Finding Lorentz Factor of a Point Particle in Curved Spacetime

[vaibhavtewari]

Hello,
Is there a way to find the lorentz factor of a point particle in a curved spacetime in terms of metric elements(diagonal)

More specifically I was trying to write energy of a point particle in a curved space.

thanks

 

[Rastapunk]

I didn't quite understand you, but I'll still answer :D

As I understand, you want to find the Lorentz factor of a point particle which moves not only on the x-apsis? If is that the question, you need to divide the path like when you do it with vectors, one on the x and other on the y. While I'm writhing this I figure that your question is not this, but I'll still post it... Just in case :D

 

[vaibhavtewari]

can you provide a prescription to finally get the answer..

All I am trying to evaluate is to write energy of a point particle in a curved space-time in terms of metric element and rest mass

 

[Mentz114]

Your question is not stated clearly but ...

There is no global quantity that represents energy in curved spacetime. Kinetic energy is obviously frame dependent even in SR. However, every particle must be following some worldline, with a local frame defined at every point along it. The rest mass of the particle is transported unchanged along any worldline ( it is a scalar ) which would lead one to conclude the only meaningful energy is mc².

If you have something else in mind try rephrasing the question.

 

[vaibhavtewari]

I actually was reading this paper

http://arxiv.org/PS_cache/physics/pdf/0409/0409064v3.pdf

and on eq(37) author comfortably writes Energy in terms of metric element. I don't though found it very convincing. I would appreciate your thoughts on this

 

[Mentz114]

I actually was reading this paper

http://arxiv.org/PS_cache/physics/pdf/0409/0409064v3.pdf

and on eq(37) author comfortably writes Energy in terms of metric element. I don't though found it very convincing. I would appreciate your thoughts on this

It looks plausible because the factor $\sqrt{g_{tt}}$ ( changing notation slightly) is generally taken to be the factor by which frequency changes between static observers at infinity and static observers in the field. So in QM where energy depends on $d\psi/dt$ it reflects the change in frequency, and so the energy of a wave function. The only thing that bothers me slightly is, he's written the relativistic Lorentz invariant energy on the right including the p-term. If p is nonzero there is also a doppler shift to consider. But at low energies it would be a close approximation.

I've saved a copy of the paper, it looks well worth a read.