Shape of the universe

[popeye85]

Right I need help with a question regarding the shape of the universe.
I get that if there is enough mass in the universe then gravity will pull the universe back on itself turning into a 'ball' shape thus making space and time finite. But, according to hawking, if there isn't enough mass then it will bend in the opposite direction into a saddle shape making s+t infinite, or if the amount of mass is somewhere in the middle then the universe will be flat and again infinite. In struggling to understand why space would bend in a different Dirrection depending on mass. Anyone able to help?

[Juicy6]

https://phys.org/news/2017-06-universe-flat-topology.html

[vlaiv]

On 3/4/2018 at 10:16, popeye85 said:

Right I need help with a question regarding the shape of the universe.
I get that if there is enough mass in the universe then gravity will pull the universe back on itself turning into a 'ball' shape thus making space and time finite. But, according to hawking, if there isn't enough mass then it will bend in the opposite direction into a saddle shape making s+t infinite, or if the amount of mass is somewhere in the middle then the universe will be flat and again infinite. In struggling to understand why space would bend in a different Dirrection depending on mass. Anyone able to help?

You might want to watch first couple of lectures on Cosmology by Leonard Susskind on youtube. We will derive FLRW model from Hubble's law and Newtonian physics (so no need of deep knowledge of GR) and will explain meaning of certain constants and their relation to observed data.

On the other hand, here is gross simplification of the things: Shape of the universe is governed by amount of energy and amount of matter (actually density of those two). Imagine just two things existing: Earth and a ball (or two balls) and nothing else. If you throw a ball of the surface of the earth 3 things can happen: it can fly a bit "in the air" and fall down back to earth due to gravity if initial velocity is small (low energy in system), or it can reach just enough speed and height to remain in orbit (equal amounts of energy and mass), or it can shoot off into infinity (high energy in system).

These three examples are related to what will happen to universe based on ratio of matter to energy density. Add to that the fact that space is bent by matter and energy in it and you can see how different shapes can arise depending on what type of energy and matter are present in it (dark energy being particularly interesting as its density remains the same with expanding space).

And on actual topic of thread - for some reason I believe that universe is flat and finite (hypertorus)

 

[ollypenrice]

I'm not sure that the geometry of the universe and the shape of the universe should be considered interchangeable phrases. The universe does not have an edge, in modern thinking, so I don't  see that it can have a shape. My amateur understanding of the geometry of the universe is that it describes the spacetime relationships between points: the gravitional interactions between separated bodies will behave and evolve according to that geometry. Sphere, saddle and flat, along with the familiar diagrams which depict them in cosmology books, are attempts to depict the possible geometries in a conceptual way but do not depict the shape of the universe.

Being all that there is, the universe doesn't have a shape. Imagine a Henry Moore sculpture. Now cause everything outside it to vanish - Air, space itself, the lot. Does the scultpure still have a shape? Personally I suspect it doesn't.  But who knows?

Olly

[vlaiv]

3 minutes ago, ollypenrice said:

I'm not sure that the geometry of the universe and the shape of the universe should be considered interchangeable phrases. The universe does not have an edge, in modern thinking, so I don't  see that it can have a shape. My amateur understanding of the geometry of the universe is that it describes the spacetime relationships between points: the gravitional interactions between separated bodies will behave and evolve according to that geometry. Sphere, saddle and flat, along with the familiar diagrams which depict them in cosmology books, are attempts to depict the possible geometries in a conceptual way but do not depict the shape of the universe.

Being all that there is, the universe doesn't have a shape. Imagine a Henry Moore sculpture. Now cause everything outside it to vanish - Air, space itself, the lot. Does the scultpure still have a shape? Personally I suspect it doesn't.  But who knows?

Olly

Both sphere (spherical surface) and torus (toroidal surface) don't have edge but they have distinct shape. Mobius strip is for example interesting entity - it is 3d object (well 2d geometry embodied in 3d space) but has only one edge and one side - and it certainly has shape to it.

Sphere, saddle and flat one are good representatives of properties that would point to general shape of universe in sense that geometry can have positive, negative and 0 curvature. With positive curvature triangles can have sum of their angles greater than 180 degrees. In flat topology - triangle will always have sum angles equal to 180. In negative curvature one can have triangle with sum of its angles less than 180 degrees.

[drjolo]

14 minutes ago, vlaiv said:

Both sphere (spherical surface) and torus (toroidal surface) don't have edge but they have distinct shape. Mobius strip is for example interesting entity - it is 3d object (well 2d geometry embodied in 3d space) but has only one edge and one side - and it certainly has shape to it.

They do have shape once you add at least one more dimension to describe it  When you are flatman crawling on the sphere, you are not really aware of it. You can assume there are more dimensions that can better explain the physics around, however it is hard to proof. And also hard to stop, because more dimensions can maybe explain more things. It is like multiplying epicycles many years ago to describe celestial movements

[vlaiv]

1 minute ago, drjolo said:

They do have shape once you add at least one more dimension to describe it  When you are flatman crawling on the sphere, you are not really aware of it. You can assume there are more dimensions that can better explain the physics around, however it is hard to proof. And also hard to stop, because more dimensions can maybe explain more things. It is like multiplying epicycles many years ago to describe celestial movements

Actually you don't need another dimension to be able to describe lower dimensional space - metric does that (mathematically).

[ollypenrice]

4 hours ago, vlaiv said:

Both sphere (spherical surface) and torus (toroidal surface) don't have edge but they have distinct shape. Mobius strip is for example interesting entity - it is 3d object (well 2d geometry embodied in 3d space) but has only one edge and one side - and it certainly has shape to it.

Sphere, saddle and flat one are good representatives of properties that would point to general shape of universe in sense that geometry can have positive, negative and 0 curvature. With positive curvature triangles can have sum of their angles greater than 180 degrees. In flat topology - triangle will always have sum angles equal to 180. In negative curvature one can have triangle with sum of its angles less than 180 degrees.

A sphere doesn't have an edge? Surely it does! It is the bit the footballer's boot hits when he kicks the sphere. Without an edge his foot would go straight through it. The surface of a sphere does not have an edge, which is the analogy widely used in cosmology conversations, but that analogy does not describe a real sphere. It specifically removes from the discussion one key characteristic of the sphere which can be defined in various ways, one of which would be 'the space around it.' Or you could say that in the analogy a dimension had been removed.

When we look at a sphere we do so from the outside and we see a round thing. We cannot look at the universe from the outside because there is no outside. That means that there is no round thing. But the geometric characteristics of a round thing's surface can still  define the spacetime relationships between two points in the environment in question.

This argument does not come with a guarantee!!!

Olly

[SteveA]

28 minutes ago, ollypenrice said:

This argument does not come with a guarantee!!!

Maybe not...but actually that is very well argued and seems to fit perfectly with my understanding...

Steve

[vlaiv]

1 hour ago, ollypenrice said:

A sphere doesn't have an edge? Surely it does! It is the bit the footballer's boot hits when he kicks the sphere. Without an edge his foot would go straight through it. The surface of a sphere does not have an edge, which is the analogy widely used in cosmology conversations, but that analogy does not describe a real sphere. It specifically removes from the discussion one key characteristic of the sphere which can be defined in various ways, one of which would be 'the space around it.' Or you could say that in the analogy a dimension had been removed.

When we look at a sphere we do so from the outside and we see a round thing. We cannot look at the universe from the outside because there is no outside. That means that there is no round thing. But the geometric characteristics of a round thing's surface can still  define the spacetime relationships between two points in the environment in question.

This argument does not come with a guarantee!!!

Olly

Just because one can define some aspects of shape in higher dimensions does not mean that those aspects / properties are not intrinsic to the shape whether or not you look at it from higher dimension.

Take sphere (spherical 2d surface) for example. When it is embedded in 3d we know fairly simple definition of the sphere: take a point in space (3d), then all the points that are exactly r distance from that chosen point will belong to a set of points. Such set we will call sphere. There is exactly one shape that corresponds to this definition.

Now consider 2d case. We can define metric what will describe exactly the same set of points without ever needing third dimension. We can do it by "distances" and "angles" (depending on metric, but for example - 3 equidistant points will form a triangle that has angles which depend on length of that triangle sides, and same relationship would hold for any 3 points). If we can find 1 to 1 mapping between set members of sphere in 3d and this structure that is defined by metric in 2d - then there is no reason what so ever not to assign them the same shape.

Shape in mathematical terms means what sort of relationships exist between members of a given set. So it can be described in different ways. And if two sets have 1-to-1 mapping between members and all relationships between each set's members have 1-to-1 mapping - those sets have same shape.

So based on metric alone we can be certain that set of points described by it can be represented in higher dimension (although we can visualize it) by appropriate surface / shape.

 

[drjolo]

I think it is very convenient from theoretical point of view - once the measurements do not agree with theory, lets add curvature to bend our space to another dimension. Do you really think that number of dimensions scales so well? We are aware of three of them, and since three is a number like any other number, we can have four or dozen? Are these new dimensions the same as three we know? I know that scientists tend to develop new theories as soon as old one met its limits (as most of theories), but oversimplifications can be really misleading. Although they work well as PR tool. 

[ollypenrice]

9 hours ago, vlaiv said:

Just because one can define some aspects of shape in higher dimensions does not mean that those aspects / properties are not intrinsic to the shape whether or not you look at it from higher dimension.

Take sphere (spherical 2d surface) for example. When it is embedded in 3d we know fairly simple definition of the sphere: take a point in space (3d), then all the points that are exactly r distance from that chosen point will belong to a set of points. Such set we will call sphere. There is exactly one shape that corresponds to this definition.

Now consider 2d case. We can define metric what will describe exactly the same set of points without ever needing third dimension. We can do it by "distances" and "angles" (depending on metric, but for example - 3 equidistant points will form a triangle that has angles which depend on length of that triangle sides, and same relationship would hold for any 3 points). If we can find 1 to 1 mapping between set members of sphere in 3d and this structure that is defined by metric in 2d - then there is no reason what so ever not to assign them the same shape.

Shape in mathematical terms means what sort of relationships exist between members of a given set. So it can be described in different ways. And if two sets have 1-to-1 mapping between members and all relationships between each set's members have 1-to-1 mapping - those sets have same shape.

So based on metric alone we can be certain that set of points described by it can be represented in higher dimension (although we can visualize it) by appropriate surface / shape.

 

While 'shape' can be defined mathematically (and you clearly know more about this than I do!) I think the OP was interested in trying to conceive of the shape in the way that we can conceive of shapes which are not 'all that there is' and I don't think this is possible.

Olly

[vlaiv]

1 hour ago, drjolo said:

I think it is very convenient from theoretical point of view - once the measurements do not agree with theory, lets add curvature to bend our space to another dimension. Do you really think that number of dimensions scales so well? We are aware of three of them, and since three is a number like any other number, we can have four or dozen? Are these new dimensions the same as three we know? I know that scientists tend to develop new theories as soon as old one met its limits (as most of theories), but oversimplifications can be really misleading. Although they work well as PR tool. 

We really don't need higher dimensions to be able to describe our 3d world as bent and twisted. If we set aside string theory and deal with classic cosmology and GR, we often use 2d model in 3d space to point out what distortion of 2d world would look like because we are able to visualize in 3d, this however does not mean that we in fact need 4d spatial universe to be able to describe curvature of our 3d universe.

33 minutes ago, ollypenrice said:

While 'shape' can be defined mathematically (and you clearly know more about this than I do!) I think the OP was interested in trying to conceive of the shape in the way that we can conceive of shapes which are not 'all that there is' and I don't think this is possible.

Olly

Not sure if OP was referring to that. I believe it was the question of trying to understand why would space bend this or that way (positive or negative curvature) depending on mass/energy density contained in it.

Again if our space was positively or negatively curved we would be able to see that without need for "space outside of our space", or "ability to visualize in 4d".

Simplest way that it would manifest would be:

In flat space we are used to perspective. So object of the same size twice away will appear to be twice as small. This is not the case in curved space. In positive curved space - galaxy that is twice more distant compared to another galaxy would not appear to be two times smaller - it would be bigger than that. In negatively curved space galaxy that is two times more distant would appear smaller than it "should normally" (meaning in flat space).

That is exactly what this image is depicting:

So features of certain size (and we know from QM what sort of fluctuations we might expect in CMB) would appear smaller in negative curvature, would appear as they should (as we expect them to appear due to perspective) in flat universe, and would appear larger than expected in positive curvature.

So far we have not been able to measure features to be bigger or smaller than expected in flat geometry to the 1% of precision. So it is most likely that we live in flat universe.

[ollypenrice]

1 minute ago, vlaiv said:

We really don't need higher dimensions to be able to describe our 3d world as bent and twisted. If we set aside string theory and deal with classic cosmology and GR, we often use 2d model in 3d space to point out what distortion of 2d world would look like because we are able to visualize in 3d, this however does not mean that we in fact need 4d spatial universe to be able to describe curvature of our 3d universe.

Not sure if OP was referring to that. I believe it was the question of trying to understand why would space bend this or that way (positive or negative curvature) depending on mass/energy density contained in it.

Again if our space was positively or negatively curved we would be able to see that without need for "space outside of our space", or "ability to visualize in 4d".

Simplest way that it would manifest would be:

In flat space we are used to perspective. So object of the same size twice away will appear to be twice as small. This is not the case in curved space. In positive curved space - galaxy that is twice more distant compared to another galaxy would not appear to be two times smaller - it would be bigger than that. In negatively curved space galaxy that is two times more distant would appear smaller than it "should normally" (meaning in flat space).

That is exactly what this image is depicting:

So features of certain size (and we know from QM what sort of fluctuations we might expect in CMB) would appear smaller in negative curvature, would appear as they should (as we expect them to appear due to perspective) in flat universe, and would appear larger than expected in positive curvature.

So far we have not been able to measure features to be bigger or smaller than expected in flat geometry to the 1% of precision. So it is most likely that we live in flat universe.

Interesting point.

Olly

[drjolo]

21 minutes ago, vlaiv said:

(...) So far we have not been able to measure features to be bigger or smaller than expected in flat geometry to the 1% of precision. So it is most likely that we live in flat universe.

Thanks for that - it is really more convenient for me not to assume we have four or more dimensions, that behave in the same way the three we know Can you point me to some sources on how these features are measured? I wonder how this comparison is being made, how do we know that object we measure are in the specific distance, and we are not measuring for example distorted images (like for gravitational lensing). 

[andrew s]

11 minutes ago, drjolo said:

Can you point me to some sources on how these features are measured?

Not sure this directly answers your question but this series is a good intro to the CMB and what we know of the Universe's geometry https://www.physicsforums.com/insights/poor-mans-cmb-primer-part-0-orientation/

Much of the discussion and disagreements above seems to center on the difference between extrinsic and intrinsic properties and geometry in particular. You can only kick a football extrinsically but not intrinsically!

Regards Andrew

[drjolo]

Thanks!

39 minutes ago, andrew s said:

Much of the discussion and disagreements above seems to center on the difference between extrinsic and intrinsic properties and geometry in particular. You can only kick a football extrinsically but not intrinsically!

This is very true

[vlaiv]

47 minutes ago, drjolo said:

Thanks for that - it is really more convenient for me not to assume we have four or more dimensions, that behave in the same way the three we know Can you point me to some sources on how these features are measured? I wonder how this comparison is being made, how do we know that object we measure are in the specific distance, and we are not measuring for example distorted images (like for gravitational lensing). 

That would be "above my paygrade"

But I can point you to "right" direction for a research on this topic. I found that wikipedia is decent starting point where you can read about basic concepts and terminology, and get basic idea of what is going on to help you in further research.

First lookup the cosmic distance ladder to see how distances are measured and what level of precision we currently have. Next thing that would be good is to get to know current cosmology models and what they imply.

After that look for different surveys that have been done. Usually surveys just gather data, and afterwards different teams of scientists come up with hypothesis to test, and use gathered data (that would for example be distances, angular sizes and apparent brightness of different galaxies) to fit to certain model. Based on how well they are able to fit the data to a specific model - model is confirmed (or rejected) to a certain degree (within a certain error margin).

Most of this stuff heavily relies on statistics and assuming different distributions (well probably normal/Gaussian distribution in most cases). It's not like you can measure things with definite precision, there are a lot of small details that need to be accounted for, and one needs to understand scope of impact of each of those things - like gravitational lensing you mention, or some other things like interstellar dust reddening and attenuation, or similar for intergalactic dust.

I know for example that CMB feature scale was done from quantum mechanical predictions calculated on supercomputers and compared to observational data.

Look at the following images:

This is sort of thing that you might expect when plotting survey data against a model - you are looking for a best fit rather than having clear cut results.

 

[goodricke1]

1 hour ago, andrew s said:

You can only kick a football extrinsically but not intrinsically!

 

1 hour ago, vlaiv said:

That would be "above my paygrade"

Which reminds me that footballers get paid a lot more than scientists