Geodesic (general relativity)

General relativity
Introduction Mathematical formulation Resources
Fundamental concepts Special relativity Equivalence principle World line · Riemannian geometry
Phenomena Kepler problem · Lenses · Waves Frame-dragging · Geodetic effect Event horizon · Singularity Black hole
Equations Linearized Gravity Post-Newtonian formalism Einstein field equations Geodesic equation Friedmann equations ADM formalism BSSN formalism
Advanced theories Kaluza–Klein Quantum gravity
Solutions Schwarzschild Reissner–Nordström · Gödel Kerr · Kerr–Newman Kasner · Taub-NUT · Milne · Robertson–Walker pp-wave · van Stockum dust
Scientists Einstein · Minkowski · Eddington Lemaître · Schwarzschild Robertson · Kerr · Friedman Chandrasekhar · Hawking others

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 force, 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 around a star is the projection of a geodesic of the curved 4-D spacetime geometry around the star onto 3-D space.

In theories such as special and general relativity, spacetime is treated as a Lorentzian manifold. Geodesics on a Lorentzian manifold fall into three classes according to the sign of the norm of their tangent vector. With a metric signature of (-+++) being used,

Timelike geodesics have a tangent vector whose norm is negative;
Null geodesics have a tangent vector whose norm is zero;
Spacelike geodesics have a tangent vector whose norm is positive.

Note that a geodesic cannot be spacelike at one point and timelike at another.

An ideal particle (ones whose gravitational field and size are ignored) not subject to electromagnetic forces (or any other non-gravitational force) will always follow timelike geodesics. Note that not all particles follow geodesics, as they may experience external forces, for example, a charged particle may experience an electric field — in such cases, the worldline of the particle will still be timelike, as the tangent vector at any point of a particle's worldline will always be timelike.

Massless particles like the photon follow null geodesics. Spacelike geodesics exist. They do not correspond to the path of any physical particle, but in a space that has space-sections orthogonal to a timelike Killing vector a spacelike geodesic (with its affine parameter) within such a space section represents the graph of a tightly stretched, massless filament.

1 Mathematical expression
2 Geodesics as extremal curves
3 Geodesic incompleteness and singularities
4 Approximate geodesic motion
5 See also
6 References

Mathematical expression

A timelike geodesic is a worldline which parallel transports its own tangent vector and maintains the magnitude of its tangent as a constant. If a curve $x^\alpha(s)$ has tangent $d x^\mu/ ds = U^\mu (s)\ ,$ then this can be expressed as

$\nabla_{U} U^\mu = 0\ ,\ \ \mathrm{or}\ \ U^\alpha \nabla_\alpha U^\mu = 0\ ,$

which says that the covariant derivative of the tangent in the direction of the tangent is zero. The above equation can be restated in terms of components of $U^\mu$ :

$\ddot x^\mu %2B \Gamma^\mu {}_{\alpha \beta} \dot x^\alpha \dot x^\beta = 0 \$

where

$\dot x^\mu = U^\mu = {d x^\mu \over ds}$

and

$\ddot x^\mu = {d^2 x^\mu \over ds^2} = \dot U^\mu = {d U^\mu \over ds} = {\partial U^\mu \over \partial x^\alpha} {d x^\alpha \over ds} = U^\mu {}_{,\alpha} U^\alpha\ .$

The full geodesic equation is therefore:

${d^2 x^\mu \over ds^2} %2B \Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds} = 0\ .$

where s is the proper time or distance and $\Gamma^\mu {}_{\alpha \beta}\$ is the Levi-Civita connection.

Proof

$\nabla_{\vec U} \vec U = 0$ ,

$U^\alpha \nabla_\alpha \vec U = 0$ ,

$U^\alpha U^\beta {}_{;\alpha} = 0$ ,

$U^\alpha (U^\beta {}_{,\alpha} %2B U^\sigma \Gamma^\beta {}_{\sigma \alpha}) = 0$ ,

$U^\alpha U^\beta {}_{,\alpha} %2B \Gamma^\beta {}_{\sigma \alpha} U^\alpha U^\sigma = 0$ ,

$\ddot x^\beta %2B \Gamma^\beta {}_{\sigma \alpha} \dot x^\sigma \dot x^\alpha = 0. \$

The parameter s typically represents proper time for a timelike curve, or distance for a spacelike curve. This parameter cannot be chosen arbitrarily. Rather, it must be chosen so that the tangent vector $d x^\alpha/ ds$ has constant magnitude. This is referred to as an affine parametrization. Any two affine parameters are linearly related. That is, if r and s are affine parameters, then there exist constants a and b such that $r = a \cdot s %2B b$ .

Geodesics as extremal curves

A geodesic between two events could also be described as the curve joining those two events which has the maximum possible length in time — for a timelike curve — or the minimum possible length in space — for a spacelike curve. The four-length of a curve in spacetime is

$l = \int \sqrt{\left|g_{\mu \nu} \dot x^\mu \dot x^\nu \right|} \, ds\ .$

Then, the Euler-Lagrange equation,

${d \over ds} {\partial \over \partial \dot x^\alpha} \sqrt{\left| g_{\mu \nu} \dot x^\mu \dot x^\nu \right|} = {\partial \over \partial x^\alpha} \sqrt{\left| g_{\mu \nu} \dot x^\mu \dot x^\nu \right|}$

becomes, after some calculation,

$2(\Gamma^\lambda {}_{\mu \nu} \dot x^\mu \dot x^\nu %2B \ddot x^\lambda) = U^\lambda {d \over ds} \ln |U_\nu U^\nu| \ .$

Proof

The goal being to extremize the value of

$l = \int d\tau = \int {d\tau \over d\phi} \, d\phi = \int \sqrt{{(d\tau)^2 \over (d\phi)^2}} \, d\phi = \int \sqrt{{-g_{\mu \nu} dx^\mu dx^\nu \over d\phi \, d\phi}} \, d\phi = \int f \, d\phi$

where

$f = \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}$

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

${d \over d\tau} {\partial f \over \partial \dot x^\lambda} = {\partial f \over \partial x^\lambda}$ .

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

${d \over d\tau} {\partial \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu} \over \partial \dot x^\lambda} = {\partial \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu} \over \partial x^\lambda}$

Now calculate the derivatives: ${d \over d\tau} \left( {-g_{\mu \nu} {\partial \dot x^\mu \over \partial \dot x^\lambda} \dot x^\nu - g_{\mu \nu} \dot x^\mu {\partial \dot x^\nu \over \partial \dot x^\lambda} \over 2 \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \right) = {-g_{\mu \nu, \lambda} \dot x^\mu \dot x^\nu \over 2 \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \qquad \qquad (1)$

${d \over d\tau} \left( {g_{\mu \nu} \delta^\mu {}_\lambda \dot x^\nu %2B g_{\mu \nu} \dot x^\mu \delta^\nu {}_\lambda \over 2 \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \right) = {g_{\mu \nu , \lambda} \dot x^\mu \dot x^\nu \over 2 \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \qquad \qquad (2)$

${d \over d\tau} \left( {g_{\lambda \nu} \dot x^\nu %2B g_{\mu \lambda} \dot x^\mu \over \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \right) = {g_{\mu \nu , \lambda} \dot x^\mu \dot x^\nu \over \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \qquad \qquad (3)$

${\sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu} {d \over d\tau} (g_{\lambda \nu} \dot x^\nu %2B g_{\mu \lambda} \dot x^\mu) - (g_{\lambda \nu} \dot x^\nu %2B g_{\mu \lambda} \dot x^\mu) {d \over d\tau} \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu} \over -g_{\mu \nu} \dot x^\mu \dot x^\nu} = {g_{\mu \nu , \lambda} \dot x^\mu \dot x^\nu \over \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}} \qquad \qquad (4)$

${(-g_{\mu \nu} \dot x^\mu \dot x^\nu) {d \over d\tau} (g_{\lambda \nu} \dot x^\nu %2B g_{\mu \lambda} \dot x^\mu) %2B {1 \over 2} (g_{\lambda \nu} \dot x^\nu %2B g_{\mu \lambda} \dot x^\mu) {d \over d\tau} (g_{\mu \nu} \dot x^\mu \dot x^\nu) \over -g_{\mu \nu} \dot x^\mu \dot x^\nu} = g_{\mu \nu ,\lambda} \dot x^\mu \dot x^\nu \qquad \qquad (5)$

$(g_{\mu \nu} \dot x^\mu \dot x^\nu) (g_{\lambda \nu ,\mu} \dot x^\nu \dot x^\mu %2B g_{\mu \lambda ,\nu} \dot x^\mu \dot x^\nu %2B g_{\lambda \nu} \ddot x^\nu %2B g_{\lambda \mu} \ddot x^\mu)$

$= (g_{\mu \nu ,\lambda} \dot x^\mu \dot x^\nu) (g_{\alpha \beta} \dot x^\alpha \dot x^\beta) %2B {1 \over 2} (g_{\lambda \nu} \dot x^\nu %2B g_{\lambda \mu} \dot x^\mu) {d \over d\tau} (g_{\mu \nu} \dot x^\mu \dot x^\nu) \qquad \qquad (6)$

$g_{\lambda \nu ,\mu} \dot x^\mu \dot x^\nu %2B g_{\lambda \mu ,\nu} \dot x^\mu \dot x^\nu - g_{\mu \nu ,\lambda} \dot x^\mu \dot x^\nu %2B 2 g_{\lambda \mu} \ddot x^\mu = {\dot x_\lambda {d \over d\tau} (g_{\mu \nu} \dot x^\mu \dot x^\nu) \over g_{\alpha \beta} \dot x^\alpha \dot x^\beta} \qquad \qquad (7)$

$2(\Gamma_{\lambda \mu \nu} \dot x^\mu \dot x^\nu %2B \ddot x_\lambda) = {\dot x_\lambda {d \over d\tau} (\dot x_\nu \dot x^\nu) \over \dot x_\beta \dot x^\beta} = {U_\lambda {d \over d\tau} (U_\nu U^\nu) \over U_\beta U^\beta} = U_\lambda {d \over d\tau} \ln |U_\nu U^\nu| \qquad \qquad (8)$

This is just one step away from the geodesic equation.

If the parameter s is chosen to be affine, then the right side the above equation vanishes (because $U_\nu U^\nu$ is constant). Finally, we have the geodesic equation

$\Gamma^\lambda {}_{\mu \nu} \dot x^\mu \dot x^\nu %2B \ddot x^\lambda = 0\ .$

Geodesic incompleteness and singularities

The notion of geodesic incompleteness is used in the study of gravitational singularities.

Approximate geodesic motion

True geodesic motion is an idealization where one assumes the existence of test particles. Although in many cases real matter and energy can be approximated as test particles, situations arise where their appreciable mass (or equivalent thereof) can affect the background gravitational field in which they reside.

This creates problems when performing an exact theoretical description of a gravitational system (for example, in accurately describing the motion of two stars in a binary star system). This leads one to consider the problem of determining to what extent any situation approximates true geodesic motion. In qualitative terms, the problem is solved: the smaller the gravitational field produced by an object compared to the gravitational field it lives in (for example, the Earth's field is tiny in comparison with the Sun's), the closer this object's motion will be geodesic.

As Einstein's field equations determine the geometry of spacetime, it should be possible to determine the geodesics of the spacetime as well. For the case of dust, the problem can be solved by using the Bianchi identities. Many attempts have been made to do the same for other matter distributions.

References

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.