Geodesic (general Relativity)

In general relativity, gravity is not a force but is instead 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.

An ideal particle (ones whose gravitational field and size are ignored) not subject to electromagnetic 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.

MATHEMATICAL EXPRESSION

A timelike geodesic is a Worldline which Parallel Transport s its own tangent vector and maintains the magnitude of its tangent as a constant. If a curve

x^\alpha(s)

has tangent

d x^\alpha/ ds =  U^\alpha(s)\ ,

then this can be expressed as
:

abla_{U} U^\beta = 0\ ,\ \ \mathrm{or}\ \ U^\alpha 
abla_\alpha U^\beta = 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^\alpha

:
:

\ddot x^\beta + \Gamma^\beta {}_{\sigma \alpha}  \dot x^\sigma \dot x^\alpha = 0 \

where
:

\dot x^\alpha = U^\alpha = {d x^\alpha \over ds}

and
:

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

The parameter ''s'' typically represents 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 + 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

U + \ddot X^\lambda) U^\lambda {d \over ds} \ln U_

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Information About

Geodesic (general Relativity)