Physics[Geodesics] - computes and solves the geodesic equations for the spacetime metric
Geodesics(tau, options = ...)
optional - a name to represent the affine parameter that parametrizes the geodesic equations; if not given, the symbol τ is used
integrabilityconditions = ...
optional - can be true (default) or false, to reduce, or not reduce, the resulting ODE output system to a canonical form (see RIF)
output = ...
optional - the right-hand-side can be equations (default) or solutions, to return the geodesic equations or also attempt solving them
parameters = ...
optional - the right-hand-side can be a set or a list of names or functions, that parameterize the spacetime metric; the system of geodesic equations (or solutions) will be split according to the different cases of these parameters
tensornotation = ...
optional - can be true or false (default), to return the geodesic equations as an algebraic equation in tensor notation instead of as a set or list of equations for the components
Geodesics computes and attempts to solve the geodesic equations for the spacetime metric g_. These equations are parametrized by an affine parameter τ typically representing the proper time for a timelike curve, or distance for a spacelike curve. These geodesic equations computed are
where Xμ is the SpaceTimeVector and Γ⁢⁢⁢α,νμ are the Christoffel symbols. As it is the case of the other commands in Physics, Geodesics operates in tensorial notation. To perform the same computation using differential forms notation see DifferentialGeometry[Tensor][GeodesicEquations].
When a name - say tau is provided as first argument, it will be used to represent the parameter for the geodesic equations; otherwise, the symbol τ itself is used.
By default, Geodesics computes only the equations, without attempting solving them. To request compute and solve the equations in one step, pass the option output = solutions. When Geodesics or dsolve fail in finding a solution for the equations, you can still look for particular solutions of different forms that may be of interest using the symmetry commands of PDEtools like FunctionFieldSolutions and PolynomialSolutions, or InvariantSolutions used together with its various options for particularizing solutions.
The ODE system returned by default by Geodesics is returned as a list of equations and automatically simplified to a canonical form (reduced involutive form - RIF and DifferentialAlgebra) using differential algebra techniques. The order of the equations in the list corresponds to a total degree, obtained with an orderly ranking (see details in casesplit). Simplifying the ODE system this way is a frequently desired but in some cases expensive mathematical computation. To avoid it, pass the optional argument integrabilityconditions = false, in which case Geodesics will return faster, a set of equations, not a list, and with no differential simplification.
When computing just the geodesic equations, it also possible to request them in tensor notation, as an algebraic equation, instead of as a set or list of equations for the tensorial components. Although this kind of output always looks just like the definition for the geodesic equations shown above, the algebraic equation returned actually contains the computable objects like the SpaceTimeVector and Christoffel symbols, and can be manipulated in different ways, including getting from it all the equations for its components.
When the metric g_ depends on parameters, either symbols or functions of spacetime variables, the equations computed with Geodesics are expected to be valid for arbitrary values of these parameters. It is sometimes of interest, however, to investigate the different kinds of solutions that may exist for different particular values of these parameters. To perform such an investigation, use the optional argument parameters = ... where the right-hand-side is a set or a list with the parameters. The system of geodesic equations will then be split into cases with respect to these parameters using differential algebra techniques. As frequently happens when splitting into cases, the resulting lists of equations may involve inequations as well. Note: passing parameters automatically forces simplifying the ODE system to a canonical (reduced involutive RIF) form regardless of the value of the keyword integrabilityconditions.
Set a system of Coordinates
⁢Default differentiation variables for d_, D_ and dAlembertian are:⁢X=x1,x2,x3,x4
⁢Systems of spacetime coordinates are:⁢X=x1,x2,x3,x4
The spacetime metric at this point is the one that loads with Physics, i.e. a Minkowski type metric
The geodesics equations for this metric, first as an algebraic equation in tensor notation, then as a list of equations for the tensorial components
The straight-lines solution computed directly
Consider now a non-galilean spacetime, for instance set the Schwarzschild metric as the current metric (see g_):
⁢Systems of spacetime coordinates are:⁢X=r,θ,φ,t
⁢Default differentiation variables for d_, D_ and dAlembertian are:⁢X=r,θ,φ,t
⁢The Schwarzschild metric in coordinates ⁢r,θ,φ,t
Compute the geodesic equations again.
This system of ODEs, as is, it is out of reach of the DE solvers of the system mainly due to the presence of non-rational objects like sin and cos having for arguments one of the unknowns of the system, θ⁡τ. On the other hand, we know the geodesics for the Schwarzschild metric describe the motion of particles of infinitesimal mass in the gravitational field of a central fixed large mass. So to investigate the solvability of these equations one can assume θ⁡τ is constant an due to the rotational symmetry choose a value for it that simplifies the equations, for example, θ⁡τ=π⁢12.
And this system is already solvable with dsolve; the first two of six solutions are:
casesplit, Coordinates, DifferentialAlgebra, DifferentialGeometry[Tensor]Geodesics], FunctionFieldSolutions, g_, InvariantSolutions, Library, pdsolve, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, PolynomialSolutions, Setup
Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.
The Physics[Geodesics] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
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