Time dilation and Length contraction in Saddle Geometries

[kmarinas86]

The way I've been reading it, elliptic geometries are due to a positive Gaussian curvature, while hyperbolic geometries are due to a negative Gaussian curvature.

Do local saddle curvatures mean local time dilation and length contraction, or do they mean local time acceleration and length expansion? Or do saddle geometries correspond to local time dilation and length expansion? Or do saddle geometries correspond to local time acceleration and length contraction?

Also, how would a localized "negative" energy density affect time and length relative to flat space?

 

[Naty1]

Fantastic questions...I hope an 'expert(s)' will come across them and reply.

I think I have a few preliminary comments to help frame a reply, but I do not know enough
to provide a complete view.

(1) In both special and general relativity, Christoffel symbols used in writing the laws of physics allow transformations (more precisely, "diffeomorphism") and preserve the form of the laws. In special relativity (flat spacetime) Lorentz transformations will also preserve the form of the laws if Christoffel symbols are not used. So I'm pretty sure we need somebody who knows how Christoffel math works.

(2) An open universe has a negative Gaussian curvature, K = -1; A closed universe has positive Gaussian curvature, K = +1. Generalizations of curvature are the scalar curvature and Ricci curvature which are used in the Einstein field equations ...so another issue is how Gaussian curvature relates to these.

[http://en.wikipedia.org/wiki/Scalar_curvature#Surfaces

says in two dimensions, scalar curvature is exactly twice the Gaussian curvature but what goes on in higher diemnsions??]

(3) Regarding negative energy density: Anti de Sitter space is a solution of Einstein's field equation with a negative (attractive) cosmological constant Λ ,corresponding to a negative vacuum energy density and positive pressure. What you may be interested in is here:

http://en.wikipedia.org/wiki/Anti_d...nguished_from_spacetime_in_general_relativity

Synopsis: Seems like just typical curvature,

Fundamentally, the key concept behind the idea of de Sitter space is that it involves a variation on the spacetime of general relativity in which spacetime is itself slightly curved even in the absence of matter or energy.

 

[Naty1]

I'm sorry no one has responded so far: I cam across this insight in my notes, but am not positive about its full interpretation:

There seem to be some really good descriptions here: [talking about the Ricci scalar]
http://en.wikipedia.org/wiki/Scalar_curvature

...the scalar curvature represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space..
When the scalar curvature is positive at a point, the volume of a small ball about the point has smaller volume than a ball of the same radius in Euclidean space. On the other hand, when the scalar curvature is negative at a point, the volume of a small ball is instead larger than it would be in Euclidean space.

 

[PeterDonis]

Do local saddle curvatures mean local time dilation and length contraction, or do they mean local time acceleration and length expansion? Or do saddle geometries correspond to local time dilation and length expansion? Or do saddle geometries correspond to local time acceleration and length contraction?

It depends on the specific geometry, and on what you define as "saddle curvature" (see below). Do you have any particular examples of "saddle curvatures" in mind?

One example of what might be called "saddle curvature" is the curvature around a spherically symmetric massive body, i.e., the Schwarzschild solution. In the radial direction, tidal gravity separates geodesics, but in the tangential direction, tidal gravity brings them closer together. So the sign of the curvature is different in different directions.

 

[PeterDonis]

(1) In both special and general relativity, Christoffel symbols used in writing the laws of physics allow transformations (more precisely, "diffeomorphism") and preserve the form of the laws. In special relativity (flat spacetime) Lorentz transformations will also preserve the form of the laws if Christoffel symbols are not used. So I'm pretty sure we need somebody who knows how Christoffel math works.

Actually, Christoffel symbols are not tensors, so they themselves are not frame independent. They can appear in equations that are frame independent overall, but they themselves act like vector or tensor components that can change from frame to frame.

(2) An open universe has a negative Gaussian curvature, K = -1; A closed universe has positive Gaussian curvature, K = +1. Generalizations of curvature are the scalar curvature and Ricci curvature which are used in the Einstein field equations ...so another issue is how Gaussian curvature relates to these.

The FRW solutions are actually very special ideal cases, where there is a well-defined "sign" to the curvature. In most cases, even very simple ones, there isn't (see my previous post).

in two dimensions, scalar curvature is exactly twice the Gaussian curvature but what goes on in higher diemnsions??

In general there isn't a unique "scalar curvature"; there are different scalars that can be derived from the curvature tensor by contracting it in different ways. The Ricci scalar is one; the Kretschmann scalar, for example, is another, distinct one:

http://en.wikipedia.org/wiki/Kretschmann_scalar

Two dimensions is a special case where, AFAIK, all of the possible scalars turn out to be the same.

(3) Regarding negative energy density: Anti de Sitter space is a solution of Einstein's field equation with a negative (attractive) cosmological constant

AdS is equivalent to a K = -1 (negative curvature) FRW model where the only energy density is due to the cosmological constant. So yes, it is one of the very special cases where the sign of the curvature is simple and well-defined.

 

[Naty1]

PeterDonis: thanks for helping out.

The Schwarzschild solution is a saddle curvature ?? I can't find an illustration...I thought a saddle shape was like the center one here:

http://en.wikipedia.org/wiki/Shape_of_the_universe


My earlier comment about Christoffel symbols was I think based in part on this reading:


http://en.wikipedia.org/wiki/Christoffel_symbols

"In mathematics and physics, the Christoffel symbols... are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds...

In general relativity, the Christoffel symbol plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor."



Seems from your posted comments above you don't agree?

I thought they behaved as tensors under linear coordinate transformations.

[well, I don't actually agree with the last sentence in the quote myself!...I'm aware no single curvature entity REALLY represents the gravitational field, per other discussions.]