-Force in Black Hole Spacetimes. In: Blanchet, L., Spallicci, A., Whiting, B. (eds) Mass and Motion in General Relativity. Fundamental Theories of Physics, vol 162. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3015-3_12

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