Phys. Rev. D 108, 084004 (2023) - Not all spacetime coordinates for general-relativistic ray tracing are created equal

Not all spacetime coordinates for general-relativistic ray tracing are created equal

Gabriele Bozzola, Chi-kwan Chan, and Vasileios Paschalidis

Phys. Rev. D 108, 084004 – Published 3 October 2023

Abstract

Models for the observational appearance of astrophysical black holes rely critically on accurate general-relativistic ray tracing and radiation transport to compute the intensity measured by a distant observer. In this paper, we illustrate how the choice of coordinates and initial conditions affect this process. In particular, we show that propagating rays from the camera to the source leads to different solutions if the spatial part of the momentum of the photon points towards the horizon or away from it. In doing this, we also show that coordinates that are well suited for numerical general-relativistic magnetohydrodynamic (GRMHD) simulations are typically not optimal for generic ray tracing. We discuss the implications for black hole images and show that radiation transport in optimal and nonoptimal spacetime coordinates lead to the same images up to numerical errors and algorithmic choices.

Received 2 August 2023
Accepted 13 September 2023

DOI:https://doi.org/10.1103/PhysRevD.108.084004

Physics Subject Headings (PhySH)

Classical black holes General relativity

Accretion disk & black-hole plasma Astronomical black holes

Numerical techniques Relativistic magnetohydrodynamics

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Gabriele Bozzola ^*

Department of Astronomy, University of Arizona, Tucson, Arizona, USA

Chi-kwan Chan ^†

Department of Astronomy, University of Arizona, Tucson, Arizona, USA; Data Science Institute, University of Arizona, Tucson, Arizona, USA; and Program in Applied Mathematics, University of Arizona, Tucson, Arizona, USA

Vasileios Paschalidis^‡

Department of Astronomy, University of Arizona, Tucson, Arizona, USA and Department of Physics, University of Arizona, Tucson, Arizona, USA

^*gabrielebozzola@arizona.edu
^†chanc@arizona.edu
^‡vpaschal@arizona.edu

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Vol. 108, Iss. 8 — 15 October 2023

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Figure 1
Example of null geodesics in ingoing Kerr-Schild coordinates. Ingoing geodesics (arrows pointing towards $x = 0$ ) are horizon-penetrating and well-behaved. Outgoing (arrows pointing towards $x = + \infty$ ) geodesics are singular at the horizon. The choice of the direction of integration (forward or backward in time) determines which of the two families is being solved for. This shows that it is impossible for observers at infinity to collect information coming from inside the horizon.
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Figure 2
Top: Time component of the photon four-velocity for a ray integrated backwards in time moving towards the camera and one integrated forward in time moving towards the horizon. (For the former, we redefined $λ \to - λ$ to enable the comparison). The filled circle indicates when the photon crosses the horizon. In the case of backward propagation, the photon never does so. The fast growth in $d t / d λ$ leads to a growth in the constraint violation as well, as shown in the bottom panel. Bottom: The constraint grows to arbitrary high values. The solution for the forward integration is constant, so the algorithm can take large steps (the entire solution took only 11 steps) and keep the error down. The filled circle indicates when the photon crosses the horizon. In the case of backward propagation, the photon never does so, and one has to impose some artificial prescription to end the integration.
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Figure 3
Conformal diagram for a maximally-extended Schwarzschild spacetime (the angular dimensions are suppressed). Null geodesics are inclined with a 45° angle, with future directed ones moving towards $I^{+}$ . The ingoing Kerr-Schild coordinates only cover regions I and II, the outgoing ones region I and IV. Our Universe is region I. Constant time and radius curves are hyperbolas (purple and blue respectively). We place our camera C at a large separation from the horizon and trace back the emission along the orange line. For the ingoing Kerr-Schild coordinates, the propagation hits a coordinate singularity at $r = 2 M$ at the past horizon (orange dot) because that part of the spacetime is not mapped in these coordinates, whereas integration proceeds uninhibited until the physical singularity in the outgoing coordinates. The diagram also shows how integrating backward and forward in time results in two different geodesics: the orange one (backward) and the black dashed one (forward). The world line of the fluid element from which we collect emission is in green. Emission that is detectable from our cameras only comes from particles in region I. The problem with performing ray-tracing in ingoing coordinates is that there is “coordinate barrier” at $r = 2 M$ at the past horizon, so numerical algorithms fail.
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Figure 4
Backward propagation in coordinate time. This scheme does not have numerical problems, and the integration can be carried on for arbitrarily long times. The photon will take an infinite amount of time to reach the horizon. $\dot{x}$ drops to zero near unstable part of the integration is pushed to infinity.
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Figure 5
Numerical violation of the null condition for a null geodesic with screen coordinates $α = - 3.559 45 M$ , $β = 0$ . The geodesic is integrated backwards in both the ingoing (solid blue line) and outgoing (red dashed line) Kerr-Schild coordinates with black hole spin $a = 0.7 M$ . Both integrations identify that the photon comes from the horizon, but one does so in a regular fashion (red dashed line), and the blue one takes an infinite amount of time (solid blue line). The filled circle indicates when the photon crosses the horizon. In the inset, we show the trajectories of the photons in their own evaluated coordinate systems.
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Figure 6
Measured intensity from a stationary fluid configuration emitting with thermal bremsstrahlung on different coordinate systems. The camera is on the equatorial plane ( $i = 90 °$ , $j = 0 °$ ) at a Euclidean distance $d$ of $1 \times 10^{3} M$ . Intensity is measured in arbitrary units (au). Error (right panel) is computed as the relative difference between the two solutions generated integrating the rays backward in time in the ingoing (left panel) and outgoing (middle panel) Kerr-Schild coordinates.
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Physical Review D

covering particles, fields, gravitation, and cosmology

Not all spacetime coordinates for general-relativistic ray tracing are created equal

Gabriele Bozzola, Chi-kwan Chan, and Vasileios Paschalidis

Phys. Rev. D 108, 084004 – Published 3 October 2023

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Physical Review D

covering particles, fields, gravitation, and cosmology

Not all spacetime coordinates for general-relativistic ray tracing are created equal

Gabriele Bozzola, Chi-kwan Chan, and Vasileios Paschalidis

Phys. Rev. D 108, 084004 – Published 3 October 2023

Abstract

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