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### Abstract

In special relativity a gyroscope that is suspended in a torque-free manner will precess as it is moved along a curved path relative to an inertial frame . We explain this effect, which is known as Thomas precession, by considering a real grid that moves along with the gyroscope, and that by definition is not rotating as observed from its own momentary inertial rest frame. From the basic properties of the Lorentz transformation we deduce how the form and rotation of the grid (and hence the gyroscope) will evolve relative to . As an intermediate step we consider how the grid would appear if it were not length contracted along the direction of motion. We show that the uncontracted grid obeys a simple law of rotation. This law simplifies the analysis of spin precession compared to more traditional approaches based on Fermi transport. We also consider gyroscope precession relative to an accelerated reference frame and show that there are extra precession effects that can be explained in a way analogous to the Thomas precession. Although fully relativistically correct, the entire analysis is carried out using three-vectors. By using the equivalence principle the formalism can also be applied to staticspacetimes in general relativity. As an example, we calculate the precession of a gyroscope orbiting a staticblack hole.

I. INTRODUCTION

II. THE GYROSCOPE GRID

A. The boost concept

B. The effect of three boosts

C. Calculating the precise turning angle

D. The uncontracted grid

E. Circular motion

F. The mathematical advantage of the uncontracted grid

G. Comments on the uncontracted grid

III. BOOSTING THE REFERENCE FRAME

IV. THREE DIMENSIONS

V. APPLICATIONS

A. Motion along a horizontal line

B. Following the geodesic photon

VI. CONCLUSIONS

VII. AXISYMMETRIC SPATIAL GEOMETRIES AND EFFECTIVE ROTATION VECTORS

VIII. CIRCULAR ORBITS IN STATIC SPHERICALLY SYMMETRIC SPACETIMES

A. The Schwarzschild black hole

B. Geodesic circular motion

IX. RELATION TO OTHER WORK

### Key Topics

- Gyroscope motion
- 87.0
- Kinematics
- 21.0
- Black holes
- 12.0
- Special relativity
- 12.0
- Photons
- 10.0

## Figures

A gyroscope transported around a circle. The vectors correspond to the central axis of the gyroscope at different times. The Newtonian version is on the left, the special relativistic version is on the right.

A gyroscope transported around a circle. The vectors correspond to the central axis of the gyroscope at different times. The Newtonian version is on the left, the special relativistic version is on the right.

A gyroscope transported along a circle at the photon radius of a static black hole. The gyroscope turns so that it always points along the direction of motion.

A gyroscope transported along a circle at the photon radius of a static black hole. The gyroscope turns so that it always points along the direction of motion.

A train at two consecutive times moving with velocity relative to a platform that accelerates upward. A gyroscope with a torque free suspension on the train will precess clockwise for .

A train at two consecutive times moving with velocity relative to a platform that accelerates upward. A gyroscope with a torque free suspension on the train will precess clockwise for .

(a) The grid at rest with respect to . (b) The grid after a pure boost with velocity to the right, relative to . Note the length contraction. (c) The grid after a pure upward boost relative to a system that moves with velocity to the right. (d) The grid after a pure boost that stops the grid relative to .

(a) The grid at rest with respect to . (b) The grid after a pure boost with velocity to the right, relative to . Note the length contraction. (c) The grid after a pure upward boost relative to a system that moves with velocity to the right. (d) The grid after a pure boost that stops the grid relative to .

The stretch-induced tilt of the two points (the filled circles) due to the final stopping of the grid. The distance between the points prior to the stretching, as measured in the direction of motion, is denoted by .

The stretch-induced tilt of the two points (the filled circles) due to the final stopping of the grid. The distance between the points prior to the stretching, as measured in the direction of motion, is denoted by .

The real grid (black thin lines) and the corresponding imagined uncontracted grid (gray thick lines) before and after a large upward boost.

The real grid (black thin lines) and the corresponding imagined uncontracted grid (gray thick lines) before and after a large upward boost.

A gyroscope grid at successive time steps. Both the grid and the gyroscope are depicted as they would be observed if they were uncontracted.

A gyroscope grid at successive time steps. Both the grid and the gyroscope are depicted as they would be observed if they were uncontracted.

Boost of the reference frame (of which the depicted thin grid is a small part) upward by a velocity . The velocity of the gyroscope grid is maintained.

Boost of the reference frame (of which the depicted thin grid is a small part) upward by a velocity . The velocity of the gyroscope grid is maintained.

Relative to the gyroscope system, the reference frame (thin lines), of which we illustrate a certain part, rotates during the boost. The reference frame is depicted as it would appear if it was not length contracted relative to the gyroscope system.

Relative to the gyroscope system, the reference frame (thin lines), of which we illustrate a certain part, rotates during the boost. The reference frame is depicted as it would appear if it was not length contracted relative to the gyroscope system.

A train with a gyroscope moving relative to an accelerating platform observed at two successive times.

A train with a gyroscope moving relative to an accelerating platform observed at two successive times.

A sketch of the rail and the train observed from an inertial system where the train is momentarily at rest.

A sketch of the rail and the train observed from an inertial system where the train is momentarily at rest.

A free photon will in general follow a curved path relative to an accelerated reference frame. A gyroscope transported along such a path will keep pointing along the path if it did so initially.

A free photon will in general follow a curved path relative to an accelerated reference frame. A gyroscope transported along such a path will keep pointing along the path if it did so initially.

Sketch of the spatial geometry of a symmetry plane outside a black hole. The local static reference frame shown (the square grid) has a proper acceleration outward. For a sufficiently small such reference frame it works just like an accelerated reference frame in special relativity.

Sketch of the spatial geometry of a symmetry plane outside a black hole. The local static reference frame shown (the square grid) has a proper acceleration outward. For a sufficiently small such reference frame it works just like an accelerated reference frame in special relativity.

Deviations from a straight line relative to a reference frame that accelerates in the direction. The plane shown is perpendicular to the momentary direction of motion (along the dashed line), and all the three vectors lie in this plane. The solid curving line is the particle trajectory. The thick line is the line that is fixed to the inertial system in question and is thus falling relative to the reference frame.

Deviations from a straight line relative to a reference frame that accelerates in the direction. The plane shown is perpendicular to the momentary direction of motion (along the dashed line), and all the three vectors lie in this plane. The solid curving line is the particle trajectory. The thick line is the line that is fixed to the inertial system in question and is thus falling relative to the reference frame.

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