An intuitive approach to inertial forces and the centrifugal force paradox in general relativity

Abstract

As the velocity of a rocket in a circular orbit near a black hole increases, the outwardly directed rocket thrust must increase to keep the rocket in its orbit. This feature might appear paradoxical from a Newtonian viewpoint, but we show that it follows naturally from the equivalence principle together with special relativity and a few general features of black holes. We also derive a general relativistic formalism of inertial forces for reference frames with acceleration and rotation. The resulting equation relates the real experienced forces to the time derivative of the speed and the spatial curvature of the particle trajectory relative to the reference frame. We show that an observer who follows the path taken by a free (geodesic) photon will experience a force perpendicular to the direction of motion that is independent of the observers velocity. We apply our approach to resolve the submarine paradox, which regards whether a submerged submarine in a balanced state of rest will sink or float when given a horizontal velocity if we take relativistic effects into account. We extend earlier treatments of this topic to include spherical oceans and show that for the case of the Earth the submarine floats upward if we take the curvature of the ocean into account.

Lux in obscuro: photon orbits of extremal black holes revisited

“…For such an orbit, a greater outward thrust is required to maintain its orbit as the orbital speed increases. (See [8] and the references therein for further discussion, and [7,9] for the generalization to stationary axisymmetric case. )…”

Inertial forces and the foundations of optical geometry

“…For such considerations we refer to a companion paper [16]. There we rely on simple principles such as time dilation and the equivalence principle and derive the relativistic three-dimensional form of the inertial force equation (68) using no four-covariant formalism at all.…”

Relativity, thermodynamics and entropic forces

“…which at first sight appears to be identical to (8), derived in flat space-time. It must be kept in mind, however, that the gradient operator in general coordinates assumes the form [19]…”

“…where the rightmost term is the entropy due to the stress-energy-momentum of all the particles, and the factor of 1/2 is included in the first term to account for duplicate pairs of particles in the sum. Equation (19) can be simplified upon noticing that the sum over λ j in the first term is just the total field potential λ (P i ) at the position P i of U i . Using this observation puts (19) in the form…”