ajb

By 'good' we mean that the theory agrees well with nature taking into account experimental errors and the domain of validity of the theory. Where there is some subjectiveness is in the 'agrees well'; we define that ourselves though typically one would set that to mean within the experimental errors. For example, there are no known phenomena that do not agree with general relativity to within the experimental errors and the theory has been tested to some huge degree of accuracy (I forget the figures, but it is on par with the standard model of particle physics). Similarly, when dealing with the lambda CDM model, which has general relativity as part of its 'backbone', we have nice fits for the parameters. The theory seems to fit nature generally very well, just not so in the very very very early universe.

This would mean abandoning general relativity as a whole. We know that under rather generic conditions singularities are an inevitable feature of the theory. Much like the infinities that arise in classical electromagnetic theory due to the electron's self-interactions. General relativity is a good approximation of reality as long as we are not near the scale of quantum gravity. Similarly, the standard model of cosmology is a good model, but we know it breaks down when we are close to the singularity.

Right, but this is according to the classical theory of gravity. The presence of such a singularity signals that the classical theory breaks down as we approach this singularity.

I would write the transformation law for the components of a metric as [math]g_{\mu' \nu'} = \frac{\partial x^{\rho}}{\partial x^{\mu'}}\frac{\partial x^{\epsilon}}{\partial x^{\nu'}}g_{\epsilon \rho}[/math], for changes of coordinates [math]x^{\mu'} = x^{\mu'}(x)[/math] (with the standard abuses of notation). I don't quite follow what you have written, but I expect this is just a notational thing.

The idea of parallel transport of vectors and curvature is easy to demonstrate in 2d. We can show this 'Blue Peter' style. Take a ball or balloon. Mark three distinct points on it and joint them with lines as directly as you possibly can. You will get a 'triangle' but now on the surface of your sphere. Next take a pencil and place it on one of the points. Now move the pencil round the path you have drawn so that it returns to the initial point. You must do this carefully without changing the pencils direction, do not add any rotation by yourself, just let the pencil follow the path. If you do this carefully, you will see that the pencil does not return to exactly the same configuration as you started. The pencil should be pointing in a different direction. It will be 90 degrees rotated as compared with the starting configuration. This angle is the curvature. They the same on a flat sheet of paper and also a cylinder. You will then have proved that a cylinder is flat! Loosley, the pencil represents a vector and moving it round the said loop without changing the vector represents parallel transport. The deficit angle is the curvature. In higher dimensions the idea is the same, just you have to make this all more mathematical and defined properly. But the basic idea is the same.

Energy is a property of physical configurations. This would include fields, massless or not. I am not quite sure what is meant by this mathematically. The general notion of a space is probability too loose to be much help, but in general relativity we are dealing with smooth manifolds with the additional structure of a (pseudo)-Riemannian metric. With this metric one can construct a canonical volume and indeed define integrals etc. However, generically I would say that a space (as we mean it here) is a collection of points together with some topology. In relativity, we can interpret these points as possible locations for 'physical events'; maybe simply the possible locations of a test particle or something. Okay, so a space-time is a pesudo-Riemannian manifold. This means that at every point the metric can be brought into the diagonal form (-1,1,1,1) (depending on your conventions). The point is one of the components has a different sign and this signals we have space and time. Note that this does not imply we have some canonical decomposition into space and time, just that it is always possible locally to do this. The solutions to the field equations are metrics on given smooth manifolds. The problem is that the field equations give no direct information on the global topology of the space-time, we can at best search for consistent topologies. There are topological obstructions to the existence of Lorentzian signature metrics. You would hope that any quantum theory of space-time will in some sensible limits reproduce the classical notions of space and time. So, provided we are not near the scale of quantum gravity, which is the Planck scale (though it is possible that this scale could be much lower), we can safely assume our classical description is good.

The cosmological principal looks good when you get to the scale of clusters of galaxies; as does the notion of Hubble expansion. This must be your 100 Mpc scale or there abouts.

Can you explain to me carefully what pure energy is and what is pure matter?

Was it ever actually thought impossible in principal, or just very difficult to achieve, maybe beyond the technology of the day?

I think so. Individuals using facebook have been arrested and prosecuted the UK, but as far as I know facebook itself has not been prosecuted. I will have to look up the details, but for sure some existing UK law on liable, copyright, child pornography and similar applies to the internet.

Maybe a serious point then I don't think international law has fully caught up with the internet. I am assuming that as this forum is hosted in the UK, British law would apply...but I have no idea what that is specifically.

A semi-serious point here... even if the first amendment held in principal, many of us are not US citizens and this forum is not hosted in the US. Taken from Forum Rules. In British law we do not have such a clear statement as the first amendment. The various laws on the subject need to be looked at, including EU human rights laws, and some of these laws may even contradict. A judge then has to see which laws trump the others in the particular circumstances in question and make a ruling. Now, I doubt one would have much of a case if booted off this forum, it is a "members only club" with its own rules. Anyway we have to accept this is an international forum, privately owned outside of the US.

You would need to have some mathematical model here before anyone will really take this seriously. In fact, without a proper mathematical model it is hard to understand what you are trying to claim could be a good description of nature.

They also miss the fact that concepts in physics are inherently mathematical as they are tied to theory. Some seem more natural than others, but when it really comes to it we are phrasing things within a model and comparing this with nature. Discussions about energy not being real, or time or whatever can more or less be applied to any concept in physics. The objections become very metaphysical quickly and rarely clarify the understanding of what physics is and what physics can really do.

The main thrust of the argument is known as Russell's paradox, which shows a contradiction in naive set theory. It was one of the motivations for axiomatic set theory, but lets not digress. The point is I can naively imagine that such a set exists. But when I really think about it and use some basic logic applied to the mathematics I reach a contradiction. But without thinking about the mathematics properly I may have never realised the contradiction. I may even have used this set to establish other results, which of course would be pants! No mathematics, or at best a poor interpretation of the mathematics, coupled with little knowledge of physics is a recipe for failure. I can imagine what I can imagine, but there is no reason why the world has to agree with my mental pictures.

Great question you pose; is there only a finite amount of mathematics and are we indeed going to be privy to all of it?

That sounds a lot like mathematics. Maybe we cannot define mathematics, but one knows when one is using mathematics!

Changing base should not be a big problem. But anyway, there are problems with notation across branches of mathematics here and now! Or would it just be mathematics that we have not discovered/invented yet?

But is it tied to the human mind in a fundamental way? Or really any mind as I assume any intelligent aliens would have mathematics and further more we would understand and agree on this mathematics. (Mod complications of notation and any time scales to actually absorb this information etc.) Sure. I think the definition in terms of "axioms -> theorems" describes how mathematicians work, rather than what is mathematics. You now have to define carefully what a mathematician is! This can be a problem with any definition. You want to avoid shifting the question to an equally as hard one.

I guess we have to be a little careful with the distinction between the objects themselves and the notation used to represent them. Though one school of thought, "formalism", would not make too much of an issue of this. Here mathematics is a formal game using symbols and have nothing to do with nature at all. It is all a game of human invention. The other extreme is "realism"; mathematical entities exist independently of the human mind, thus mathematics is discovered and not invented. Personally I have no idea which end of the spectrum here is closer to being right or wrong. I don't think we can ever actually decide this. But, given how mathematicians work I am tending towards realism. Any thoughts? This is a great question. See my comments above. At the fundamental level that is how mathematicians work. You take some starting statements and use whatever tools you have, maybe even inventing some, to derive new statement. My opinion is that this is a too simple point of view that undersells the beauty and power of mathematics.

Not at all, any humor is welcomed. I think just about all dictionary definitions of mathematics are equivalent to this. What is meant by arrangment? I guess this is the same as "structure".

Stripped to its bare bones, mathematics can be viewed in that light. One proves theorems given some starting axioms that we agree on. But that, to me, seems to undersell what mathematics really is. Okay, mathematics grew out of the need of commerce to keep track of things. Indeed multipliction most likely has its origins in tedious additions and so on. But there is more to mathematics than just basic operations on numbers. It is true that the real numbers, or collections of real numbers can be useful in representing more abstract ideas, mathematics in no way is constrained to just numbers. Simple plane geometry is a good example here. The Greeks developed this mathematical theory synthetically. That is "without reference to anything else using logical deductions". Then Decartes applied numbers to geometry by defining the notion of local coordinates and so analytical geometry was born. But even in anaytical geometry, points and lines "exist" independent to the "numbers" used to represent them. As mathematics became more and more abstract a simple connection with numbers was generally lost. This is more a question of the usefulness of mathematics. This is a seperate question, and one we could go into in another thread. That said, I doubt many people here will need convincing of the usefulness and applications of mathematics. Right I take your point. But this is now a question of mathematics education, which is an awkward thing. Not everyone will become pure mathematicians and this needs relfecting in any syllabus. My own experinces are in post compulsary education, so 6th form and university undergrads. Here one still needs to take care of the wide range of motivations each student has for studying mathematics. But again, this is really another question for another thread. Old joke, but still funny! Thank you for the link.

A quick "google" with give you several definitions of mathematics. To me they tend to be necessary, but far from sufficent. As example; Mathematics is the abstract study of numbers, shape, structure and change. Numbers are part of mathematics, so are simple geometric shape, by structures we mean patterens and relations between them, and differential calculus ("rates of change") are all parts of mathematics. This definition seems okay, but a little vague and it may not be very clear what we mean by a structure. Also, as mathematics evolves, solutions to problems often come from outside the area the problem was initially defined, any defintion must be able to include any future mathematics. It maybe better not to define mathematics by what it studies by how it studies them. A first attempt at this would be to reduce all mathematics to logic, but this fails. Mathematics as a science is also a tricky issue. There are many parallels between how a mathematician works and the philosophy of the scientififc method. Anyway... My question to you is what is mathematics?

Back in 2010 the Science is Vital set up a petition in favour of protecting the UK research budget. Thankfully funding for science and engineering was ring-fenced and frozen instead of being reduced. Now Science is Vital has another active petition, that I invite you all to sign, to urge the UK Government to increase its research and development spend in 2015-16. Link Science is Vital latest petition

The CERN gift shop? Seriously now, antimatter can be found quite naturally all around us, just not in any great quantity. One has to use artificial methods to create antiparticles and this produces not exactly huge amounts. For example, it has been estimated that it would take 100 billion years to produce 1 gram antihydrogen using the methods employed at CERN's ALPHA experiment.