Geodesic Deviation in 2D: Is There Directional Dependence?

AcidRainLiTE
Messages
89
Reaction score
2
In 2 dimensions, is the geodesic deviation equation governed by a single scalar, independent of the direction of the geodesics? That is, if ξ is the separation of two nearby geodesics, do we have d^2 \xi/ds^2 + R\xi = 0 where R is a scalar that is completely independent of the direction of the geodesics?

If so, how can we see that there can be no directional dependence in 2d?
 
Physics news on Phys.org
AcidRainLiTE said:
In 2 dimensions, is the geodesic deviation equation governed by a single scalar, independent of the direction of the geodesics? That is, if ξ is the separation of two nearby geodesics, do we have d^2 \xi/ds^2 + R\xi = 0 where R is a scalar that is completely independent of the direction of the geodesics?

If so, how can we see that there can be no directional dependence in 2d?
By 2D do you mean ##t,x## ? So if you have a metric ##ds^2=-g_{00} dt^2 + g_{11}dx^2##, can you work it out explicitly ? If the velocities of the geodesics depend on ##t## or ##x##, is that directional dependence ?
 
Thread 'Can this experiment break Lorentz symmetry?'

Thread 'Can this experiment break Lorentz symmetry?'

1. The Big Idea: According to Einstein’s relativity, all motion is relative. You can’t tell if you’re moving at a constant velocity without looking outside. But what if there is a universal “rest frame” (like the old idea of the “ether”)? This experiment tries to find out by looking for tiny, directional differences in how objects move inside a sealed box. 2. How It Works: The Two-Stage Process Imagine a perfectly isolated spacecraft (our lab) moving through space at some unknown speed V...
View full post »

Thread '1-Way Speed of Light'

Does the speed of light change in a gravitational field depending on whether the direction of travel is parallel to the field, or perpendicular to the field? And is it the same in both directions at each orientation? This question could be answered experimentally to some degree of accuracy. Experiment design: Place two identical clocks A and B on the circumference of a wheel at opposite ends of the diameter of length L. The wheel is positioned upright, i.e., perpendicular to the ground...
View full post »

Thread 'How can a black hole absorb matter?'

According to the General Theory of Relativity, time does not pass on a black hole, which means that processes they don't work either. As the object becomes heavier, the speed of matter falling on it for an observer on Earth will first increase, and then slow down, due to the effect of time dilation. And then it will stop altogether. As a result, we will not get a black hole, since the critical mass will not be reached. Although the object will continue to attract matter, it will not be a...
View full post »

Similar threads

Insights Geodesic Congruences in FRW, Schwarzschild and Kerr Spacetimes
Replies
8
Views
3K
A Minimal property of Spacelike geodesics in GR/curved spacetime?
2
Replies
76
Views
3K
A Re-writing the geodesic deviation eqn in matrix notation (3d only)
Replies
0
Views
1K
A Does the EP imply that a free-falling observer follows a geodesic?
Replies
3
Views
484
I A question about geodesics and notion of parallelism
Replies
19
Views
2K
A Curvature with different connections on the two-sphere
Replies
15
Views
1K
I Curved spacetime bulging in time direction
Replies
30
Views
2K
A Computing Null Geodesics in Schwarzschild Geometry
Replies
4
Views
2K
A Two Questions about Novikov Coordinates
Replies
8
Views
2K
I Geodesics using 2 Variables: Time & Radius from Mass
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top