What is time? (Again)

[studiot]

12 minutes ago, joigus said:

Mmmm...

Remindful of, suggestive of, rather than equal.

Reminds more of Cauchy's integral theorem of complex calculus.

And even more of Stokes' theorem for differential forms.

Because we always use analytic functions, things on the inside are determined by things on the surface. But I'm getting hopelessly vague and metaphorical.

Although the FTC is a particular case.

Last time I tried to write this out here, the site's crappy Latex defeated me.

I will have another go sometime.

It really is a wondrous thing that for many situations that actually concern us all the information about what is going on inside a closed boundary is available by just considering the boundary.

But yes this is tied up with Gauss, Stokes and Green's and Riemann.

Even better there is a digital or discrete version that is in great use as boundary elements v finite elements.

[joigus]

1 minute ago, studiot said:

Last time I tried to write this out here, the site's crappy Latex defeated me.

I will have another go sometime.

It really is a wondrous thing that for many situations that actually concern us all the information about what is going on inside a closed boundary is available by just considering the boundary.

But yes this is tied up with Gauss, Stokes and Green's and Riemann.

Even better there is a digital or discrete version that is in great use as boundary elements v finite elements.

Let me help you. All of them are contained here,

\[\int_{M}d\omega=\int_{\partial M}\omega\]

[studiot]

1 minute ago, joigus said:

Let me help you. All of them are contained here,

 

∫Mdω=∫∂Mω

 

Yes I might try that, I wanted the little circles round the integral signs.

I do remember when I first came here the site's latex couldn't display the triple bar identity sign or the proper limits of a definite integral or summation sign.

At least these have improved.

[michel123456]

3 hours ago, Markus Hanke said:

Indeed - but it isn't so much the finer details of how exactly the observer moves, but rather the very fact itself that the observed vacuum depends on the motion of the observer at all. You travel in an accelerating rocket and see a sea of particles around you; then you stop, and pop! - they are all gone, though you are still in the same region of spacetime. If you really think about this, it poses very serious questions about what is really fundamental, and what is not.

But we are accelerated, through gravitation (see equivalence principle). And we do see a sea of particles around us (the universe). And it is impossible to stop. Our sliding in time cannot stop.

8 hours ago, joigus said:

Then you've got Maldacena's mind-blowing mathematical result that gravity inside a ball is describing classical gauge field theory on the surface of that ball, but at the price of having the metric be anti-DeSitter (something like an anti-universe or exponentially-contracting universe). This strongly suggests that any new physics should be capable of relating inside-outside quantities for any observers (that naturally perceive some kind of inside-outside distinction), which is what I was trying to connect with before.

Exponentially-contracting universe, yes that crossed my mind but geometrically how do you make this? With one universal center of contraction? Or multiple centers? I was completely blocked by the incapacity of figuring the idea.

13 hours ago, michel123456 said:

The Unruh effect: under the prism of what I posted you in PM, it may be (although I thought it would be impossible) that the observer under ridiculously high acceleration is "kicked out" of his own time line in such a way that he can observe particles of his own future (otherwise unobservable) or belonging to his own past, depending on the direction of acceleration.

I have erased the "depending on the direction of acceleration", after much thought it must be wrong.

Edited August 13, 2020 by michel123456

[joigus]

23 minutes ago, michel123456 said:

But we are accelerated, through gravitation (see equivalence principle). And we do see a sea of particles around us (the universe). And it is impossible to stop. Our sliding in time cannot stop.

Exponentially-contracting universe, yes that crossed my mind but geometrically how do you make this? With one universal center of contraction? Or multiple centers? I was completely blocked by the incapacity of figuring the idea.

Well, Maldacena's initial idea wasn't motivated by a realistic model of the universe. So the so-called AdS (anti-DeSitter Space) was not meant to represent the real universe. Although the AdS space-time is an exact solution of the Einstein field equations.

But exponentially expanding or contracting universes don't have to have a center. Our universe is a DeSitter universe (exponentially expanding) AFAWK and it's not doing it around any particular point. Everything is expanding with respect to everything else.

[michel123456]

10 hours ago, joigus said:

Well, Maldacena's initial idea wasn't motivated by a realistic model of the universe. So the so-called AdS (anti-DeSitter Space) was not meant to represent the real universe. Although the AdS space-time is an exact solution of the Einstein field equations.

But exponentially expanding or contracting universes don't have to have a center. Our universe is a DeSitter universe (exponentially expanding) AFAWK and it's not doing it around any particular point. Everything is expanding with respect to everything else.

How do you describe this geometrically? if I expand, and you expand, at some time we will touch each other. In order to remain in place (relatively to each other) some displacement must take place. Otherwise it will look (from our expanding point of view) as if something was pushing us against each other.

[Markus Hanke]

14 hours ago, studiot said:

Indeed there is.

It's called the fundamental theorem of calculus.

I don't think it is that simple. The duality we are referring to here is a duality between distinct types of physical theories - geometric theories of spacetime (on the bulk) on the one hand, and conformal field theories (on the boundary) on the other side. How does this relate to the FTC?

13 hours ago, joigus said:

Let me help you. All of them are contained here,

 

∫Mdω=∫∂Mω

 

Try this:

\[\oint _{M} d\omega =\oint _{\partial M} \omega \]

Even for this, I am struggling to make a connection to the AdS/CFT correspondence.

[joigus]

1 hour ago, michel123456 said:

How do you describe this geometrically? if I expand, and you expand, at some time we will touch each other. In order to remain in place (relatively to each other) some displacement must take place. Otherwise it will look (from our expanding point of view) as if something was pushing us against each other.

Spaces with intrinsic curvature can expand or contract without any coordinate points touching each other, like when you paint dots on a balloon and start to blow. They separate but they never touch.

8 minutes ago, Markus Hanke said:

I don't think it is that simple. The duality we are referring to here is a duality between distinct types of physical theories - geometric theories of spacetime (on the bulk) on the one hand, and conformal field theories (on the boundary) on the other side. How does this relate to the FTC?

Exactly. "Remindful of", "suggestive of". And that's the reason. +1

Unless anybody comes up with a closer analogy that one theory is like an exterior differential of the other in some sense.

[Markus Hanke]

12 hours ago, michel123456 said:

But we are accelerated, through gravitation (see equivalence principle)

It's not so much the equivalence principle, but the principle of least action.

12 hours ago, michel123456 said:

And we do see a sea of particles around us (the universe).

Well, you can also go into a stable orbit (no acceleration once there), and the universe will look just the same. This is not due to Unruh effect.

1 hour ago, michel123456 said:

How do you describe this geometrically?

By making the metric time-dependent, i.e. by reducing the distance between all points on the manifold.

[studiot]

1 hour ago, Markus Hanke said:

I don't think it is that simple. The duality we are referring to here is a duality between distinct types of physical theories - geometric theories of spacetime (on the bulk) on the one hand, and conformal field theories (on the boundary) on the other side. How does this relate to the FTC?

Obviously I must have misunderstood you, perhaps you would like to clarify ?

(Particularly FTC, ADS and CFT ?)

 

I thought that Gauss' theorem was a link between the bulk physical theory and a field theory on the boundary ?

 

Are we not talking about solution methods to the 'boundary value problem' ?

Have you come across the method of Prager and Synge linking the geometry to boundary value solutions

"The Hypercircle in Mathematical Physics." ?

[Markus Hanke]

48 minutes ago, studiot said:

Obviously I must have misunderstood you, perhaps you would like to clarify ?

(Particularly FTC, ADS and CFT ?)

FTC = Fundamental theorem of calculus
AdS = Anti-deSitter Space
CFT = Conformal field theory

So essentially the AdS/CFT correspondence relates two formally completely different theories - a geometric theory of gravity (such as M-Theory, or Loop Quantum Gravity) in the bulk, and a conformal field theory (a special type of quantum field theory) on the boundary of that bulk. The point here is that two very different theories over different domains can describe the same physics.

53 minutes ago, studiot said:

Have you come across the method of Prager and Synge linking the geometry to boundary value solutions

No, I haven't heard of that - I'm not sure if that is really the same thing.

[joigus]

1 hour ago, studiot said:

Obviously I must have misunderstood you, perhaps you would like to clarify ?

(Particularly FTC, ADS and CFT ?)

 

I thought that Gauss' theorem was a link between the bulk physical theory and a field theory on the boundary ?

 

Are we not talking about solution methods to the 'boundary value problem' ?

Have you come across the method of Prager and Synge linking the geometry to boundary value solutions

"The Hypercircle in Mathematical Physics." ?

Allow me to answer, and Markus, feel free to correct mistakes/imprecissions and/or add info as you see fit.

The AdS/CFT dualities are more sophisticated. For example:

On the inside (bulk) you have a gravitational theory, which has a metric connection (a rule to parallel-transport vectors from a metric).

On the boundary you have a gauge theory (a non-metric connection or affine connection, which stands on its own). They are completely different animals, and there is no simple way to relate the degrees of freedom. Also, some solutions of the theory without metric may be completely devoid of meaning as solutions of the theory in the bulk. There are topological aspects in the spectrum of solutions on the boundary that have no easy interpretation (or non at all) in the bulk. And so on... People who work on these type of dualities generally speak of a dictionary (a set of rules to translate boundary conditions, etc. from one theory to the other).

Edit: x-posted with @Markus Hanke .

3 hours ago, Markus Hanke said:

Try this:∮

\[\oint_{M}d\omega=\oint_{\partial M}\omega\]

Even for this, I am struggling to make a connection to the AdS/CFT correspondence.

I think you have an extra circle there. It'd be,

\[\int_{M}d\omega=\oint_{\partial M}\omega\]

I personally prefer not to use the circle because there's no simple way to iterate the operation. So, for example, if you want to express Hodge duality's simple result "the boundary of a boundary is empty":

\[\int_{M}d^{2}\omega=\int_{\partial^{2}M}d\omega\]

\[d^{2}\omega=0\Rightarrow\partial^{2}M=\textrm{Ø}\]

The boundary of a boundary has no points.

Edit: And the \partial symbol already implies you're looping around. But that's a matter of taste.

Edited August 14, 2020 by joigus
bad rendering of eq./minor addition/correct mistake in formula

[studiot]

47 minutes ago, Markus Hanke said:

FTC = Fundamental theorem of calculus
AdS = Anti-deSitter Space
CFT = Conformal field theory

So essentially the AdS/CFT correspondence relates two formally completely different theories - a geometric theory of gravity (such as M-Theory, or Loop Quantum Gravity) in the bulk, and a conformal field theory (a special type of quantum field theory) on the boundary of that bulk. The point here is that two very different theories over different domains can describe the same physics.

No, I haven't heard of that - I'm not sure if that is really the same thing.

Thank you for the amplification.

Quote

The point here is that two very different theories over different domains can describe the same physics.

Yes of course they can. Knowledge of this goes way back a simple example being energy methods v force methods. Fourier transforms are another.

Another way of putting this is that much of our physics is really just drawing glorified graphs.
And as we have noted before, one can always plot the same geometric result using different axes (domains).

 

Quote

No, I haven't heard of that - I'm not sure if that is really the same thing.

Hopefully you have heard of Synge. He was a famous 20th century professor of Maths and Physics at Dublin (died in 1995) who did a lot of work on Mechanics and Relativity, amongst other things.

The hypercircle method combines two infinite dimensional function spaces  (domains) to generate a boundary hypercircle to/of solutions to a physical situation when on F space is mapped to the other.

[Markus Hanke]

3 hours ago, joigus said:

I think you have an extra circle there

Quite possible, I am not sure what the convention for the notation is, on this one. I thought I have seen people use circles on both integrals...?

3 hours ago, joigus said:

The boundary of a boundary has no points.

Indeed. Did you know that the form of the GR field equations follows from this seemingly simple topological principle? 
Both this principle, and the generalised Stoke's theorem above, are IMHO among the most beautiful results in all of mathematics  

2 hours ago, studiot said:

Hopefully you have heard of Synge

The name kind of rings a bell somewhere, but I wouldn't be intimately familiar with what he did (even though I live in Ireland).
This is probably a good time to reiterate that all my maths are entirely self-taught, so there are large holes in my mathematical knowledge. I really only ever looked at those areas that are directly relevant to the areas of physics I am interested in.

3 hours ago, joigus said:

Allow me to answer, and Markus

Gladly  I'm somewhat out of my depths on this one, since I've never really studied QFT in any detail. That's a shortcoming I am intending to rectify when I have the time and inclination.

[joigus]

15 minutes ago, Markus Hanke said:

Indeed. Did you know that the form of the GR field equations follows from this seemingly simple topological principle? 

I didn't!! +1 Please elaborate or give me a reference, if you don't mind.

15 minutes ago, Markus Hanke said:

Both this principle, and the generalised Stoke's theorem above, are IMHO among the most beautiful results in all of mathematics  

I agree.

-----------

I would like to think, and comment, and read more comments, about how all this Stokes' theorem story could have some bearing on the question of time, as integrating on the boundary of a set requires orientation, while integrating on a bulk that is not a boundary, doesn't. Same as time.

I googled for the circle, as I wasn't sure. Here's what I found:

Quote

The circle on an integral generally means the integral is performed on a space which has lower dimension than the ambient space and is a "closed loop" which is informal language to say it's compact (finite in size) and without boundary.

 

20 hours ago, studiot said:

But yes this is tied up with Gauss, Stokes and Green's and Riemann.

My apologies, @studiot, because I've been talking all the time about exterior calculus without mentioning, explaining, or giving a proper reference:

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

It's the technique to wrap it all up (Stokes, Green, Gauss, etc.) into one unified description. Thus, for example, Stokes' theorem (or is it Green's?) can be obtained by exterior differenciating a line element (1-form):

\[d\left(v_{x}dx+v_{y}dy\right)=\]

\[=v_{x,x}dx\wedge dx+v_{x,y}dy\wedge dx+v_{y,x}dx\wedge dy+v_{y,y}dy\wedge dy=\]

\[=v_{x,y}dy\wedge dx+v_{y,x}dx\wedge dy=\]

\[=\left(v_{y,x}-v_{x,y}\right)dx\wedge dy\]

\[\int_{\partial\Gamma}\textrm{curl}\boldsymbol{v}\cdot d\boldsymbol{S}=\oint_{\Gamma}\boldsymbol{v}\cdot d\boldsymbol{l}\]

It's powerful because you can do it for any dimensions, surface element, line elements, volume elements, and in general, n-surface and (n-1)-surface elements.

Edited August 14, 2020 by joigus
Addition

[studiot]

5 hours ago, joigus said:

It's powerful because you can do it for any dimensions, surface element, line elements, volume elements, and in general, n-surface and (n-1)-surface elements.

Good stuff, +1.

However you missed the zero dimension.

In one dimension the 1 dimensional differentiable manifold is the interval [a , b] , a function of one variable f(x) and the boundary comprises the two points a and b.

so the theorem says that

[math]\int\limits_a^b {df = f(b) - f(a)} [/math]

 

IOW the Riemann integral.

The link to time (I am trying to avoid the word connection) is that in space-time, time is somehow one of the variables in the manifold.

 

Edited August 14, 2020 by studiot

[joigus]

 

6 hours ago, joigus said:

It's the technique to wrap it all up (Stokes, Green, Gauss, etc.) into one unified description. Thus, for example, Stokes' theorem (or is it Green's?) can be obtained by exterior differenciating a line element (1-form):

<wrong formula>

It's powerful because you can do it for any dimensions, surface element, line elements, volume elements, and in general, n-surface and (n-1)-surface elements.

My apologies again. I made a mistake here (as usual):

\[\int_{S}\textrm{curl}\boldsymbol{v}\cdot d\boldsymbol{S}=\oint_{\partial S}\boldsymbol{v}\cdot d\boldsymbol{l}\]

The circuit must be the boundary of the surface.

30 minutes ago, studiot said:

However you missed the zero dimension.

I didn't mention it, but it was always on my mind.  

[Markus Hanke]

20 hours ago, joigus said:

I didn't!! +1 Please elaborate or give me a reference, if you don't mind.

The only textbook I know of that explicitly mentions this is Misner/Thorne/Wheeler "Gravitation". It's a very beautiful connection between GR (and also electromagnetism!) and that topological principle. If you don't have access to this text, I can try and summarise the derivation here, if I have time over the next few days. 

20 hours ago, joigus said:

 about how all this Stokes' theorem story could have some bearing on the question of time, as integrating on the boundary of a set requires orientation, while integrating on a bulk that is not a boundary, doesn't. Same as time.

Interesting observation, but...isn't the AdS/CFT duality the exact opposite of this? Only spacetime within the bulk would have a time dimension, whereas the boundary on which the CFT lives is purely spatial. However, I may have this wrong, so please correct me if necessary.

Edited August 15, 2020 by Markus Hanke

[joigus]

1 hour ago, Markus Hanke said:

The only textbook I know of that explicitly mentions this is Misner/Thorne/Wheeler "Gravitation". It's a very beautiful connection between GR (and also electromagnetism!) and that topological principle. If you don't have access to this text, I can try and summarise the derivation here, if I have time over the next few days. 

Looking forward to it. Don't feel any pressure.

I think I can try and look it up. So, again, no rush. The observation is interesting enough so that I won't forget.

1 hour ago, Markus Hanke said:

Interesting observation, but...isn't the AdS/CFT duality the exact opposite of this? Only spacetime within the bulk would have a time dimension, whereas the boundary on which the CFT lives is purely spatial. However, I may have this wrong, so please correct me if necessary.

You're probably right. I'd say I'm learning about this probably much more slowly than you are. I trust you far more than I trust myself, and I don't say that out of courtesy. I wouldn't dare to correct you at this point.

I've been watching 3 ICTS lectures by one of the authors of the Kerr/CFT duality you told me about (Alejandra Castro) whom Lubos Motl (wrongly, probably) mentions as Alejandra Fidel Castro (LOL). Curiously enough, she avoids mentioning anything about future timelike and past timelike at some point. Those are AdS/CFT lectures. The sound is terrible in the 3rd lecture.

Chern-Simons gravity is very nearly unfathomable to me. And all of it, I think, is in 1+2 gravity, so...

Apparently you can formulate the equivalent of geodesic equations by prescribing a gauge choice in the SL(2) or SL(N) (higher spin) group. That's the most interesting idea I got from the lectures.

Anyway, it was never my intention to suggest that there's a direct relationship between Stokes' theorem and AdS/CFT. It was somebody else's.  

[MigL]

Don't know much about AdS/CFT, but always thought that although the boundary of AdS space has a lower dimensionality than the space itself, it must still encode all the higher dimensional information on that surface ( holographic principle ), and the boundary, locally around any point, reduces to Minkowsky space. IOW, still include time.

Or have I misunderstood AdS/CFT all this time ?

[Markus Hanke]

1 hour ago, joigus said:

I think I can try and look it up

That would probably help things - it's in chapter 15 of the book. The essential train of thought is this - suppose you have an elementary 4-cube of spacetime \(\Omega\). We know that energy-momentum within that cube is conserved, so (in differential forms language):

\[\int _{\partial \Omega } \star T=0\]

If we want to obtain a metric theory of gravity, the question becomes - what kind of object can we couple to energy-momentum, that obeys the same conservation laws, in order to obtain the field equations? For this, consider that the boundary of our 4-cube consists of 8 identical 3-cubes, each of which is in turn bounded by 6 faces. We can now associate a moment of rotation with each of the 3-cubes; to provide the link to energy-momentum, we then associate that moment of rotation with the source density current in the interior of each 3-cube.

Let \(\bigstar \) define a duality operation that acts only on contravariant vectors, but not on differential forms (the Cartan dual). The moment of rotation is then

\[\bigstar ( dP\ \land R) \]

wherein R denotes the curvature operator. We also find that this expression is just the dual of the Einstein form:

\[\bigstar ( dP\ \land R) = \star G\]

So let's put this all together. First, we create a moment of rotation in our 4-cube of spacetime:

\[\int _{\Omega } d\star G\]

Apply Stoke's theorem:

\[\int _{\Omega } d\star G=\int _{\partial \Omega } \star G \]

Rewrite in terms of the curvature operator:

\[\int _{\partial \Omega } \star G=\bigstar \int _{\partial \Omega } ( dP\ \land R) \]

To associate this with total net energy-momentum (at the moment it's associated with source current density), we must sum not just over the boundary of the 4-cube (which are 3-cubes), but also over the faces of each 3-cube; so we must apply Stoke's theorem again, to get

\[ \bigstar \int _{\partial \Omega }( dP\ \land R) =\bigstar \int _{\partial \partial \Omega }( P\ \land R) \]

But because \(\partial \partial =0 \), this automatically yields

\[\bigstar \int _{\partial \partial \Omega }( P\ \land R) =0 \]

But because the bracketed expression is just the dual of the Einstein form, this implies

\[\int _{\partial \Omega } \star G=0 \]

This is the exact same as the conservation law for energy-momentum given above. So we can associate the two:

\[G=T\]

So, using the concept of a moment of rotation, some elementary geometric considerations, and the "boundary of a boundary is zero" principle, the form of the Einstein equations is uniquely fixed up to a proportionality constant. MTW even obtain this constant somehow, though I don't quite follow their thoughts on this minor detail.

Anyway, that's the general idea. If you can get access to the text, it is all described in much more detail there.

[studiot]

 

32 minutes ago, Markus Hanke said:

That would probably help things - it's in chapter 15 of the book. The essential train of thought is this - suppose you have an elementary 4-cube of spacetime Ω. We know that energy-momentum within that cube is conserved, so (in differential forms language):

...................................

Anyway, that's the general idea. If you can get access to the text, it is all described in much more detail there.

Thank you for posting that. +1

On 8/14/2020 at 3:41 PM, joigus said:

It's powerful because you can do it for any dimensions, surface element, line elements, volume elements, and in general, n-surface and (n-1)-surface elements.

I think that an underlying reason you can do all these mappings (Grassman and Hodge algebras etc)  successfuly  from dimension m to dimension n and vice versa goes back to Cantor since they all have the dimension (cardinality) of the continuum.

Once you have R, you have everything else.

Edited August 15, 2020 by studiot

[Markus Hanke]

17 hours ago, studiot said:

Thank you for posting that. +1

No problem. It is far more understandable though if one has access to the textbook, where all of this is described in much more detail.

[studiot]

@joigus   and @Markus Hanke

So how would you gentlemen regard the following simple example

Consider a sphere divided into  many conducting lands, insulated from each other, and each charged to some different electrical potential.

The Field within the sphere can be determined from a knowledge of the position and potentials of the surface lands alone.

Edited August 16, 2020 by studiot

[Markus Hanke]

15 minutes ago, studiot said:

The Field within the sphere can be determined from a knowledge of the position and potentials of the surface lands alone.

No, it also depends on the metric of the underlying manifold, and obviously permeability and permittivity of the vacuum. 
But provided the above things are given, an interesting question arises - are the relative positions and potentials on the surface of the sphere sufficient boundary conditions in order to fix a unique solution to Maxwell's equations? I don't know the answer, but my guess would be no, because there is a gauge freedom in where the zero point of the electric potential is, so I think we need precisely one additional boundary condition that specifies whether or not there is a background potential/field permeating the space. The relative positions of the charges alone don't uniquely fix this. But that's just a guess now, without having done any actual maths.