Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic.
In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress–energy tensor (representing matter, for instance).
The quantity on the left-hand-side of this equation is the acceleration of a particle, so this equation is analogous to Newton's laws of motion, which likewise provide formulae for the acceleration of a particle.
The Christoffel symbols are functions of the four spacetime coordinates and so are independent of the velocity or acceleration or other characteristics of a test particle whose motion is described by the geodesic equation.
This formulation of the geodesic equation of motion can be useful for computer calculations and to compare General Relativity with Newtonian Gravity.
Notice that both sides of this last equation vanish when the mu index is set to zero.
If the particle's velocity is small enough, then the geodesic equation reduces to this:
This equation simply means that all test particles at a particular place and time will have the same acceleration, which is a well-known feature of Newtonian gravity.
For example, everything floating around in the International Space Station will undergo roughly the same acceleration due to gravity.
Physicist Steven Weinberg has presented a derivation of the geodesic equation of motion directly from the equivalence principle.
which is a form of the geodesic equation of motion (using the coordinate time as parameter).
The geodesic equation of motion can alternatively be derived using the concept of parallel transport.
[4] We can (and this is the most common technique) derive the geodesic equation via the action principle.
There is a negative sign inside the square root because the curve must be timelike.
Integrating by-parts the last term and dropping the total derivative (which equals to zero at the boundaries) we get that:
(Note: Similar derivations, with minor amendments, can be used to produce analogous results for geodesics between light-like[citation needed] or space-like separated pairs of points.)
Albert Einstein believed that the geodesic equation of motion can be derived from the field equations for empty space, i.e. from the fact that the Ricci curvature vanishes.
He wrote:[5] It has been shown that this law of motion — generalized to the case of arbitrarily large gravitating masses — can be derived from the field equations of empty space alone.
According to this derivation the law of motion is implied by the condition that the field be singular nowhere outside its generating mass points.and [6] One of the imperfections of the original relativistic theory of gravitation was that as a field theory it was not complete; it introduced the independent postulate that the law of motion of a particle is given by the equation of the geodesic.
On the basis of the description of a particle without singularity, one has the possibility of a logically more satisfactory treatment of the combined problem: The problem of the field and that of the motion coincide.Both physicists and philosophers have often repeated the assertion that the geodesic equation can be obtained from the field equations to describe the motion of a gravitational singularity, but this claim remains disputed.
Other assumptions are needed to derive the theorems in question.”[8] Less controversial is the notion that the field equations determine the motion of a fluid or dust, as distinguished from the motion of a point-singularity.
[9] In deriving the geodesic equation from the equivalence principle, it was assumed that particles in a local inertial coordinate system are not accelerating.
However, in real life, the particles may be charged, and therefore may be accelerating locally in accordance with the Lorentz force.
This last equation signifies that the particle is moving along a timelike geodesic; massless particles like the photon instead follow null geodesics (replace −1 with zero on the right-hand side of the last equation).
The letter g with superscripts refers to the inverse of the metric tensor.
In curved spacetime, it is possible for a pair of widely separated events to have more than one time-like geodesic between them.
Any curve that differs from the geodesic purely spatially (i.e. does not change the time coordinate) in any inertial frame of reference will have a longer proper length than the geodesic, but a curve that differs from the geodesic purely temporally (i.e. does not change the space coordinates) in such a frame of reference will have a shorter proper length.
Substituting the expression of f into the Euler–Lagrange equation (which makes the value of the integral l stationary), gives
The geodesic equation can be alternatively derived from the autoparallel transport of curves.
The derivation is based on the lectures given by Frederic P. Schuller at the We-Heraeus International Winter School on Gravity & Light.