Electromagnetism is a property of spacetime itself, study finds
sciencex.com
sciencex.com
Non technical version (ELI25):
In quantum mechanics the wavefunction Ψ is has complex values. If you multiply everything in the universe by -1, nothing changes because all the physical results use ΨΨ* (where * is the complex conjugation). You can also multiply everything by i or -i. Moreover by any other complex number of modulo 1 because ΨΨ* does not change. (The technical term for this is U(1) global gauge symmetry.)
But you can be more ambitious and want to multiply each point of the universe by a different complex number of modulo 1. ΨΨ* does not change but the derivatives of Ψ change and they are also important. (When you use the same complex number everywhere, the derivatives is just a multiple of the original derivative. When you use a different number in each point, it changes.)
The only way to fix the problem with the derivative is to add a new field A. When you and multiply each point of the universe by a different complex number of modulo 1, then A changes in a simple to calculate but not obvious way. The change in A fix the problem with the derivatives of Ψ.
So now the equations of the universe with Ψ and A don't change when you make this change. (The technical term for this is U(1) local gauge symmetry.) When you write carefully how a universe like this look like, the new field A is electromagnetism. (Actually, you can get the electric field and magnetic field using the derivatives of A.)
This explanation looks more complicated than the explanation of the article, but the article is full of technical terms that you really don't want to know, like:
> Riemann curvature tensor is more than just Ricci curvature—electromagnetic fields stretch and bend the spacetime
Er, I'm not sure that things like dressed particles [1] and off-shell matter [2] violating E=mc^2 [3] could be described as even remotely "natural".
Perhaps they are valid theories, but "natural" certainly isn't an appropriate description of most of what unavoidably follows from particle assumptions.
[1] https://en.m.wikipedia.org/wiki/Dressed_particle
[2] https://en.m.wikipedia.org/wiki/On_shell_and_off_shell
[3] https://en.m.wikipedia.org/wiki/Virtual_particle#Properties
If you do the calculations in another way (e.g. by discretizing stuff on a lattice) no virtual particles appear but your calculations become a lot harder.
http://www.claymath.org/millennium-problems/yang–mills-and-m...
It's a complex number for QED but in general, it's a unitary matrix: a rotation which preserves the magnitude of a wavefunction (a complex vector).
In physics, symmetries play a fundamental role.
In the case of gauge theory, the idea that we should consider the most general case has been well proven. As GP pointed out, all observable phenomena ultimately depend only on the absolute modulus of the field Ψ, so a theoretical physicist naturally wonders, what happens if you allow its complex phase to vary. Turns out nothing interesting happens if you apply a global phase, but if you allow the phase to vary at every point in spacetime, it ends up breaking the theory. That is, unless you include an additional field at every point in spacetime that precisely cancels out the change induced by the gauge freedom.
In other words, the motivation is that we can't simply look and "see" whether or not there is a locally varying phase on the wavefunction Ψ, since we only can measure |Ψ|^2. So we have to assume there is, until proven otherwise. Since a local phase would imply the existence of an extra field to cancel it out, we can indirectly check for this scenario by looking for the corresponding field. As pointed out by GP, in the case of a U(1) gauge, it turns out there is such a field, and electromagnetism (and all of its laws) exactly fit the bill.
There are other "unmeasurable" symmetries you could apply to the wave function as well, beyond just a complex phase. SU(2) is a Lie group symmetry which would mean that the measurable properties of certain tuples of fields (Ψ, ϕ) are indistinguishable under a sort of complex-valued rotation of Ψ->ϕ and ϕ->Ψ. Again, if you assume a such symmetry is locally varied at every point in spacetime, you end up requiring not one but three new fields to cancel out the effects on the SM Lagrangian. It turns out that the vector bosons W+, W-, and Z, which mediate weak nuclear forces exactly fit the bill.
The work of theoretical physicists always seemed like the other side of an ocean of math away, but thanks to this explanation it feels like I can at least make out a lighthouse in the distance.
Could you elaborate on this a bit? To a layperson this sounds like a hack. "Things get screwy when you screw with them, UNLESSSSS we add a magic thing that undoes our work". Well yeah.
Here's an analogy: You find a box, and you can't see inside it. You have no reason to think there's something inside it. But also, boxes have stuff in them sometimes. So, you shake the box, and hear something clinking around. Therefore, you infer there's something in the box.
Somebody next to you says "This sounds like a hack. There was a box and you had to go shake it until it started making sounds that it wasn't making before. UNLESSSSS we now magically have to agree that there's something in the box."
It's a perfectly reasonable question, and I'm just turning your words on you in good faith :)
To take the technical discussion a bit further, it's exactly this kind of reasoning that led to the discovery of the Higgs boson. Strictly speaking, it's impossible for gauge bosons (that's what the particles are called that show up when you add these locally-varying symmetries) to have nonzero mass. The photon and gluons (from the SU(3) strong force) are massless, but the W and Z bosons are VERY massive. This was a big problem with the Standard Model; the vector gauge bosons had every property expected from the gauge theory, except for this one point about their mass, which was experimentally incontrovertible.
That is, until Brout/Englert/Higgs came along. They said "Yeah the vector bosons must be massless UNLESSSSSSSS you assume there's this magic additional field that couples to every particle's mass, in which case it perfectly cancels out all the problems and allows the W and Z bosons to be heavy". It took 50 years but we found that particle eventually.
The argument isn't against physicists inventing new fields or interactions to fit data. The argument is about why you can motivate it as "something that has to be added out of pure logic" :)
In a Physics degree, the order is quite historical. Like one full course for the three first next items, and all the other together. I'm not sure if there is a book with all of them, you probably need 4 or 5 books.
1) Non-Quantum Non-Relativistic Electromagnetism
2) Quantum Non-Relativistic Electromagnetism
3) Non-Quantum Relativistic Electromagnetism
4) Quantum Relativistic Electromagnetism
5) By the way, you can interpret the Quantum Relativistic Electromagnetism as a U(1) symmetry. (my comment)
6) It looks like a good idea. Let's use other groups to explain other known forces: the weak and strong force. (The G...GP comment about SU(2) and SU(3).)
7) ???
[See note 1]
For some reason, popular science articles love to show something almost magical and prefer to present something like the "5)". It makes it easier to hide the math and use hand waving.
Also, Physicist working in physic particle also believe that "5)" and "6)" are the correct approach, and the other are just useful for teaching and for historical reasons. But to discover "7)" it's better to think about some weird new symmetry group [2].
For examples, a few years ago, it was popular to think the next step "7)" was using a new group SU(5) that combines SU(2) and SU(3). The problems is that the experiments gave different results than then new proposed theory, not too bad but like 1% off. I still remember my professor talking about how great was SU(5) and how the experiment disagree, and he looked heartbroken because he really liked SU(5).
[1] You should add some material about the historical discovery of the weak and strong forces between "5)" and "6)".
[2] Other's prefer superstrings for "7)", there are other approach, but all are weird.
The first option might be true, no real way to measure it and that's sort of it, no explanation for all the other stuff going on. The second option is a little more complex, it requires an extra construct to make it work. And apparently when we do the math and work out this construct, it exactly maps to things we can measure in reality, that were not explained by the other simpler option.
So it's not backwards because it's simply the first full match in a depth first search through the possible realities that follow from this wave function theory.
So it turns out this happens quite often that there is some kind of constant that can be divided out. In case of particle physics a whole framework has been developed out of it that has really close relations to Lie group theory. (The experimentally confirmed parallels are just astonishing with group generators and elements corresponding to interaction particles and the normal particles.)
For example the laws of physics are time translation invariant, i.e. it does not matter what point in time you call t equals zero which essentially means that the equations do not contain time but only time differences so that you can add the same constant to all your times and nothing changes as the constant cancels out when calculating a difference between two times.
Its the same with voltage, where you put your reference potential does not matter but this is again a global symmetry and you have to use the same reference potential everywhere, you will obviously get nonsense if you use different reference potentials at different points.
Local symmetries on the other hand are kind of defects in the mathematics of physical theories, they are the expression of redundancies in the mathematical description. Say you want to describe the orientation - but not the strength - of the magnetic field on earth and for simplicity lets assume the earth is flat and the magnetic field parallel to the surface, then you could do this by associating a two dimensional vector with each point on earth that describes the tip of a compass needle placed at that point.
But there is a problem, there are longer and shorter compass needles but that is irrelevant for the orientation of the magnetic field, the length of the vector does not actually matter. You could multiply this vector field with a different constant at every point, i.e. independently change the length of all the compass needles and you would still describe the same magnetic field orientation across earth. What you do to fix this is to declare that all fields are physically equivalent if they only differ by a constant factor - which may depend on the point - at each point.
The other solution is to use a better mathematical representation without the redundancy, instead of compass needle tip vectors you use the bearing angle and take the compass needle length out of the equation to begin with. Problem solved. Now you can no longer change the field value, i.e. the angle, at every point independently and still describe the same physical situation. Also note that there is still a global symmetry, you can still change all the angles by the same constant, you are free to pick which direction you label with angle zero, which is again physically meaningful and expresses that space is isotropic, i.e. there is no preferred direction and the equation therefore do not depend on the direction but only on the angle between directions.
And you can do the same thing with the known physical laws, you can for example get rid of the U(1) symmetry - which should better be called a redundancy - in quantum electro dynamics. The price you have to pay is that the resulting equations lack some other properties often considered desirable, for example they are no longer obviously local.
Looking at my comment, it says "multiply each point of the universe by a different complex" it's actually a "event" like in relativity, i.e. (ct, x, y z). This is an easy case where the function you use to multiply does not depend on x, y, z, but only on t.
What you describe as a "hack" is actually the empirical (look/probe and see what happens) nature of physics as a science.
It seems like a hack because there's no a priori theoritical reason to do it. But physics is not based on verifying empirically a set of a priori rules (who would give them?), but by building theories and rules by empirically looking at things at seeing what model fits, and if a changed model fits better - we then verify those empirically with more experiments.
And this is not "Things get screwy when you screw with them, UNLESSSSS we add a magic thing that undoes our work", but more like a reverse engineering session:
"Behavior X appears to be described by this formula with parameter p. What if we changed the parameter to -p? Hmm, the results would still be consistent with X, if only there was an additional factor v in the formula.
Would this (-p,v) combo buy us anything over our previous (p)?
Wow, yeah, v would then perfectly match the behavior we see for this other thing Y too. So (-p, v) seems to describe both X and Y, whereas before with p we could only describe X.
The origin is set at an arbitrary point so this "space shift invariance" is saying that it doesn't matter what point we set for the origin (and mathematically this corresponds to the conservation of momentum - see Noether's theorem[0])
Hmm maybe the "zero" for the quantum states is arbitrary, so you should be able to add anything to it for the whole universe, and this merely changes the zero state in the opposite direction.. and since this should be a conversation law, pretty sure this is equivalent to the conservation of electric charge
Now that may purely a choice of convention for Ψ at different points in space/time (a choice of "gauge" in the jargon), but where it gets interesting is if your successive nearby points in space/time trace out a closed loop. If your A is such that the phase of Ψ ends up different as a result of going round the loop, you have an electromagnetic field!
A is quantum, the electromagnetic potential is classical, so they are not really the same thing.
https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect#P...
Given that "light" is fundamentally a electromagnetic wave and its propagation in spacetime is constant, and this results in time slowing down when you go faster to maintain this property, it isn't unreasonable to hypothesize a more fundamental basis here.
Personally, I think adding in the time component will be essential to completing this puzzle but all in all it makes for an avenue of investigation which is interesting.
Basically "the speed of light" should be called "the speed of massless things" or possibly "the speed of causality" or something. We just call it "the speed of light" because light is the first thing we discovered that travels at this special speed.
*the star is because everything about quantum-chromodynamics is terrible so gluons don't really ever exist as particles themselves. If they did they would travel at the speed of light.
* not actually infinite, because reality itself propagates at a finite speed
(I’d love to know if this is wrong - this is my best attempt to make sense of it from college classes)
Not a physicist, but I've often wondered if the basis of QFT got off on the wrong foot by making time a privileged coordinate instead of a quantum operator like it does for position.
Speaking of which, time(-of-arrival) measurements in quantum mechanics have recently attracted quite some interest: The classic Copenhagen formalism doesn't seem to give an answer here (or at least not a unique one – it depends on how you perform the calculation). Meanwhile, Bohmian mechanics does seem to make a precise prediction. It will be interesting to see what experiments will yield.
Also, it's not so immediate, because you must be stubborn enough to think that a global obvious symmetry "must" be extended to a local symmetry. And in any case, it took like 40 years a few brilliant persons to discover it.
Right. It seems to me that both approaches make sense. Perhaps with some cleaver yet-to-be-determined math both ideas can finally be mated.
I've never been convinced that the æther doesn't exist. Sure, it's been long debunked in the luminiferous æther sense but as the article points out "...the aether hypothesis was abandoned, and to this day, the classical theory of electromagnetism does not provide us with a clear answer to the question in which medium electric and magnetic fields propagate in vacuum." It is this aspect of the abolition of the æther that has always worried me.
For starters, any new model of the æther would have to exhibit Lorentz-invariant properties. Then there's the matter of vacuum permittivity ε0 and vacuum permeability μ0 to consider as the speed of light/aka 'electromagnetism' is directly linked to these physical constants via the expression c = 1/(μ0 ε0)^0.5. If one constant were to change then so too would the others including α Sommerfeld's fine structure constant, RK the von Klitzing constant, and Z0 the vacuum (free space) impedance, etc., etc. (Anyway, one would expect them to change—not that we'd ever know as we'd likely not exist if they did). ;-)
But I digress a little. We know that ε0 and μ0 have actual non-zero values and cannot be equated out (as we sort of tried to do in the days when we expressed electromagnetism in cgs units). In essence, physical constants ε0 and μ0 are absolutely intrinsic to electromagnetism, and whilst I cannot prove the fact, it seems to me they would be just as intrinsic to any new definition of the æther. Moreover, similar reasoning makes me think that QFT, ZPE/Zero-point energy/quantum vacuum state, ε0 and μ0 are all inextricably linked to GR.
It seems to me that whilst matters such as whether the spacetime manifold is Ricci-flat, etc. are extremely important principally from the perspective that when properly dovetailed into any new theory they'll provide proof thereof—are secondary to the proposition (note, I'm not saying they're secondary aspects of physics, only that they're secondary to the initial proposition).
Moreover, an equally important question to ask is why the constants ε0 and μ0 have the values they do given the quantum vacuum state, etc. Of course, the same logic applies to both α and c. Finally, we base just about everything on c it being the fundamental immutable constant (despite the the perennial emphasis/importance of α ≈ 1/137). The question is, is it in fact so, or is it that underlying physics first determines ε0 and μ0 and thus these constants could be considered more fundamental to any new formulation of the æther than that of c, it being the consequential resultant of the properties of those constants. (Heresy I know, but it would seem to make sense to view c in this context if or when we end up with new definition for the æther.)
The article looks a lot like word salad... too many specific technical terms for the average person to manage, yet lacking in the specifics that would be needed by someone capable of understanding their theory.
If it's written for their audience, then the only audience they seem to be targeting is average people who won't understand their assertions may be a load of crap.
Or is there someone here with a deep understanding of these topics that would care to chime in?
https://news.ycombinator.com/item?id=27944642
(Warning: Rather technical and full of math)
But I think the article is well written. It essentially says, "We think this interesting thing is true. It has these nice properties, and should be provable/falsifiable. Please help us prove it!"
For example, I used that if k is a complex number with modulus 1, and k* is the conjugate, then kk*=1. It's a subject from a course of the first years of the university of a technical degree and perhaps a high school. Adding a detailed explanation in the middle makes the explanation too long. Also, it's a very important part of the correct technical complete version of the explanation, it's not a side comment or a metaphor.
I think it could be explained better, or with different tradeoff to make it easier to understand. Anyway, it's my best effort with my personal taste.
If you tell me the part that confused you, I can try to explain that part more. (And if you provide some personal background, like age range and what you studied, I can try to tailor it more.)
Besides, how can you talk about unifying general relativity and electromagnetism without mentioning Kaluza-Klein theory[1]? And what about one of the most beautiful principles in physics, gauge invariance[2]?
I don't want to be rude, but I'm very curious as to how this got through peer review.
0: https://iopscience.iop.org/article/10.1088/1742-6596/1956/1/... 1: https://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory 2: https://en.wikipedia.org/wiki/Gauge_theory
No offense but the very first paragraph of the article's introduction mentions Kaluza's work:
> The earliest attempts can be reasonably traced back to the German physicist Gustav Mie (1868-1957) and the Finnish physicist Gunnar Nordström (1881-1923). Fruitful efforts came, for example, from David Hilbert (1862-1943), Hermann Weyl (1885-1955), Theodor Kaluza (1885-1954), Arthur Eddington (1882-1944) and of course also from Albert Einstein (1879-1955). It is less well-known that, for example, Erwin Schrödinger (1887-1961) had such inclinations as well, see [1]. For a thorough historical review, see [2].
https://iopscience.iop.org/article/10.1088/1742-6596/1956/1/...
Higher acceptance rate, but more reviewers per paper.
Kaluza-Klein theory is the archetype of expressing electromagnetism purely through curvature, and I'd expect any paper doing the same to refer to it, as well as explain how the work in the paper differs from or expands on it.
The fact that the metric proposed in the paper corresponds to the term added to the 4D spacetime part of the Kaluza-Klein metric is already suspicious. It makes me think they're either repeating Kaluza and Klein's work, or aren't properly citing it when they should have.
But they do:
> The strength of the present approach is simplicity, there is no need for higher dimensions, torsion tensors, asymmetric metrics or the like.
(emphasis mine)
> The fact that the metric proposed in the paper corresponds to the term added to the 4D spacetime part of the Kaluza-Klein metric is already suspicious.
Why? There aren't many ways to write down a symmetric 2-tensor starting with a vector field.
> It makes me think they're either repeating Kaluza and Klein's work, or aren't properly citing it when they should have.
They're doing neither. Sure, they might have mentioned Kaluza's work more explicitly (AFAIK Klein's contribution was the suggestion to compatify the 5th dimension) but I suppose 1) they assumed that readers are familiar with it and 2) they wanted to mention several approaches to marry GR and ED and Kaluza's approach was just one of them.
At some level, this feels like a vacuous result. If you start with a rank-2 massless field theory, and constrain it to act on rank-1 fields, is it surprising that you get a rank-1 massless field theory? Is this not just an elaborate form of completing the square?
> The metric tensor of spacetime tells us how lengths determine in spacetime. The metric tensor also thus determines the curvature properties of spacetime. Curvature is what we feel as "force." In addition, energy and curvature relate to each other through the Einstein field equations. Test particles follow what are called geodesics—the shortest paths in the spacetime.
>> The metric tensor of spacetime tells us how lengths determine in spacetime. The metric tensor also thus determines the curvature properties of spacetime. Curvature is what we feel as "force." In addition, energy and curvature relate to each other through the Einstein field equations. Test particles follow what are called geodesics—the shortest paths in the spacetime.
Could you elaborate on why you think this is gibberish? I mean, I agree that the article is giving off a pseudo science vibe and the authors should work on their style. (Instead of presenting their results in a matter-of-fact manner, they should rather dedicate more time to explaining their assumptions and their reasoning in a step-by-step manner.) But the paragraph you quoted seems perfectly fine.
I agree, the article is not very coherent. However, neither of the authors is an English native speaker, so maybe that is playing a role here?
OTOH, the website where the article is being hosted does look somewhat sketchy, so maybe you're right and the article was not written by the paper's authors but indeed strung together by an algorithm.
In the view of GR, all objects follow straight lines absent acceleration, and the force of gravity is actually a result of curvature of spacetime. That and the rest of your points that follow are more an issue of not being familiar with GR. I agree the article could have done a better job of elaborating these to a wider audience, but if you've read about GR a bit these concepts will be quite familiar.
https://en.m.wikipedia.org/wiki/Metric_tensor_(general_relat...
Maybe the UAPs really are just secret warp drive tech we made 20 or 30 years ago.
I can somewhat see how to interpret the mathematics in free space. But what about when there are massive bodies in the picture? They will result in a non-flat metric... does that imply they create their own electromagnetism?
Here, I feel the authors are not entirely clear who the audience is supposed to be. At first, they seem to target people who need the difference between Einstein and Maxwell explained. The section is titled:“ Maxwell's equations and general relativity—what are these all about?“
Then when they reveal the missing link, the uninformed reader is presented with a logical progression that is obviously written towards somehow for whom the statement:“the Lagrangian of electrodynamics is just the Einstein-Hilbert action“ is self explanatory. You know, people who say, yes of course, if you say:“ keep the spacetime manifold Ricci-flat.“
Writing about science is hard
Turns out I could learn almost nothing from it, although I really tried: The first half of the book was basic stuff I already knew, but I could understand nothing of the second half, it passed so quickly from the elementary to the super-advanced. I ended up wondering who it was written for—I imagined everyone would have a similar experience, whatever their level—it's stuff you already know until suddenly it's stuff you can't understand, and no way of passing beyond. It covers so much ground so quickly, with no time for enough explanation—if you didn't already understand the current topic. Maybe that was just me! But it was extremely surprising putting so much (money,) time and effort into a book with rave reviews by a leading physicist, and learning virtually nothing.
That reminds me of Roger Penrose's 1100pp The Road to Reality: A Complete Guide to the Laws of the Universe, which I was very excited about reading when I bought it years ago. Lots of lovely diagrams. Turns out I could learn almost nothing from it, although I really tried: The first half of the book was basic stuff I already knew, but I could understand nothing of the second half, it passed so quickly from the elementary to the super-advanced. I ended up wondering who it was written for—I imagined everyone would have a similar experience, whatever their level—it's stuff you already know until suddenly it's stuff you can't understand, and no way of passing beyond."
This is nearly universally an answer to the request" tell me you don't understand something you think you understand without telling me that you don't understand something that you think you understand"
The ramp isn't steep. It's just that if you think you're on it but aren't, then the second floor looks like a wall
I have no idea what this means, sorry.
> The ramp isn't steep. It's just that if you think you're on it but aren't, then the second floor looks like a wall
I'm not sure what this is meant to mean. Could you be less cryptic? Thanks.
I did not see anything novel that would warrant further attention - did I miss something?
Another commenter https://news.ycombinator.com/item?id=27943428 talks what basically looks to me as an emergence of EM field from rotation - "to multiply each point of the universe by a different complex number of modulo 1" - ie. as an artefact emerging by changing the frame to the one where the system is rotating (ie. gets a spin). Kind of similar how magnetic field is just emergent artefact in the frame where charge is linearly moving.
Maxwell's equations are the key linear partial differential equations that describe classical electromagnetism. The equations relate the electromagnetic field to currents and charges. On the other hand, in general relativity, the Einstein field equation is a set of nonlinear partial differential equations describing how the metric of spacetime evolves, given some conditions, such as mass density in the spacetime. Both equations are ultimately of second order, if seen properly.
Therefore, we thought that perhaps we are talking about the same governing equation, which could describe both electromagnetism and gravitation. Indeed, it becomes clear that Maxwell's equations hide inside the Einstein field equations of general relativity. The metric tensor of spacetime tells us how lengths determine in spacetime. The metric tensor also thus determines the curvature properties of spacetime. Curvature is what we feel as "force." In addition, energy and curvature relate to each other through the Einstein field equations. Test particles follow what are called geodesics—the shortest paths in the spacetime.
“This is aesthetically pleasing, as nature seems to strive for harmony, efficiency and simplicity.”
it makes me think they are not being the most objective evaluators of reality.
Nope, that was hubris.
https://aeon.co/ideas/beauty-is-truth-truth-is-beauty-and-ot...
The acceptance of heliocentricity was an aesthetic judgement, one which favored simplicity, and predated the work on elliptical orbits which gave the Copernican theory, thusly modified, equivalent predictive power.
Generally in physics and in maths researchers tend to spend time evaluating solutions that are symmetrical or otherwise elegant. I suppose that does make us biased but we’re searching for answers in a very large space of possible answers. Beaming towards elegant solutions seems like a reasonable heuristic.
On the other hand, some our aesthetics could be founded in an adaptive sense for reality.
g_{\mu \nu} = A_\mu A_\nu
looks a little weird to me. The left side is a tensor, but isn't the electromagnetic four-potential is a gauge field, not a tensor?
Although it does seems strange for other reasons. Primarily because it seems very unlikely that everyone else who developed this field wouldn't have considered the possibility and then discarded it.
A -> O A O^{-1} - dO O^{-1} (O is the Jacobian matrix)
The second term is absent for tensors (such as the left side of their equation 4). It also vanishes on a flat spacetime where one only considers linear (Lorentz) coordinate transformations, but based on my cursory reading they don't seem to be making that assumption.
Let me address curt15's (and your) question in two ways:
####### The physicist's argument #######
In classic electrodynamics it doesn't make much sense to talk about a U(1) gauge symmetry (whether global or local) – because there's no electron field exhibiting such a symmetry in the first place. Gauge symmetries really only become important when talking about quantum mechanics, Dirac's equation and, more generally, coupling particle fields to force-carrying boson fields in a QFT. There it turns out that imposing local gauge invariance would guarantee you a way to accomplish the coupling while preserving the conservation of the Noether current associated with the global gauge symmetry.
But in classical electrodynamics, none of that is needed and the vector potential is simply a vector field on Minkowski space. In fact, the value of the vector field and the choice of gauge are not even important because there's no classic analogue of the Aharonov-Bohm effect.
Now pretty much the same applies to Einstein-Maxwell theory: There is no electron field and no U(1) gauge symmetry. It's all classical. Indeed, in my experience relativists just view the vector potential as a vector field on spacetime and ignore the QFT-inspired classical gauge theory stuff (principal bundles, associated vector bundles, etc.) altogether. You might disagree with this approach but given that no one has yet managed to write down any 4-dimensional, coupled quantum field theory in a mathematically coherent fashion (let alone on a curved background), I wouldn't say there's too much pressure to incorporate results from QED into GR.
Back to the paper: The authors basically start from the same perspective, i.e. the foundations of Special Relativity and classic electrodynamics where A was simply viewed as a vector field. They then propose an alternative to General Relativity and, in fact, electrodynamics(!) Like true physicists, they first worry about the functorial nature of their objects (what do they do & how do they relate to one another), not about the category the objects live in. If that means redefining[2] what A is in a mathematical sense, so be it. At the end of the day, experiments are what matters.
####### The mathematician's argument #######
Could it be that you and curt15 are thinking of the transformation behavior of A under local gauge transformations[0], not coordinate transformations? Because, unless I'm mistaken[1], the transformation of A under coordinate transformations is the same as for a vector fields, see definition 5.4.1 on p. 270 of [3]. The relevant portion reads:
Let p: P -> M be a principal G-bundle on the manifold M and A a connection 1-form on P, let s: U -> P be a local gauge of the principal bundle on an open subset U of M. Then we define the *local connection 1-form* (or *local gauge field*) A_s on U by
A_s := A ○ Ds = s* A
If we have a manifold chart on U and {∂_μ} (μ=1,…,n) are the local basis vector fields on U, we set
A_μ := A_s(∂_μ)
---
[0]: See the transformation formula in https://en.wikipedia.org/wiki/Gauge_theory#An_example:_Scala... or theorem 5.4.2 of [3].
[1]: It's been a while since I last did mathematical gauge theory, so please let me know in case I'm committing any errors here.
[2]: Since g is defined in terms of A, A must necessarily be a globally defined vector field and cannot be a local gauge field.
[3]: Mark J.D. Hamilton: "Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics" (Springer, 2017)
As for the math: the key point is that the metric is globally defined on the spacetime manifold M. I agree that the A_\mu transform as the coefficients of a differential form (A is a connection after all), but the notation elides the fact that the A_\mu are defined only on U. They depend, in particular, on the choice of local gauge s:U \to P. So the question about covariance (or globalization or coordinate-independence or whatever you want to call it) of both sides of the equation g_{\mu\nu}=A_\mu A\nu very much involves the question of local gauge transformations.
Either the principal bundle is assumed to be trivial, in which case there exists a global gauge and the connection A can be identified with a global 1-form on M (connections form an affine space modeled on such 1-forms; a gauge effectively converts this affine space to a vector space by choosing an origin), or we need to check that the right-hand side A_\mu A_\nu is indeed a symmetric two-tensor on M. The latter is not clear to me.
I admit I haven't looked at the paper carefully, and physicists typically don't approach things in such an explicitly mathematical way. So perhaps there's some (physical?) justification for why that equation typechecks. I don't quite see it though.
I don't think there is. The local U(1) gauge symmetry really comes from the (complex-valued) Dirac field and its coupling to the photon field (i.e. A). In classic electrodynamics you can add the 4-gradient of any function to the 4-potential A without changing the equations of motion, so the space of valid gauge transformations is infinite-dimensional. (Which is not that interesting – given that you can't measure the potential A –, so all those degrees of freedom are non-physical.)
> I agree that the A_\mu transform as the coefficients of a differential form (A is a connection after all), but the notation elides the fact that the A_\mu are defined only on U.
That's a very good point indeed! Though I think the authors simply interpreted A as a regular vector field on the manifold, meaning that it is defined everywhere. But following your train of thought for a moment: Could you solve this issue through a partition-of-unity argument? I.e. cover the manifold with neighborhoods where you have local gauges and then construct the metric locally as a finite sum of those A_{\mu,s} (s being the gauge).
The issue of a metric constructed this way not necessarily being a Lorentz metric (let alone non-degenerate) of course nonwithstanding. Then again, the authors didn't worry too much about this, either… :)
The paper's calculation reminds me of Kaluza-Klein theory, which uses a similar construction as part of extending the metric from four dimensions to five:
This somehow reminds me of C.S. Lewis’s “Out of the Silent Planet”, where the narrator says regarding the protagonist
“He wondered how he could ever have thought of planets, even on Earth, as islands of life and reality floating in a deadly void. Now, with a certainty which never after deserted him, he saw the planets - the 'earths' he called them in his thought - as mere holes or gaps in the living heaven - excluded and rejected wastes of heavy matter and murky air, formed not by addition to, but by subtraction from, the surrounding brightness.”
Yet somehow, I still find myself on Wolfram's side, suspecting that the entire QM/GR formulation is going to turn out to be the wrong way to understand the universe and that something quite different (e.g. cellular automata) will eventually be seen as much better.
The situation seems quite analogous to the one with fluid mechanics: we can use the continuously valued models that involve concepts like rate, turbulence, friction, etc. and be happy that we can reason about the system even though we fundamentally do not know what these properties are and can reasonably doubt that they are real. Or we can use cellular automata models, which use discrete math and have no "properties" that we can reason about. The outcomes are broadly similar in terms of prediction, but the approaches are totally different.
I will not be alive in 300 years, but if I were to be, I'd be willing to wager right now that most of the QM/GR formalisms will have been replaced by something entirely different.
This is also the Platonic-Pythagorean perspective, that the world is literally made of math.
Wonder if somebody could explain this?
Charge becomes an area of higher or lower air-density.
EM Waves become areas of changed air-density, propagated.
This is just pulled fresh from my butt, mind you.
Let g be the spacetime metric, and A be the electromagnetic 4-potential.
1. Suppose you could write the metric as g_{μν} = A_μ A_ν, i.e. the symmetric product of A with itself.
2. Conclude that the Einstein-Hilbert action is just the electromagnetic action plus a correction term A_μ ∇^μ ∇_ν A^ν = g(A, grad(div A)).
3. Assume that J^μ = ∇^μ ∇_ν A^ν = grad(div A).
4. The correction term from step 2 then becomes the usual electromagnetic coupling A_μ J^μ. As a consequence, the Einstein-Hilbert action for g is just the usual full (non-vacuum) action of electrodynamics on a background curved by g = Sym(A⊗A).
5. Consider the vacuum Einstein equations and, thus, a Ricci-flat spacetime. Show that this is equivalent to ∇² A^μ = J^μ which are the inhomogeneous Maxwell equations in Lorentz gauge. The fact that we're in a vacuum spacetime but still considering electromagnetism seems odd but I guess their idea is that if electromagnetism is a purely geometric property of spacetime, then the electromagnetic action (including any potential electric current) shouldn't appear on the right-hand side of Einstein's equations in the first place – because the Einstein equations are Maxwell's equations.
6. Identify the 4-current J^μ with terms involving the electromagnetic field tensor and the metric's Weyl curvature. (Meaning, once again, that J^μ can be non-trivial even though we're considering a vacuum/Ricci-flat spacetime.)
7. Identify the remaining (homogeneous) Maxwell equations with the first Bianchi identity for the Riemann tensor.
8. Impose the continuity equation ∇_μ J^μ = 0, i.e. assume conservation of charge.
9. Conclude from 3) and 8) that div(A) fulfills a homogeneous wave equation.
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Comments and observations:
- In step 5, I don't see how Maxwell's equations (18) are supposed to follow from equation (17). But it's late, maybe I'm just being blind.
- As other comments have already pointed out, step 1 seems unreasonable because the metric will no longer be of Lorentzian type (with determinant -1) but instead will be positive-semi-definite. (To see this, diagonalize the metric at a given point => g = (A_μ)² (dx^μ)².) In particular, the metric might even be degenerate(!) It seems section 2.1 in their paper is supposed to address the signature issue but from my POV it's insufficient.
- The paper basically claims that gravity is just the theory of a vector 4-potential. That doesn't seem right, given that much effort was spent in the past 100 years to find such a theory. AFAIK it's pretty much ruled out these days.
- Given step 5 and the fact that the EM field no longer seems to contribute to the field equations, I have even more doubts this theory could ever turn out to be true. There are lots of solutions to the Einstein-Maxwell equations and I'm sure some of them have been confirmed experimentally by now. (I'm thinking of black hole jets etc.)
- For instance, IIRC there's a paper showing that from Einstein-Maxwell's equations it follows that photons move along null geodesics (which in the beginning of GR was merely an axiom of the theory). I wonder what would happen to this result. Hypothetically, photons might no longer move along geodesics in this new gravito-electromagnetic theory but the theory might still reproduce gravitational lensing. I don't think that's very likely, though.
- More generally, I think their theory is even difficult to reconcile with classic electrodynamics in the first place. In the absence of strong gravitational and quantum effects, we know that Maxwell's equations describe ED very well. However, the equations ∇² A^μ = J^μ above no longer are the classic (linear!¹) vacuum Maxwell equations we know – they are now highly non-linear since the covariant derivative ∇ now also involves the vector potential A. To reobtain classic ED in flat space one would basically need to ensure that in every-day situations A is "constant enough" not to produce any significant curvature through g = Sym(A⊗A) but still dynamic enough to reproduce the classic wavey nature of light. This doesn't seem likely. Plugging any known (experimentally proven) solution to the Maxwell equations into the equations here should invalidate the theory.
¹) in the absence of charges, i.e. J^μ = 0
You can see how special relativity is true by just taking Maxwell's equations seriously. They show the speed of light is the same in every reference frame.
Some interesting potential tie-ins https://en.wikipedia.org/wiki/Abraham–Minkowski_controversy and https://en.wikipedia.org/wiki/Casimir_effect
> The earliest attempts can be reasonably traced back to the German physicist Gustav Mie (1868-1957) and the Finnish physicist Gunnar Nordström (1881-1923). Fruitful efforts came, for example, from David Hilbert (1862-1943), Hermann Weyl (1885-1955), Theodor Kaluza (1885-1954), Arthur Eddington (1882-1944) and of course also from Albert Einstein (1879-1955). It is less well-known that, for example, Erwin Schrödinger (1887-1961) had such inclinations as well, see [1]. For a thorough historical review, see [2].
https://iopscience.iop.org/article/10.1088/1742-6596/1956/1/...
I thought it was the other way around. Gravity is not well understood.
(emphasis mine)
As someone with a background in mathematical relativity, I would like to note that GR actually is not very well understood at all. Physicists seem to focus on the few simple solutions to Einstein's field equations that people have found through educated guessing, but there a ton of questions about the field equations that are open to this day.
wrote two very well cited articles [1,2] on this in the context of string theory. but this was just extending kaluza klein, where by adding one compact dimensions (with some assumptions) you get maxwell out of einstein for free. this result is from 1919, 102 years old.
i'm glad i can write all gravitation, electromagnetism, yang mills and string theory in two equations (1.6 in [2]]) but honestly i don't that's a breakthrough.
i might be wrong, yet it smells more like PR than important discovery.
btw you can as as well find solutions of GR that are dual to navier stokes equs.
[0] https://en.m.wikipedia.org/wiki/Kaluza–Klein_theory
There really is no plausible theory for FTL of any kind. Maybe there will be in the future. Personally I'd put everything I have on the speed of light being an absolute limit to causality.
Look at it this way: if FTL travel were possible we probably wouldn't exist. Whoever came first would probably colonize the Universe and sterilize it of competition. So there's that.
FTL doesn't necessarily mean infinite/instant travel. Also the scenario you propose could be part of the Great Filter, and we just haven't been culled yet. Or FTL civilizations tend to get so large they fracture & turn to in-fighting or encounter other resource bottlenecks that limit exponential expansion.
Or C is the law and going FTL spawns a cosmic traffic cop like the meatball head things in Rick & Morty.
Or FTL is simply impossible, but perhaps not constant and various factors impact the local C
Oh yeah, and you'd also be annihilated as you were completely turned to energy as you tried to go through it of course. But you'd sure fire a hell of a lot of randomized high energy particles out the other side after the amount of time it would have taken light to get there.
Not my preferred kind of "travel", I'll just freeze myself and go the slow way or something.
We haven't observed any Reissner-Nordstrom/Kerr-Newman black holes iirc. That said, _anything_ with stress-energy-momentum is held to affect spacetime curvature in GR.
The most obvious inclusion of it for me is the radiative (also known as the photon) component of the matter density in Lambda CDM models (that is, they need to account for the proportion of the energy density of the universe which is related to electromagnetic fields to rule it out as dark matter).