To transform from flat to curved space

[friend]

Many of the formulas of physics assume a background of flat spacetime. But it's natural to think that this is just a special case of a more general formulation in curved spacetimes. So I'd like to develop a general procedure to transform flat space formulas and integrals, etc, into their curved space counterparts.

It seems that the Jacobian matrix can be used to transform volume elements of curvilinear coordinates to the equivalent volume element in flat space. This is done by transforming differential moves in the curved space to differential moves in flat space. The Jacobian matrix does not seem to care what kind of space (curved or flat) is being transformed into flat space. One just does the differentiation of the curvilinear coordinates with respect to the flat space coordinates. So I have to wonder if the Jacobian matrix also applies when transforming from spaces with curvature and torsion to flat spaces. If it does, then does that mean that every coordinate system whatsoever is locally flat? I assume every coordinate system has coordinate lines that are locally linear independent. And using the Gram-Schmidt procedure this can be made into an orthogonal coordinate system. So is every coordinate system locally flat?

But more to the point, I wonder how to transform from flat to curved spaces. I assume that the inverse Jacobian can be used to transform volume elements in flat space to volume elements in curved space, right? And does this mean that it is just a matter of transforming differential moves in flat space to differential moves in curved space?

Any insight that anyone can give would be appreciated. Thanks.

 

[Mute]

I'm not sure if it will help you much, but you can try looking at Sean Carroll's General Relativity book "Spacetime and Geometry". I think you can get a look at the rough pdf version here: http://arxiv.org/abs/gr-qc/9712019

Somewhere in there, chapter 3 perhaps, it mentions the "recipe" for converting laws of physics in flat spacetime to curved spacetime (I remember one of them was to replace partial derivatives $\partial_\mu$ with covariant derivatives $\nabla_\mu$).

 

[CompuChip]

I also recall one of the problems was the ordering, as
$$[\partial_\mu, \partial_\nu] \equiv 0$$
but
$$[\nabla_\mu, \nabla_\nu] \neq 0$$
in general. So a priori the result depends on what order you choose.

But indeed, Carroll says some useful things about it.

 

[HallsofIvy]

Yes, every surface is "locally" flat. Given a "flat coordinate system" at a point, however, there exist an infinite number of global coordinates that reduce to the given coordinate system at that point so there is no general method of going from a local coordinate system to a global system.

 

[friend]

Yes, every surface is "locally" flat. Given a "flat coordinate system" at a point, however, there exist an infinite number of global coordinates that reduce to the given coordinate system at that point so there is no general method of going from a local coordinate system to a global system.

So this may prove a requirement of the metric to be a dynamic thing, and not just a prescribed background.

Thanks for the link, Mute, I take a look sometime today.

 

[friend]

I'm not sure if it will help you much, but you can try looking at Sean Carroll's General Relativity book "Spacetime and Geometry". I think you can get a look at the rough pdf version here: http://arxiv.org/abs/gr-qc/9712019

Somewhere in there, chapter 3 perhaps, it mentions the "recipe" for converting laws of physics in flat spacetime to curved spacetime (I remember one of them was to replace partial derivatives $\partial_\mu$ with covariant derivatives $\nabla_\mu$).

From that reference, page 106, "Take a law of physics in flat space, traditionally written in terms of partial derivatives and the flat metric. According to the equivalence principle this law will hold in the presence of gravity, as long as we are in Riemannian normal coordinates (RNC). Translate the law into a relationship between tensors; for example, change partial derivatives to covariant ones. In RNC’s this version of the law will reduce to the flat-space one, but tensors are coordinate-independent objects, so the tensorial version must hold in any coordinate system.

This procedure is sometimes given a name, the Principle of Covariance. I’m not sure that it deserves its own name, since it’s really a consequence of the EEP plus the requirement that the laws of physics be independent of coordinates."

So it seems only those flat space formulas that can be expressed in terms of tensors can be transformed to curved space(time) coordinates. That brings up the question of what kinds of functions are they. Are they only those expressions that involve differentials and metrics? Must they be expressible as exact differentials (in the language of differential forms)?



Prof Hagen Kleinert has an effort to transform formulas from flat space to curved space. He calls them nonholonomic transformations. But he seems to use them only for transforming Feynman path integrals from flat to curved spacetimes. But I was hoping for something more general. Prof Hagen Kleinert's efforts can be found at:

http://www.physik.fu-berlin.de/~kleinert/kleiner_re252/quequpw.html

His nonholonomic transformation is defined as:

$$\[\dot q^\mu = e_i ^\mu \cdot \dot x^i \]$$ found here: http://www.physik.fu-berlin.de/~kleinert/kleiner_re252/node7.html#SECTION00025000000000000000

where the $$\[q^\mu \]$$ are coordinates in the curved space, the $$\[x^i \]$$ are coordinates in the flat space, and $$\[e_i ^\mu = \frac{{\partial q^\mu }}{{\partial x^i }}\]$$ is the inverse jacobian matrix from flat to curved space (assuming the dimensionality of the two spaces is the same.)

He then goes on to transform the $$\[\Delta x_n \]$$ into q-space using this nonholonomic transformation, see details at: http://www.physik.fu-berlin.de/~kleinert/kleiner_re252/node10.html#SECTION00031000000000000000
I wonder if his transformation of $$\[\Delta x_n \]$$ is more complicated than it needs to be. For isn't
$$\[\Delta x_n = \int_{x_n }^{x_{n + 1} } {dx = \int_{x_n (q)}^{x_{n + 1} (q)} {e_\mu ^i \cdot \dot q^\mu dt} } \]$$
,where the last equation is in terms of q-space coordinates? Or am I missing something and the complication of that page is really necessary? Thanks.

 

[friend]

Or what would be the covariant version of

$$\[ \Delta x_n = \int_{x_n }^{x_{n + 1} } {dx = \int_{x_n (q)}^{x_{n + 1} (q)} {\frac{{\partial q^\mu }}{{\partial x^i }}\cdot\dot q^\mu dt} } \]$$