Explanation of Wiki regarding Geodesics as Hamiltonian Flows:

1. Nov 5, 2009

runner108

In the article from Wikipedia called: Geodesics as Hamiltonian Flows at:

http://en.wikipedia.org/wiki/Geodesics_as_Hamiltonian_flows" [Broken]

It states the following:

It is frequently said that geodesics are "straight lines in curved space". By using the Hamilton-Jacobi approach to the geodesic equation, this statement can be given a very intuitive meaning: geodesics describe the motions of particles that are not experiencing any forces. In flat space, it is well known that a particle moving in a straight line will continue to move in a straight line if it experiences no external forces; this is Newton's first law. The Hamiltonan describing such motion is well known to be H = mv2 / 2 = p2 / 2m with p being the momentum. It is the conservation of momentum that leads to the straight motion of a particle.

Under the wiki article regarding momentum it states:

Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry) of space (position in space is the canonical conjugate quantity to momentum). So, momentum conservation can be philosophically stated as "nothing depends on location per se".

My understanding of relativity is fairly basic but I feel I intuitively understand most of this. My question is if we insert the second bold text into the first it basically says that straight line motion is a result of shift symmetry. Can someone explain this further? Or am I fishing for a connection between two related but ancillary points?

Last edited by a moderator: May 4, 2017
2. Nov 5, 2009

Ben Niehoff

In Special Relativity, you would be right on. Momentum conservation is a result of translation symmetry.

However, in General Relativity, the analog to "translation symmetry" doesn't always exist, because spacetime is curved. The concept you are looking for in this case are Killing vectors, which are vector fields with certain properties that allow you to define globally conserved quantities. Killing vectors cannot always be found, however. Hence, momentum conservation as a global symmetry does not always apply in General Relativity; however, it does apply locally (that is, over portions of the spacetime manifold that are not too large).

3. Nov 6, 2009

pervect

Staff Emeritus
I'm pretty sure it follows from Hamilton's equations that a body obeying the principle of least action will move in a straight line if its isolated, i.e. if its Hamiltonian is not a function of position. I couldn't really come up with a proof I was happy with, however.

Note that I'm staying callssical here, to make it easier to talk about...

4. Nov 6, 2009

runner108

I just read an interesting book called "Relativity and the Nature of Spacetime" by Vesselin Petkov, I don't know if it's legitimate or not. He seemed to say that inertial force is the resistance a worldtube feels towards deviation. I don't know what if his argument is accepted or not, seems logical to me but I am a novice with no physics background.:

http://www.fqxi.org/data/essay-contest-files/Petkov_PetkovFQXi.pdf