Showing papers by "Samuel E. Gralla published in 2020"

Lensing by Kerr Black Holes

Abstract: Interpreting horizon-scale observations of astrophysical black holes demands a general understanding of null geodesics in the Kerr spacetime. These may be divided into two classes: ``direct'' rays that primarily determine the observational appearance of a given source, and highly bent rays that produce a nested sequence of exponentially demagnified images of the main emission: the so-called ``photon ring.'' We develop heuristics that characterize the direct rays and study the highly bent geodesics analytically. We define three critical parameters $\ensuremath{\gamma}$, $\ensuremath{\delta}$, and $\ensuremath{\tau}$ that respectively control the demagnification, rotation, and time delay of successive images of the source, thereby providing an analytic theory of the photon ring. These observable parameters encode universal effects of general relativity, independent of the details of the emitting matter.

The shape of the black hole photon ring: A precise test of strong-field general relativity

Abstract: We propose a new test of strong-field general relativity (GR) based on the universal interferometric signature of the black hole photon ring. The photon ring is a narrow ring-shaped feature, predicted by GR but not yet observed, that appears on images of sources near a black hole. It is caused by extreme bending of light within a few Schwarzschild radii of the event horizon and provides a direct probe of the unstable bound photon orbits of the Kerr geometry. We show that the precise shape of the observable photon ring is remarkably insensitive to the astronomical source profile and can therefore be used as a stringent test of GR. We forecast that a tailored space-based interferometry experiment targeting M87* could test the Kerr nature of the source to the sub-subpercent level.

Null geodesics of the Kerr exterior

Abstract: The null geodesic equation in the Kerr spacetime can be expressed as a set of integral equations involving certain potentials. We classify the roots of these potentials and express the integrals in manifestly real Legendre elliptic form. We then solve the equations using Jacobi elliptic functions, providing the complete set of null geodesics of the Kerr exterior as explicit parametrized curves.

Observable shape of black hole photon rings

Abstract: Motivated by the prospect of measuring a black hole photon ring, in previous work we explored the interferometric signature produced by a bright, narrow curve in the sky. Interferometric observations of such a curve measure its ``projected position function'' $\mathbit{r}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{\mathbit{n}}$, where $\mathbit{r}$ parametrizes the curve and $\stackrel{^}{\mathbit{n}}$ denotes its unit normal vector. In this paper, we show by explicit construction that a curve can be fully reconstructed from its projected position, completing the argument that space interferometry can in principle determine the detailed photon ring shape. In practice, near-term observations may be limited to the visibility amplitude alone, which contains incomplete shape information: for convex curves, the amplitude only encodes the set of projected diameters (or ``widths'') of the shape. We explore the freedom in reconstructing a convex curve from its widths, giving insight into the shape information probed by technically plausible future astronomical measurements. Finally, we consider the Kerr ``critical curve'' in this framework and present some new results on its shape. We analytically show that the critical curve is an ellipse at small spin or inclination, while at extremal spin it becomes the convex hull of a Cartesian oval. We find a simple oval shape, the ``phoval,'' which reproduces the critical curve with high fidelity over the whole parameter range.

Measuring the shape of a black hole photon ring

Abstract: General relativity predicts that gravitational lensing near black holes will produce narrow "photon rings" on images. Building on recent work of Johnson, Lupsasca et al. focusing on circular rings, I show that the full detailed shape and intensity profile of any prescribed ring is directly measurable on long interferometric baselines. The shape information is contained entirely in the periodicity of the visibility amplitude as a function of baseline angle, which encodes the projected diameter at that angle. These results raise the possibility of model-independent measurements of the shape of a thin ring, which can then be compared to the precise predictions of general relativity.

Consistent Blandford-Znajek expansion

Abstract: The Blandford-Znajek mechanism is the continuous extraction of energy from a rotating black hole via plasma currents flowing on magnetic field lines threading the horizon. In the discovery paper, Blandford and Znajek demonstrated the mechanism by solving the equations of force-free electrodynamics in a perturbative expansion valid at small black hole spin. Attempts to extend this perturbation analysis to higher order have encountered inconsistencies. We overcome this problem using the method of matched asymptotic expansions, taking care to resolve all of the singular surfaces (light surfaces) in the problem. Working with the monopole field configuration, we show explicitly how the inconsistencies are resolved in this framework and calculate the field configuration to one order higher than previously known. However, there is no correction to the energy extraction rate at this order. These results confirm the basic consistency of the split monopole at small spin and lay a foundation for further perturbative studies of the Blandford-Znajek mechanism.

Horizon instability of the extremal BTZ black hole

Abstract: We study real-time propagation of a massive scalar field on the extremal BTZ black hole spacetime, focusing on the Aretakis instability of the event horizon. We obtain a simple time-domain expression for the AdS3 retarded Green function with Dirichlet boundary conditions and construct the corresponding time-domain BTZ retarded Green function using the method of images. The field decays at different rates on and off the horizon, indicating that transverse derivatives grow with time on the horizon (Aretakis instability). We solve the null geodesic equation in full generality and show that the instability is associated with a class of null geodesics that orbit near the event horizon arbitrarily many times before falling in. In an appendix we also treat the problem in the frequency domain, finding consistency between the methods.

Consistent Blandford-Znajek Expansion

Abstract: The Blandford-Znajek mechanism is the continuous extraction of energy from a rotating black hole via plasma currents flowing on magnetic field lines threading the horizon. In the discovery paper, Blandford and Znajek demonstrated the mechanism by solving the equations of force-free electrodynamics in a perturbative expansion valid at small black hole spin. Attempts to extend this perturbation analysis to higher order have encountered inconsistencies.We overcome this problem using the method of matched asymptotic expansions, taking care to resolve all of the singular surfaces (light surfaces) in the problem. Working with the monopole field configuration, we show explicitly how the inconsistencies are resolved in this framework and calculate the field configuration to one order higher than previously known. However, there is no correction to the energy extraction rate at this order. These results confirm the basic consistency of the split monopole at small spin and lay a foundation for further perturbative studies of the Blandford-Znajek mechanism.