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ajb

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Everything posted by ajb

  1. 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.
  2. Think about the cylinder and the Mobius strip.
  3. A fibre bundle is a manifold that locally looks like the product of two manifolds - but not globally. A trivial fibre buble is a fibre bundle that is globally a product. For example, the (finite) cylinder is a trivial bundle of the form IxS, where I is an interval and S is the circle. The Modius band is also a fibre bundle, but this is only locally of the form IxS as there is a twist - it is a non-trivial fibre bundle. You should think in terms of attaching an interval to each point on the circle and how many ways you can do this in a smooth way. Another non-trivial (in general) example is the tangent bundle of a manifold. This is is made by building up all the tangent spaces of all the points on the manifold. In fact, this is an example of a vector bundle as each fibre - ie, each tangent space - is a vector space. A principle bundle is a little more complicated, but again they are examples of fibre bundles. Locally principle bundles looks like XxG, where X is a manifold and G is a Lie group that acts on X. The frame bundle of a manifold is the principle bundle associated with the tangent bundle. It consists of all the ordered bases of the tangent spaces and these come with a natural action of GL(n) - which is just a change of bases. So, for general relativity, the mathematical setting can be understood as the frame bundle, which locally looks like MxGl(n) - though as we have a metric we can reduce the group structure here by picking just the othogonal bases. For electromagnetism, we have a U(1) principle bundle, so this locally looks like MxU(1). It may also be worth saying something about connections. A fibre bundle is a collection of 'fibres' - we have another space attached to each point of the 'base' manifold. Although these fibres are all 'the same' we don't have a canonical way of mapping the fibre at one point to the fibre at another near by point. This extra information that give or construct from other structures is a connection. I will just say that there are a few ways of understanding connections, but the most intuative is that connections connect near by fibres.
  4. People who work in cosmology do question it - the principle is the backbone of our cosmological models, however people are looking at inhomogeneous cosmologies, usually to see if dark energy can be removed in this way. Anyway, I think it maybe a question of the scale at which we expect the principle to hold well. Other evidence from observational cosmology suggest that we don't want to throw this principle away completley. It would be hard to work with for sure. I think the CMBR fits okay here - maybe not if we don't include inflation. I think it is more a question of scales again. Anyway, we need to wait and see what further analysis and observations reveals before getting ahead of ourselves.
  5. I am not sure how much we can turn it down and still say something meaningful. If you really want to get to grips with what the EM field 'is' and what gravity 'is', then you will need to learn the maths. Or just be happy with what has been said - gravity is the local 'shape' of space-time and that electromagnetism can be understood in a similar, but different way.
  6. Yes. You point your telescope at the sky. You will have to read the paper carefully - I would need some more context to answer that properly. Yes, both models and observations give us a structure like a spounge or something like that on quite large scales. Only on larger scales again does the Universe really start to look uniform. I forget the numbers here, you can google for details. I am not sure in this case. In general, lots of interesting science is just too specialised for this forum. In this case, I guess it is the fact that people here don't like to jump on things that are outside of there interests or expertese. I mean, it is not totally clear if this ring is real- if it is do we have a mechanism for its formation? Or do the cosmological models need more work?
  7. No, this is the scale of the differences in temperature as compared with the average. This means that the CMBR is quite uniform until you get to 1 part in 100,000. There are also some hot and cold spots in the CMBR that are unexpected and may just be down to not subtracting the local environment properly. On the subject to this ring structure, you can read the paper here http://arxiv.org/abs/1507.00675 The point is that it seem to be too be larger than the scales that we would expect the Universe to appear homogeneous and isotropic. This violates the cosmological principle, which is of course an approximation - though thought to be a good one.
  8. The temperature fluctuations are something like 1:100,000. We think this is quite small - but what we expect from theory.
  9. Sure - the anisotropies are small but very important, for instance they are linked to structure seeding in the early Universe. I think that Strange also knows this.
  10. I am talking about the CMBR and the scale of the fluctuations. The ring of gamma ray bursts is something else. If these bursts really are all connected then they form a larger structure that the current models really allow - and the details of the CMBR support the current models. So, either it is chance that they formed in this way they have, or some of the data on their positions is poor, or we have to think more about the models. Interesting stuff either way.
  11. Yes - the CMBR has very small fluctuations in temperature when compared to the average. This is well established science. To carefully measure these small fluctations... and remember there has also been balloon made measurements. Because the details of these small fluctuations are important in understanding models of cosmology. They are like the fingerprint of the Universe and allow us to rule out certain cosmological models. So right now the best model is the Lambda CDM model with inflation. I think you are misunderstanding the fact that these fluctuations are small with the idea that they are not important. None of us has said that these fluctuations are not important - just that they are small and that the CMBR is near homogeneous.
  12. Space-time has 'two parts' i) the structure of a smooth manifold of dimension 4 - meaning locally it looks like R^4. ii) on that manifold we have a metric of signature (-1,1,1,1) (or (1,-1,-1,-1) depending on conventions) The manifold structure we think of as the collection of all possible events - we don't dwell too much on what that means. The metric structure gives us a notion of the distence between two near by points, which we can also use to denfine the length of paths joint two points. This really encodes the causal structure. Gravity we can think of in terms of this metric - the metric is like the gravitational potential found in Newtonian gravity. When there is no gravity the space-time is Minowski space-time, but when we have gravity the metric is different to the Minkowski metric. Now, we like to measure how this is different. To do this we construct a connection - which gives us a way of moving vectors from one point to another near by point - and then we look at the curvature of this connection. We can think of the gravitational degree of freedom in terms of this connection rather than the metric. We can do something similar in the case of EM, but we start with a connection (on a specific fibre bundle over the space-time) which we understand as the potential A. The field strength is the curvature of this connection (all mod possible plus or minus the complex unit).
  13. It is very nearly homogeneous and the scale of the fluctuations is very small compared to the average temperature. It is all a question of scales. As for the possible structure you ask about - I have no idea, it is far from my area of knowledge.
  14. You are right - so I we change my statement to 'one could look up Galois theory to understand the reasons why the answer is no'.
  15. Both the EM and gravitational field need space-time thought of as the underlying manifold of 'events'. But be careful here, gravity is really the local geometry of space-time and not 'just' space-time. Gravitational waves are ripples in the local geometry of space-time - or really the frame bundle thereof - and electromagnetic waves are ripples in the local geometry of a U(1) principle bundle over space-time.
  16. Dedicated a preprint to my father - http://arxiv.org/abs/1608.01585

    1. jimmydasaint

      jimmydasaint

      A touching gesture. I couldn't understand a word but it is a fitting way to show your respect and love.

    2. StringJunky

      StringJunky

      Yes, a lovely gesture.

    3. Theoretical

      Theoretical

      Very nice indeed!

  17. It is impossible in general for polynomials of order 5 and above - see the Abel–Ruffini theorem which states there are no algebraic solutions of such polynomial equations. Note that this does not mean there are no solutions (real or complex), just that you cannot write then in an algebraic form (in terms of radicals). Also it does not mean that you cannot find algebraic solutions to some polynomials of higher order. You should also look up Galois theory which is the theory that deals with this question properly.
  18. My father died last night - so I won't be posting here as regular

    1. Show previous comments  15 more
    2. CharonY

      CharonY

      So sorry to for your loss and my sincerest condolences to you and your family.

    3. Function

      Function

      Sorry to hear that. My sincere condolences, ajb.

    4. Ant Sinclair

      Ant Sinclair

      My condolences and best wishes to you and your family ajb

  19. You take a peice of paper and a pen, and do some calculation that say I will measure some value of something to be X. I then do some experiment and I see that I do, near enough measure the value of that thing to be X. We can claim no more that this.
  20. A great cover... I won't make you suffer the original
  21. It is like good porn - you know it when you see it, but can't quite describe it to others.
  22. I am not sure we can decide what metaphysical view of space-time is correct. All we can really do is put our models to the test - and with that general relativity works. I don't think one can say a lot more..
  23. You will have to think carefully about this. As all our interpretations of experiements and observations require a mathematical model, can you really separate the two? Without introduces the other forces at this stage, lets say the electromagnetic field and the gravitational field. The problem is we don't know what dark energy is and so saying that it gave rise to the fields of the standard model - or just gravity and electromagnetism - is just a story. You need so show some mechanisms here. As to where the fields came from, we don't really know. What we do know is that with supersymmetry all the running couplings ('interaction strengths') of the standard model come together at a high enough energy. This suggests, in the same sort of way as electromagnetism and the weak force can be unified, all the fields of the standard model an be unified. This leaves out gravity, however maybe at or near the Planck sale a full unification is possible. We just don't really know.
  24. Well, what is the problem? Do you have a real objection to the electromagnetic field as a mathematical object? You can understand this field in terms of some differential geometry - it is related to a connection on a U(1) principle bundle. Not that details matter at all, just that we do understand EM in terms of geometry. In a similar, but different way, we can understand the gravitational field to be a mathematical object that encodes all that we need. This object we understand, again in terms of differential geometry - in the standard formulation we understand the gravitational field to be a metric tensor. But we can also understand it it terms closer to EM as a connection on a principle bundle known as the frame bundle. Again, details are less important than the ideas. You will have to think about the objections you have to gravity and if these are really any different to your objections to electromagnetic theory.
  25. I don't think that Einstein was greatly interested in metaphysics and so I would not be surprised if you can find differing quotes. I know for sure that he was not impressed by some philosophical views on time. He is free to hold that view. The problem is that as our understanding of space-time changes, and it must do at some level when we take quantum effects into account, will the classical view of space-time still be 'physical'? I am not sure, other than mathematically we should have a good understanding of how the classical notion comes from the quantum one - otherwise we will be at a complete loss as to why classical general relativity has worked so well so far. All the tests of general relativity tell us that the physical theory matches nature very very very well. I don't think one can really say much more.
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