# What is the relationship between expansion and curvature?

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• RogerWaters
In summary: This is known as an open universe."2. But why should a universe that expands forever be negatively curved like a saddle? Why can't a flat universe continually expand?In summary, the expansion and curvature of the universe are linked by the amount and distribution of matter within it. More mass leads to more curvature and potentially slower expansion, while less mass can result in a flatter universe that expands forever. This relationship is described by general relativity and is influenced by the presence of dark energy. However, it is possible for a flat universe to continually expand, depending on its density and the amount of matter present.
RogerWaters
TL;DR Summary
Untangling the effects of mass on the expansion and the curvature of the universe

Premise 1: Whether or not expansion is slowing down or speeding up depends on a battle between two phenomena: the attractive gravitational pull of matter and the repulsive gravitational push of dark energy. What counts in this contest is the density of each. (Also, the density of matter decreases as the universe expands because the volume of space increases). Basically: More mass, less/slower expansion.

Premise 2: The spatial curvature of the universe is related to general relativity, which describes how spacetime is curved and bent by mass and energy. If all forms of dark energy are ignored, then the curvature of the universe can be determined by measuring the average density of matter within it, assuming that all matter is evenly distributed. The distribution of mass in the universe determines the curvature of space in Einstein's theory. Basically: More mass (per unit of space), more curvature.

To my mind this means that the expansion and the shape of the universe are intrinsically linked? On first thought, this seemed strange to me as surely the universe could be expanding regardless of whether it is positive curved, negatively curved, or flat.

Question time:

"If the actual density of the universe is less than the critical density, then there is not enough matter to stop the expansion of the universe, and it will expand forever. The resulting shape is curved like the surface of a saddle. This is known as an open universe."

2. But why should a universe that expands forever be negatively curved like a saddle? Why can't a flat universe continually expand?

Thanks.

Last edited:
RogerWaters said:
But why should a universe that expands forever be negatively curved like a saddle? Why can't a flat universe continually expand?
It can.

This image shows the Lambda-CDM model. As you can see, the dividing lines for open/flat and expand forever/recollapse are very different. The statement is particular for a Universe containing only matter.

Edit: Note that the observational data (implications are shown as coloured regions) suggest that we are firmly inside the expand forever regime while still being compatible with a flat universe.

RogerWaters
RogerWaters said:
Physics novice here reading pop sci cosmology.
You should not be trying to learn actual science from pop science sources.

RogerWaters said:

That article is a good example of why we are not keen on popular science sources. It may be unfair to say so, but it appears the author is writing about something that she herself does not fully understand.

The Wikipedia pages are considerably better, although still not considered a valid source for discussion here.

https://en.wikipedia.org/wiki/Expansion_of_the_universe

https://en.wikipedia.org/wiki/Ultimate_fate_of_the_universe

Orodruin said:
It can.
View attachment 293296
This image shows the Lambda-CDM model. As you can see, the dividing lines for open/flat and expand forever/recollapse are very different. The statement is particular for a Universe containing only matter.

Edit: Note that the observational data (implications are shown as coloured regions) suggest that we are firmly inside the expand forever regime while still being compatible with a flat universe.
Thanks for this - what's the source out of interest?

RogerWaters
RogerWaters said:
Summary:: Untangling the effects of mass on the expansion and the curvature of the universe

Premise 1: Whether or not expansion is slowing down or speeding up depends on a battle between two phenomena: the attractive gravitational pull of matter and the repulsive gravitational push of dark energy. What counts in this contest is the density of each. (Also, the density of matter decreases as the universe expands because the volume of space increases). Basically: More mass, less/slower expansion.

Premise 2: The spatial curvature of the universe is related to general relativity, which describes how spacetime is curved and bent by mass and energy. If all forms of dark energy are ignored, then the curvature of the universe can be determined by measuring the average density of matter within it, assuming that all matter is evenly distributed. The distribution of mass in the universe determines the curvature of space in Einstein's theory. Basically: More mass (per unit of space), more curvature.

To my mind this means that the expansion and the shape of the universe are intrinsically linked? On first thought, this seemed strange to me as surely the universe could be expanding regardless of whether it is positive curved, negatively curved, or flat.

Question time:

"If the actual density of the universe is less than the critical density, then there is not enough matter to stop the expansion of the universe, and it will expand forever. The resulting shape is curved like the surface of a saddle. This is known as an open universe."

2. But why should a universe that expands forever be negatively curved like a saddle? Why can't a flat universe continually expand?

Thanks.
The expansion is a manifestation of space-time curvature.

If you calculate the Ricci curvature scalar for a uniform, expanding universe, you get a result which is just a function of the expansion and the spatial curvature:

$$R = 6 \left( \dot{H} + 2 H^2 + {k \over a^2}\right)$$

So you have terms for the rate of expansion ##H##, its time derivative ##\dot{H}##, and the spatial curvature ##k##. You can literally understand the expansion as being space-time curvature.

Edit: I'm really not sure why the forum isn't picking up the LaTeX in this post, but hopefully it's still legible.

kimbyd said:
I'm really not sure why the forum isn't picking up the LaTeX in this post
It looks OK to me. You might need to reload the page and/or log out of PF and log back in again, and/or clear cookies.

kimbyd said:
Edit: I'm really not sure why the forum isn't picking up the LaTeX in this post, but hopefully it's still legible.
It's a known problem. If you are the first to use Latex in a thread (except in the OP), then it does not render until you refresh the page.

hmmm27
kimbyd said:
The expansion is a manifestation of space-time curvature.

If you calculate the Ricci curvature scalar for a uniform, expanding universe, you get a result which is just a function of the expansion and the spatial curvature:

$$R = 6 \left( \dot{H} + 2 H^2 + {k \over a^2}\right)$$

So you have terms for the rate of expansion ##H##, its time derivative ##\dot{H}##, and the spatial curvature ##k##. You can literally understand the expansion as being space-time curvature.

Edit: I'm really not sure why the forum isn't picking up the LaTeX in this post, but hopefully it's still legible.
Well, there is the well known exception of the limit of a universe without mass, which still has expansion but no curvature (Milne universe). It is just Minkowski spacetime in funny coordinates. However, it has all the major features of realistic cosmologies - arbitrarily large superluminal recession rates, cosmological redshift, etc.

Noting that H(t) and a(t) are related (##H(t)=a'(t)/a(t)##) , if a(t) is taken to be simply t, then H(t) is required to be 1/t, and taking k=-1, R vanishes. k=-1 implies the spatial slices are hyperbolic.

PeroK said:
It's a known problem. If you are the first to use Latex in a thread (except in the OP), then it does not render until you refresh the page.
Ahhh, I did not know this! Thanks!

I think that, at the end, it is absolutely necessary to understand what "space" is or what "space-time" is, and we need to stop the abuse of terminology like "matter curve spacetime" or "space is expanding" or "spacetime is expanding" or things like that.

pabloweigandt said:
I think that, at the end, it is absolutely necessary to understand what "space" is or what "space-time" is, and we need to stop the abuse of terminology like "matter curve spacetime" or "space is expanding" or "spacetime is expanding" or things like that.

pabloweigandt said:
I think that, at the end, it is absolutely necessary to understand what "space" is or what "space-time" is, and we need to stop the abuse of terminology like "matter curve spacetime" or "space is expanding" or "spacetime is expanding" or things like that.
"Matter curves spacetime" doesn't seem to me to be a terrible abuse. Strictly it's the stress-energy associated with the matter (and, indeed, other things like radiation) that is the source term for curvature, but I don't really see the issue. "Space is expanding" is kind of an abuse, yes, in that it's more like that later surfaces of constant cosmological time have increased scale factors compared to earlier ones, but that's a lot harder to say. And "spacetime is expanding" isn't an abuse of terminology, it's just wrong.

"Space is expanding" will mean that I know what space is. And that there is no difference between two galaxies that are getting apart at superluminal speeds (if it is true that I can use the term "speed" or "velocity" in this case) and two cars moving apart each other.

pabloweigandt said:
"Space is expanding" will mean that I know what space is. And that there is no difference between two galaxies that are getting apart at superluminal speeds (if it is true that I can use the term "speed" or "velocity" in this case) and two cars moving apart each other.
In any case, this is what's called "hijacking" a thread.

RogerWaters
Ok. I am new to this. I found the definition of "hijacking". You are right and I am so sorry. I would be grateful If anyone can tell me where to post this kind of topics.

PeroK
Go to the front page (physicsforums.com), scroll down until you find the Cosmology forum, click on that, and then click "Post thread". Do make sure you've read the posting guidelines.

pabloweigandt said:
Ok. I am new to this. I found the definition of "hijacking". You are right and I am so sorry. I would be grateful If anyone can tell me where to post this kind of topics.
This is a "B" level thread, which suggests that the OP (original poster) is not a university student. The answers to his question were at a level that hopefully he could understand.

The thread is nearly a month old and the OP has not responded in some time.

pabloweigandt said:
"Space is expanding" will mean that I know what space is.
And in the case where that term is used, FRW spacetimes, there is a definition of "space" that is picked out by the symmetries of the spacetime (namely, the spacelike surfaces that are orthogonal to the comoving worldlines).

pabloweigandt said:
And that there is no difference between two galaxies that are getting apart at superluminal speeds (if it is true that I can use the term "speed" or "velocity" in this case) and two cars moving apart each other.
No. There is a difference. In a curved spacetime, there is no such thing as a well-defined "relative speed" for objects that are separated by significant distances. That issue doesn't arise for two cars near each other on Earth, but it does for galaxies distant from each other in the universe. It is not true that you can use the term "speed" or "velocity" in the latter case with their usual meanings. Nor do you need to to give a well-defined meaning to "space is expanding".

pabloweigandt said:
I would be grateful If anyone can tell me where to post this kind of topics.
Before starting a new thread, I would suggest taking some time to learn what cosmologists actually mean by "space is expanding" as a description of our universe. If you have questions about what you read, as @Ibix has said, you can start a new thread in the Cosmology forum (the forum this thread is in). His advice about reading the posting guidelines first is also good.

PeroK said:
The thread is nearly a month old and the OP has not responded in some time.
And that being the case, this thread is now closed.

## 1. What is the definition of expansion in relation to curvature?

The expansion of the universe refers to the increase in the distance between galaxies over time. This is due to the fact that the universe is continuously expanding, causing galaxies to move away from each other.

## 2. How does expansion affect the curvature of the universe?

The expansion of the universe does not directly affect the curvature of the universe. However, the curvature of the universe can affect the rate of expansion. In a positively curved universe, the expansion will eventually slow down and stop, while in a negatively curved universe, the expansion will continue to accelerate.

## 3. What is the relationship between the rate of expansion and the curvature of the universe?

The rate of expansion is closely linked to the curvature of the universe. In a flat universe, the expansion rate will remain constant over time. In a positively curved universe, the expansion rate will decrease over time, and in a negatively curved universe, the expansion rate will increase over time.

## 4. Can the curvature of the universe change over time?

Yes, the curvature of the universe can change over time. In the early stages of the universe, the curvature was likely very different from what it is today. As the universe continues to expand, the curvature may also change.

## 5. How do scientists measure the curvature of the universe?

Scientists use various methods to measure the curvature of the universe, including studying the cosmic microwave background radiation, observing the distribution of galaxies, and measuring the shape of the universe through gravitational lensing effects. These methods provide evidence for a flat universe with a very small curvature.

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