Mathematical Expression
A timelike geodesic is a worldline which parallel transports its own tangent vector and maintains the magnitude of its tangent as a constant. If a curve has tangent then this can be expressed as
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 :
where
and
The full geodesic equation is therefore:
where s is the proper time or distance and is the Levi-Civita connection.
Proof- ,
- ,
- ,
- ,
- ,
The parameter s typically represents proper 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 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 .
Read more about this topic: Geodesic (general Relativity)
Famous quotes containing the words mathematical and/or expression:
“An accurate charting of the American womans progress through history might look more like a corkscrew tilted slightly to one side, its loops inching closer to the line of freedom with the passage of timebut like a mathematical curve approaching infinity, never touching its goal. . . . Each time, the spiral turns her back just short of the finish line.”
—Susan Faludi (20th century)
“There exists in a great part of the Northern people a gloomy diffidence in the moral character of the government. On the broaching of this question, as general expression of despondency, of disbelief that any good will accrue from a remonstrance on an act of fraud and robbery, appeared in those men to whom we naturally turn for aid and counsel. Will the American government steal? Will it lie? Will it kill?We ask triumphantly.”
—Ralph Waldo Emerson (18031882)