Geodesic (general relativity)

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In physics, and specifically general relativity, geodesics are the world lines of a particle free from all external force. In this theory, gravity is not a force but is instead a curved spacetime geometry where the source of curvature is the stress-energy tensor. Thus, for example, the orbital path of a planet 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 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, and
spacelike geodesics have a tangent vector whose norm is positive.

Note that a geodesic cannot be spacelike at one point and timelike at another since parallel transport preserves the norm of the vector (since the metric is parallel transported along any curve).

Ideal particles (ones whose gravitational field is ignored) in free fall and any particle not subject to electromagnetic or pressure forces (or the like) 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 will 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.

Contents

1 Mathematical expression
2 Geodesic as maximal curve
3 Geodesic incompleteness and singularities
4 Approximate geodesic motion
5 References

Mathematical expression

A timelike geodesic is a worldline which parallel transports its own tangent. If a worldline [\vec x (\tau) ] has tangent [\vec U(\tau)] then this can be expressed as

[ \nabla_ \vec U = 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 [ \vec U]:

[ \ddot x^\beta + \Gamma^\beta _ \dot x^\sigma \dot x^\alpha = 0 \ ]

where

[ \dot x^\alpha = U^\alpha = ]

and

[ \ddot x^\beta = \dot U^\beta = = = U^\beta _ U^\alpha ],

τ being proper time (an affine parameter which makes the curve a unit-speed curve).

Geodesic as maximal curve

A geodesic between two events could also be described as the curve joining those two events which has the maximum possible length. The four-length of a curve in spacetime is

[ l = \int f \, d\phi]

where

[ f = \sqrt \dot x^\mu \dot x^\nu} ].

Then the Euler-Lagrange equation,

[ = ]

becomes, after some calculation,

[ 2(\Gamma^\lambda _ \dot x^\mu \dot x^\nu + \ddot x^\lambda) = U^\lambda \ln |U_\nu U^\nu| \ ]

If parameter τ is chosen to be affine (so that the tangent with respect to it has constant magnitude), then the right side the above equation vanishes (because [ U_\nu U^\nu = \vec U \cdot \vec U ] is constant, thereby its derivative is zero),

[ \Gamma^\lambda _ \dot x^\mu \dot x^\nu + \ddot x^\lambda = 0. \ ]

An affine parameter is directly proportional to proper time, and the above equation remains true (geodesically) for any linear reparameterization

[ m \tau + b \mapsto \phi ].

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 idealisation 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 the Einstein 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.

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