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GR and integration

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,104
The Astral plane
The Earth and Moon are considered to be too massive and too close to find a metric. In other words we can't use GR to find the gravitational energy of the Moon. What can be done is to break the Moon into tiny little pieces and do the calculation and add up the whole thing. The problem is that I don't know how to do that.

When we take derivatives we use covariant derivatives to keep everything nice and tensorial. But how do you take a "covariant integral?" All I know how to do is the calculation of the metric for one little piece. I don't know how to add them together into a whole.

(Hopefully I've explained myself well enough. I really have no idea where to find this information and my jargon might not be correct.)

Thanks!

-Dan
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,104
The Astral plane
I pulled out my Gravitation and Cosmology text (by Weinberg). I think what I need is found with differential forms. I have information in this text and one other about forms but I really have little idea how to use it for my needs. What I am trying to learn how to do is an integral of the following format:
\(\displaystyle \int _M f(x_{\mu})~dx^{\sigma}\)
where M is a smooth manifold and \(\displaystyle f(x_{\mu})\) is a tensor.

Any suggestions or maybe any references?

Thanks!

-Dan

Addendum: It's possible that I need to do an integral of the form: \(\displaystyle \int _M f(x_{\mu}) ~ Dx^{\sigma}\) but I don't even know if that formula even makes any sense to talk about.
 

Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,890
Hi Dan,

I'm afraid your question is not clear enough for me to give the answer you want, but the potential energy of the Earth-Moon system is derived in a 1973 article by Milan Burša and J. Pícha using methods of spherical harmonics:

Stud Geophys Geod (1973) 17: 272. https://doi.org/10.1007/BF01576717

Stokes' theorem and the Green identities apply when dealing with integration of tensor fields over compact oriented manifolds with boundary. For noncompact manifolds, partitions of unity or smooth cut-off functions are used.

Is the second integral you wrote meant to be a type of function integral, like a Feynmann or Polyakov path integral?
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,104
The Astral plane
Hi Dan,

I'm afraid your question is not clear enough for me to give the answer you want, but the potential energy of the Earth-Moon system is derived in a 1973 article by Milan Burša and J. Pícha using methods of spherical harmonics:

Stud Geophys Geod (1973) 17: 272. https://doi.org/10.1007/BF01576717

Stokes' theorem and the Green identities apply when dealing with integration of tensor fields over compact oriented manifolds with boundary. For noncompact manifolds, partitions of unity or smooth cut-off functions are used.

Is the second integral you wrote meant to be a type of function integral, like a Feynmann or Polyakov path integral?
That's the problem. I really don't know what approach to this needs to be made. I can easily do the non-Relativistic problem but I'm looking for some "inverse operation" to a covariant derivative. (Similar to the idea of the "inverse operation" to a Calc I derivative being an integral.)

I've read that the gravitational potential of the Earth-Moon system can be done by the standard approach of breaking the Moon up into small pieces and summing those up. Briefly speaking, I'm looking to a way to "integrate" this. But I've never seen a method to do it covariantly. If it can be done by "ordinary" integration I'm totally lost of the concept. Either way I'm stuck.

Here's some probably better notation than the last post:
In terms of my GR text I think I'm looking for the following analogy:

\(\displaystyle \int \dfrac{dA}{dt}~dt\)

and defining \(\displaystyle \dfrac{DA^{\mu}}{D \tau} = \dfrac{dA^{\mu}}{d \tau} + \Gamma _{\nu \lambda}^{\mu}~\dfrac{dx^{\lambda}}{d \tau}~A^{\nu}\) (\(\displaystyle \tau\) is the proper time, \(\displaystyle \Gamma\) is the affine connection, and \(\displaystyle A^{\mu}\) is a 4-vector.)

Then I'm looking for some kind of \(\displaystyle \int \dfrac{DA^{\mu}}{D \tau}~D \tau\) or something.

-Dan
 

Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,890
Are you trying to recover $A^\mu$ from the covariant derivative? Doing $\int D\tau$ wouldn't make sense. If you used normal coordinates, the covariant derivative would be simplified, and then you could integrate with respect to proper time. However, this technique does not always work. More generally though, you would solve the covariant differential equation with certain boundary conditions -- which may not be feasible without a numerical method.
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,104
The Astral plane
Are you trying to recover $A^\mu$ from the covariant derivative? Doing $\int D\tau$ wouldn't make sense. If you used normal coordinates, the covariant derivative would be simplified, and then you could integrate with respect to proper time. However, this technique does not always work. More generally though, you would solve the covariant differential equation with certain boundary conditions -- which may not be feasible without a numerical method.
Okay. Well at least I know a few more things about it. I'll have to work on the coordinate approach and see what I can come up with.

Thanks for the help!

-Dan