Self-force and the Schwarzschild star

TL;DR

This work analyzes the self-force on a static charge in the spacetime of a Schwarzschild star, a constant-density interior matched to a Schwarzschild exterior. By exploiting a conformal mapping of the interior to a three-sphere (Maxwell fisheye lens), the authors obtain closed-form interior mode solutions for scalar and electromagnetic fields and implement two regularization schemes—direct and difference—within both the exterior vacuum and interior nonvacuum regions. They derive and validate new representations: (i) far-field and centre-series for the self-force, (ii) a leading-logarithmic divergence near the star surface, and (iii) comprehensive numerical data across the radial domain, including exact mode-sum identities that simplify sums. The results show a repulsive self-force throughout and reveal a universal logarithmic divergence as the charge approaches the boundary, highlighting the influence of global spacetime structure and boundaries on self-interactions in curved spacetimes.

Abstract

We consider the self-force acting on a pointlike (electromagnetic or conformal-scalar) charge held fixed on a spacetime with a spherically-symmetric mass distribution of constant density (the Schwarzschild star). The Schwarzschild interior is shown to be conformal to a three-sphere geometry; we use this conformal symmetry to obtain closed-form expressions for mode solutions. We calculate the self-force with two complementary regularization methods, direct and difference regularization, showing agreement. For the first time, we show that difference regularization can be applied in the non-vacuum interior region, due to the vanishing of certain regularized mode sums. The new results for the self-force come in three forms: series expansions for the self-force near the centre of the star and in the far field; a new approximation that describes the divergence in the self-force near the star's boundary; and numerical data presented in a selection of plots. We conclude with a discussion of the logarithmic divergence in the self-force in the approach to the star's surface, and the effect of boundaries.

Figures (7)

• Figure 1: This figure shows the behaviour of the partial mode-sum as we increase the value of $\ell$ where we truncate the sum. The data $\Sigma_i$, $i=1,2,3$ denote the BH scalar sum \ref{['eq:FrBHsum']}, the scalar interior sum \ref{['eq:Fr0interior']} and the EM interior mode-sum respectively. In all cases, we see that the partial sums decrease according to a power law as we truncate the mode-sums at higher values of $\ell_{\text{max}}$, implying that the infinite sum must vanish, in accordance with the results in Secs. \ref{['sec:ident-ext']} and \ref{['sec:ident-int']}.
• Figure 2: Modes of the self force $F_r^\ell$ on a log-log scale showing the expected polynomial decay after subtracting regularization terms in the interior (left) and the exterior (right). For the above plots we choose a star of radius $R = 11 M$ with the charge placed at $r_0 = 6 M$ (left) and $r_0 = 16M$ (right) in \ref{['intregcurves']} and \ref{['extregcurves']} respectively. In all figures units such that $M=q=1$ are adopted.
• Figure 3: A comparison of the numerical SF data obtained by evaluating the mode-sum results [solid] for the scalar and electromagnetic SFs, with the series expansions [dashed] in Eq. \ref{['scalarexp']} and Eq. \ref{['EMexp']} respectively, for a star of radius $R = 50M$.
• Figure 4: Numerical data for the $\ell$-modes of the SF (dots) on a log-log plot, for a particle close to the star surface at $R=3M$, with $\Delta r = r_0 - R = 0.005 M$. The expected exponential ($(l+1/2)^{-2}$) behaviour for the difference (direct) regularised modes only starts after an initial regime with a slower decay. The slow-decay behaviour is compared with the green dashed guideline, which is proportional to $(l+1/2)^{-1}$ and with a coefficient taken from Eq. \ref{['surfexp']}.
• Figure 5: The modes of the SF calculated via direct regularisation (upper) and via the difference method (lower) for a particle at $r_0 = R + \Delta r$, near the surface of a Schwarzschild star of radius $R=3M$. Plots (a) and (b) show that, as the particle gets closer to the surface of the star, the number of $\ell$ modes required to reach a regime where the bare modes show the expected $\left(\ell + 1/2\right)^{-2}$ fall off (in plots (a) and (b)) increases linearly with $1/|\Delta r|$. Plot (c) shows the difference-regularized $\ell$-modes, for a range of $\Delta r$. The dashed lines show the large-$\ell$ asymptotic approximation in Eq. \ref{['surfexp']}. The exponent $\Omega$ in that approximation approaches zero as $\Delta r \rightarrow 0$, and thus the total self-force diverges as $\Delta r \rightarrow 0$ in a logarithmic manner.