Understanding Geodesic Deviation: Solving for Equations (7) to (8)

In summary, equations (7) to (8) provide a way to solve for geodesic deviation, which is a measure of the separation between two geodesic paths in curved spacetime. These equations involve the Riemann curvature tensor and its derivatives, and can be used to understand how gravity affects the motion of particles in spacetime. They are an important tool in the study of general relativity and can help us better understand the behavior of objects in the presence of curved spacetime.
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  • #2
Originally posted by Arcon
Can someone take a look at

http://wps.aw.com/wps/media/objects/500/512494/supplements/Ch21.pdf

and tell me how they go from Eq. (7) to Eq. (8)? I've tried this and keep getting additional terms.

i took a look at it, and did the calculation. i thought it was pretty straightforward. where did you get stuck? what extra terms do you have?

remember that x is a geodesic. so there is a geodesic equation in x, and it therefore vanishes. and remember that χ is very small; drop any term with more than one χ in it.
 
  • #3


Originally posted by lethe
i took a look at it, and did the calculation. i thought it was pretty straightforward. where did you get stuck? what extra terms do you have?

remember that x is a geodesic. so there is a geodesic equation in x, and it therefore vanishes. and remember that χ is very small; drop any term with more than one χ in it.

I fingered it out :smile:

One has to drop not only the term &chi*&chi but the term which is the product of &chi and a derivative of &chi. That was what I was missing.
 
  • #4


Thank you

I believe that I've fingered it out :smile:

One has to drop not only the term χ*χ but the term which is the product of χ and a derivative of χ. That was what I was missing.

Again - thanks for the response

Arcon
 
  • #5


Originally posted by lethe
i took a look at it, and did the calculation. i thought it was pretty straightforward. where did you get stuck? what extra terms do you have?

remember that x is a geodesic. so there is a geodesic equation in x, and it therefore vanishes. and remember that χ is very small; drop any term with more than one χ in it.

Seems that this small detail (drop term with χdχdT) has always tripped me up in that derivation. I guess I was just blind to it. But now that I know it then the derivation is simple.

Just to make sure I understood the approximation can you check this for me?

http://www.geocities.com/physics_world/gr/geodesic_deviation.htm

I commented on the terms to drop right after Eq. (14) and right after Eq. (15)

Thanks

I don't know how I missed this before but the equation of geodesic deviation clearly shows that tidal forces are velocity dependant!

Arcon
 

1. What is geodesic deviation?

Geodesic deviation is the measurement of the change in the distance between two points along a geodesic curve, which is the shortest path between two points on a curved surface. It is a fundamental concept in the study of curved spaces, such as in general relativity.

2. How is geodesic deviation related to understanding curved spaces?

Geodesic deviation is a key concept in understanding curved spaces because it allows us to measure the effects of curvature on the behavior of objects moving through that space. By studying geodesic deviation, we can gain a better understanding of how gravity and other forces act on objects in curved spaces.

3. What are equations (7) to (8) used for in geodesic deviation?

Equations (7) to (8) are used to solve for the geodesic deviation in a specific situation. These equations take into account the curvature of the space, the velocity of the object, and the forces acting on the object to calculate the change in distance between two nearby points along the geodesic curve.

4. Can geodesic deviation be measured in real-world situations?

Yes, geodesic deviation can be measured in real-world situations. For example, in the field of astrophysics, geodesic deviation is used to study the behavior of light and other objects in the presence of massive objects, such as black holes. Geodesic deviation can also be observed in the motion of planets and other celestial bodies.

5. How does understanding geodesic deviation help us understand the universe?

Understanding geodesic deviation is crucial in understanding the universe because it allows us to study the curvature of space and the effects of gravity and other forces on objects in that space. By understanding geodesic deviation, we can better explain phenomena such as the bending of light around massive objects and the motion of planets in our solar system.

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