Geometry of the Universe: Euclidean or Hyperbolic?

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In summary, the conversation discusses the concept of the universe being Euclidean and how this relates to the theory of gravity and black holes. It is mentioned that the universe is believed to be flat within our current ability to measure it, but there is also the possibility of it being slightly open or closed. The idea of infinite fine-tuning is also mentioned, as well as the role of inflation in driving the total density fraction close to one. The consensus is that it would be an amazing coincidence for the universe to be exactly flat, and it is suggested that pop-science journals may have perpetuated the idea that it is likely very close to one instead.
  • #1
neginf
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I've read that the universe is Euclidean and have also read that space is bent by gravity. Descriptions of geometry near black holes almost sounds like hyperbolic geometry.

1. Is this so?
2. If it is, does it mean we're in a universe that is Euclidean overall but has non Euclidean regions ?
 
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  • #2
neginf said:
I've read that the universe is Euclidean and have also read that space is bent by gravity. Descriptions of geometry near black holes almost sounds like hyperbolic geometry.

1. Is this so?
2. If it is, does it mean we're in a universe that is Euclidean overall but has non Euclidean regions ?

I think that's exactly right, although I believe that it is NOT absolutely known that the U is perfectly flat, just that it is flat to within our current ability to measure it and assumed to be very likely perfectly flat.
 
  • #3
A spherically symmetric, static black hole is, well, just that. If you look at the schwarzchild metric, the [itex]r^{2}d\Omega ^{2}[/itex] term indicates that the geometry outside the black hole is spherically symmetric as well. Where did you read that it was hyperbolic by the way? If it was hyperbolic then it wouldn't have [itex]\frac{\partial }{\partial \phi }[/itex] as a killing field. Even for a kerr black hole, which isn't spherically symmetric, the geometry outside the black hole isn't hyperbolic.
 
  • #4
I didn't read that it was hyperbolic, just misinterpreted it. Although I can't remember where, I read that if something was close enough to a black hole to fall in, from a certain distance away, that thing would appear never to fall all the way, just fall ever more slowly.

That reminded me of something I learned a long time ago about the upper half plane the hyperbolic metric. If I remember right, a disc moving towards the real line would just decrease in size and never quite get there.
 
  • #5
While it is true that if you start on the y axis, then any other point on the upper half plane would have an infinite distance from the original point for that metric, the story of observes getting closer and closer to the EH but never actually reaching it is another story. While it does seem to take infinite coordinate distance to get to the EH for the schwarzchild black hole, remember that coordinate related quantities are secondary to geometric quantities. If you compute the proper distance, instead of the coordinate distance, you will see that it is finite. Therefore, in your frame you will fall past the EH.
 
  • #6
phinds said:
I think that's exactly right, although I believe that it is NOT absolutely known that the U is perfectly flat, just that it is flat to within our current ability to measure it and assumed to be very likely perfectly flat.
I don't think that's generally assumed. I'm pretty sure most cosmologists assume the opposite: that it is very likely to be either slightly open or slightly closed, with closed being preferred. Though in many models it is very likely so incredibly flat that we'd never be able to measure the overall curvature.
 
  • #7
Chalnoth said:
I don't think that's generally assumed. I'm pretty sure most cosmologists assume the opposite: that it is very likely to be either slightly open or slightly closed, with closed being preferred. Though in many models it is very likely so incredibly flat that we'd never be able to measure the overall curvature.

Why would "slightly not flat" be assumed when the theoretical motivation would seem to be for [itex]\Omega_\textrm{tot} = 1 [/itex]? Is this assumption of slightly closed because inflation is thought to have produced something very very close to flat but not necessary exactly so?
 
  • #8
cepheid said:
Why would "slightly not flat" be assumed when the theoretical motivation would seem to be for [itex]\Omega_\textrm{tot} = 1 [/itex]? Is this assumption of slightly closed because inflation is thought to have produced something very very close to flat but not necessary exactly so?
Yes to the second question. But there's also the point that this is a geometrical factor that would require infinite fine-tuning to be made identically flat.
 
  • #9
Chalnoth said:
Yes to the second question. But there's also the point that this is a geometrical factor that would require infinite fine-tuning to be made identically flat.

Interesting point. Thanks.

I believe I had read, and was following this in my logic (apparently incorrectly, from what you are saying, which I had not thought about), that the consensus was that it would be an amazing coincidence for it to be VERY close to 1 but not actually 1, out of all the possible values.
 
  • #10
phinds said:
Interesting point. Thanks.

I believe I had read, and was following this in my logic (apparently incorrectly, from what you are saying, which I had not thought about), that the consensus was that it would be an amazing coincidence for it to be VERY close to 1 but not actually 1, out of all the possible values.
I've never heard that argument. Especially since we do have a mechanism to drive the total density fraction exponentially-close to one.
 
  • #11
phinds said:
I believe I had read, and was following this in my logic (apparently incorrectly, from what you are saying, which I had not thought about), that the consensus was that it would be an amazing coincidence for it to be VERY close to 1 but not actually 1, out of all the possible values.

You might have read it in pop-science journals, I recall having read something like that there.
The real fact is rather the opposite, it would be an amazing coincidence for it to be exactly 1 instead of the infinity of values close to 1 either above or below.
Besides the only spatial geometry that can never be empirically proved is the flat limit case as it could always be suspected that the fact one measures a flat space is due to the lack of precision of the measuring tools to measure a very tiny curvature so that it goes undetected. That is not the case with the open and closed universes that could actually be shown to be the real ones thru experiment.
 

FAQ: Geometry of the Universe: Euclidean or Hyperbolic?

What is the difference between Euclidean and Hyperbolic geometry?

Euclidean geometry is the study of flat or two-dimensional surfaces, where parallel lines remain at a constant distance from each other. Hyperbolic geometry, on the other hand, is the study of curved or non-flat surfaces, where parallel lines eventually intersect.

How do we determine which geometry the universe follows?

Scientists use various methods, such as measuring the angles of triangles formed by distant objects and analyzing the curvature of space, to determine the geometry of the universe. Currently, the general consensus is that the universe follows a Euclidean geometry on a large scale.

What implications does the geometry of the universe have on our understanding of space and time?

The geometry of the universe has significant implications for our understanding of space and time. For example, in a hyperbolic geometry, the angles of a triangle would add up to less than 180 degrees, which challenges the traditional understanding of Euclidean geometry. It also affects concepts such as parallel lines and the speed of light in different directions.

How does the expansion of the universe relate to its geometry?

The expansion of the universe is closely related to its geometry. In a Euclidean geometry, the expansion would continue at a constant rate, while in a hyperbolic geometry, the expansion would eventually slow down and stop. The current observations and measurements of the universe's expansion support a Euclidean geometry.

Could the geometry of the universe change over time?

While it is possible for the geometry of the universe to change over time, it is not currently supported by scientific evidence. The geometry of the universe is believed to be a fundamental property that remains constant, much like the laws of physics. However, there are theories, such as the inflationary universe model, that suggest the possibility of a temporary change in the geometry of the universe in its early stages.

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