Geodesic Equation
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Derivation
1: Straightest Possible Line in a Curved Space
The first definition of a geodesic defines them as the straightest possible lines on a smooth manifold. Here we use affine geometry in which the space is explored by parallel transport. The term affine means connected and refers to how parallel lines are connected. Let l be an affine parameter that parameterizes the curve, which is expressed in the generalized coordinates qa. A geodesic in this case is a to the current definition of a geodesic can be defined by demanding parallel transport of the tangent vector along the curve. I.e.
Where
The chain rule for partial derivatives is
Multiplying Eq. (2)
through by Ua
allows Eq. (3) to express Eq. (1) as
Substituting
into Eq. (4) and yields
Eq. (6) is known as the geodesic equation.
Derivation
2: Path of Extremal Length
The second definition states that a geodesic is the path of extremal length. Here we use metric geometry in which the space is explored by measuring distances. The term metric means 'to measure' and we use it to measure the path which has an extreme length. Let l be an affine parameter that parameterizes the curve that is to be expressed in generalized coordinates qa. Consider the length of a curve as defined as the integral of ds, where ds2 = gabdqadqb and
An integral is said to be extremal when its first variation vanishes, i.e. dS = 0. As such the Lagrangian L satisfies the Euler-Lagrange Equations
Carrying out the
differentiation on the first term on the left side in braces (note: gab is
not a function of Um) gives
where
is the Kronecker delta. Which is one for a = b and zero otherwise. Therefore Eq. (9) simplifies to
If l is chosen to
be arc length then ds = dl.
Then
We take, for simplicity,
the value of L = +1 since either value will work in the variation of the
integral. This value results in
Taking the total derivative of Eq. (9) with respect to l which, after, relabeling indices, gives
In order to find the second term on the left side of Eq. (8) take the partial derivative of the Lagrangian with respect to qm (note: Ua is not a function of qm)
Substituting Eq. (14) and (15) into Eq. (8) gives
To put this in a more familiar and compact form rewrite Eq. (15) as
Multiply Eq. (17) through
by gnm
to give (note: gnmgam
= dna)
Since
Eq. (18) becomes
Recall the definition of
the components of the affine connection
Substituting the
components into Eq. (20) gives, upon replacing n
with m
Which, again, is the geodesic equation as derived at the top but now with a different, equivalent, definition.
Derivation 3: Change Coordinates from Locally Flat Coordinates
The third definition is similar to the first in that we define a geodesic as a path that is locally flat everywhere.
In a locally inertial frame particles move uniformly. When defined in terms of the affine l parameter this condition can be expressed in Cartesian coordinates which then becomes
The Cartesian coordinates xm can be expressed in generalized coordinates as xn =xn(qm). Substituting into Eq. 16 and applying the chain rule
Multiplying through by ¶qm/¶xn and noting that
reduces Eq. (25) to
It can be shown that
Substituting this expression into Eq. (27), relabeling indices, once again yields