Markus Hanke

If you want to know the gravitational potential for 2D, just solve Laplace’s equation - you’ll find that the potential is logarithmic, as it needs to be, since the force now follows a 1/r law. However, the story is actually more complicated - the more general law of gravity isn’t Newton, but GR. If you apply GR two a 2D universe, you find that the Weyl tensor identically vanishes; in vacuum, the Ricci tensor vanishes as well, as per the Einstein equation. Since the Riemann tensor decomposes into Weyl and Ricci, the result is that in 2D there is no gravity in vacuum at all, outside a mass distribution. You only have gravity in the interior of masses.

Yes, they are apparent in the LIGO data. Searching for additional polarisation states (vector and scalar states in addition to the tensor states) is also under way, as this provides a way to test for modifications to the laws of gravity. I’m sorry, I don’t quite understand what you mean by this? Gravitational waves are always transverse; their dynamics are also nonlinear, so I doubt it is possible to decompose them in simple ways. This is fine - but again requires that such a unique decomposition is possible, which brings us back to the gravitational potential issue. So yes, your approach works - but only given the boundary conditions I mentioned. Another way to look at this is via the action, which is the difference of kinetic and potential energies (Langrangian), integrated over time. In Newtonian gravity, the Lagrangian is just a sum of three terms, so you can easily see the kinetic and potential parts. In GR however the action is of the form \[S=\frac{1}{2\kappa} \int{R \sqrt{-g}}d^4x \] How do you decompose this into T and V parts? You can’t, except under very special circumstances. And again, this is only sort-of true under special circumstances. In general, in the geodesic equation, the geodesic depends on all components of the metric tensor in a rather non-trivial way, each one of which may depend on several coordinates. If, however, you introduce the symmetry conditions I mentioned, and in addition assume low velocities and weak fields, then the geodesic depends as an approximation only on initial conditions and g00 - which, in effect, plays the role of a potential. Well, you then have to acknowledge that in the general case, the kinematics of test particles depend on more than a single quantity. How does that work, exactly? Remember that it isn’t position that determines the difference in that example, but orientation (ie motion in direction of rotation, or against it).

If the source of the field were a 4-potential (or any combination of vector+scalar potentials), then gravitational radiation would have polarisation states offset by a 90 degree angle - just like electromagnetism. This is not the case though, as in reality the two polarisation states are offset by 45 degrees, meaning such radiation fields can only couple to rank-2 tensors as source. I have mentioned this further down - because the work done to go between points in the field must be independent of the path taken, or else the difference between those points cannot be a single unique quantity (=potential). This is how gravitational potential energy is defined - as a path integral between points within the field. If the value of this integral explicitly depends on the path, the definition becomes meaningless. This is true also in Newtonian theory. The potential must also vanish at infinity, or else you are left with a degree of freedom that cannot be fixed from the theory. Taken together, that gives you the symmetry requirements I mentioned. Why would it do that? The distance to the central body is the same, so your gravitational potential would be the same. Also, relativistic mass is not a source of gravitational time dilation. No, what I am saying is that your refraction model only works in a subset of particular spacetimes with particular symmetries. It cannot be generalised to describe all gravitational degrees of freedom, as GR does. So it’s not wrong as such, just limited. I’m afraid I lost you now - I thought your entire idea is based on the concept of a gravitational potential? If no such thing is required, and you don’t accept GR (ie no curvature), then what is it about spacetime that yields your refractive index? Hm...Im not so sure. How would you show this mathematically? On a very general note, I’m curious - why do you feel the need to replace GR in this way? It works perfectly well as it stands, and we have already known for a long time that relativistic gravity cannot be a scalar or vector theory, or any combination thereof (ref Misner/Thorne/Wheeler). I also don’t see how your idea would locally yield Special Relativity.

The problem isn’t the mathematics, but the basic premise - namely that you assume the existence of the notion of ‘gravitational potential’, as a generally applicable concept. If such a potential exists, then the following boundary conditions must apply: 1. The spacetime is asymptotically flat 2. It is spherically symmetric 3. It is stationary 4. It is static And vice versa - if any of these conditions do not hold, you cannot meaningfully define a gravitationally potential in a self-consistent way. This is basically to say that the work done to get from one point in a gravitational field to another must not depend on which path you take through spacetime, or else the difference between these points can’t be consistently described by a single number. You are right in one respect - if, and only if, you are in a spacetime that admits a consistent definition of gravitational potential, then you can describe gravitational light deflection as a refraction-like process, analogous to some variation of Snell’s Law. An example would be any system that can approximately be described as Schwarzschild. The issue is that it doesn’t generalise; violate any of the above conditions, and it will no longer work. A rotating body is the simplest example. So what I am saying is not that you are categorically wrong; it’s just that your formalism works only for a small subset of gravitational scenarios. It is not a general description of gravitational degrees of freedom. But for some special cases, I grant you that such an approach may come in handy. Two more examples to illustrate the point; each of these violates one or more of the above conditions: 1. In free space far from other sources, emit two parallel rays of light in the same direction - despite the energy-momentum they carry, they remain parallel and don’t gravitationally deviate. Now repeat the experiment, but emit the same parallel rays in opposite directions - they now gravitationally converge! (This is an example of a pp-wave spacetime) 2. Consider two intersecting beams of light at right angles, but in the same plane (a gravitational wave detector). Now expose this setup to a gravitational wave front - as the wave passes, the relative lengths of the beams will contract and expand relative to one another, even though their distance to the source (which is very far away!) is identical. Furthermore, comparing two or more such setups at different orientations in space (ie at different points on Earth) reveals the nature of these waves to be quadrupole, with two polarisation states at 45 degree angles - which necessarily implies that the field must couple to a rank-2 tensor. This is why gravity needs at least a rank-2 tensor description. I could give more examples where the gravitational potential approach doesn’t work, but I think you can see my point - ‘gravitational potential’ can only be meaningfully defined under certain conditions, it is not a general concept.

That’s not true. Instead of introducing a quantised field on smooth spacetime, you can quantise spacetime itself, giving you a set of models that do not require a graviton. One example of this is Loop Quantum Gravity. Higgs bosons do not have the required properties to be carriers of gravity, as Grenady correctly pointed out.

Suppose you have a rotating massive body of some kind. Further suppose you have two identical rays of light that pass this body in the equatorial plane, such that one passes along the central body’s direction of rotation, while the other one passes opposite its direction of rotation. The light rays are identical in all other aspects, ie they pass the rotating body at the same distance. GR predicts that these light rays are frequency-shifted by slightly different amounts as they pass the body, even though they both pass at the same distance, and in the same equatorial plane. Alternatively, you can put two sensitive clocks in the same orbit around the rotating body, but let them move in opposite directions. After one orbital period, even though they will have traversed the exact same region of space on the exact same orbit, their clocks will have recorded slightly different times. How do you explain these using Snell’s Law, and a single scalar field, respectively? I have said it many many times on here before - you can not describe all degrees of freedom of gravity using either scalar or vector fields alone, or any combination of these, for fundamental reasons. You need at least a rank-2 tensor field.

It’s the conserved current associated with spatial translation invariance under Noether’s theorem - as Grenady has pointed out. I say it again - it is not helpful to speak of the ‘mass of an EM field’.

...which of course vanishes for photons. Given all that has been said, in what way is it meaningful or consistent to talk about the ‘mass of an EM field’? Such a concept creates far more problems than solutions. It’s simply not helpful.

You’re right of course, my terminology was sloppy. Let’s, for simplicity’s sake, say it was a homogenous ‘soup of particles’ (though this is problematic too, but you get my drift). Ah, I get you now. I hadn’t looked at it quite in this way...something to think about. Great reference +1 I was not aware of this field of maths. Goes on my reading list!

I think it is not particularly helpful to think about the situation in this way, because if you look at the Lagrangian that describes the electromagnetic field, you will find that it does not contain any mass terms - neither in classical field theory nor in QED. Saying that “the field has mass” is thus misleading at best.

Perhaps I have simply overthought all this - ultimately it comes down to gravity. You start off with an early universe that is basically a homogenous soup of energy. If you just naively scale this up, then it is difficult to understand how gradients of entropy could evolve. The problem though is that a state like this is in an unstable equilibrium - even the tiniest fluctuation is enough, and gravity will kick in, drawing the ever so slightly denser regions together in themselves, creating in homogeneities. Over time this leads to what we see now - clusters, galaxies, and stars. And once you get gradients of energy like this, from which work can be extracted, it is no surprise that entropy increases at different rates in different places. It is not a big step from here to local complex systems. I had made an attempt to connect this to the stationary action principle, but I now think this might have been ill-conceived. The trouble is that the action principle is not well defined for all types of systems; in particular, in most cases it does not apply to dissipative systems, which is what a biosphere would be. So this is problematic. Another issue is that, even if an action principle exists, the action itself might not be unique. So to make a long story short, I no longer think there is necessarily an issue with entropy and action principles. Whether or not this is sufficient to explain the sheer degree of local complexity we see is another question again. Could you elucidate on this a bit? It’s the variation of a line integral, in the form it’s usually given. But I think I may just be missing the point you are attempting to make. This may be related to my above point about not all systems admitting well-defined action principles. Also, the action is a global property of an entire region in phase space (ie an integral over time), whereas the second law is a local statement (in time). If the region has no clear boundaries, then it will be difficult to make sense of the concept.

Yes, certainly. This is one of the standard solutions that can be written in closed analytic form. I don’t know what you mean by “quantum charge”. GR is a purely classical model, and the Q parameter is not quantised. Either way, charge (along with spin and mass) is just a parameter within the metric, it’s not a function of coordinates. So it can’t be localised anywhere - it’s a global property. The paper you referenced introduces an additional charged scalar field, so we are no longer dealing with a vacuum solution. The original RN metric (the topic of this thread) contains no such extra fields, so I’m not sure what ref 7 has to do with this thread.

To be honest, I’m not sure what the main point of this discussion actually is - in Schrödinger’s scenario, the radioisotope that sets of the mechanism is never in any superposition relative to the cat; at any given time it has either decayed or it has not. Likewise, the cat is not in any superposition, it’s just that its state is unknown until the box is opened. This was not intended as a real-world example of quantum superposition, but merely as an analogy to demonstrate the basic idea - the uncertainty here is merely epistemic, but not ontological. I still maintain that for all intents and purposes, in the real world, decoherence prevents a system the size of a cat to be in a superposition for any reasonable amount of time (I’d guesstimate no longer than perhaps 10^-30s or so). And that’s not even considering how you establish such a superposition in the first place, before decoherence.

Because a system the size of a cat cannot be isolated from its environment by any practical means. When you scale up a quantum system, you also scale up the degree by which it will undergo decoherence due to interactions with its environment. For a system the size of a cat, decoherence happens so quickly that any quantum effects become entirely negligible almost instantaneously (not that you could even establish such a superposition in the first place!). Colloquially put, the system’s quantumness - if there is any to begin with - bleeds out into the environment, and this happens the more quickly the bigger the system is. In principle it is possible for such a superposition to exist, if you could somehow find a way to completely prevent the cat from interacting with its environment; in practice this is not possible by any conceivable means. Even superpositions in very small systems - like on atomic scales - are difficult to maintain for any length of time for this same reason; this is eg one of the fundamental issues in quantum computing.

Two quick comments on this: 1. Mass, charge and spin are global properties of the entire spacetime - you cannot localise these quantities at any particular place 2. None of the metrics in the Kerr-Newman family will realistically appear in the real world in an exact way, because all four of them require asymptotic flatness - meaning these require an otherwise completely empty universe. A more realistic - yet still idealised - family of solutions would be the Vaidya spacetimes.

I think it’s more complicated than even this - because in my opinion language is more than a simple mapping into the outside world. It is strongly contextual, and meaning isn’t inherent (as it would be in a mapping), but given only through its actual use by people. Thus, language is more than an abstract set of rules and maps - it’s a cultural and social convention, and as such it is fluid and permanently evolving. You cannot separate language from the context of its users. I’m pretty firmly with Wittgenstein’s philosophy of language in this regard. I’m not saying that comparative philology isn’t a worthwhile endeavour (it’s quite interesting!); only that there are inherent limitations to such a project. I don’t believe this is true. Consider the example I gave earlier of เกรงใจ in Thai - this is a very subtle social concept that is quite specific to Thai culture. It is a real ‘thing’ in the outside world (an aspect of culture), but there exists no adequate translation for this in English or any other European language. Even trying to explain this concept in all its subtleties requires an entire paragraph of text at least, and even then it isn’t guaranteed that the reader will understand. Whole guide books have been written about it! Another example of such a thing is the word fa’alavelave in Samoan, which roughly refers to a social obligation created by something that has happened in the extended family, and for which material resources need to be raised so as not to loose face in the community (it also means simply ‘trouble’ or ‘problem’). You can verbally understand the explanation, but you won’t understand what fa’alavelave truly means to a Samoan person, unless you have lived in Samoa (it took me a long time to fully understand all implications of this when I lived there). The concept simply does not exist outside its cultural context, so no other language has any way to adequately express it in all its subtlety.

A language consists of much more than object-nouns, though. Furthermore, not all languages cleanly distinguish between word types. For example, I live in Thailand at the moment, and the Thai language does not, in many cases, make a distinction between noun, verb, and adjective (there are ways of marking a word to be of a specific type, though, if necessary). Furthermore, there are many cultural concepts here that have no equivalent in any other language - eg เกรงใจ, which means something like not wanting to cause an unnecessary burden for someone else. There is also a large number of personal pronouns that denote subtle differences in social status between speaker and listener, and have again no equivalent in any other language. Also, you have language registers - meaning you use different vocabulary for some things depending on who you are talking to (monks, people with status, members of royal family etc). How would you handle such things? Basically what I’m trying to say is that languages are not generally a 1-1 mapping into each other, unless they are very closely related already. Some things will be like this in many languages, but you will always have words that cannot be mapped like this, because language always reflects local culture.

Well, there simply won’t be a B field if the situation is a static one; the energy then is just the integral given by joigus. You can’t increase this without adding more electric charge, which in the real world implies moving charges into the spatial region in question (and thus the temporary existence of B). Sure. Nonetheless, it is often helpful to go to the full covariant formalism simply to illuminate the underlying physics. In this case, the point is that splitting the EM field into E and B fields is an arbitrary (mostly historical) choice; in reality though there is just one electromagnetic field that permeates all of spacetime, and the energy stored in it, as captured by the energy-momentum tensor, does not in any way depend on which observer defines it. Thus, whether the source distribution is static or not relative to any given observer is irrelevant for the underlying physics. The only way to increase total field energy is to move extra charges into it, ie increase its source density - which requires work to be done. Relative motion alone doesn’t qualify. I think while it may be formally possible to attribute some notion of ‘mass’ to an EM field (as an equivalent to its total energy), in practice this would be a fairly useless quantity, since it would be a global property of the entire field - which stretches into infinity. No observer could ever measure this mass. This is why in relativistic EM theory (the OP did reference the energy-momentum relation) we use the energy-momentum tensor of the EM field instead - which is a local quantity.

Indeed, I didn’t. Thanks for clarifying. And I had to make an edit to my post, as I was typing it in haste, and it got all muddled up and imprecise. Yes, it is interesting that the path integral formalism in QFT gives the same results as the action principle; but I’m not sure if this can be considered a derivation. I rather think these are different formulations of the same principle (but I’m open to correction on this)?

Could you further specify precisely which quantity you wish to obtain? Is it the total energy stored in an EM field? If so, then you can work this out using Poynting’s theorem - this should help: https://www2.ph.ed.ac.uk/~mevans/em/lec14.pdf This formalism can be straightforwardly generalised to the relativistic case, where it is written in terms of invariants of the EM energy-momentum tensor - meaning it applies no matter what reference frame you work from. I don’t think looking at this in terms of mass and momentum is very helpful. But maybe that’s just me

The principle of least action is a general principle of nature, which applies both in the classical and the quantum domain. It says that a given system will evolve such that the variation of the ‘action’ - a quantity which equals the time integral of the Lagrangian of the system (being the difference between kinetic and potential energy) - vanishes, ie it is stationary. This is equivalent to the Euler-Lagrange equation. Hence, to find the evolution equation of a system, you can first work out its Lagrangian, and then make the variation of the action vanish. For example, the Einstein equations emerge in this way from the Hilbert action. This is amongst the most fundamental and most powerful known principles in physics.

That’s true. But so far as GR is concerned, it doesn’t exist in a vacuum - one has to also look at the wider context. For example, if, in addition to GR, we also take into account QFT, then we know that event horizons of Schwarzschild black holes carry entropy (as a function of horizon area). This implies that the bulk enclosed by the surface has a finite, well defined number of non-trivial degrees of freedom associated with it - which physically means that the bulk must have some kind of ‘structure’, or is granular, or is topologically non-trivial. This of course runs counter to a naïve application of GR alone, under which spacetime in the Schwarzschild bulk is everywhere smooth and continuous, and singly connected outside the singularity. More recent results (ref Netta Engelhardt et al) support this. This is one of the indications that make me think that GR isn’t fundamental. In fact, I’ll eat my monk’s robes should it turn out to be - without salt and pepper, but perhaps with a bit of chilli 🌶 sauce.

I would tend to agree. Good point!

My interpretation of Newton’s words is that ultimately this is what he is asking for, hence my response. They’re measurements of space and time, but that doesn’t mean they don’t have physical consequences, so in that sense they are also a physical object. I don’t think one can cleanly distinguish between these. Energy is simply the conserved quantity associated with time-translation symmetry under Noether’s theorem, so it is itself a measurement of space and time. I would say that the fundamental dynamical quantity of GR - the metric tensor - is in fact a tensor field; would this not make it a field theory? Furthermore, one can write GR as a gauge theory, with the Lanczos tensor being the gauge potential. This should make it a field theory for sure, no?

Yes, but spacetime and its degrees of freedom is - at least in my opinion - only an effective description of gravity. I do not think it is fundamental at all, and the underlying mechanisms that give rise to the appearance of smooth spacetime aren’t yet known (though there are of course some interesting hypotheses out there). To put it slightly differently - the domain of applicability of GR is limited, at least in a ‘downwards’ direction.