Can Geodesics Cross? - Exploring GR Proofs with Maki

In summary, the conversation discusses the focusing theorem and the concept of geodesics crossing in relation to it. The speakers mention examples of geodesics crossing and the role of gravity in this phenomenon. They also touch on the idea of non-crossing geodesic wordlines and the relationship between the focusing theorem and the area of black hole event horizons.
  • #1
mtak0114
47
0
Hi,

I am currently doing a course in GR and have just gone over a proof of the focusing theorem..
now this relied on the fact that geodesics do not cross. But I could not see clearly the contradiction if geodesics did happen to cross?

any help would be greatly appreciated.

Maki :smile:
 
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  • #2
I don't know the focusing theorem, but there can definitely be more than one geodesic between the same two events. A simple example is if you drill a hole through a planet with no atmosphere, along the rotational axis, and drop something into it. The path of the object can intersect the path of something in orbit on both sides of the planet.
 
  • #3
? Yes, of course, geodesics cross. At any point you can have geodesics in every direction. Surely there is more to this question? Are you asking if geodesics can cross twice? That would depend on the geometry.
 
  • #4
Consider geodesics on a sphere, they all cross twice.
 
  • #5
Geodesics can cross, since at any point there is a different geodesic in each direction. However, for a given region in spacetime, one can choose a bunch of non-crossing geodesic wordlines (congruence of geodesics) that are all headed into the future. The focusing theorem says that under certain circumstances, at least some of these geodesics are going to cross anyway, because gravity is attractive.
 
  • #6
mtak0114 said:
have just gone over a proof of the focusing theorem..
now this relied on the fact that geodesics do not cross.
There is something similar in the proof that the area of black hole event horizons never decreases.
 

FAQ: Can Geodesics Cross? - Exploring GR Proofs with Maki

Can geodesics cross in general relativity?

Yes, geodesics can cross in general relativity. According to the theory of general relativity, the shortest path between two points in spacetime is a geodesic. This means that geodesics can cross and intersect with each other.

How does general relativity explain the crossing of geodesics?

In general relativity, the curvature of spacetime is caused by the presence of massive objects. As a result, the geodesics that particles follow are affected by this curvature and can cross or intersect with each other. The bending of spacetime is what allows geodesics to cross.

Are there any real-life examples of geodesics crossing?

Yes, one example of geodesics crossing can be seen in the bending of light around massive objects, known as gravitational lensing. In this phenomenon, the geodesics of light are bent by the curvature of spacetime caused by massive objects, resulting in the crossing of light paths.

Can geodesics cross in other theories of gravity?

Geodesics can also cross in other theories of gravity, such as Newtonian gravity and Einstein's theory of gravity. However, the concept of geodesics and their behavior may differ in these theories compared to general relativity.

How is the crossing of geodesics relevant in understanding the universe?

The crossing of geodesics is relevant in understanding the behavior of particles and light in the universe. It allows us to study the effects of gravity on the paths of objects and helps us understand the nature of spacetime and its curvature. This understanding is crucial in explaining various phenomena in the universe, such as black holes and the expansion of the universe.

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