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Abstract:

We present an extremely simple derivation of the general relativistic periastron advance for Schwarzschild geometry. The method lends itself to iteration, and so in principle may achieve an arbitrary level of accuracy with relative ease. Our technique involves no analysis of differential equations beyond that of a simple harmonic oscillator. The practical utility of the calculation is limited by its neglect of frame-dragging rotational effects.

Abstract:

A set of analytic solutions for the plunging region thermodynamics has been developed recently under the assumption that the fluid undergoes a gravity-dominated geodesic plunge into the black hole. We test this model against a dedicated 3D global general relativistic magnetohydrodynamics simulation of a thin accretion disc around a Schwarzschild black hole using the code athenak . Provided that we include the effects of non-adiabatic heating (plausibly from grid-scale magnetic dissipation), we find excellent agreement between the analytic model and the simulated quantities. These results are particularly important for existing and future electromagnetic black hole spin measurements, many of which do not include the plunging fluid in their emission modelling. This exclusion typically stems from the assumption of a zero-stress boundary condition at the innermost stable circular orbit (ISCO), forcing all thermodynamic quantities to vanish. Instead, we find a non-zero drop in the angular momentum over the plunging region, which is consistent with both prior simulations and observations. We demonstrate that this stress is small enough for the dynamics of the fluid in the plunging region to be well-described by geodesic trajectories, yet large enough to cause measurable dissipation near to the ISCO – keeping thermodynamic quantities from vanishing. In the plunging region, constant -disc models are a physically inappropriate framework.

Abstract:

We show by direct calculation that the common equivalence principle explanation for why gravity must deflect light is quantitatively incorrect by a factor of three in Schwarzschild geometry. It is, therefore, possible, at least as a matter of principle, to tell the difference between local acceleration and a true gravitational field by measuring the local deflection of light. We calculate as well the deflection of test particles of arbitrary energy and construct a leading-order coordinate transformation from Schwarzschild to local inertial coordinates, which shows explicitly how the effects of spatial curvature manifest locally for relativistic trajectories of both finite and vanishing rest mass particles.