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I vote for the Minkowski metric for the Universe in general, but I think that locally there exists other metrics, for example the Schwarzschild metric around black holes

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marcus

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that's interesting. Do you allow for the universe to expand?Originally posted by meteor

Minkowski coordinates are those of Special Relativity, are they not? I didn't know anyone supposed that the universe was non-expanding and had a global set of Minkowski coordinates. It would be the space of Special Relativity! A lot of things would be different!

Forgive me if I misunderstand what you mean by the Minkowski metric. Perhaps you could say explicitly in this one case?

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I want a metric that is flat.I know that Einstein-De Sitter metric is a flat metric, but it's not for a expanding Universe. I will look in Internet.

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Labguy

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Number three.

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marcus

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Hi meteor, good luck in the search! Astronomers cause everyone much confusion by using "flat" in two separate ways.Originally posted by meteor

I want a metric that is flat.I know that Einstein-De Sitter metric is a flat metric, but it's not for a expanding Universe. I will look in Internet.

The spacetime of cosmology has a distinguished space----that of an observer at rest relative to the expansion----relative to the CMB.

This SPACE can be flat without spacetime being flat.

Current observations of the CMB provide evidence of SPATIAL flatness. Spatial flatness has come to be generally accepted. There is a consensus that on the large scale space is flat or very very close to flat. Of course there is local curvature around stars and black holes etc.

Inflation scenario also predicts this spatial flatness, so the observation of spatial flatness is a lucky thing for the inflation scenario.

Observations indicate however that spacetime is not flat---because it expands---and therefore spacetime cannot have global Minkowski coordinates (that would be "flat-flat" really flat-out flat )

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marcus

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One of the mentors should explain-----e.g. Tom.Originally posted by Jack

But anyone else can also explain. That way all the pieces of the puzzle will show up. I will write a PM to Tom alerting hm of yr question but I will also try to contribut a bit of the anser.

In GR it must be possible to describe 4-dim geometry

How can you describe the geometry inhabiting a 4D manifold or 4D set of points or 4D "space" as one says.

A metric is a machine for finding the length of paths in the manifold from one point to another.

You can have several paths paths from A to B and the metric alows you to go in baby steps along of each path and add up teensy segments and in the end find the length of each path.

the machinery is very clunky and annoying but after using it enough one hardly knows how to get along without it.

a metric gives you also the idea of paths which are the shortest ones between points they connect-----like the great circle routes planes fly----'geodesics'

a metric can give a notion of "parallel transport" of tangent vectors from one point to another. this is an elegant and fertile idea.

See this animated film:

http://mathworld.wolfram.com/HolonomyGroup.html

See this still picture of the same thing (ball with tangent vectors)

http://www.math.ucr.edu/home/baez/einstein/node2.html

Remember, above all, that intuitive idea of parallel transport of a tangent vector from one point to another depends on which path.

If you take a tangent vector and run with it around a loop, it will come back different, depending on the loop. As long as there is curvature. On flat things it doesnt happen. So loops sense curvature.

This is called, disgustingly enough, "Holonomy". It is an interesting fact about manifolds and the metrics on them.

Ultimately it is why LOOPS can explore the metric on a manifold for you and tell you all about it.

Passage around any loop defines a transformation of the tangent space.

that is enough for a start about metrics

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I want to correct myself. The Einstein-de Sitter metric is the metric of an expanding universeI want a metric that is flat.I know that Einstein-De Sitter metric is a flat metric, but it's not for a expanding Universe. I will look in Internet.

Yeah, I remember that the results of WMAP indicated that the Universe was flat, but now I'm not very sure in which of the two kind of spaces that you are mentioning they referred. I searched a flat metric for that reason.You know, a metric that makes that the universe,or has a boundary, or it's infinite and unlimited.If I can't take the Minkowski metric I thin I will get the Einstein-de Sitter metricHi meteor, good luck in the search! Astronomers cause everyone much confusion by using "flat" in two separate ways.

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marcus

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This is, I believe, an excellent choice!Originally posted by meteor

I want to correct myself. The Einstein-de Sitter metric is the metric of an expanding universe

Yeah, I remember that the results of WMAP indicated that the Universe was flat, but now I'm not very sure in which of the two kind of spaces that you are mentioning they referred. I searched a flat metric for that reason.You know, a metric that makes that the universe,or has a boundary, or it's infinite and unlimited.If I can't take the Minkowski metric I thin I will get the Einstein-de Sitter metric

This will provide a good approximation of what you probably have

in mind----an expanding universe that does not eventually re-collapse on itself.

As may have occurred to you, these things are idealized models and the real universe cannot perfectly match any model.

Assuming GR is right then the actual geometry of the universe depends on the distribution of energy in it (matter, light, possibly dark energy, etc.) The geometry is not a given, but is a dynamic outcome.

The real universe is likely, in my opinion, to be very similar to the Einstein-deSitter pattern. But it has the little detail of accelerated expansion. So reality does not exactly match the EdS model!

Nothing is ever an entirely perfect fit. So there is always something left to do---an additional detail to consider. However you have made a good start.

Personally I decided to vote [?] for the Robertson-Walker because it seemed to me to have a little more flexibility. One could include a non-zero cosmological constant---dark energy, accelerated expansion. The EdS model has zero cosmological constant, in the discussions I have seen. Not as adjustable to whatever observations come in.

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to think of a "Metric of the Universe". More and more our picture of the Universe is that space is dynamic as well as matter, and a metric is by nature a static property.

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I don't think we have found the correct one yet.

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I agree.

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