Curved Space-Time: Books & Research on Mass & Force Curve

[bohtauri]

Im looking for books or research that contains data on the actual curve of space (for example, the curve a specific mass or force creates) e.t.c

If anyone knows any books or research done on this topic would be highly appreciated.

 

[atyy]

The curvature of spacetime is information contained in the metric tensor. Curvature is a set of numbers in the Riemann curvature tensor, that can be obtained by differentiating the metric tensor.

The metric tensor near the Earth used by astronomers can be found in http://arxiv.org/abs/astro-ph/0303376.

Aspects of the metric tensor near the Earth used by the global positioning system can be found in http://relativity.livingreviews.org/Articles/lrr-2003-1/ .

 

[Jonathan Scott]

Im looking for books or research that contains data on the actual curve of space (for example, the curve a specific mass or force creates) e.t.c

If anyone knows any books or research done on this topic would be highly appreciated.

There's actually more than one sort of curvature involved. Masses locally curve space-time within their own location in a way which is a bit like the curvature of a ball, but they curve space-time around them in a way which is more like the curvature of the surface of a cone, and it is that curvature which conveys the gravitational field, so I assume that is what you mean.

For the simple case of a central mass with a gravitational field that is not extremely strong, the curvature of space and the curvature with respect to time are both simply equal to the Newtonian acceleration with appropriate factors of c to give the right units.

For example, if the Newtonian acceleration is g, then the curvature in terms of angle turned in radians per distance traveled is given by g/c². For the sort of accelerations which occur in the solar system, the curvature is extremely small and barely detectable.

The curvature with respect to time (that is, the rate at which the velocity turns in the direction of the central object per amount of time) affects all objects including those at rest and is equivalent to normal Newtonian gravity. However, the curvature with respect to space only has a significant effect on fast-moving objects, proportionally to v²/c².

For light, moving at c, the effect of the curvature of space is equal to the effect of the curvature with respect to time and the total effect is twice the Newtonian acceleration. This was originally confirmed directly by the way in which the deflection of the light meant that images of stars appear to be moved very slightly in the sky when they are very closely to the sun, which could however only be confirmed during a solar eclipse. We can now confirm this deflection in various other ways such as using radio frequency images of quasars instead, which can be observed without the need for an eclipse.

 

[pervect]

Im looking for books or research that contains data on the actual curve of space (for example, the curve a specific mass or force creates) e.t.c

If anyone knows any books or research done on this topic would be highly appreciated.

In general it's space-time that's curved,not just space. However, one of the PPN parameters, gamma, specifically measures the curvature of space. (To even distinguish between "space curvature" and "time curvature", one needs to adopt a specific coordinate system, of course - but the PPN formalism does this.)

http://en.wikipedia.org/w/index.php?title=Parameterized_post-Newtonian_formalism&oldid=428288286 has some basic info on the PPN parameters, including gamma, which is defined there as

How much space curvature gij is produced by unit rest mass ?

The PPN parameters are important because they're what are measured by experiment to test GR, so if you're looking for experiments that show space is curved, in particular,measurements of the PPN parameter gamma will be just what you need.

Light bending experiments are one of the tests that measure gamma. One of the earliest tests performed , initial results were the subject of some debate,but later results are pretty much unambiguously in agreement with GR and in disagreement with Newtonian gravity. For instance, Wiki ,quoting Will, mentions that the Cassini tracking experiments put gamma at 1 with an error bound of .00002.

Gamma would have to be zero for space (in the PPN coordinate system) to have no curvature.

 

[bcrowell]

There's actually more than one sort of curvature involved. Masses locally curve space-time within their own location in a way which is a bit like the curvature of a ball, but they curve space-time around them in a way which is more like the curvature of the surface of a cone, and it is that curvature which conveys the gravitational field, so I assume that is what you mean.

This is incorrect. The surface of a cone has zero intrinsic curvature, so the Riemann tensor vanishes. The Riemann tensor doesn't vanish in the vacuum surrounding a spherical mass.

For the simple case of a central mass with a gravitational field that is not extremely strong, the curvature of space and the curvature with respect to time are both simply equal to the Newtonian acceleration with appropriate factors of c to give the right units.

This is incorrect. Curvature is a tensor, and the Newtonian gravitational field (acceleration) isn't. Therefore we can always pick coordinates such that, at any given point in spacetime, the Newtonian gravitational field is zero. Curvature tensors can never be made zero by a change of coordinates, because they're tensors, and a tensor that is zero in one set of coordinates is zero in all sets coordinates.

However, the curvature with respect to space only has a significant effect on fast-moving objects, proportionally to v²/c².

No, this is incorrect. Spatial curvature does not only affect fast-moving objects.

-Ben

[EDIT] Fixed an error pointed out by Passionflower.

 

[bcrowell]

Hi, bohtauri,

Welcome to PF!

Atyy has given some links with correct information in them, but it may not be at the right level for you to understand. Could you tell us a little about your background in math and physics, specifically special and general relativity, so we can give you an answer at the right level?

-Ben

 

[Passionflower]

This is incorrect. The surface of a cone has zero intrinsic curvature, so the Ricci tensor vanishes. The Ricci tensor doesn't vanish in the vacuum surrounding a spherical mass.

Perhaps you meant Riemann tensor, as the Ricci scalar vanishes in vacuum solutions.

 

[bcrowell]

Perhaps you meant Riemann tensor, as the Ricci scalar vanishes in vacuum solutions.

Oops -- thanks for the correction! I've edited my post to fix the mistake.

-Ben

 

[bohtauri]

Could you tell us a little about your background in math and physics, specifically special and general relativity, so we can give you an answer at the right level?

Yeah, sorry, I should've added why i need to know in my original post. I am writing a book on causality and I have a section in it that would explain that the rate of causality is identical to the curvature of space-time. This is a very generalistic explanation and its still in draft form; so bear with me.

I am lacking information on "how curved is space-time" to help me develop an equation on how the curve is affecting the rate of causality. Everything I've researched so far shows they have a strong link but I haven't found any specific data on the curvature part of the idea yet. Due to causality affecting everything, i need data on applicable curvature or curvatures if they are separated into separate categories of mass, energy or forces.

My background is I am an avid enthusiast, far better than basic knowledge and some professionally advanced concepts. I am a fast learner though.

 

[bcrowell]

Yeah, sorry, I should've added why i need to know in my original post. I am writing a book on causality and I have a section in it that would explain that the rate of causality is identical to the curvature of space-time. This is a very generalistic explanation and its still in draft form; so bear with me.

OK. Please keep in mind that if you have an original theory about this, PF's rules say that PF is not the place to discuss it.

I am lacking information on "how curved is space-time" to help me develop an equation on how the curve is affecting the rate of causality. Everything I've researched so far shows they have a strong link but I haven't found any specific data on the curvature part of the idea yet. Due to causality affecting everything, i need data on applicable curvature or curvatures if they are separated into separate categories of mass, energy or forces.

My background is I am an avid enthusiast, far better than basic knowledge and some professionally advanced concepts. I am a fast learner though.

This is kind of a vague description of your level of knowledge. Do you know calculus? Have you taken a college-level introductory physics course? What books have you read on special relativity? General relativity?

 

[bohtauri]

OK. Please keep in mind that if you have an original theory about this, PF's rules say that PF is not the place to discuss it.

Yes, I know. I am looking for data of existing research, not discussions on personal theories.

This is kind of a vague description of your level of knowledge. Do you know calculus? Have you taken a college-level introductory physics course? What books have you read on special relativity? General relativity?

Currently I'm in my first year at university studying civil engineering so I would say I'm a bit above introductory calculus and physics.

books-wise i have read a lot of internet content but not so much physical books. I don't usually avoid the complex materials out there (it just takes a bit longer to understand the concepts and still enjoy it).

 

[pervect]

The only thing I can imagine that _might_ be a link between the "rate of causality", which is a bit vague, and curvature, is the metric coefficient g_00. This is not generally regraded as "curvature", but it's possible that it might be what you had in mind. If I'm understanding what you mean by "rate of causality" at all.

I'm guessing that you are ascribing or attempting to ascribe some physical significance to g_00 as some sort of "rate at which time passes", in accordance with the way in which it's used in popularizations as a measure of "gravitational time dilation", and that you are in addition calling g_00 a measure of curvature. Before I make more comments on the merits of this, we should probably find out if I'm even on the right track as far as guessing what's going through your mind.

I'd recommend reading http://www.eftaylor.com/pub/chapter2.pdf online, or the source book, "Exploring black holes", to get a better idea of what "curvature" might mean to people in the field of General Relativity.

The value of the PPN parameter gamma that I mentioned before is a measure of curvature, specifically the curvature of space, but doesn't have anything to do with the rate at which time passes. The Riemann curvature that other posters have mentioned is much more fundamental than the PPN parameter gamma, but also related to space-time rather than specifically to space, which is what you originally asked about.

A lot of people in the field will automatically think "Riemann curvature tensor" whenever the word "curvature" is used, I'm not convinced this is necessarily good for communication.