Geodesic Equation

Given a metric tensor $\boldsymbol{\it g}$, the mathematical definition of a geodesic is that of a curve that parallel transports its tangent vector, i.e. $\nabla_{_{\!\!\bf V}}{\bf V}=0$, where

$$\nabla_{_{\!\!\bf V}}{\bf V}= 0 = V^{\alpha }_{  ;\beta } V^{\bet... ...a }_{  ,\beta } V^{\beta } + \Gamma^{\alpha }_{\beta \mu} V^{\beta } V^{\mu} ,$$ (1)

Figure: A generic curve ${\cal C}(\tau )$ in a 4-dimensional spacetime and its tangent 4-vector ${\bf V}$ which is parallel transported along ${\cal C}(\tau )$, i.e. $\nabla_{_{\!\!\bf V}}{\bf V}=0$.

and where $\nabla$ is the covariant derivative with respect to $\boldsymbol{\it g}$ (see Fig. 1).

If $\tau$ is a parameter along the curve ${\cal C}$ having ${\bf V}$ as tangent vector, then $V^\mu = {dx^\mu}/{d\tau}$ and

$$V^\mu,_\beta = \frac{\partial}{\partial x^\beta}\left(\frac{dx^... ...ial x^\beta}\left(\frac{dx^\mu}{d\tau}\right) = \frac{d^2x^\mu}{d\tau^2} .$$ (2)

so that the geodesic equation (1) can be written as a second-order differential equation

$$\frac{d^2x^\mu}{d\tau^2}+\Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0 ,$$ (3)

having $x^\mu(P_0)$ and ${dx^\mu}/{d\tau}(P_0)$ as initial conditions. If the particle has a nonzero mass, $\tau$ can then be associated to the proper time measured by the particle along ${\cal C}$.

On the other hand, a physical definition of a geodesic is that of a curve along which a freely falling particle moves. In this sense, then, a deviation from a purely geodetic motion must be interpreted as the result of the loss of a ``free-fall'' or, in other words, as the result of the application of a force. In this Section we will show how the deviation from a geodesic motion can indeed be related to a nonzero curvature of the spacetime, or, equivalently, to the presence of (tidal) force.

However, before we consider the concept of geodesic deviation in full General Relativity, it is instructive to first look at this problem in Newtonian gravity, where the basic features are also present.