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


Image of Fig. 1.
Fig. 1.

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.

Image of Fig. 2.
Fig. 2.

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.

Image of Fig. 3.
Fig. 3.

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.

Image of Fig. 4.
Fig. 4.

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.

Image of Fig. 5.
Fig. 5.

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.

Image of Fig. 6.
Fig. 6.

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.

Image of Fig. 7.
Fig. 7.

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.

Image of Fig. 8.
Fig. 8.

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.

Image of Fig. 9.
Fig. 9.

(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 .

Image of Fig. 10.
Fig. 10.

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.

Image of Fig. 11.
Fig. 11.

(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 .

Image of Fig. 12.
Fig. 12.

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.

Image of Fig. 13.
Fig. 13.

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.

Image of Fig. 14.
Fig. 14.

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.

Image of Fig. 15.
Fig. 15.

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 .

Image of Fig. 16.
Fig. 16.

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.

Image of Fig. 17.
Fig. 17.

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

Image of Fig. 18.
Fig. 18.

A symmetry plane through a black hole.

Image of Fig. 19.
Fig. 19.

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.

Image of Fig. 20.
Fig. 20.

(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).

Image of Fig. 21.
Fig. 21.

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


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: An intuitive approach to inertial forces and the centrifugal force paradox in general relativity