# Force equilibrium in GR

1. Jul 7, 2012

### unchained1978

When considering the Newtonian forces, it's somewhat intuitive to think about the equilibration of forces, e.g. two charged particles overcoming the gravitational attraction to repel each other. Nothing new. What puzzles me though is to consider the same case within the context of GR and QED. Now gravity isn't a "force" and neither is the photon field. The gravitational acceleration is simply a trajectory through curved spacetime, and the electromagnetic force arises from the exchange of photons. So now how am I to imagine this "force" equilibrium? Does the electromagnetic energy counteract the gravitational mass energy of our two particle system in such a way that space is flat? Or is the electromagnetic "force" resisting the curvature of space? I have a hard time conceptualizing the intuitive content of classical physics within the framework of more fundamental theories.

2. Jul 7, 2012

### Mentz114

I've never seen work where GR and QED are mixed, but this question of 'forces' balancing is addressed in the solution of the Einstein-Maxwell equations.

I did this for the unphysical but interesting maximally-charged matter spacetime where the charge of the matter exactly offsets the effect of the gravitational field. The SET is that of an electric field + static matter.

The gravitational acceleration vector of a static particle is given by

αUμUα

where ∇ is the covariant derivative and Uμ=(1/√(-g00),0,0,0) for the stationary particle.

The acceleration caused by the electric field is

(Q/ρ)FαβJβ

where Q is the charge density and ρ is the matter density. The Faraday tensor F has been calculated taking the gravitational field into account ( ie in curved spacetime).
It turns out that with Q=ρ the two acceleration vectors are both non-zero, but sum to zero.

This has a straightforward interpretation as force cancellation, and shows the internal consistency of the E-M equations.

Here's a ref for the maximally charged solutions

http://arxiv.org/PS_cache/gr-qc/pdf/9805/9805002v1.pdf

and my force calculation is in this

http://www.blatword.co.uk/space-time/charged_dust_lut.pdf

Last edited: Jul 7, 2012
3. Jul 7, 2012

### Bill_K

It's not a force equilibrium, because as you said, there's only one force. Rather it's an example of F = ma, where F is the (unbalanced) electromagnetic force, and a is the gravitational acceleration that a particle undergoes even when it seems to us to be "sitting still".