Markus Hanke

The modern formalism of the Maxwell equations uses differential forms. This formalism is fully covariant, and all of classical electrodynamics reduces to two very simple statements: [math]\displaystyle{dF=0}[/math] [math]\displaystyle{d\star F=4 \pi \star J}[/math]

Just to clarify my earlier comment a bit more - they are scale related in the sense that you won’t see quantum effects on large (i.e. macroscopic) scales. If you have a single quantum object, then quantum effects are always apparent, but if you have a very large ensemble of quantum objects (like a mascroscopic body), then the statistical average of that ensemble’s dynamics becomes classical. In other words - it’s actually classicality that is an emergent, scale-dependent phenomenon, whereas ‘quantumness’ is a fundamental property of the universe.

No, it refers to the fact that quantum objects sometimes behave like waves, and sometimes like particles, depending on what kind of setup you use to observe them. They have both wave and particle characteristics (which are complimentary descriptions of the same entity), hence the name “wave-particle duality”.

Yes. We fully understand the classical behaviour of the electromagnetic field - we can say this with some confidence, because we know not only the Lagrangian from which it arises, but also the topological principles that underlie the structure of the electromagnetic field. And of course we know that it is the classical limit of QED. So we can check its completeness from several different angles. It’s the other way round - you deduce classical electrodynamics from quantum electrodynamics. The quantum description is more fundamental, and exhibit phenomena that do not exist in the classical world. It is a much broader framework, so no, it cannot be deduced purely from Maxwell. What you can do though is work backwards, and apply the tools of quantum field theory to perform a full quantisation of the classical field theory, to arrive at QED. But this quantisation procedure does not follow from Maxwell’s equations, it’s an external tool based on the principles of quantum physics.

The energy-momentum tensor remains the same (more precisely: covariant) regardless of the state of relative motion of your test particle. Also, remember that GR is a purely classical theory, so strictly speaking it has no concept of photons - it would be more correct to speak either of an electromagnetic field, or of null dust. Either way, the energy-momentum tensor is always covariant.

Not really, except of course that it would make conditions on Earth largely unobservable. We would have to rely on indirect evidence (such as spectral analysis of light etc) - which is precisely what we are now doing in the case of all those many exoplanets we have found around other stars. No. What would change is only our observation methods (see example of exoplanets), but there would be no doubt that what we observe is a purely classical system. Quantum effects are intrinsic to the nature of quantum objects, not just an observational artefact.

It would not be the same - what one would see is a very tiny (relative to the observer), but purely classical system. In other words, quantum effects are not just an artefact of observation scale. Also, there are no “problems” in quantum physics - it’s just different from the classical world at larger scales.

Maxwell‘s electrodynamics is the purely classical limit of quantum electrodynamics - hence it does not and cannot account for any quantum effects.

Why would that be a problem?

Gravity is geodesic deviation - and in the geodesic deviation equation, it is not straightforwardly possible to separate the “time” from the “space” contributions. I don’t think such an approach is very meaningful. What we can do, however, is look at the metric itself, and sometimes (like e.g. in the case of Schwarzschild) we see that the time part plays a bigger role than the space part, due to the presence of a factor of c squared. It should be noted though that this is not always true - for example, in the immediate vicinity of a black hole with a very small mass, the spatial contributions would be at least on the same order of magnitude than the time contributions. And then of course you have cases where the metric has non-vanishing off-diagonal terms...

This is a science forum, so the benchmark for whether something is appropriate and constructive is the scientific method. Unfortunately your idea fails on that hurdle (its failure is akin to claiming that leaves are actually blue, not green), and quite spectacularly so. Pointing out to you that it does fail when the scientific method is applied is constructive feedback. The trouble is not the feedback you have received, but rather your own perception of it. You are looking only for positive comments, because you aren’t open to criticism; this is a very common thing, and not exclusive to you by any means.

What is the actual question or point?

No problem at all Crossed wires happen, so no offence was taken. +1 for a very kind response!

This sounds like you are claiming I edited your post - I am not an admin, so I do not have the power to do that. And even if I did, I still wouldn’t, because that’s wrong. All quotes in this thread are accurately labelled with the correct names. Hence I also never implied that my words were in fact yours. Looking back over the thread, it is not clear to me at all how you could even think that. I never claimed that the electron is never in the nucleus, and I don’t think the OP did either. I do concede though that the way I formulated my reply was not precise enough - it would have been better to say that the greatest probability of finding the electron is within a certain area located outside the nucleus, which does not preclude it being found someplace else. I also concede that my comment about tunnelling was incorrect, as - having read up on it a bit more - that is not what happens. I was wrong on this one. You clearly have good knowledge of quantum mechanics, and I appreciate your input on this thread - but I would also appreciate if you could be a little less hostile. There is no need for this at all. It reads like you think I have some sort of beef going against you, which is not the case at all - I was merely trying to help the OP with his question.

When I first learned about GR, the notion of some free parameter in the metric (such as mass, charge, angular momentum) being a property of spacetime itself rather than any material body in it, took me quite some time to get my head around. But it is what it is.

The basic problem is that if you only have two isolated points which do not belong to any underlying manifold of at least dimension 1, then it is not possible to establish any kind of meaningful relationship between them. You can consider a point as being a 0-dimensional topological space, but such spaces are fully disconnected.

If you have only two points that are not part of the same manifold, then there is no spacetime, and it is not possible to establish any relationship between those points. Notions of space and time are thus meaningless in such a scenario.

I may be wrong on this (someone correct me, if so), but I think there has not actually been any observations of electrically charged black holes (or any other body with substantial net charge, for that matter). It may still be possible for this to happen during short periods of time, but I think such net charge would be neutralised fairly quickly. Actually, the Schwarzschild solution is just a 1-parameter family of metrics that arises from the field equations for a certain set of boundary conditions - one amongst which is asymptotical flatness. It is also a vacuum solution, so the energy-momentum tensor vanishes everywhere in this spacetime. We identify the one free parameter that appears as “mass”, but that is not a source that is part of the manifold, and that identification with mass arises from boundary conditions (specifically, from the demand that Newtonian gravity is reproduced at infinity). As such, the mass is actually a global property of the entire spacetime, not of any body within it.

I honestly don’t understand what it actually is you are asking - could you reformulate your question in a different way? At the moment you are bringing up many different concepts which are not really related, for example SR mixed with wave-particle duality, or the evolution of the universe.

I wouldn’t think so, since all objects we observe around us carry some form of angular momentum. It doesn’t seem as if such a cancelling out is happening. The thing is that the Schwarzschild solution requires spacetime to be empty and asymptotically flat, neither of which is a situation we actually find in the real world (except as an approximation). It is also stationary, which excludes any evaporation processes, or any changes at all to the black hole over time. I think in terms of real-world finding, a Vaidya-type spacetime is far more realistic - specifically, the Vaidya-Bonnor-Kerr metric could be a good model of the natural end state of a black hole.

Actually, this is not correct. The charge that is associated with the weak interaction is a quantum number called “weak isospin”. All particles possess this, and it is conserved in all types of interactions - unlike flavour, which is carried only be quarks, and is not conserved anyway. Thus, the weak interaction concerns more (elementary) particles than just quarks. To be honest, I don’t think there is an intuitive way to really understand the weak interaction. It’s really quite a complicated mechanism, and requires quite a bit of background knowledge in quantum field theory to fully understand.

I think what you are asking is whether solutions to the gravitational field equations linearly superimpose (like EM waves do). The answer is no, unfortunately. Unlike electromagnetism, gravity is a non-linear affair - this means that the effects of the rotating bodies will combine in some way to yield an overall metric of spacetime, but this combination is not a linear superposition, but something much more complicated. Having four charged, rotating bodies in General Relativity is a scenario so complex that it could only be solved numerically (and would require substantial computing power to do so). Yes, it generally would. I’m not so sure about this one. But what mechanism would it get neutralised? In either case, this does not appear to be what we observe around us. The Schwarzschild solution requires spacetime to be everywhere empty and asymptotically flat. This is not a situation that we will ever find in the real world, except as an approximation. No black hole can ever be perfectly Schwarzschild.

No, but you are tacitly assuming it in the way you are thinking about the situation. For example in your next sentence: Different times according to who? Since time in relativity is a purely local concept, simultaneity becomes relative too. The same is true for “always” - it’s an observer-dependent concept. Actually, it isn’t. The metric is a tensorial quantity, so the relationship between events in spacetime does not depend on your choice of coordinates. In the specific case of Special Relativity, you don’t even need to pick any coordinate system at all - you need to know only that the metric is constant. This is a valid question, but it is not how I understood your OP. So I may have misunderstood you. Perhaps you can clarify again exactly what it is that you are asking, maybe in more general terms? In relativity there is a causal order to things, but not a chronological one (i.e. one that is based on clock readings), because time is a relative concept that not all observers agree on. The notion of “at the same time” is thus not shared by all observers. What is shared between all observers is the spacetime interval between events, specifically whether it is a timelike, spacelike, or null interval. This imposes a causal structure on spacetime, which everyone agrees on.

This is true, but it is not what the OP has asked. The original question was why the electron does not fall into the nucleus, i.e. how is an atom different from a purely classical system of a charge in free fall towards another (opposite) charge, which of course is not a stable situation in the classical domain. So the OP wanted to know how this is possible, so I have attempted to answer the question. The spontaneous tunnelling through the nucleus - or any other classically forbidden region - is not the same as the electron “falling in”.

I’m having trouble understanding your point, I’m afraid. There is no absolute time nor is there absolute space, so it is irrelevant just where you place your point of origin. Either observer can choose any suitable coordinate system he/she likes, as this makes no difference to the physics. What is important is only the relationship between the observers, and this is something they both agree on. Specifically, and assuming they are inertial observers in flat spacetime, their two reference frames are related by a simple hyperbolic rotation about some angle, and the spacetime interval AB is the same as the spacetime interval BA. So there are no contradictions. More generally, you can actually prove that no matter how you set up your observers, there are never any inconsistencies. It should be mentioned also that spacetime is the collection of all events, i.e. of all points in space at all instances in time; there is not actually any concept of “flow of time”, there are only geometric relationships between events in spacetime. The theory of relativity cannot answer the question as to why we experience a “flow” from past to the future, as this is outside its scope.