Talk:PlanetPhysics/Geodesic

Original TeX Content from PlanetPhysics Archive

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\begin{document}

 A \emph{geodesic} is generally described as the shortest possible, or topologically allowed, path between two points in a curved space.


\begin{remark}
Given a curved space $\S_C$ one can find the geodesic by writing the equation for the length $l_v$ of a {\em curve}-- which is defined as a \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} $f: (R) \to \S_C$ from an open interval $(R)$ of $\R$ to the \htmladdnormallink{manifold}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} $\S_C$-- and then by using the calculus of variations minimizing this length. In physical
applications, however, to simplify the calculation one may also require the minimization of \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} as well as the length of the curve.

However, in \htmladdnormallink{Riemannian geometry}{http://planetphysics.us/encyclopedia/NonabelianAlgebraicTopology3.html} geodesics are not coinciding with the ``shortest length curves'' joining two points, even though a close connection may exist between geodesics and the shortest paths; thus, moving around a great circle on a Riemann sphere the `long way round' between two arbitrary, fixed points on a sphere is a geodesic but it is not obviously the shortest length curve between the points (which would be a straight line that is not permitted
by the topology of the surface of the Riemann sphere).

\begin{example}
The orbits of satellites and planets are all geodesics in curved
\htmladdnormallink{spacetime}{http://planetphysics.us/encyclopedia/SR.html}. As a more general physical example in general relativity theory, \emph{relativistic geodesics} describe the \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} of \htmladdnormallink{point particles}{http://planetphysics.us/encyclopedia/CenterOfGravity.html} in a spacetime with a curvature determined only by gravity.

Consider such a point particle $ z^{\mu}$ that moves along a trajectory or ``track'' in physical spacetime; also assume that the track is parameterized with the values of $ \tau $. Then, the \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} pointing in the direction of motion of the point particle in spacetime can be written as:

$$ u^{\mu} = { dz^{\mu} \over d\tau }. $$

If there are no \htmladdnormallink{forces}{http://planetphysics.us/encyclopedia/Thrust.html} acting on a point particle, then its velocity is unchanged along the trajectory or `track' and one has the following
\emph{geodesic equation}:

$$ { d u^{\nu} \over d \tau} + \Gamma^{\nu}_{\mu \sigma} u^{\mu} u^{\sigma} \quad = \quad { d^2 z^{\nu} \over d \tau^2} + \Gamma^{\nu}_{\mu \sigma} { d z^{\mu} \over d \tau} { d z^{\sigma} \over d \tau} \quad = \quad 0 . $$

\end{example}

\end{remark}

\begin{definition}
More generally, a \emph{geodesic} in \htmladdnormallink{metric}{http://planetphysics.us/encyclopedia/MetricTensor.html} geometry is defined as a
a curve $\Gamma: I \to M$ from an interval $I \subset \R$ to the \htmladdnormallink{metric space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $M$ for which there exists a constant $v \leq 0$ such that for any $t \in I$ there is a neighborhood $J$ of $t \in I$ such that for any $t_1, t_2 \in J$ one has that

$$d(\Gamma(t_1),\Gamma(t_2)) = v|t_1-t_2|.\,$$

\end{definition}

When the equality
$$ d(\Gamma(t_1),\Gamma(t_2))=|t_1-t_2| \, $$
is satisfied for all $t_1, t_2 \in I$, the geodesic is called the shortest path or a {\em minimizing geodesic}.

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