^{1,a)}

### 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 observer’s 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.

I. INTRODUCTION

II. THE TRAIN AND THE PLATFORM

III. THE STATICBLACK HOLE

IV. A MORE QUANTITATIVE ANALYSIS

V. FOLLOWING THE GEODESIC PHOTON

VI. THE DIFFERENCE BETWEEN THE GIVEN AND THE RECEIVED FORCE

VII. THE RELATIVISTIC SUBMARINE

A. A flat ocean in special relativity

B. A real spherical ocean

VIII. THE WEIGHT OF A BOX WITH MOVING PARTICLES

IX. GENERALIZING TO THREE DIMENSIONS

X. PARALLEL ACCELERATIONS

A. Combining the force equations

B. The given parallel force

XI. ROTATING REFERENCE FRAME

XII. DISCUSSION

### Key Topics

- Kinematics
- 42.0
- Photons
- 33.0
- Black holes
- 15.0
- Special relativity
- 12.0
- General relativity
- 8.0

## Figures

Rockets orbiting a static black hole. The solid arrows correspond to the force (the rocket thrust) necessary to keep the rocket in circular orbit. Inside of the photon radius (the dashed circle), the required force increases as the orbital velocity increases.

Rockets orbiting a static black hole. The solid arrows correspond to the force (the rocket thrust) necessary to keep the rocket in circular orbit. Inside of the photon radius (the dashed circle), the required force increases as the orbital velocity increases.

A train on a platform with a constant proper acceleration upward. (a) The train is at rest; (b) the train is moving relative to the platform. The force required of a man on the train to keep an apple at a fixed height is higher when the train moves than when it is at rest relative to the platform.

A train on a platform with a constant proper acceleration upward. (a) The train is at rest; (b) the train is moving relative to the platform. The force required of a man on the train to keep an apple at a fixed height is higher when the train moves than when it is at rest relative to the platform.

Two apples on an upward accelerating line (the solid line). The apples were initially at the position of the unfilled apple, one at rest and the other moving horizontally to the right. Both apples have to move up the same amount for a given coordinate time. But the one that moves horizontally has less proper time to do it. It must therefore experience a greater acceleration.

Two apples on an upward accelerating line (the solid line). The apples were initially at the position of the unfilled apple, one at rest and the other moving horizontally to the right. Both apples have to move up the same amount for a given coordinate time. But the one that moves horizontally has less proper time to do it. It must therefore experience a greater acceleration.

A freely falling frame (the grid) accelerating relative to the spatial geometry of a black hole (Ref. 13). We consider circular motion along the dashed line. The bottom edge of the depicted surface corresponds to the horizon. At this edge the embedding approaches a cylinder and the circle at the horizon is thus straight in the sense that it does not curve relative to the surface.

A freely falling frame (the grid) accelerating relative to the spatial geometry of a black hole (Ref. 13). We consider circular motion along the dashed line. The bottom edge of the depicted surface corresponds to the horizon. At this edge the embedding approaches a cylinder and the circle at the horizon is thus straight in the sense that it does not curve relative to the surface.

A zoom-in on the circular motion observed from a static system (with proper upward acceleration). The trajectory curves slightly downward, which will decrease the upward acceleration of the object relative to a freely falling system. In the limit that the acceleration of the freely falling frames is infinite, we can disregard the small curvature.

A zoom-in on the circular motion observed from a static system (with proper upward acceleration). The trajectory curves slightly downward, which will decrease the upward acceleration of the object relative to a freely falling system. In the limit that the acceleration of the freely falling frames is infinite, we can disregard the small curvature.

A particle moving along a trajectory of curvature relative to the accelerating reference frame. The thick line is freely falling and is initially aligned with the dashed line. For forces perpendicular to the direction of motion, only the perpendicular part of the acceleration is relevant.

A particle moving along a trajectory of curvature relative to the accelerating reference frame. The thick line is freely falling and is initially aligned with the dashed line. For forces perpendicular to the direction of motion, only the perpendicular part of the acceleration is relevant.

Trajectories of geodesic photons (dashed curves) relative to circles around a black hole. Inside the photon radius a circle curves outward relative to a locally tangent photon trajectory.

Trajectories of geodesic photons (dashed curves) relative to circles around a black hole. Inside the photon radius a circle curves outward relative to a locally tangent photon trajectory.

A simple model where small particles bounce elastically on the object. (a) The scenario as observed from a system where the impulse giving particle has no horizontal velocity. (b) The corresponding scenario as observed from a system comoving with the object.

A simple model where small particles bounce elastically on the object. (a) The scenario as observed from a system where the impulse giving particle has no horizontal velocity. (b) The corresponding scenario as observed from a system comoving with the object.

(a) An idealized (rectangular) submarine submerged in water at rest relative to the water. (b) As the submarine moves, it will be length contracted and thus the given force from the water will decrease by a factor of .

(a) An idealized (rectangular) submarine submerged in water at rest relative to the water. (b) As the submarine moves, it will be length contracted and thus the given force from the water will decrease by a factor of .

A submarine submerged in a balanced state of rest in a flat ocean with proper upward acceleration will sink due to relativistic effects if it starts moving horizontally.

A submarine submerged in a balanced state of rest in a flat ocean with proper upward acceleration will sink due to relativistic effects if it starts moving horizontally.

(a) Observed from the water system the submarine is length contracted by a factor of . (b) Observed from the submarine the water columns are length contracted and thus denser by a factor of .

(a) Observed from the water system the submarine is length contracted by a factor of . (b) Observed from the submarine the water columns are length contracted and thus denser by a factor of .

Submarines moving at different depths in the ocean of an imaginary very dense planet. The dashed line is the photon radius. The submarines outside the photon radius will float upward if they are given an azimuthal velocity; the opposite holds within the photon radius.

Submarines moving at different depths in the ocean of an imaginary very dense planet. The dashed line is the photon radius. The submarines outside the photon radius will float upward if they are given an azimuthal velocity; the opposite holds within the photon radius.

A black box (transparent for clarity) containing a pair of balls that (a) are at rest and (b) are moving. The force needed to hold the box at a fixed height on Earth is greater when the balls are moving than when they are at rest. The force is proportional to the total relativistic energy of the box.

A black box (transparent for clarity) containing a pair of balls that (a) are at rest and (b) are moving. The force needed to hold the box at a fixed height on Earth is greater when the balls are moving than when they are at rest. The force is proportional to the total relativistic energy of the box.

Deviations from a straight line relative to the (properly) accelerated reference system. The direction is chosen to be antiparallel to the local . The plane in which we study the deviations is perpendicular to the momentary direction of motion (the dashed line) and the three vectors lie in this plane. The solid curve is the particle trajectory as observed in the (properly) accelerated reference system. The thick line is a freely falling line that was aligned with the dashed line (and at rest relative to the reference frame) at the time when the particle was at the origin.

Deviations from a straight line relative to the (properly) accelerated reference system. The direction is chosen to be antiparallel to the local . The plane in which we study the deviations is perpendicular to the momentary direction of motion (the dashed line) and the three vectors lie in this plane. The solid curve is the particle trajectory as observed in the (properly) accelerated reference system. The thick line is a freely falling line that was aligned with the dashed line (and at rest relative to the reference frame) at the time when the particle was at the origin.

An object moving relative to an accelerated reference frame. At the velocity of the particle is . In a time the reference frame is accelerated to a velocity , and the velocity of the particle relative to the accelerated reference frame is .

An object moving relative to an accelerated reference frame. At the velocity of the particle is . In a time the reference frame is accelerated to a velocity , and the velocity of the particle relative to the accelerated reference frame is .

A particle (the black dot) moving along a rotating straight line (depicted at two successive time steps—the dashed and the solid line), as observed relative to an inertial system . Relative to the particle trajectory (the dotted line) curves.

A particle (the black dot) moving along a rotating straight line (depicted at two successive time steps—the dashed and the solid line), as observed relative to an inertial system . Relative to the particle trajectory (the dotted line) curves.

A path on a plane always corresponds locally to a circle as far as direction and curvature are concerned.

A path on a plane always corresponds locally to a circle as far as direction and curvature are concerned.

A symmetry plane through a black hole.

A symmetry plane through a black hole.

A sketch of the apparent geometry of a symmetry plane through a black hole. The innermost circle is at the surface of the black hole.

A sketch of the apparent geometry of a symmetry plane through a black hole. The innermost circle is at the surface of the black hole.

(a) Dropping an apple inside an elevator on Earth gives the same motion relative to the elevator as (b) dropping it inside a (properly) accelerated elevator in outer space. In both cases we can introduce an inertial (fictitious) gravitational force—but there is (in either case) no real gravitational force (in Einstein’s theory).

(a) Dropping an apple inside an elevator on Earth gives the same motion relative to the elevator as (b) dropping it inside a (properly) accelerated elevator in outer space. In both cases we can introduce an inertial (fictitious) gravitational force—but there is (in either case) no real gravitational force (in Einstein’s theory).

A coordinate system accelerating (falling) relative to the curved spatial geometry of a black hole.

A coordinate system accelerating (falling) relative to the curved spatial geometry of a black hole.

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