Geodesic Deviation
Curvature of spacetime can be measured when we observe the geodesics of two freefall test particles deviating relative to each other. We will derive this deviation equation below. By a consideration of how this equation relates to the Newtonian gravitational theory we can deduce what the field equations of General Relativity (i.e. the differential equations that determine the gravitational field) should be.
Consider two nearby geodesics and
. In terms of the coordinates and the curve parameter u, these geodesics are represented respectively as:
and
. Small coordinate differences can always be thought of as vectors. Hence we can let
be the small vector joining points on the two geodesics having the same u value. That is
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It can be shown that we can always use the affine freedom available for geodesic curves to arrange the geodesics in such a way that there will always be some range of the curve value u giving small values for
. We want to investigate the case where the magnitude of
is a small quantity of order one.
The following differential equations must be true.
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The tilde symbol ~ on the connection in equation (14.2) implies that it is evaluated at a point with coordinate while the connection in the 2nd equation is to be evaluated at the point
.
Equation (14.1) can be rearranged as follows
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In Taylor series calculations that follow we will treat as our expansion parameter and we will only keep terms of 1st order in this parameter.
The two affine connections can be linked through such an expansions as follows:
| (14.5) |
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Using this expression and the previous relation (14.4) we now subtract the two geodesic equations (14.2) and (14.3). Keeping only 1st order terms we find that this subtraction implies the following.
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We are using the following definition for the dot derivative
We now rewrite this equation so as to connect it with the Riemann curvature tensor. It can be shown (exercise) that the following equation is equivalent to (14.6)
(14.7)
To get rid of the x double dot term we use the geodesic equation itself in the following form:
| (14.8) |
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Rearranging the terms and with the dummy indices suitably relabeled we get the following
We have introduced the absolute derivative definition which in our case is
| (14.10) |
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The quantity in the round brackets in equation (14.9) is the Riemann curvature tensor so (14.9) can be written as:
| (14.11) |
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This equation is called the Equation of Geodesic Deviation.
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In a flat space the Riemann curvature tensor is zero and the geodesic deviation equation reduces to the differential equation
| (14.12) |
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This equation gives rise to deviations that are straight lines.
| (14.13) |
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where and
are constants. Hence in flat space the separation vector increases linearly with the parameter u. In a curved manifold this simple linear relationship is not present for the deviation.
We will now compare the situation in GR with the Newtonian case. Assume that we have two slowly moving particles moving under Newtonian gravity conditions. These particles follow paths and
given by the following differential equations :
(14.14)
| (14.15) |
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We now subtract these equations to get the deviation equation in this Newtonian case. We have similar relations to the GR case to use:
(14.16)
| (14.17) |
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The difference of the equations of motion is
| (14.18) |
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This is the Newtonian analog of the General Relativity Geodesic Deviation Equation. We introduce the quantity
| (14.19) |
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The Newtonian deviation equation is then given as:
| (14.20) |
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In General Relativity we can also define analogous K quantity. It would be obtained from
| (14.21) |
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where we have chosen the affine parameter to be the proper time. We see that the General Relativity K can be defined as follows:
| (14.22) |
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The empty space field equations of Newton say that
| (14.23) |
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In GR if we likewise take our gravity equation from
| (14.24) |
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we get
| (14.25) |
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