How can a big bang expand to an infinite size?

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5 hours ago, geordief said:

is the topology not a function of spacetime curvature? (perhaps the "curvature" description of the topology gave me the wrong idea)

What might cause different topologies to arise?

Take a 2D flat surface, like a flat sheet. Its intrinsic curvature is identically 0 everywhere. Roll it into a cylinder. You get a different topology, but it has the same, 0 everywhere, intrinsic curvature.

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1 hour ago, Airbrush said:

How can one assume that photons have any mass at all?  I thought photons were energy and that is why they travel the speed of light.  How could any mass travel light speed?

As an extremely small mass ( compared to the mass of the Sun ) makes no difference whatsoever in Newton's equation, we can make that 'assumption' to ease the calculation.
That doesn't mean light does not have the property of  mass; it is massless, otherwise it could not move at c .
And light is NOT energy, but it does have that property.

2 hours ago, MigL said:

and could not form a 'closed' topology like a hypersphere or a flat torus.

Ooops. My mistake.
As genady correctly points out above, a flat torus has no curvature, and even a 'regular' torus is only curved along one axis.

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Take a 2D flat surface, like a flat sheet. Its intrinsic curvature is identically 0 everywhere. Roll it into a cylinder. You get a different topology, but it has the same, 0 everywhere, intrinsic curvature.

How many different kinds  of topology are possible for the universe?

Would these topologies depend on more than the spacetime curvature?

Are they all expanding ,contacting ,steady state?

Any other ways to distinguish between different candidate topologies?

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2 hours ago, geordief said:

How many different kinds  of topology are possible for the universe?

I don't know.

2 hours ago, geordief said:

Would these topologies depend on more than the spacetime curvature?

Absolutely. Metric determines curvature. It does not determine topology.

2 hours ago, geordief said:

Are they all expanding ,contacting ,steady state?

Depends on other properties, e.g. homogeneity and isotropy.

2 hours ago, geordief said:

Any other ways to distinguish between different candidate topologies?

I don't know.

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I don't know.

Absolutely. Metric determines curvature. It does not determine topology.

Depends on other properties, e.g. homogeneity and isotropy.

I don't know.

The difficulty I have is of picturing  a topology for a universe that has no edge or boundary.

It seems others don't have that problem or perhaps they have enough of a mathematical understanding of the possibilities   that they don't need to have a mental picture

Would I be right to think that a topology  doesn't require any particular shape  but it just describes the way parts of the whole connect with each other?

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4 minutes ago, geordief said:

Would I be right to think that a topology  doesn't require any particular shape  but it just describes the way parts of the whole connect with each other?

I think so.

5 minutes ago, geordief said:

The difficulty I have is of picturing  a topology for a universe that has no edge or boundary.

For an edgeless but bounded space imagine a 2D version, a sphere.

For an edgeless and unbounded space - I don't see a problem. It's just like a number line but in 3D.

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I think so.

For an edgeless but bounded space imagine a 2D version, a sphere.

For an edgeless and unbounded space - I don't see a problem. It's just like a number line but in 3D.

Is this a space with no time component?

A space that is ready to be populated with objects?

I think I am more familiar with a space that is created by objects reconfiguring themselves(in an overall expansionist way as per observations)

I think that is the orthodox view even if I am not up to speed with it.

For an edgeless but bounded space imagine a 2D version, a sphere

It still seems difficult for me to imagine  the 3d universe existing on the 2d surface of the sphere

There doesn't seem to be room for the 3 dimensions.

Is it just an analogy?

And the universe is not hollowed out ,is it?)unless the "hollow" is somehow the past history-surely not that)

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56 minutes ago, geordief said:

Is this a space with no time component?

A space that is ready to be populated with objects?

I think I am more familiar with a space that is created by objects reconfiguring themselves(in an overall expansionist way as per observations)

I think that is the orthodox view even if I am not up to speed with it.

This does not affect geometry or topology.

57 minutes ago, geordief said:

It still seems difficult for me to imagine  the 3d universe existing on the 2d surface of the sphere

There doesn't seem to be room for the 3 dimensions.

Is it just an analogy?

And the universe is not hollowed out ,is it?)unless the "hollow" is somehow the past history-surely not that)

It is not an analogy. It is a model.

2D sphere is not hollow. 2D spherical surface in 3D space, is.

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This does not affect geometry or topology.

It is not an analogy. It is a model.

2D sphere is not hollow. 2D spherical surface in 3D space, is.

Can we populate the surface  of that 2d  model of a sphere  with 3d objects of an imaginary universe which is posited to be edgeless and bounded?

How would I see in the model that the objects were 3d if they were embedded in the 2d surface?

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10 minutes ago, geordief said:

Can we populate the surface  of that 2d  model of a sphere  with 3d objects of an imaginary universe which is posited to be edgeless and bounded?

How would I see in the model that the objects were 3d if they were embedded in the 2d surface?

2D sphere is a geometrical model, not a physical model of the 3D space of our universe. I've suggested it as a tool to help in understanding how it is geometrically possible to be edgeless and bounded. It was not to suggest that our space is a 2D sphere.

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2D sphere is a geometrical model, not a physical model of the 3D space of our universe. I've suggested it as a tool to help in understanding how it is geometrically possible to be edgeless and bounded. It was not to suggest that our space is a 2D sphere.

Are there any physical models that do that with posited versions of an actual universe?

Are there any models that picture such a universe "from  a bird 's eye view"?

Or is that just an illogical question to ask?(the  "bird"/observer  being part of the universe could not take a "bird's eye view"  of itself as a part of the whole)

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1 minute ago, geordief said:

Are there any physical models that do that with posited versions of an actual universe?

Are there any models that picture such a universe "from  a bird 's eye view"?

Or is that just an illogical question to ask?(the  "bird"/observer  being part of the universe could not take a "bird's eye view"  of itself as a part of the whole)

Yes, to take such a view the "bird" would need to get into a higher-dimensional embedding space.

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You're having problems imaging the topology of 4D space-time by comparing it to 3D or 2D topologies ?
Welcome to the club.

Mathematics, however, has no such limitations.

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Exactly. This is the solution:

1 minute ago, MigL said:

Mathematics has no such limitations.

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14 hours ago, geordief said:

They must have thought light had some mass,I suppose.

They didn’t know whether light has mass or not, it was an unanswered question at the time. But since Newtonian gravity can only handle test particles with mass, they had to assume that it did in order to derive any kind of prediction at all. The deflection angle doesn’t depend on the exact value of the mass - it just can’t be zero in Newton’s theory.

12 hours ago, Airbrush said:

How can one assume that photons have any mass at all?

You have to assume this to derive a deflection angle from Newton’s theory, which was the only theory available back then - it can’t handle massless test particles.

12 hours ago, Airbrush said:

I thought photons were energy and that is why they travel the speed of light.  How could any mass travel light speed?

You are absolutely right of course - this is one of the areas where Newtonian theory fails, and GR is needed. One must remember though that this wasn’t well understood prior to Einstein.

13 hours ago, MigL said:

Yes, the topology depends on curvature.

Actually, it doesn’t - they are separate concepts. GR determines only local geometry, but not global topology. For example, the maximally extended Schwarzschild metric could describe both two separate, singly-connected regions of spacetime, or a single multiply-connected spacetime. Geometry is the same in both cases, but the global topology isn’t.

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5 hours ago, Markus Hanke said:

Actually, it doesn’t - they are separate concepts. GR determines only local geometry, but not global topology. For example, the maximally extended Schwarzschild metric could describe both two separate, singly-connected regions of spacetime, or a single multiply-connected spacetime. Geometry is the same in both cases, but the global topology isn’t

How do we know that a global topology exists at all and that we don't just have a patchwork of local topologies?

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9 minutes ago, geordief said:

How do we know that a global topology exists at all and that we don't just have a patchwork of local topologies?

We see the light coming from very distant sources in the observable universe. It appears to be very homogenous and isotropic.

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We see the light coming from very distant sources in the observable universe. It appears to be very homogenous and isotropic.

very distant= still local ?

That wouldn't  be evidence of a global topology, would it?

The observable universe is still  "local" isn't it?

8 hours ago, MigL said:

You're having problems imaging the topology of 4D space-time by comparing it to 3D or 2D topologies ?
Welcome to the club.

Mathematics, however, has no such limitations.

Does the maths you are referring to make any physical predictions or is it entirely " theoretical"? ("theoretical" in layman's speak)

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48 minutes ago, geordief said:

very distant= still local ?

That wouldn't  be evidence of a global topology, would it?

The observable universe is still  "local" isn't it?

No. "Local" is space-time volume where the metric can be approximated by Minkowski's one with vanishing first derivatives.

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No. "Local" is space-time volume where the metric can be approximated by Minkowski's one with vanishing first derivatives.

OK .So can subsets of  the physical universe (eg the solar system)  be described as "local" or "global"?

Does "local" only describe the model and not the physical objects it attempts to model?

And are we talking about spacetime curvature or the topology of the physical (subsets of the) universe?

I think I was talking about just topology in that last post(even if I used terminology  that applies normally to spacetime curvature) ie is there such a thing as local topography versus global topography?

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5 minutes ago, geordief said:

are we talking about spacetime curvature or the topology

The former.

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So, if I'm following any of this, curvature is defined in a way that is intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces.  Like, if I'm a flat bug on the shore of a calm lagoon I can measure a curvature of the "flat" water as flat bug-ships drop over the horizon, without any reference to an earth interior or global topology?

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Posted (edited)
29 minutes ago, TheVat said:

So, if I'm following any of this, curvature is defined in a way that is intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces.  Like, if I'm a flat bug on the shore of a calm lagoon I can measure a curvature of the "flat" water as flat bug-ships drop over the horizon, without any reference to an earth interior or global topology?

Yes ,you draw  a triangle  on the flat water and measure the 3 angles.

If they add up to 180 it is flat .

Less and it is a positive curvature ;more and it is negative (like a saddle)

And ,less obviously you can do the same in 3d(perhaps using a 2d surface embedded in the 3d in the same way)

Not sure how you would do it in  4d spacetime.

Edited by geordief
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Posted (edited)
1 hour ago, geordief said:

Less and it is a positive curvature ;more and it is negative (like a saddle)

A minor correction here: one measures Δ=(sum-of-the-angles-of-a-triangle)π .

If Δ>0 , i.e., sum of the angles is more then 180o , then the curvature is positive (like a sphere).

If Δ<0 , i.e., sum of the angles is less then 180o , then the curvature is negative (like a saddle).

1 hour ago, geordief said:

Not sure how you would do it in  4d spacetime.

You would not. It works only for space, more precisely, for a metric signature +++... .

The spacetime metric signature is -+++ (or +---, depending on convention). One calculates Riemann curvature tensor in a general case.

2 hours ago, TheVat said:

if I'm a flat bug on the shore of a calm lagoon I can measure a curvature of the "flat" water as flat bug-ships drop over the horizon

No, that would be an extrinsic curvature.

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No, that would be an extrinsic curvature

I am confused.Wouldn't it be extrinsic curvature  if the boat fell off an  imagined edge but intrinsic curvature  if it just dropped out of view when it reached the horizon?

Or does the light ray  with which we see the boat disappear exist  in  an embedding 3rd dimension making the surface extrinsic after all?

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