- Solving the geodesic equations
Solving the geodesic equations is a procedure used in
mathematics , particularlyRiemannian geometry , and inphysics , particularly ingeneral relativity , that results in obtaininggeodesic s. Physically, these represent the paths of (usually ideal) particles with no proper acceleration, their motion satisfying the geodesic equations. Because the particles are subject to no four-acceleration, the geodesics generally represent the straightest path between two points in a curvedspacetime .The geodesic equation
On an n-dimensional
Riemannian manifold , the geodesic equation written in acoordinate chart with coordinates is::
where the coordinates "x""a"("s") are regarded as the coordinates of a
curve γ("s") in and are theChristoffel symbol s. The Christoffel symbols are functions of the metric and are given by::
where the comma indicates a
partial derivative with respect to the coordinates::
As the manifold has dimension , the geodesic equations are a system of
ordinary differential equation s for the coordinate variables. Thus, allied withinitial conditions , the system can, according to thePicard-Lindelöf theorem , be solved.Heuristics
As the laws of physics can be written in any
coordinate system , it is convenient to choose one that simplifies the geodesic equations. Mathematically, this means, acoordinate chart is chosen in which the geodesic equations have a particularly tractable form.Effective potential sWhen the geodesic equations can be separated into terms containing only an undifferentiated variable and terms containing only its
derivative , the former may be consolidated into an effective potential dependent only on position. In this case, many of theheuristic methods of analysingenergy diagram s apply, in particular the location of turning points.olution techniques
Solving the geodesic equations means obtaining an
exact solution , possibly even the general solution, of the geodesic equations. Most attacks secretly employ the point symmetry group of the system of geodesic equations. This often yields a result giving a family of solutions implicitly, but in many examples does yield the general solution in explicit form.In general relativity, to obtain
timelike geodesics it is often simplest to start from the spacetime metric, after dividing by to obtain the form:
where the dot represents differentiation by ds. Because timelike geodesics are maximal, one may apply the
Euler-Lagrange equation directly, and thus obtain a set of equations equivalent to the geodesic equations. This method has the advantage of bypassing a tedious calculation ofChristoffel symbols .Examples
ee also
*
Geodesics of the Schwarzschild vacuum *
Mathematics of general relativity *
Transition from special relativity to general relativity References
: [1] cite book | author=Einstein, A. | title=Relativity: The Special and General Theory | location= New York | publisher=Crown| year=1961 | id=ISBN 0-517-02961-8
: [2] cite book | author=Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald | title=Gravitation | location=San Francisco | publisher=W. H. Freeman | year=1973 | id=ISBN 0-7167-0344-0
: [3] cite book | author=Landau, L. D. and Lifshitz, E. M.| title=Classical Theory of Fields (Fourth Revised English Edition) | location=Oxford | publisher=Pergamon | year=1975 | id=ISBN 0-08-018176-7
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