# Geodesic hypothesis

The statement that determines the motion of a free test particle in Einstein's theory of gravity (i.e. in general relativity theory). In Newtonian physics a particle is called free if there are no forces acting on it (even no gravitational forces). In general relativity theory the concept of a gravitational force as a four-dimensional vector is absent, and gravitational properties are defined by the Riemannian space-time structure. Accordingly, the motion of a particle in a gravitational field (provided it is not acted upon by any non-gravitational forces) is considered as free in general relativity theory. The precise formulation of the geodesic hypothesis is as follows: The world line of a free test particle with non-zero rest mass is a non-isotropic time-like geodesic line of space-time; the world line of a free test particle with zero rest mass (a photon, a neutrino) is an isotropic geodesic line of space-time.

The geodesic hypothesis is a natural generalization of the law of inertia in classical mechanics. The differential equations of geodesic lines (cf. Geodesic line) are the equations of motion in general relativity theory.

The concept of a test particle appearing in the formulation of the geodesic hypothesis means that the effects related to the finite dimensions of these particles and their internal structure are ignored, and the gravitational field produced by them is considered to be negligibly small. A test particle is an idealized limiting case of a real particle, and it is possible to obtain the geodesic hypothesis as a consequence of the Einstein equations [3] in a broad and natural class of idealized test particles. This fact represents a substantial difference between general relativity theory and other field theories, in which it appears impossible to obtain the equations of motion from the field equations.

#### References

[1] | V.A. [V.A. Fok] Fock, "The theory of space, time and gravitation" , Macmillan (1954) (Translated from Russian) |

[2] | J.L. Synge, "Relativity: the general theory" , North-Holland & Interscience (1960) pp. Chapt. II |

[3] | L. Infeld, B. Hoffmann, "Gravitational equations and problems of motion" Ann. of Math. , 39 (1938) pp. 65–100 |

#### Comments

#### References

[a1] | L. Infeld, J. Plebański, "Motion and relativity" , Pergamon & Polish Acad. Sci. (1960) |

**How to Cite This Entry:**

Geodesic hypothesis. D.D. Sokolov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Geodesic_hypothesis&oldid=12683