Geodesic
Deviation
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Consider
two test particles in free-fall. Each test particle moves on a geodesic in
spacetime. Spacetime curvature manifests itself through geodesic deviation, the
relative acceleration of the test particles, also known as tidal acceleration.
This page describes the meaning of this term and provides a calculation of
geodesic deviation. Geodesic deviation is defined as the relative
separation of two test particles moving on closely spaced diverging geodesics.
Thus two particles in free-fall for which there are deviating geodesics have a
relative acceleration. Let x
be the relative position
4-vector of the two test particles. This vector originates on one test particle
and terminates on the other nearby particle each of which is moving on a nearby
geodesic. Two such nearby geodesics are shown below in Figure 1 with the
relative position 4-vector shown
In Fig. 1 there are two geodesics, X and Y,
described by the coordinates xm
and ym
respectively, each of which satisfies the geodesic equation. The
4-vector, x(t),
shown in the figure points from xm(t)
to ym(t)
and represents the relative acceleration of the two test particles. Let
the components of the 4-vector x
be xm.
Then
The relative
acceleration of the two test particles is then defined as
Our task
is then to evaluate the second total derivative in Eq. (1). The total derivative
of x is
given by
The
second total derivative is found by applying the total derivative once to Eq.
(3)
Substituting
Eq. (3) gives
Carrying
out the derivative on the first term on the right hand side and multiplying the
terms through on the second term results in
This is
further simplified by expanding the second term on the right side using the
product rule for derivatives. The result is
The
second derivative may be obtained from the geodesic equation
Substitute
this last expression into Eq. (7) to obtain
The goal
at this point is to find an expression for the first term on the right hand side
of Eq. (8). Employ the geodesic equation for each geodesic.
The Christoffel symbols are related to each other by a Taylor expansion. To first order in be xl we find
Subtract
Eq. (10a) from Eq. (10b) to obtain
Substitute
in Eq. (1) to obtain
To
simplify this expression expand the second term
Discard
the second term order terms, i.e. the second term on the left side of Eq. (14)
and substitute Eq. (11) to yield the approximation
The
third term from the left is also a second order term that may be discarded to
yield the final result
Substitute
Eq. (16) into Eq. (9) to obtain
Cancel
the common term to give
Re-label
indices so as to obtain a common factor
Factor
out like terms
Recall
the definition of the Riemann tensor
Substituting
Eq. (21) into Eq. (20) yields the final result
Eq. (22)
is known as the equation of geodesic deviation. This may also be written
in a more familiar form if we define the 4-vector
Than Eq.
(22) becomes
Compare
this to the Newtonian relation for tidal acceleration
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