arXiv subjects

Matthew S. Fox

Publications and source records attributed to Matthew S. Fox.

4 recordsLinked to original sources

arXiv2020

On Ramanujan Sums of a Real Variable and a New Ramanujan Expansion for the Divisor Function

We show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization, which is obtained by replacing the integer argument in the Ramanujan sums with a real number. We also construct a new Ramanujan expansion for the divisor function. While our expansion is amenable to a continuous and absolutely convergent real variable generalization, it only interpolates the divisor function locally on $\mathbb{R}$.

math.NT↗

arXiv2020

Electric Multipole Fields of Higher-Dimensional Massive Bodies

It was shown in a recent paper [J. Math. Phys. 60, 102502 (2019)] that slowly lowering an electric charge into a Schwarzschild-Tangherlini (ST) black hole endows the final state with electric multipole fields, which implies the final state geometry is not Reissner-Nordström-Tangherlini in nature. This conclusion departs from the four-dimensional case in which the no-hair theorem (NHT) requires the final state to be a Reissner-Nordström black hole. To better understand this discrepancy clearly requires a deeper understanding of the origin of the multipole hair in the higher-dimensional case. In this paper, we advance the conjecture that charged, static, and asymptotically-flat higher-dimensional black holes can acquire electric multipole hair only after they form. This supposition derives from studying the asymptotic behavior of the field of a multipole charge onto which a massive and hyperspherical shell with an exterior ST geometry is collapsing. In the mathematical limit as the shell approaches its ST radius, we find that the multipole fields (except the monopole) vanish. This implies that the only information of an arbitrary (but finite) charge distribution inside the collapsing shell that is available to an asymptotic observer is the total electric charge. Our results yield considerable insight into how higher-dimensional black holes acquire electric multipole hair, and also imply that, in four dimensions, the fadeaway of multipole moments during gravitational collapse is not strictly because of the NHT.

gr-qc↗

arXiv2019

Multipole Hair of Schwarzschild-Tangherlini Black Holes

We study the field of an electric point charge that is slowly lowered into an $n + 1$ dimensional Schwarzschild-Tangherlini black hole. We find that if $n > 3$, then countably infinite nonzero multipole moments manifest to observers outside the event horizon as the charge falls in. This suggests the final state of the black hole is not characterized by a Reissner-Nordström-Tangherlini geometry. Instead, for odd $n$, the final state either possesses a degenerate horizon, undergoes a discontinuous topological transformation during the infall of the charge, or both. For even $n$, the final state is not guaranteed to be asymptotically-flat.

gr-qc↗

arXiv2018

Palatini $f(\mathcal{R}, \mathcal{L}_m, \mathcal{R}_{μν}T^{μν})$ gravity and its Born-Infeld semblance

We investigate Palatini $f(\mathcal{R},\mathcal{L}_m, \mathcal{R}_{μν}T^{μν})$ modified theories of gravity wherein the metric and affine connection are treated as independent dynamical fields and the gravitational Lagrangian is made a function of the Ricci scalar $\mathcal{R}$, the matter Lagrangian density $\mathcal{L}_m,$ and a "matter-curvature scalar" $\mathcal{R}_{μν}T^{μν}$. The field equations and the equations of motion for massive test particles are derived, and we show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy-momentum dependent metric, related to the physical metric by a matrix transformation. Similar to metric $f(\mathcal{R}, T, \mathcal{R}_{μν}T^{μν})$ gravity, the field equations impose the non-conservation of the energy-momentum tensor, leading to an appearance of an extra force on massive test particles. We obtain the explicit form of the field equations for massive test particles in the case of a perfect fluid, and an expression for the extra force. The nontrivial modifications to scalar fields and both linear and nonlinear electrodynamics are also considered. Finally, we detail the conditions under which the present theory is equivalent to the Eddington-inspired Born-Infeld (EiBI) model.

gr-qc↗