Static gravity as a two form

[pervect]

I've been thinking recently that for static systems (it won't work in general), you can treat gravity as being a two-form if you use the scaled "force at infinity" rather than the locally measured gravitational force.

The "local force of gravity" is the acceleration of a worldline of a stationary observer (as measured by an accelerometer at the observer's worldline). The "force at infinity" is the "local force" multiplied by the time dilation factor between the observer's local clock and a clock at infinity.

I was wondering if this tensor had a name, or if there were any papers exploring this analogy. It's more or less hinted at in Wald, where he mentions that you can integrate the "force at infinity" around a sphere to get the enclosed Komar mass, but he doesn't actually define the corresponding two-form |f@inf_a u_b|, where f@inf is the "force at infinity" or give it a name.

 

[TrickyDicky]

I've been thinking recently that for static systems (it won't work in general), you can treat gravity as being a two-form if you use the scaled "force at infinity" rather than the locally measured gravitational force.

The "local force of gravity" is the acceleration of a worldline of a stationary observer (as measured by an accelerometer at the observer's worldline). The "force at infinity" is the "local force" multiplied by the time dilation factor between the observer's local clock and a clock at infinity.

I was wondering if this tensor had a name, or if there were any papers exploring this analogy. It's more or less hinted at in Wald, where he mentions that you can integrate the "force at infinity" around a sphere to get the enclosed Komar mass, but he doesn't actually define the corresponding two-form |f@inf_a u_b|, where f@inf is the "force at infinity" or give it a name.

Are you sure it would be a two-form?, I would picture it more like a vector-valued one-form, at least I think the stress-energy tensor can be viewed as a bivector-valued one-form but I'm not really sure.
Maybe Ben Niehoff can shed some light here.

 

[WannabeNewton]

What page in Wald's book may I ask?

 

[pervect]

about pg 288 - got to run

 

[WannabeNewton]

Is the corresponding two form you are referencing $\alpha _{ab} = \epsilon _{abcd}\triangledown ^{c}\xi ^{d}$? Wald says if you integrate this over the 2- sphere it gives you the force needed to keep in place a unit surface mass density that has already been distributed over the 2 - sphere. Is that what you are talking about treating as the gravitational force? What would that mean physically since $\alpha _{ab} = -\alpha _{ba}$.

 

[pervect]

Is the corresponding two form you are referencing $\alpha _{ab} = \epsilon _{abcd}\triangledown ^{c}\xi ^{d}$? Wald says if you integrate this over the 2- sphere it gives you the force needed to keep in place a unit surface mass density that has already been distributed over the 2 - sphere. Is that what you are talking about treating as the gravitational force? What would that mean physically since $\alpha _{ab} = -\alpha _{ba}$.

That's the one. Looks like re-reading Wald all my questions, except for whether it has a name, though it would be good to know if has a similar or perhpas simpler treatment.

 

[TrickyDicky]

Are you sure it would be a two-form?, I would picture it more like a vector-valued one-form

Ok, nevermind this.
For some reason I ignored the static setting and was thinking in terms of the general case.

After reading Wald's pertinent page I can see that gravity force is indeed a 2-form the way it is constructed as the integrand of a surface integral.
I do not know whether this particular static construction of the curvature form has a name, I doubt it, since usually curvature tensors are named for the general case given our universe is not static.
I'm curious, why are you interested in a static force?

 

[Ben Niehoff]

So it's true that there is a 2-form that you integrate around spheres to give the Komar mass.

But in order to say "static gravity is represented by a 2-form force", you have a second non-trivial thing to prove. You must show that the acceleration measured by a stationary observer can be obtained from the 2-form via some Lorentz-force formula. Does this actually work out? I haven't checked. You might need to convert it to "force at infinity" first, like you hinted.

Also, what about boosted observers? Does the 2-form in question correctly give gravitomagnetic effects? I strongly suspect it will be off by factors of 2 or something.

 

[pervect]

I think Wald already shows the first part, you get the force at infinity by computing the acceleration of a static observer, and multiply it by the redshift factor to infinity. And that's what you integrate to get the Komar mass.

The locally measured force can't be a two-form, but the "force at infinity" seems like it should be.

I'll have to think about the second part, it's a good point and it might well be a problem.

 

[TrickyDicky]

Regarding the gravitomagnetic effects, isn't the 2-form gravity force at infinity constructed in the context of a static spacetime? In this case there wouldn't be any gravitomagnetic effects right?

It is a bit confusing because in the Wald refernce the Komar mass obtained from that 2-form is later generalized to stationary spacetimes.

 

[pervect]

If we're in a static space-time and choose the right coordinates,there shouldn't be any gravitomagnetic effects. But if we're really saying the force is a tensor (I hadn't really thought about the issue, perhaps because it was computed via a tensor formula), then it has to transform like one.

Even in a static space-time, we can ask if the expression tells us the "force on a moving mass" - I wasn't originally expecting it to.

But I don't think it can transform via the standard Lorentz transform. So, um...mumble mumble. Well, I need to think about it, and I just hope I get some time to do that.