The Nature of Time

[Genady]

6 hours ago, Mordred said:

Our universe is extremely close to flat

Yes, it is extremely close to flat spatially, but the discussion here is about flatness of spacetime, and the spacetime of our universe is not flat.

[Mordred]

the point is that even if k=0 precisely the universe can still expand due to the kinetic energy terms from radiation and the cosmological constant term. A good way to understand that is to examine the single component toy model universes such as radiation or Lambda only universes alternatively the DeSitter and anti-Desitter universes. It is a good way to examine individual each comp0sition of our universe.

 

Edited January 12, 2023 by Mordred

[Genady]

The point of the above discussion was that even with k=0 precisely, the spacetime is not flat although the space is flat.

[Mordred]

Regardless of whether your examining space or spacetime isn't doesn't matter in this case a truly flat space or spacetime can still expand or contract when you include radiation or Lambda. You can experiment with this formula which will show this

\[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\]

or use the cosmological calculator in my signature link. It will allow you to set the cosmological parameters

Edited January 12, 2023 by Mordred

[Genady]

I think that the issues has been already resolved, multiple times, e.g.,

3 hours ago, Markus Hanke said:

The FLRW metric is an interior solution to the Einstein equations - it represents the geometry of spacetime in the interior of a homogenous, isotropic distribution of dust. Since this “dust” is certainly a gravitational source, spacetime here cannot be flat.

When it comes to questions of geometry, one must carefully distinguish between space and spacetime. These are not the same things at all. In the FLRW scenario, spacetime is always curved in a particular way; but, for the right choice of parameters, each 3D hypersurface of space within that spacetime can be flat.

 

[Mordred]

You should reexamine the FLRW metric in particular the ds^2 line element you will find the k=0 metric Euclidean which can be considered space even though the metric includes spacetime with the addition of the scale factor. Besides one of the lessons Minkowskii taught is that you cannot separate space and spacetime.

[studiot]

 

Again I must emphasise something Markus said.

3 hours ago, Markus Hanke said:

When it comes to questions of geometry, one must carefully distinguish between space and spacetime. These are not the same things at all.

 

But it is even more complicated than this.

To fully appreciate the depth of this statement - in relation to the nature of time - we need to bring together several disciplines.
In doing so we need to avoid the trap of introducing extraneous concepts and matters from these several disciplines that are associated with material we need.

 

I suggest the disciplines are at least and in no particular order:-   language (semantics), general philosophy, physics, pure mathematics, applied mathematics.

We need to consider carefully the meaning of at least the terms, again in no particular order:-  movement, virtual, geometry, space and dimension, embedded.

As a matter of interest an example of movement that occurs without time in physics is called a virtual displacement.

Further examples occur in mathematics, particularly in geometry. You cannot fully consider congruence, similarity and transformations without some sort of movement-without-time.

Finally a warning about mathematics.

Nearly all courses and texts on 'the geometry of n dimensions' assume an underlying embedding in n+1 dimensions.
Furthermore most such geometries are static. That is they are without movement. It is physics that introduces the movement in it's efforts to model the 'real' world around us.

 

This is an absolutely huge subject.

 

23 minutes ago, Genady said:

The point of the above discussion was that even with k=0 precisely, the spacetime is not flat although the space is flat.

I would like to see the mathematics supporting this statement.

Edited January 12, 2023 by studiot

[Genady]

41 minutes ago, Mordred said:

You should reexamine the FLRW metric in particular the ds^2 line element you will find the k=0 metric Euclidean which can be considered space even though the metric includes spacetime with the addition of the scale factor. Besides one of the lessons Minkowskii taught is that you cannot separate space and spacetime.

 

33 minutes ago, studiot said:

I would like to see the mathematics supportng this statement.

It goes like this. For the 3+1 dimensional spacetime to be flat, i.e., to have a vanishing curvature it has to be Minkowski, i.e., to allow coordinates in which the metric is Minkowski. For a spacetime with FLRW metric to allow this, the scale factor needs to be constant. Thus, with a non-constant scale factor, the spacetime with FLRW metric is not flat.

We can keep one dimension, namely time, constant and consider the 3D slice separately. This 3D space can be Euclidean, hence flat.

Edited January 12, 2023 by Genady

[joigus]

2 minutes ago, Genady said:

 

It goes like this. For the 3+1 dimensional spacetime to be flat, i.e., to have a vanishing curvature it has to be Minkowski, i.e., to allow coordinates in which the metric is Minkowski. For a spacetime with FLRW metric to allow this, the scale factor needs to be constant. Thus, with a non-constant scale factor, the spacetime with FLRW metric is not flat.

We can keep one dimension, namely time, constant and consider the 3D slice separately. This 3D space can be Minkowski, hence flat.

What about a 4=1+3 cilinder? A cilinder is flat, it's S¹xR³, but it's not R⁴.

[Genady]

Just now, joigus said:

What about a 4=1+3 cilinder? A cilinder is flat, it's S¹xR³, but it's not R⁴.

Yes, so? It is not a spacetime. The metric signature is different.

[Mordred]

12 minutes ago, Genady said:

 

It goes like this. For the 3+1 dimensional spacetime to be flat, i.e., to have a vanishing curvature it has to be Minkowski, i.e., to allow coordinates in which the metric is Minkowski. For a spacetime with FLRW metric to allow this, the scale factor needs to be constant. Thus, with a non-constant scale factor, the spacetime with FLRW metric is not flat.

We can keep one dimension, namely time, constant and consider the 3D slice separately. This 3D space can be Euclidean, hence flat.

No in the FLRW metric if k=0 then \[T_{\mu\nu}=0\] you still have the scale factor though as that universe can still expand or contract. If the factor is constant at zero you have a static universe which is a special class of solution (Einsteins biggest blunder) and Einstein needed to add a cosmological term to get a= constant zero.

 

Edited January 12, 2023 by Mordred

[Genady]

If the scale factor in FLRW metric is a constant, the space cannot expand or contract.

[studiot]

48 minutes ago, Genady said:

 

It goes like this. For the 3+1 dimensional spacetime to be flat, i.e., to have a vanishing curvature it has to be Minkowski, i.e., to allow coordinates in which the metric is Minkowski. For a spacetime with FLRW metric to allow this, the scale factor needs to be constant. Thus, with a non-constant scale factor, the spacetime with FLRW metric is not flat.

We can keep one dimension, namely time, constant and consider the 3D slice separately. This 3D space can be Euclidean, hence flat.

That is not mathematical.

Curvature in 1 dimension had no parameters and does not exist.

Curvature in 2 dimensions involves one parameter usually called kappa, and refers to  lines.

Curvature in 3 dimensions involves two parameters usually called kappa and tau and refers to surfaces.

When we move to 4 dimensions this involves abother parameter, usually called sigma and involves blocks of 3D space.

Mathematically I would expect to see a discussion of this., in terms of the Gaussian or Riemanian curvature matrices.

The parameters correspond to directions and expanded they correspond to 4 scalar equations

[math]\frac{{dT}}{{ds}} = \kappa N[/math]

[math]\frac{{dT}}{{ds}} =  - \kappa N + \tau B[/math]

[math]\frac{{dB}}{{ds}} =  - \tau N + \sigma D[/math]

[math]\frac{{dD}}{{ds}} =  - \sigma B[/math]

 

 

Edited January 12, 2023 by studiot

[Genady]

6 minutes ago, studiot said:

Mathematically I would expect to see a discussion of this., in terms of the Gaussian or Riemanian curvature matrices.

You are correct. The discussion here is in terms of Riemann curvature, which is relevant for discussions on GR, usually.

 

8 minutes ago, studiot said:

That is not mathematical.

You are correct again. The mathematical treatment can be found in textbooks on differential geometry that include discussion on the spacetime of GR. Also, in bits and pieces, in articles on Internet.

PS. The Riemann curvature is of course tensor rather than a matrix.

[studiot]

7 minutes ago, Genady said:

You are correct again. The mathematical treatment can be found in textbooks on differential geometry that include discussion on the spacetime of GR. Also, in bits and pieces, in articles on Internet.

Yes, but I didn't make the claim about 'flatness', you did.
So it is up to you to support it properly.

 

Note I have added some useful equations to my last post.

It should also be noted that GR does not embody or use all aspects of the Nature of Time (the topic of this thread), although I think we are all agreed that a 'timelike' variable incorporating some of these are necessary for GR.
So discussion should not be limited to GR

Edited January 12, 2023 by studiot

[Genady]

9 minutes ago, studiot said:

It should also be noted that GR does not embody or use all aspects of the Nature of Time (the topic of this thread), although I think we are all agreed that a 'timelike' variable incorporating some of these are necessary for GR.
So discussion should not be limited to GR

Agreed. This detour was just a little exchange between @sethoflagos and me and should've been over with the @Markus Hanke's post above. However, some posters have picked up some comments as if they were standalone statements and have used them to educate the audience in an unrelated material.

[joigus]

1 hour ago, Genady said:

Yes, so? It is not a spacetime. The metric signature is different.

Why not? Minkowski is R⁴ with signature (-+++), while R⁴ with signature (++++) is 4-dimensional Euclidean space. S¹xR³ with signature (-+++) would be a Minkowskian cilinder, while with signature (++++) would be Euclidean. Metric signature and topology are quite independent.

[Genady]

3 minutes ago, joigus said:

Why not? Minkowski is R⁴ with signature (-+++), while R⁴ with signature (++++) is 4-dimensional Euclidean space. S¹xR³ with signature (-+++) would be a Minkowskian cilinder, while with signature (++++) would be Euclidean. Metric signature and topology are quite independent.

Right. The only issue is that I don't have any idea what we are discussing now and why. Yes, cylinder is flat. A cone is flat everywhere except the very tip. A bagel is not flat except along the very top and the very bottom. Etc.

[joigus]

1 hour ago, Genady said:

Right. The only issue is that I don't have any idea what we are discussing now and why. Yes, cylinder is flat. A cone is flat everywhere except the very tip. A bagel is not flat except along the very top and the very bottom. Etc.

I don't know either.

[Mordred]

3 hours ago, Genady said:

If the scale factor in FLRW metric is a constant, the space cannot expand or contract.

Correct however as I mentioned above you need a cosmological term in combination to the EFE to keep the scale factor constant. Hence Einsteins blunder a static eternal solution is unstable as I mentioned previously 

Edited January 12, 2023 by Mordred

[Genady]

13 minutes ago, joigus said:

I don't know either.

Maybe, to keep focus, threads should be limited to, say, 63 or 127 replies. If members want to continue a discussion beyond that, somebody will start a new thread with a clear formulation of the issue from scratch.

[joigus]

1 hour ago, Genady said:

Maybe, to keep focus, threads should be limited to, say, 63 or 127 replies. If members want to continue a discussion beyond that, somebody will start a new thread with a clear formulation of the issue from scratch.

You're probably right. We got lost in geometry.

I  suspect there's something about time that's not entirely geometric. To me, it has the unmistakable flavour of abstract algebra. Suggestions from QFT are clear. Deeply involved in microcausality, operator-ordering questions, and the CPT theorem.

[TheVat]

The first page and OP seemed less about Friedmann equations (which later chat left us nonphysicists barely treading water) and more about interrogating our intuitions about time as anything more than a part of a geometric description of how matter/energy behaves in space.  

There is matter/energy.  To observing biological entities, it changes - position, energy state, decay, etc.  This is due to changes in the biological entities, as well. These changes can be described (and predicted) with geometric descriptions that include a measurable elapsed time, t.  The OP prompts the question: is there an ontology of time?  Or is time simply how we sentient beings process the universe as we move across a landscape of "events"?  Could there be an entity that could see the universe and all events at once, like some sort of cosmic jewel?  (A question that invites incoherence, for sure)

[studiot]

1 hour ago, Genady said:

Maybe, to keep focus, threads should be limited to, say, 63 or 127 replies. If members want to continue a discussion beyond that, somebody will start a new thread with a clear formulation of the issue from scratch.

There are quite a few more popular threads with over 1000 responses.

These tend to be less technical.

 

37 minutes ago, joigus said:

You're probably right. We got lost in geometry.

I  suspect there's something about time that's not entirely geometric. To me, it has the unmistakable flavour of abstract algebra. Suggestions from QFT are clear. Deeply involved in microcausality, operator-ordering questions, and the CPT theorem.

To be precise the theory of algebraic varieties.

Although

strangely enough this has a great deal to do with geometry.

The most interesting thing about the geometry of n dimensions is that it has few applications except in the realm of statistics where it is considered vital knowledge for higher level study.

 

32 minutes ago, TheVat said:

The first page and OP seemed less about Friedmann equations (which later chat left us nonphysicists barely treading water) and more about interrogating our intuitions about time as anything more than a part of a geometric description of how matter/energy behaves in space.  

 

Indeed it is the aspects of Time that are non geometric that set it apart from the other variables in GR.

What did you make of my recent comments about the connections between time and movement ?

Edited January 12, 2023 by studiot

[sethoflagos]

33 minutes ago, TheVat said:

Could there be an entity that could see the universe and all events at once, like some sort of cosmic jewel?  (A question that invites incoherence, for sure)

If we created an image of a star from not just its emr transmissions but all concurrent arrivals of its cosmic rays in a range of velocities, we'd obtain a trace of it's historic worldline as viewed from our current location in space. Is that what you mean?