arXiv subjects

Adrian C. Ottewill

Publications and source records attributed to Adrian C. Ottewill.

At least 19 recordsLinked to original sources

arXiv2026

Quantum fluxes and $\langle\hatΦ^2\rangle$ for a non-minimally coupled scalar field: ringdown and tail on approaching the polar Kerr inner horizon

We compute $\langle\hatΦ^{2}\rangle_\text{ren}$ as well as the energy fluxes $\langle \hat{T}_{uu}\rangle_\text{ren}$ and $\langle \hat{T}_{vv}\rangle_\text{ren}$ (where $u$ and $v$ are the standard Eddington-Finkelstein coordinates) associated with a quantum massless real scalar field $\hatΦ$, with a general curvature coupling constant $ξ$, near the inner horizon (IH) of a Kerr black hole, along the axis of rotation. The quantum field is in the Unruh state, corresponding to an evaporating black hole. We renormalize these quantities by the state-subtraction method. We drop the assumption of minimal coupling to the curvature, thereby generalizing the results of arXiv:2203.08502 for the fluxes at the IH. This requires understanding the asymptotic behavior of $\langle\hatΦ^{2}\rangle_\text{ren}$ neat the IH. State subtraction allows us to push the computation of $\langle\hatΦ^{2}\rangle_\text{ren}$ along the axis of rotation in the Kerr interior in arXiv:2409.17464 deeper into the near-IH region, exposing their final asymptotic behavior on approaching the IH. For $\langle\hatΦ^{2}\rangle_\text{ren}$ (a $ξ$-independent quantity in the Kerr case), we find that the approach to its finite asymptotic IH value is given, per $\ell$-mode, by a ringdown phase (namely exponentially damped oscillations), followed by an inverse-power tail, both in the tortoise coordinate $r_{*}$ (which diverges at the IH). Interestingly, in the regime where the ringing dominates, the ringing's complex frequencies are (numerically) found to match twice the well-known classical quasinormal-mode frequencies in Kerr, and the inverse-power tails are found to be $r_{*}^{-2\ell-3}$ (resembling Price's law in the classical black hole exterior, upon replacement $t\to r_*$). [Abridged]

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arXiv2025

Computation of $\langle Φ^2\rangle$ and quantum fluxes at the polar interior of a spinning black hole

Renormalization of physical quantities for quantum field theories in curved spacetimes can be achieved via the consistent subtraction of counterterms within a regularization scheme such as a point-splitting method. Pragmatic mode-sum regularization (PMR) is a point-splitting method which is particularly suitable for rotating black hole spacetimes. We extend and tailor the t-splitting variant of PMR specifically for the interior of a Kerr black hole on the axis of rotation, focusing on a minimally-coupled massless scalar field in the physically-motivated Unruh state. The method addresses unique challenges in the black hole interior that do not occur outside. In particular, while the infinite sum over multipolar number l converges in the black hole exterior, it diverges in the interior, necessitating the subtraction of a so-called intermediate divergence which includes introducing an additional "small" split in the direction of the polar angle. This procedure is outlined and justified, along with the standard PMR method's counterterms subtraction. We apply this method to calculate the renormalized energy-momentum fluxes $\langle T_{uu}\rangle^U_\text{ren}$, $\langle T_{vv}\rangle^U_\text{ren}$ (where u and v are the standard Eddington coordinates) and the renormalized field square $\langle Φ^2\rangle^U_\text{ren}$ throughout the black hole interior, spanning from (just off) the event horizon to (just off) the inner horizon. Special emphasis is placed on the inner horizon vicinity, where our t-splitting results for the fluxes asymptote to those obtained directly at the inner horizon using a different method in a previous work. In an Appendix, we develop an alternative t-splitting PMR variant which does not include the intermediate divergence subtraction. We utilize it for independent computations that are used to verify the standard t-splitting variant presented in the main text.

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arXiv2022

Quantum fluxes at the inner horizon of a spinning black hole

Rotating or charged classical black holes in isolation possess a special surface in their interior, the Cauchy horizon, beyond which the evolution of spacetime (based on the equations of General Relativity) ceases to be deterministic. In this work, we study the effect of a quantum massless scalar field on the Cauchy horizon inside a rotating (Kerr) black hole that is evaporating via the emission of Hawking radiation (corresponding to the field being in the Unruh state). We calculate the flux components (in Eddington coordinates) of the renormalized stress-energy tensor of the field on the Cauchy horizon, as functions of the black hole spin and of the polar angle. We find that these flux components are generically non-vanishing. Furthermore, we find that the flux components change sign as these parameters vary. The signs of the fluxes are important, as they provide an indication of whether the Cauchy horizon expands or crushes (when backreaction is taken into account). Regardless of these signs, our results imply that the flux components generically diverge on the Cauchy horizon when expressed in coordinates which are regular there. This is the first time that irregularity of the Cauchy horizon under a semiclassical effect is conclusively shown for (four-dimensional) spinning black holes.

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arXiv2022

Two-point function of a quantum scalar field in the interior region of a Kerr black hole

Quantum field effects on a classical background spacetime may be obtained from the semiclassical equations of General Relativity with the expectation value of the stress-energy tensor of the quantum field as a source. This expectation value can be calculated from Hadamard's elementary two-point function, which in practice is given in terms of sums of products of field modes evaluated at two spacetime points. We derive expressions for the two-point function for a massless scalar field in the Unruh state on a Kerr black hole spacetime. Our main result in this paper is a novel expression valid when the two points lie inside the black hole; we also (re-)derive, using a new method, the known expression valid when the two points lie outside the black hole. We achieve these expressions by finding relationships between Unruh modes, defined in terms of the retarded Kruskal coordinate, and Eddington modes, defined in terms of the Eddington coordinates. While our starting expression for the two-point function is written in terms of the Unruh modes, we give our final expression in terms of the Eddington modes, which have the computational advantage that they decompose into factors that obey ordinary differential equations. In an appendix we also derive expressions for the bare mode contributions to the flux components of the stress-energy tensor for a minimally-coupled massless scalar field inside the black hole. Our results thus lay the groundwork for future calculations of quantum effects inside a Kerr black hole.

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arXiv2021

Analytical high-order post-Newtonian expansions for extreme mass ratio binaries

We present analytic computations of gauge invariant quantities for a point mass in a circular orbit around a Schwarzschild black hole, giving results up to 15.5 post-Newtonian order in this paper and up to 21.5 post-Newtonian order in an online repository. Our calculation is based on the functional series method of Mano, Suzuki and Takasugi (MST) and a recent series of results by Bini and Damour. We develop an optimised method for generating post-Newtonian expansions of the MST series, enabling significantly faster computations. We also clarify the structure of the expansions for large values of $\ell$, and in doing so develop an efficient new method for generating the MST renormalised angular momentum, $ν$.

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arXiv2021

Analytical high-order post-Newtonian expansions for spinning extreme mass ratio binaries

We present an analytic computation of Detweiler's redshift invariant for a point mass in a circular orbit around a Kerr black hole, giving results up to 8.5 post-Newtonian order while making no assumptions on the magnitude of the spin of the black hole. Our calculation is based on the functional series method of Mano, Suzuki and Takasugi, and employs a rigorous mode-sum regularization prescription based on the Detweiler-Whiting singular-regular decomposition. The approximations used in our approach are minimal; we use the standard self-force expansion to linear order in the mass ratio, and the standard post-Newtonian expansion in the separation of the binary. A key advantage of this approach is that it produces expressions that include contributions at all orders in the spin of the Kerr black hole. While this work applies the method to the specific case of Detweiler's redshift invariant, it can be readily extended to other gauge invariant quantities and to higher post-Newtonian orders.

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arXiv2020

Transport Equation Approach to Calculations of Hadamard Green functions and non-coincident DeWitt coefficients

Building on an insight due to Avramidi, we provide a system of transport equations for determining key fundamental bi-tensors, including derivatives of the world-function, σ(x,x'), the square root of the Van Vleck determinant, Δ^{1/2}(x,x'), and the tail-term, V(x,x'), appearing in the Hadamard form of the Green function. These bi-tensors are central to a broad range of problems from radiation reaction to quantum field theory in curved spacetime and quantum gravity. Their transport equations may be used either in a semi-recursive approach to determining their covariant Taylor series expansions, or as the basis of numerical calculations. To illustrate the power of the semi-recursive approach, we present an implementation in \textsl{Mathematica} which computes very high order covariant series expansions of these objects. Using this code, a moderate laptop can, for example, calculate the coincidence limit a_7(x,x) and V(x,x') to order (σ^a)^{20} in a matter of minutes. Results may be output in either a compact notation or in xTensor form. In a second application of the approach, we present a scheme for numerically integrating the transport equations as a system of coupled ordinary differential equations. As an example application of the scheme, we integrate along null geodesics to solve for V(x,x') in Nariai and Schwarzschild spacetimes.

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arXiv2019

Regularized calculation of the retarded Green function in Schwarzschild spacetime

The retarded Green function for linear field perturbations of black hole spacetimes is notoriously difficult to calculate. One of the difficulties is due to a Dirac-$δ$ divergence that the Green function possesses when the two spacetime points are connected by a "direct" null geodesic. We present a procedure which notably aids its calculation in the case of Schwarzschild spacetime by separating this direct $δ$-divergence from the remainder of the retarded Green function. More precisely, the method consists of calculating the multipolar $\ell$-modes of the direct $δ$-divergence and subtracting them from the corresponding modes of the retarded Green function. We illustrate the usefulness of the method with some specific calculations in the case of the scalar Green function and self-field for a point scalar charge in Schwarzschild spacetime.

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arXiv2019

High-order asymptotics for the Spin-Weighted Spheroidal Equation at large real frequency

The spin-weighted spheroidal eigenvalues and eigenfunctions arise in the separation by variables of spin-field perturbations of Kerr black holes. We derive a large, real-frequency asymptotic expansion of the spin-weighted spheroidal eigenvalues and eigenfunctions to high order. This expansion corrects and extends existing results in the literature and we validate it via a numerical calculation.

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arXiv2018

Accelerated motion and the self-force in Schwarzschild spacetime

We provide expansions of the Detweiler-Whiting singular field for motion along arbitrary, planar accelerated trajectories in Schwarzschild spacetime. We transcribe these results into mode-sum regularization parameters, computing previously unknown terms that increase the convergence rate of the mode-sum. We test our results by computing the self-force along a variety of accelerated trajectories. For non-uniformly accelerated circular orbits we present results from a new 1+1D discontinuous Galerkin time-domain code which employs an effective-source. We also present results for uniformly accelerated circular orbits and accelerated bound eccentric orbits computed within a frequency-domain treatment. Our regularization results will be useful for computing self-consistent self-force inspirals where the particle's worldline is accelerated with respect to the background spacetime.

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arXiv2017

Spin-orbit precession along eccentric orbits for extreme mass ratio black hole binaries and its effective-one-body transcription

In this work we present an analytical gravitational self-force calculation of the spin-orbit precession along an eccentric orbit around a Schwarzschild black hole, following closely the recent prescription of Akcay, Dempsey, and Dolan. We then transcribe this quantity within the Effective-One-Body (EOB) formalism, thereby determining several new, linear-in-mass-ratio, contributions in the post-Newtonian expansion of the spin-orbit couplings entering the EOB Hamiltonian. Namely, we determine the second gyro-gravitomagnetic ratio $g_{S_*}(r,p_r,p_ϕ)$ up to order $p_r^2/r^4$ included.

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arXiv2017

Spin-1 Quasi-normal Frequencies in Schwarzschild space-time for Large Overtone Number

We analytically investigate the spin-1 quasinormal mode frequencies of Schwarzschild black hole space-time. We formally determine these frequencies to arbitrary order as an expansion for large imaginary part (i.e., large-n, where n is the overtone number). As an example of the practicality of this formal procedure, we explicitly calculate the asymptotic behaviour of the frequencies up to order $n^{-5/2}$.

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arXiv2016

High-order tail in Kerr spacetime

We investigate the late-time tail of the retarded Green function for the dynamics of a linear field perturbation of Kerr spacetime. We develop an analytical formalism for obtaining the late-time tail up to arbitrary order for general integer spin of the field. We then apply this formalism to obtain the details of the first five orders in the late-time tail of the Green function for the case of a scalar field: to leading order we recover the known power law tail $t^{-2\ell-3}$, and at third order we obtain a logarithmic correction, $t^{-2\ell-5}\ln t$, where $\ell$ is the field multipole.

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arXiv2015

Analytic self-force calculations in the post-Newtonian regime: eccentric orbits on a Schwarzschild background

We present a method for solving the first-order field equations in a post-Newtonian (PN) expansion. Our calculations generalize work of Bini and Damour and subsequently Kavanagh et al., to consider eccentric orbits on a Schwarzschild background. We derive expressions for the retarded metric perturbation at the location of the particle for all $\ell$-modes. We find that, despite first appearances, the Regge-Wheeler gauge metric perturbation is $C^0$ at the particle for all $\ell$. As a first use of our solutions, we compute the gauge-invariant quantity $\langle U \rangle$ through 4PN while simultaneously expanding in eccentricity through $e^{10}$. By anticipating the $e\to 1$ singular behavior at each PN order, we greatly improve the accuracy of our results for large $e$. We use $\langle U \rangle$ to find 4PN contributions to the effective one body potential $\hat Q$ through $e^{10}$ and at linear order in the mass-ratio.

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arXiv2015

High-order Tail in Schwarzschild Space-time

We present an analysis of the behaviour at late-times of linear field perturbations of a Schwarzschild black hole space-time. In particular, we give explicit analytic expressions for the field perturbations (for a specific multipole) of general spin up to the first four orders at late times. These expressions are valid at arbitrary radius and include, apart from the well-known power-law tail decay at leading order ($\sim t^{-2\ell-3}$), a new logarithmic behaviour at third leading order ($\sim t^{-2\ell-5}\ln t$). We obtain these late-time results by developing the so-called MST formalism and by expanding the various MST Fourier-mode quantities for small frequency. While we give explicit expansions up to the first four leading orders (for small-frequency for the Fourier modes, for late-time for the field perturbation), we give a prescription for obtaining expressions to arbitrary order within a `perturbative regime'.

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arXiv2015

Vacuum polarization on the brane

We compute the renormalized expectation value of the square of a massless, conformally coupled, quantum scalar field on the brane of a higher-dimensional black hole. Working in the AADD brane-world scenario, the extra dimensions are flat and we assume that the compactification radius is large compared with the size of the black hole. The four-dimensional on-brane metric corresponds to a slice through a higher-dimensional Schwarzschild-Tangherlini black hole geometry and depends on the number of bulk space-time dimensions. The quantum scalar field is in a thermal state at the Hawking temperature. An exact, closed-form expression is derived for the renormalized expectation value of the square of the quantum scalar field on the event horizon of the black hole. Outside the event horizon, this renormalized expectation value is computed numerically. The answer depends on the number of bulk space-time dimensions, with a magnitude which increases rapidly as the number of bulk space-time dimensions increases.

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arXiv2015

Tidal invariants for compact binaries on quasi-circular orbits

We extend the gravitational self-force approach to encompass `self-interaction' tidal effects for a compact body of mass $μ$ on a quasi-circular orbit around a black hole of mass $M \gg μ$. Specifically, we define and calculate at $O(μ)$ (conservative) shifts in the eigenvalues of the electric- and magnetic-type tidal tensors, and a (dissipative) shift in a scalar product between their eigenbases. This approach yields four gauge-invariant functions, from which one may construct other tidal quantities such as the curvature scalars and the speciality index. First, we analyze the general case of a geodesic in a regular perturbed vacuum spacetime admitting a helical Killing vector and a reflection symmetry. Next, we specialize to focus on circular orbits in the equatorial plane of Kerr spacetime at $O(μ)$. We present accurate numerical results for the Schwarzschild case for orbital radii up to the light-ring, calculated via independent implementations in Lorenz and Regge-Wheeler gauges. We show that our results are consistent with leading-order post-Newtonian expansions, and demonstrate the existence of additional structure in the strong-field regime. We anticipate that our strong-field results will inform (e.g.) effective one-body models for the gravitational two-body problem that are invaluable in the ongoing search for gravitational waves.

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arXiv2014

Static detectors and circular-geodesic detectors on the Schwarzschild black hole

We examine the response of an Unruh-DeWitt particle detector coupled to a massless scalar field on the (3+1)-dimensional Schwarzschild spacetime, in the Boulware, Hartle-Hawking and Unruh states, for static detectors and detectors on circular geodesics, by primarily numerical methods. For the static detector, the response in the Hartle-Hawking state exhibits the known thermality at the local Hawking temperature, and the response in the Unruh state is thermal at the local Hawking temperature in the limit of a large detector energy gap. For the circular-geodesic detector, we find evidence of thermality in the limit of a large energy gap for the Hartle-Hawking and Unruh states, at a temperature that exceeds the Doppler-shifted local Hawing temperature. Detailed quantitative comparisons between the three states are given. The response in the Hartle-Hawking state is compared with the response in the Minkowski vacuum and in the Minkowski thermal state for the corresponding Rindler, drifted Rindler, and circularly accelerated trajectories. The analysis takes place within first-order perturbation theory and relies in an essential way on stationarity.

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