Gravity waves and the aether

[Strange]

I think (and I'm sure ajb will correct me if I am wrong!) that the reason that the geometry is described in terms of bundles is, at least partly, because we are talking about INTRINSIC curvature.

 

So, the images you have posted show 2D surfaces curved in 3D space. But when we are talking about the geometry of space-time, the curvature of the 4D manifold(surface) does not require a higher dimensional space. This is almost impossible to visualise and can really only be described mathematically. (The nearest thing I can think of as an analogy is bump mapping used in 3D graphics.)

[MigL]

In the two lowest pics of the Cylinder and Mobius, notice the vertical lines with arrowheads pointing up ?

Those would be equivalent to fibres.

Notice what happens in transforming the cylindrical manifold to the Mobius manifold; about halfway around the circumference of the cylinder, the fibres are flipped so that the arrowhead is pointing down, and this results in the Mobius.

 

At least that is my (very limited ) understanding.

[Mike Smith Cosmos]

I think (and I'm sure ajb will correct me if I am wrong!) that the reason that the geometry is described in terms of bundles is, at least partly, because we are talking about INTRINSIC curvature.

 

So, the images you have posted show 2D surfaces curved in 3D space. But when we are talking about the geometry of space-time, the curvature of the 4D manifold(surface) does not require a higher dimensional space. This is almost impossible to visualise and can really only be described mathematically. (The nearest thing I can think of as an analogy is bump mapping used in 3D graphics.)

.

Take this as a compliment " you are sounding more like AJB , every day! Lol.

 

Well I get the tiniest bit of a glimmer , in what you just said . ( only a glimmer ) .

 

I would prefer for my sanity's sake to come up from the bottom . BY asking neive questions ,

 

Firstly . . Which of those pictures most represents ( even if wrongly, or the wrong way up, or not quite , but has something to do with . ELECTRO - MAGNETIC FIELDS . ) even if ' not at all ' or a small part, or whatever ?? .

 

Secondly . ...Which of those pictures most represents ( even if wrongly, or the wrong way up, or not quite , Or has fibres shoved in either end of the cylinder, ( sorry ' Bundled in either end ) but has something to do with GRAVITATIONAL FIELDS ,and SPACE TIME , even if ' not at all ' or a small part, or HOW , or whatever.???

 

Thirdly ....where quite , does the picture of the MANEFOLD fit , or contain , in all this ?

 

Fourthly ... in/on which surface/s would the Gravitational Waves , show up ?

 

Fifly... What quite do the fibres represent ? ' field lines of force? , or if they are only a suppose, what are they supposing?

 

Thanks in anticipation!

 

 

Mike

Edited August 20, 2016 by Mike Smith Cosmos

[MigL]

Remember the definition of a field ?

A scalar value, vector or tensor associated with each point in space.

 

A field, say a vector field like the EM field, is defined on a differentiable manifold, and belongs to a certain symmetry group, U(1). The tangent vector at each point would then be called the tangent bundle, for example.

And, it can be used to perform operations on the manifold.

 

So in answer to your question, none of those pictures represent the EM or gravitational field.

And no, I'm not really sure about all the above, so we'll have to wait on AJB

[disarray]

I would prefer for my sanity's sake to come up from the bottom . BY asking neive questions ,

 

Yes, learning in general is best done from bottom up, of course, and having someone answer questions as one goes along is helpful, though sometimes a bit of outside reading/studying by oneself certainly accelerates the process. This is especially true, I would imagine, when it comes to counterintuitive subjects such as Relativity.

[ajb]

Classical fields are sections of fibre bundles - but it gets more complicated to describe this.

 

The gravitational field - so the metric - is a particular kind of section of the vector bundle bundle [math] (TM \otimes TM)^{*}[/math]. That is a particular map from the manifold to the dual of two copies of the tangent bundle.

 

Or if you want to think about connections - such as the Levi-Civita connection or the electromagnetic potenial - then you need to think in terms of sections of cetrian first jet bundles. But this is a complicated story for another place.

 

The point is that all classical fields are sections of fibre bundles build over the space-time manifold. These bundles and various sections thereof are what we mean by geometry.

[Mordred]

Or as Sean Carroll wrote " Fibre bundles can be thought of as the internal vectors within a group".

[Mike Smith Cosmos]

Classical fields are sections of fibre bundles - but it gets more complicated to describe this.The gravitational field - so the metric - is a particular kind of section of the vector bundle bundle [math] (TM \otimes TM)^{*}[/math]. That is a particular map from the manifold to the dual of two copies of the tangent bundle.Or if you want to think about connections - such as the Levi-Civita connection or the electromagnetic potenial - then you need to think in terms of sections of cetrian first jet bundles. But this is a complicated story for another place.The point is that all classical fields are sections of fibre bundles build over the space-time manifold. These bundles and various sections thereof are what we mean by geometry.

AJB .

 

Is the ' Manifold ' on the inside, on which the bundles are built ? Or is the 'Manifold ' on the outside ( as it were ) .

. In other words , the whole of Spacetime and everything else in the universe , is contained , " , within the manifold " ?

 

And are you saying " one slice through this ' bundle of fibres ' will give you the ' Electro-magnetic field ' and a different slice through a double ended bundle of fibres , will give you the ' Gravitational field ' , or as named ' metric ' ?

 

Or have I got it ' all screwed up !

 

 

Mike

Edited August 20, 2016 by Mike Smith Cosmos

[ajb]

In other words , the whole of Spacetime and everything else in the universe is contained , " , within the manifold " ?

Space-time is a manifold and you add extra geometry to this in terms of fibre bundles 'build on top' - you attach more manifolds to each point of space-time. The 'reality' of these fibre bundles is another philosophical question - fibre bundles are used because their sections give us the right objects for physics. All tensors and tensor-like objects can be seen as sections of particular fibre bundles.

 

Or as Sean Carroll wrote " Fibre bundles can be thought of as the internal vectors within a group".

Does he explain that a bit more?

[disarray]

 

The point is that all classical fields are sections of fibre bundles build over the space-time manifold. These bundles and various sections thereof are what we mean by geometry.

 

I gather that, in terms of gravitation, "waves" refer as much to the mathematical model (which may or may not have some sort of Platonic existence depending upon whom you are talking to), as to actual waves that can be detected on instruments. It sounds to me that in this sense the word "waves" is sometimes used in a rather metaphorical sense. If you have an advanced understanding of the mathematics (e.g., geometry of spacetime manifolds, etc.), the use of such a term makes sense, while, to someone less advanced, the term may be confusing, as it is only natural to immediately wonder what sort of medium, (e.g., water) gravitational waves are traveling through, and to wonder why there are waves at all if there is no medium (but rather some sort of direct orthogonal transfer of whatever it is that is being transferred), and to wonder how such waves could "pull" when most waves seem to push things away.

 

As physicists often tend to use common everyday terms in scientific ways that are not entirely consistent with their every day use, I think that, in some cases, more carefully defining the terms one is using to those less versed in the advanced maths involved would simplify matters and eliminate prolonged confusion, particularly to those who are unable, or who will never have the wherewithal to understand the advanced maths that makes it all seem so plain and clear.

[ajb]

It sounds to me that in this sense the word "waves" is sometimes used in a rather metaphorical sense.

Not so much metaphorically, but mathematically.

[disarray]

Space-time is a manifold and you add extra geometry to this in terms of fibre bundles 'build on top' - you attach more manifolds to each point of space-time

Don't know if I am at all on the same page here, but I have wondered whether everything (to some minute degree) is deteriorating/decaying/ageing at different rates with respect to everything else once one took into account the seemingly infinite differences in reference frames caused by differences in acceleration and gravitational distances with respect to just one thing, such as that lamp post in my front yard. Even in the mind of some Einsteinian God, the possibility of ever keeping track of the various time dilations seems an impossible task.

Not so much metaphorically, but mathematically.

Yes, of course.....but I was presuming that I could speak about math as being a metaphor/model. After all, I presume that spacetime is more than the geometries we make to describe it.

[ajb]

After all, I presume that spacetime is more than the geometries we make to describe it.

I have no idea - all I do know is that the mathematical notions here are very useful in our understanding of the Universe.

[Mike Smith Cosmos]

Well! I am familiar with the mathmatical manipulation as a transform, whether that be :-

Logarithms , Fourier transforms , la place transforms , and whatever formulae one wishes to generate a particular transform such that in the new 'form ' a clearer picture may be derived, or an easier ability to manipulate data becomes possible .

 

But surely , somewhere in all these tiers of information , SOMEWHERE, in one of these tiers A REALITY , MUST EXIST .

And if not , then certainly the REALITY that we exist in , is the one we are primarily interested in . As we walk around in it , breath, live and exist . IF. at the same time ' a transform ' of all that information

 

' is doing piroets in A vaporous purple existence ,upside down .'

 

That is not going to help us terribly well ? Surely ? Other than , doing some otherwise impossible calculation , in the transformed state , doing a reverse transform and getting a digestable picture , out of the process?

 

Mike

Edited August 20, 2016 by Mike Smith Cosmos

[ajb]

...SOMEWHERE, in one of these tiers A REALITY , MUST EXIST .

This is rather a metaphysical question. Pragmatically, the best we can do is match our models to what we can measure/observe. Space-time 'exists' and is 'real'in the sense that we can use the mathematical structure to make predictions that match nature very well.

 

One can ask the same of the electromagnetic field or any other concept in physics.

[Mike Smith Cosmos]

This is rather a metaphysical question. Pragmatically, the best we can do is match our models to what we can measure/observe. Space-time 'exists' and is 'real'in the sense that we can use the mathematical structure to make predictions that match nature very well.One can ask the same of the electromagnetic field or any other concept in physics.

Yes, but this is paramount to our understanding of the Universe , we live and move and think and exist in .

 

What's going on up there , or down there , in the transformed state , be they ,

 

,' doing piroets in a vaporous purple existence ,upside down .'

 

may be mathematically beautiful , but it's not going to help me catch a fish and keep my family alive ? So to speak ?

 

Mike

Edited August 20, 2016 by Mike Smith Cosmos

[ajb]

... but it's not going to help me catch a rabbit and put it in a stew pot ?

I disagree with this. Much of our modern technology is founded on physics, which itself requires mathematics and sometimes quite advanced mathematics.

[Strange]

 

I gather that, in terms of gravitation, "waves" refer as much to the mathematical model (which may or may not have some sort of Platonic existence depending upon whom you are talking to), as to actual waves that can be detected on instruments. It sounds to me that in this sense the word "waves" is sometimes used in a rather metaphorical sense.

 

 

While it is true that the mathematical description is some sort of abstraction and not reality itself. It seems to be describing something real as the doctor does pick up a wave-like signal.

[Mike Smith Cosmos]

I disagree with this. Much of our modern technology is founded on physics, which itself requires mathematics and sometimes quite advanced mathematics.

.

 

 

Yes, I agree , I perhaps sounded disrespectful . Which was not my intention , and I do thoroughly understand . I have experienced working in transforms to understand otherwise impossible calculations , leading to inventions , serving the world population , with whole body scanners , etc etc.

 

I am rather caught up in the real problem for me. That is trying to understand , what is going on in the universe at large. And how it is working the way it does. If the ONLY way , Is , by mathematical manipulation , then I am SUNK .

 

I just really hope , in the end , the mathmatical contortions , can be translated back into something I can understand and relate to ?

 

Mike

Edited August 20, 2016 by Mike Smith Cosmos

[Strange]

Yes, but this is paramount to our understanding of the Universe , we live and move and think and exist in .

 

What's going on up there , or down there , in the transformed state , be they ,

 

,' doing piroets in a vaporous purple existence ,upside down .'

 

may be mathematically beautiful , but it's not going to help me catch a fish and keep my family alive ? So to speak ?

 

Mike

 

 

The computer you are suing to communicate with us depends on similar mathematical abstractions describing quantum theory; including some very counterintuitive concepts such as tunnelling. So this mathematics can produce useful results.

[ajb]

That is trying to understand , what is going on in the universe at large. And how it is working the way it does. If the ONLY way , Is , by mathematical manipulation , then I am SUNK .

I think mathematics is the only way.

 

Every experiment or observation made today is interpreted in terms of a physical theory. You need this mathematical background to make any sense of what you are looking at - well if you want some deeper understanding that is. One can make observations of the Moon and planets and doing so can be wonderful, but it is not really science until the maths hits in.

[Mike Smith Cosmos]

The computer you are suing to communicate with us depends on similar mathematical abstractions describing quantum theory; including some very counterintuitive concepts such as tunnelling. So this mathematics can produce useful results.

.

 

I agree with you all . I will go and sit in a corner and beat myself with thin bamboo canes of half an hour as penance for my disgraceful statements about ' mathmatical abstractions '

 

When you return from your transformed , abstracted state. Can you do me the favour , of coming over to my ' penance corner , where I will be coming ,more and more contrite , and telling me in simple English language , and digestable engineering terms

 

". What is making the Universe work the way it does "

 

Mike

[ajb]

". What is making the Universe work the way it does "

Maybe your local priest has an opinion on that - physics cannot really answer your question.

[disarray]

But surely , somewhere in all these tiers of information , SOMEWHERE, in one of these tiers A REALITY , MUST EXIST .
And if not , then certainly the REALITY that we exist in , is the one we are primarily interested in . As we walk around in it , breath, live and exist . IF. at the same time ' a transform ' of all that information

' is doing piroets in A vaporous purple existence ,upside down .'

That is not going to help us terribly well ? Surely ? Other than , doing some otherwise impossible calculation , in the transformed state , doing a reverse transform and getting a digestable picture , out of the process?

Mike

 

Well we all experience spacetime in ways other than just mathematical models. Football fans who make a group wave by swaying their hands above their heads certainly have the impression that they are interacting with spacetime.

 

I think that it is unclear whether gravity is a force per se...My understanding is that there is no such animal as "gravity" as a force per se, but rather that everything in the universe is moving in a straight line (or rather geodesic in spacetime curvature) as the universe expands so that gravity can be explained in these terms rather than in terms of a force.

 

But in any case, humans only have so many senses...and they can detect waves in air as sound and certain light waves as visual images, but they just ain't got senses for most EM waves or for time dilation or gravitational waves at all, so that, as far as everyday experience goes, we can only rely upon the math and stop speculating about what things beyond our perception might be like IF we could perceive them. (That is basically the Kantian view). By the by, I recall reading somewhere that humans can only perceive a 10 trillionth of the known EM spectrum. So no, I don't think that it is something that the mind can digest in terms of some big picture, and indeed we can't even 'imagine' what it means when it is suggested that the universe has no boundaries, but rather "curves" back onto itself.

Edited August 20, 2016 by disarray

[geordief]

@ ajb

 

Are these fibre bundles infinitesimally small and can they be "stuck together" or perhaps "spliced" like a 4-stranded rope to give the larger picture?

 

Are they like a 4D bicycle wheel with only 4 spokes centred at the local point in spacetime.?

 

I am just trying to imagine this as a mathematical model.

 

 

..My understanding is that there is no such animal as "gravity" as a force per se, but rather that everything in the universe is moving in a straight line (or rather geodesic in spacetime curvature) as the universe expands so that gravity can be explained in these terms rather than in terms of a force.

 

I don't think there is any connection between the universe expanding and things moving straightly along geodesics is there.?

 

It would help the progress of my understanding if that was the case but I think I have learned that it is not true.

Edited August 20, 2016 by geordief