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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