Geodesic (general Relativity)

Recall that Spacetime in general relativity is 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.

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.

MATHEMATICAL EXPRESSION

A timelike geodesic is a Worldline which Parallel Transport s its own tangent. If a worldline

ec x (	au)

has tangent

ec U(	au)

then this can be expressed as
:

abla_{ec U} ec 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

ec U

:
:

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

where
:

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

and
:

\ddot x^\beta = \dot U^\beta = {d U^\beta \over d	au} = {\partial U^\beta \over \partial x^\alpha} {d x^\alpha \over d	au} = U^\beta {}_{,\alpha} 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{-g_{\mu 
u} \dot x^\mu \dot x^
u}

.
Then the Euler-Lagrange Equation ,
:

{d \over d	au} {\partial f \over \partial \dot x^\alpha} = {\partial f \over \partial x^\alpha}

becomes, After Some Calculation ,

:

2(\Gamma^\lambda {}_{\mu

u} \dot x^\mu \dot x^

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Information About

Geodesic (general Relativity)