Does the shape of the Universe contribute to the gravitational path?

[lsimpson1943]

Reading just enough to be dangerous concerning Riemann geometry, choices of geometry regarding the shape of the Universe, etc., and the geodesic mappings to different geometric models (flat, spherical, and hyperbolic), I have an unanswered question in my mind. Acknowledging the affect of large masses distorting the space-time fabric, should this also be coupled with the shape of the Universe to calculate orbital paths? I guess I am asking are these two issues that have no connection or are they connected?

When reading about Einstein's conception of gravity, I see in textbooks the "abbreviated" pictures of the space-time fabric being distorted by large balls representing celestial objects and realize that it is only a three-dimensional depiction, given our visual capabilities. Trying to take my mind one layer higher to where I am viewing the shape of our Universe, I am wondering if this fabric is wrapped around some strange shape that is the "surface" of our Universe. I conclude in my mind that if it is wrapped around something that is not flat, then the gravitational paths would always be curved to some extent. Further using the same reasoning, world-lines would always be curved to some degree and never straight world-lines. Am I off base?

I sure would appreciate some help on this since I have never taken a physics course outside of my high school fifty years ago, and I am not taking a course now. I am pretty good at math, however; so you include the appropriate math in any answers you might submit.

 

[Mentz114]

The short answer to your question is 'yes'. Once the spacetime is defined by writing the metric tensor, all the geodesics are defined also.

As you've said, the rubber sheet analogy is not accurate. Spacetime curvature is intrinsic to the metric so it might not be useful to think about spacetime being 'curved around a surface'. Rather, curvature is a property that may vary from place to place and is affected by mass and energy in accordance with the Einstein field equations.

 

[lsimpson1943]

I assume you are making reference to the Riemann metric. Correct? As for the tensor, are we talking the Riemann curvature tensor or the Ricci tensor? If I understand the difference between the two, one is more "local" and the other averages the curvature for a defined part of a manifold.

I hope this doesn't sound stupid, but even if it does, could you further clear this up for me? These are the perils of being self-taught.

 

[pervect]

Reading just enough to be dangerous concerning Riemann geometry, choices of geometry regarding the shape of the Universe, etc., and the geodesic mappings to different geometric models (flat, spherical, and hyperbolic), I have an unanswered question in my mind. Acknowledging the affect of large masses distorting the space-time fabric, should this also be coupled with the shape of the Universe to calculate orbital paths? I guess I am asking are these two issues that have no connection or are they connected?

While in theory the expansion of the universe would affect the shape of orbits in our solar system, in practice the effect is negligible.

See http://www.astro.ucla.edu/~wright/cosmology_faq.html#SS Why doesn't the Solar System expand if the whole Universe is expanding? and the associated references, particularly the Cooperstock reference cited by Ned Wright. http://xxx.lanl.gov/abs/astro-ph/9803097, "The influence of the cosmological expansion on local systems".

There are some simple cases to consider that give more insight, though you will probalby need more background to fully understand them. I'll mention them anyway. The first is the Schwarzschild-de Sitter space time. This is a space-time that has a black hole at the center of a universe with an accelerating expansion, the de Sitter spacetime.

The Schwarzschild-de Sitter space-time (and also the de Sitter space-time) looks at first glance like it isn't static, but, when you look closely at it's Killing Vectors, it turns out it actually is. Therefore, while there is some effect on planetary orbits, the effect doesn't change with time.

If one is familiar with Birkhoff's theorem, one might expect this result. Spherical symmetry implies that only matter inside the sphere contributes to the gravity - the rest of the universe "doesn't matter".

Using this approach provides a fairly simple way to figure out what's important. What's important is the gravitating matter and energy inside a planetary orbit. Significant things to worry about are 1) normal matter. 2) dark matter inside the sphere, and 3) dark energy, (or the contribution of the cosmological constant) in the sphere.

The effect of the cosmological constant is the easiest to quantify. It effectively makes the mass of the central body look lower, the further away one goes, because one includes the positive mass of the central body with a negative Komar mass due to the amount of "dark energy" enclosed also in the sphere, an effect that does not change with time unless the cosmological constant changes with time, but does change with the radius of the enclosing sphere.

Well, there is also 4) - the possible effects of a lack of perfect spherical symmetry. The rest of the universe doesn't have any effect on the inside of the shell IF the expansion is symmetrical, but if it's not quite there's some possibility of a small, suppresed effect.

Some other papers that might be of interest:

http://uf.fpf.slu.cz/rag/.tpapps/v1/tpa010-E.pdf "Some properties of the Schwarzschild–de Sitter and Schwarzschild–anti-de Sitter spacetimes"

http://prd.aps.org/abstract/PRD/v78/i2/e024035, "Geodesic equation in Schwarzschild-(anti-)de Sitter space-times: Analytical solutions and applications" also looks interesting, if you can get a hold of it.

 

[lsimpson1943]

Thanks, Pervect. Maybe I am just not smart enough to understand what you are saying or I don't have enough physics education. I really appreciate your effort.

When I posed my question, I was not really concerned about the affect of an expanding Universe. Maybe I was not clear with what I was asking.

Let me try to clarify. Suppose my brain was wired to see a four axes of the fourth-dimension. Of course, I am not, but let's pretend. Assuming that the cosmos is not boundless and therefore possibly has some shape; then unless that shape is more than four dimensions, I would be able to see it providing I could somehow enter the fourth dimension and get outside this shape that is expanding.

Since Einstein, Minkowski, Poincare' and others, have shown time as an axis orthogonal to our three-dimensional space, I am supposing space-time is like a four dimensional object, with boundaries. Further, somehow our solar system is embedded in this this "shape". Let's further assume that this in not some shape flat on all it's sides (or outer surface) and there are potholes everywhere. If all this is true, then not only would the mass of large objects affect the terrain, but also the typology of the terrain itself.

In many ways however my explanation of getting "outside" space-time to view a shape does not make much sense to me, since time would follow me as I tried to get to a proper viewing perspective. That is, it seems to be impossible to get outside time. Perhaps I am reading too much into this. As I understand, the time axis of Minkowski diagrams is really a product of the speed of light and time as we know it. By convention, adjusting the units of measurement so that c works out to equal one, it leaves only t as defining the fourth axis. We in affect use this convention to change something that is temporal into something that appears measurable as spatial. Maybe that is where I am getting mixed up. Maybe this is just a math trick to show a synergy between time and space and has no physical value.

I may be hopelessly confused with all of this. I sure wish someone could sort it out for me. I don't think, somehow, the answer lies inside black holes though.

 

[Mentz114]

I assume you are making reference to the Riemann metric. Correct? As for the tensor, are we talking the Riemann curvature tensor or the Ricci tensor? If I understand the difference between the two, one is more "local" and the other averages the curvature for a defined part of a manifold.

I hope this doesn't sound stupid, but even if it does, could you further clear this up for me? These are the perils of being self-taught.

There is no 'the tensor'. The metric that defines the spacetime is a tensor. The Ricci and Riemann tensors are curvature tensors.

The relationship between the main tensors is - the Riemann tensor can be found from the metric tensor ( by a tedious calculation usually done by a program these days) and the Ricci is a contraction of the Riemann.

You should get an introductory book on GR. There are some free web-books available, including here -
www.lightandmatter.com

 

[lsimpson1943]

Thanks, Mentz114. I love these free books. I have downloaded them and will start trying to catch up with your knowledge. Ha. Ha. I appreciate your help.