Markus Hanke

This is not what happens. All objects fall at the same rate (in vacuum).

Yes. I should note here though that the equivalence is between uniform acceleration, and a uniform gravitational field. This means the equivalence holds only locally, in a small enough region. On larger scales, a gravitational field due to a mass distribution (such as a planet) is not uniform.

As I said before, the electromagnetic field (and it’s associated energy-momentum tensor) is always the same, it’s just that different observers see different aspects of it, depending on their own perspective in spacetime. That is why some see radiation, whereas others do not. But breakage of a macroscopic object isn’t like that - it either happens or it doesn’t. Everyone agrees on the outcome, they just won’t agree on the where and when.

No, because all observers agree on the value of proper acceleration of that object, no matter what reference frame they are in. This proper acceleration is calculated in a way that is independent of the coordinate system chosen.

Actually, the elemental charge itself depends on both vacuum permittivity and Planck’s (barred) constant, the numerical values of which have to be determined experimentally. The fine structure constant can also be obtained in a number of other ways, so while this article is interesting, it does not make any unique predictions as such.

To some degree, yes. Nonetheless, whatever the outcome, it must fit into all the rest of what we already know about physics, so it is by no means an arbitrary process.

I don’t know whether quantum foam (of spacetime) would suffice as a countermeasure to gravitational collapse. I guess one would have to bite the bullet and do the math on this, which most certainly wouldn’t be easy. What I do know though is that some models that quantise spacetime itself (i.e. where space and time cannot be infinitely subdivided) naturally and explicitly lead to such mechanisms. Case in hand is Loop Quantum Gravity (I’m picking this because the spin foam of LQG is mathematically related to Wheeler’s original quantum foam idea) - in this model, there is a smallest unit of area and volume, and if you do the math, you find that no singularity forms during gravitational collapse. Instead, the discreteness of space counteracts the collapse, and infalling matter “rebounds” in that non-classical region, and begins to move outwards again (presumably in the form of black body radiation at the horizon). So essentially, a black hole is a cyclical process - infalling matter, then a bouncing outward. The bouncing out bit, however, happens in the far future, due to the extreme time dilation in the non-classical region. We are talking very large time scales here, many times the current age of the universe - which is why we don’t see a bunch of exploding black holes around us. The same mechanism would apply for the universe as a whole - there is no singularity at the BB, but rather a “Big Bounce”, meaning we’d be the result of a “previous” (a dangerous term to use in this context, but you get the drift) universe collapsing. NOTE: This is not a proven theory, just an example amongst many different hypotheses that are currently under investigation. I’m not claiming that this is a valid description of what really happens - we don’t know this just yet.

In very/overly simple terms, a physical (as opposed to coordinate) singularity in GR is defined as a region of geodesic incompleteness. This means that time-like geodesics cannot be extended into the future past the singularity region. This definition does not assume any specific topology or geometry for such regions; for example, you can also have ring singularities, and naked singularities (whether these exist in the real universe is a different question, of course).

Because the QCD Lagrangian is so complicated and so highly non-linear, most calculations in QCD cannot be performed in closed analytical form (i.e. with pen and paper, in principle). Numerical methods need to be employed, using powerful computers. And even then, obtaining exact results is often very difficult, because it is simply too computationally resource-intensive. For that reason, various approximation (i.e. non-perturbative) methods have been developed to enable us to run numerical computations in QCD, the most common one probably being a model called “Lattice QCD”: https://en.m.wikipedia.org/wiki/Lattice_QCD

There are two types of measuring acceleration - there is proper acceleration, which is what an accelerometer in the accelerated frame would physically record. This is “absolute” in the sense that all observers agree on it. And then there is coordinate acceleration, which is what a distant observer would calculate using his own set of clocks and rulers; this cannot be measured with any kind of accelerometer, and it is an observer-dependent quantity, and hence relative. The two are not the same at all. Consider the case of the charge in free fall again - an accelerometer attached to the charge will read zero at all times (since it is in free fall), so there is no proper acceleration. However, a distant observer who looks on from “the outside” sees the charge fall faster and faster as it approaches the Earth; using his own clock and his own way to measure velocities, he thus calculates a non-zero coordinate acceleration. What the magnitude of that coordinate acceleration will be depends on the observer - not everyone agrees on it.

Interesting article, thanks for the reference!

Very true, thanks for the reminder!

I spoke about rotations in the post you quoted me on, as did you yourself in the post I quoted you on. You then said that this has never been studied, which is evidently wrong. As for electrodynamics, of course this can be formulated using quaternions - even Maxwell himself did this. You need to realise that there is nothing special at all about the quaternion formulation - using quaternions just means you are using different language to describe the same physics. Here’s a sample text of how electrodynamics are done using quaternions: https://www.scribd.com/document/91253848/quaternionic-electrodynamics As a matter of fact, quaternions are a very common tool in mathematical physics, and are used extensively in all different areas. To prove that point, here’s a further 1382 (!) references to texts and papers employing the quaternion formalism in one form or another: https://arxiv.org/pdf/math-ph/0510059.pdf Are you still so sure that they have never been studied (and not just in electrodynamics)?

Using quaternions for rotations - as well as comparing that method to other methods - is pretty basic material, and has hence been well studied and understood. You’d be discussing this stuff on an undergrad level of a math degree. There’s even a Wiki page on it: https://en.m.wikipedia.org/wiki/Quaternions_and_spatial_rotation

You are right, I misread this. My bad. In this case, what I had written in the third paragraph of my earlier post applies - as you correctly say, the earth-bound detector will detect radiation.

A couple of points: Vector algebra stipulates no such thing The curl and divergence operators are differential operators; they are not part of vector algebra, but rather vector calculus You are trying to contrast vector algebra with quaternions, but quaternions and their inner product are an early type of vector algebra To avoid any of these types of issues, I have presented the differential forms formalism, which supersedes both quaternions and vector calculus, since it is both covariant and more general than either of these. Differential forms are in fact antisymmetric tensors, so there is a strong link between exterior calculus, and tensor calculus The curl (being a differential operator) does not actually represent any rotating bodies, but infinitesimal rotations of a vector field about some point. You can indeed use quaternions to represent rotations (such objects are called versors), but this method is not any more or less advantageous than more common methods such as rotations matrices. In fact, I’d say an argument can be made that quaternions are more cumbersome, and less transparent, in terms of computational effort.

I would strongly advice any readers of this to not open attachments from unknown and potentially hazardous sources. Doing so presents a substantial security risk. Post reported.

What exactly do you mean by “topological difference”? If you smoothly deform the region in question (no breaks, holes etc), then all topological invariants remain the same, and you end up with something that is topologically equivalent to what you started with. This is just a diffeomorphism, and GR is diffeomorphism invariant, so you will get the same physics in that region. As a practical example - you start off with a region of spacetime that has the topology of a torus. You then transform the “shape” of that region into that of a teacup - they look different at first glance, but all topological invariants remain the same, so this transformation is just a diffeomorphism, which leaves the physics untouched. You do the same physics on a teacup as you do on a torus.

In a general scenario where you combine two spin-½ particles, the overall system would have either total spin 0, or total spin 1, with (I believe) equal probability.

This is not correct. There is no relative motion between detector and electric charge, so, from the perspective of the detector, there is no magnetic field. Without a magnetic field, there can of course be no radiation (which is an oscillating electromagnetic field) - despite the fact that this is a “supported” (i.e. accelerated) charge. Hence, in condition A), the radiation detector detects nothing, because there is no magnetism there to form any radiation. More generally, a “supported” observer in the same frame as a “supported” charge will never detect the charge emitting any radiation. The same is also true when you put the entire assembly into free fall - again there is no relative motion between detector and charge, and hence there is also no radiation detected in condition B). This preserves the equivalence principle - it’s in fact just a simple charge without any forces acting on it locally. It is very important to understand though that, in a curved spacetime background, the answer to the question of whether or not there is radiation detected is a purely local one; if the detector were placed (e.g.) on the surface of the Earth, as opposed to in a co-moving frame with the charge, the outcome would be different. For condition A) you still would not have radiation, but for condition B) you would now detect a radiation field, because there is accelerated motion between detector and charge. Freely falling charges will always appear to radiate from the point of view of a “supported” observer. The paradox is only an apparent one though, because observers in curved spacetimes are not related by Lorentz transformations, so they do not necessarily agree on which aspects of the electromagnetic field they detect, and which ones not. It’s not a contradiction, just a question of perspective. The other thing of course is that an observer who experiences proper (!) acceleration (a supported observer) has a horizon associated with his reference frame, so there are regions of spacetime that are inaccessible to him - simply put, the entirety of the electromagnetic field is always there, but not all observers are able to detect all aspects of it. Hence under the right conditions, some observers see radiation, whereas others do not. Since energy is also observer-dependent, there is also no contradiction in that regard.

This force is the result of dipole-dipole interaction, so the total force between two macroscopic magnets will depend on how these dipoles are spatially distributed. In other words - it depends on the shape, location, and orientation of the magnets. For two single, point-like dipoles, the force indeed falls off with the 4th power of distance; but for other types of dipole distributions the force law will be more complex: https://en.m.wikipedia.org/wiki/Force_between_magnets This is not a new insight, since it directly derives from Maxwell’s laws.

The OP didn’t, but member quiet brought up the question of whether quantum effects can be deduced from Maxwell’s classical equations. The comment was aimed at that particular sub-discussion.

That’s a good question. I would tend to answer yes to this.

What I wrote there isn’t primitive - it’s the full covariant Maxwell equations in 4-dimensional spacetime. It encompasses all of classical electrodynamics. Also, the curl and divergence operators are defined in 3-dimensional (Euclidean) space, they are not 2-dimensional.

No, because the general public does not usually read arXiv, since few people have the technical knowledge needed to understand the documents published there. Also, arXiv is a pre-print server, so these are publications that have not yet been subjected to peer review, they have only undergone a basic screening. Their scientific validity is hence not yet established.