Geodesic (general Relativity)
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Geodesic General Relativity

In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a 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). Thus, for example, the path of a planet orbiting a star is the projection of a geodesic of the curved 4-D spacetime geometry around the star onto 3-D space.

Mathematical expression

The full geodesic equation is


Equivalent mathematical expression using coordinate time as parameter




Derivation directly from the equivalence principle


The next step is to employ the multi-dimensional chain rule. We have:

Differentiating once more with respect to the time, we have:


Multiply both sides of this last equation by the following quantity:

Consequently, we have this:

Using (from Christoffel symbols#Change of variable and the fact that the Christoffel symbols vanish in an inertial frame of reference)

it becomes

Applying the one-dimensional chain rule gives

As before, we can set . Then the first derivative of x0 with respect to t is one and the second derivative is zero. Replacing λ with zero gives:

Subtracting d xλ / d t times this from the previous equation gives:

which is a form of the geodesic equation of motion (using the coordinate time as parameter).


Deriving the geodesic equation via an action


Let the action be

where is the line element. There is a negative sign inside the square root because the curve must be timelike. To get the geodesic equation we must vary this action. To do this let us parameterize this action with respect to a parameter . Doing this we get:


Using the product rule we get:


Simplifying a bit we see that:


multiplying this equation by we get:

So by Hamilton's principle we find that the Euler-Lagrange equation is

Multiplying by the inverse metric tensor we get that

Thus we get the geodesic equation:

with the Christoffel symbol defined in terms of the metric tensor as

(NOTE: Similar derivations, with minor amendments, can be used to produce analogous results for geodesics between light-like[] or space-like separated pairs of points.)

Equation of motion may follow from the field equations for empty space


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 [5]

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.

A complete field theory knows only fields and not the concepts of particle and motion. For these must not exist independently from the field but are to be treated as part of it.

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.[6] 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.[7]

Extension to the case of a charged particle



The Minkowski tensor is given by:





Geodesics as curves of stationary interval




The interval of a curve in spacetime is

Then, the Euler-Lagrange equation,

becomes, after some calculation,



The goal being to find a curve for which the value of

is stationary, where

such goal can be accomplished by calculating the Euler-Lagrange equation for f, which is


Substituting the expression of f into the Euler-Lagrange equation (which makes the value of the integral l stationary), gives

Now calculate the derivatives:

This is just one step away from the geodesic equation.

If the parameter s is chosen to be affine, then the right side of the above equation vanishes (because is constant). Finally, we have the geodesic equation

See also


  • Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, (1972) John Wiley & Sons, New York ISBN 0-471-92567-5. See chapter 3.
  • Lev D. Landau and Evgenii M. Lifschitz, The Classical Theory of Fields, (1973) Pergammon Press, Oxford ISBN 0-08-018176-7 See section 87.
  • Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0.
  • Bernard F. Schutz, A first course in general relativity, (1985; 2002) Cambridge University Press: Cambridge, UK; ISBN 0-521-27703-5. See chapter 6.
  • Robert M. Wald, General Relativity, (1984) The University of Chicago Press, Chicago. See Section 3.3.


  3. ^ Pleba?ski, Jerzy and Krasi?ski, Andrzej. An Introduction to General Relativity and Cosmology, p. 34 (Cambridge University Press, 2006).
  5. ^ Einstein, A.; Rosen, N. (1 July 1935). "The Particle Problem in the General Theory of Relativity". Physical Review. 48 (1): 76. Bibcode:1935PhRv...48...73E. doi:10.1103/PhysRev.48.73. and ER - Einstein Rosen paper ER=EPR
  6. ^ Tamir, M. "Proving the principle: Taking geodesic dynamics too seriously in Einstein's theory", Studies In History and Philosophy of Modern Physics 43(2), 137-154 (2012).
  9. ^ Charles W. Misner; Kip Thorne; John Archibald Wheeler (1973). Gravitation. W. H. Freeman. pp. 316, 318-319. ISBN 0-7167-0344-0.




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