- #1

- 9,777

- 997

## Main Question or Discussion Point

To avoid hijacking an existing thread, I wanted to start a new one on how "gravitational forces" are represented in GR.

There doesn't seem to be a lot on this in the intro textbooks, alas, which mostly deal with the issue by avoiding it. Which suggests there could be some non-obvious problems, or at least grounds for long arguments.

But, forging ahead, nonetheless, I would like to propose an idea, which seems reasonable to me, to see if it will stay afloat, and perhaps the discussion (if we get one) will clarify things a bit.

The basic idea is that gravitational forces in GR are represented by Chrsitoffel symbols - or rather a subset of them.

This does represent the forces one experiences in an accelerating elevator as 'real' forces - but this is exactly the goal we want to achieve, I think, from the principle of equivalence.

To sketch the mathematical, if we consider the reference frame of an accelerating observer in flat space time, ala MTW we find that the metric is

ds^2 = (1 + 2g x) dt^2 - dx^2 - dy^2 - dz^2

see for instance MTW, pg 331, or, with some sign differences, http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3A0901.4465 [Broken], above (20), where we've simplified things by assuming no rotation.

and we can identify [itex]\Gamma^{x}{}_{tt} = -g[/itex] from the usual expression

[tex]\Gamma^{c}{}_{ab} = \frac{1}{2} g^{cd} \left( \partial_{a} g_{bd} + \partial_{b} g_{ad} - \partial_{d} g_{ab} \right) [/tex]

Curved space-time introduces second order corrections to the metric, which won't affect the values of the Christoffel symbols.

So the basic idea is that "gravitational forces" are represented by [itex]\Gamma^{x}{}_{tt}, \Gamma^{y}{}_{tt}. \Gamma^{z}{}_{tt}[/itex], three of the Christoffel symbols.

We can also note, ala MTW pg 330, that the Fermi-Walker transport law _requires_ that [itex]\Gamma^{\hat{x}}{}_{\hat{t}\hat{t}} = \Gamma^{\hat{t}}{}_{\hat{x}\hat{t}} = a^j[/itex] along the worldline of the accelerating observer, where [itex]a^j[/itex] is the 4-acceleartion of the observer.

So, basically, Christoffel symbols carry information about the acceleration and the rotation of the worldline, and no information about the curvature of the space-time (that information is in second order terms in the metric, and comes from the Riemann tensor).

There doesn't seem to be a lot on this in the intro textbooks, alas, which mostly deal with the issue by avoiding it. Which suggests there could be some non-obvious problems, or at least grounds for long arguments.

But, forging ahead, nonetheless, I would like to propose an idea, which seems reasonable to me, to see if it will stay afloat, and perhaps the discussion (if we get one) will clarify things a bit.

The basic idea is that gravitational forces in GR are represented by Chrsitoffel symbols - or rather a subset of them.

This does represent the forces one experiences in an accelerating elevator as 'real' forces - but this is exactly the goal we want to achieve, I think, from the principle of equivalence.

To sketch the mathematical, if we consider the reference frame of an accelerating observer in flat space time, ala MTW we find that the metric is

ds^2 = (1 + 2g x) dt^2 - dx^2 - dy^2 - dz^2

see for instance MTW, pg 331, or, with some sign differences, http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3A0901.4465 [Broken], above (20), where we've simplified things by assuming no rotation.

and we can identify [itex]\Gamma^{x}{}_{tt} = -g[/itex] from the usual expression

[tex]\Gamma^{c}{}_{ab} = \frac{1}{2} g^{cd} \left( \partial_{a} g_{bd} + \partial_{b} g_{ad} - \partial_{d} g_{ab} \right) [/tex]

Curved space-time introduces second order corrections to the metric, which won't affect the values of the Christoffel symbols.

So the basic idea is that "gravitational forces" are represented by [itex]\Gamma^{x}{}_{tt}, \Gamma^{y}{}_{tt}. \Gamma^{z}{}_{tt}[/itex], three of the Christoffel symbols.

We can also note, ala MTW pg 330, that the Fermi-Walker transport law _requires_ that [itex]\Gamma^{\hat{x}}{}_{\hat{t}\hat{t}} = \Gamma^{\hat{t}}{}_{\hat{x}\hat{t}} = a^j[/itex] along the worldline of the accelerating observer, where [itex]a^j[/itex] is the 4-acceleartion of the observer.

So, basically, Christoffel symbols carry information about the acceleration and the rotation of the worldline, and no information about the curvature of the space-time (that information is in second order terms in the metric, and comes from the Riemann tensor).

Last edited by a moderator: