Variational Approach to Gravity Field Theories by Alberto Vecchiato

Author:Alberto Vecchiato
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

8.2 The Geodesic Equations

If, by the equivalence principle, the motion of a test particle in a gravity field can be modeled as that of a free particle in a curved geometry characterized by the metric , i.e., by the line invariant of Eq. 8.1.1, we can find its equations of motion by means of the usual variational technique, considering an action

(8.2.1)

which is formally the same as Eq. 6.​3.​5 used for special relativity. Such formal analogy conceals the decisive fact that the preferred geometry is not the Minkowski one anymore, but a general pseudo-Riemaniann one using more “general” spacetime tensorial objects. Scalars have to be invariant with respect to any transformation and thus the actual form of the (invariant) line element is determined by instead of . Only the former, in fact, ensures the essential condition of the variational approach that the action is a scalar in the appropriate geometry.

Another fundamental difference with respect to our usual procedure of “assembling” the action is that we do not need any interaction part as long as only gravity is concerned inasmuch as the latter is automatically absorbed in the free part because of the equivalence principle. The action needs to be complemented with an interaction part when it is necessary to describe the motion in the presence of gravity and any additional field, such as the electromagnetic one, which will introduce an extra “force” term in the geodesic equations.

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