Ghost of vector fields in compact stars
journals.aps.org
journals.aps.org
Looking at collisions of neutron stars, perhaps it's possible to find difference from the predictions of General Relativity that may be helpful to discover an extension of the theory.
How bad is that? I'd love to read any correction or improvement from someone that is working in a related topic.
> Lastly, we stress that our results are relevant for most known extensions of spontaneous scalarization to other fields, not just the vectors, and our study can be considered as a first step to obtain a no-go theorem for extending spontaneous scalarization to other fields. For vector fields, Garcia-Saenz et al. [51] has identified the presence of ghost and gradient instabilities in the background of compact objects in a broad class of generalized Proca theories [84,85]. Similar concerns were also raised in the context of cosmology in Ref. [86]. Going beyond vector fields, all known formulations of nonminimally coupled spin-2 fields that could spontaneously grow are known to lead to ghost instabilities as well [75]. Likewise, p-form fields also have the same constraint structure we discussed in Sec. V, hence they suffer from similar divergent terms [87]. Spontaneous growth of spinor fields as it was introduced in Ref. [88] also contains divergent terms.
Essentially, build a theory (perturb the theory), and make a testable prediction that works on the large scale, but would be hard or impossible to observe on a weak gravity environment (for some definition of weak, I think they mean something less massive than a star).
They ask whether or not you can do something like this with a more complex, vector field (i.e. a new force/particle).
It's all very theoretical.
But it may give an avenue to find new physics, maybe even explain dark matter/energy.
> ... may be helpful to discover an extension of the theory.
The authors have a set of families of alternatives to General Relativity, and are exploring whether they are viable as a physical description of gravitation in our universe, and if not (spoiler: they're not; they admit this up front) whether any of them can be adapted into a good physical description.
The second author of the (purely theoretical) headline paper in [(headline)paper ref [43]] explored variations on Brans-Dicke (BD) scalar-tensor gravitation.
Briefly, BD adds an additional scalar field to the familiar Einstein curvature tensor of General Relativity. The field is physically interpretable as a gravitational potential, or more precisely, a value that takes the global value of the gravitational constant G to a local effective gravitational constant G_{eff}. One as a result has to adapt the relationship between the curvature and the distribution of stress-energy, by introducing a coupling between matter and the scalar field. In BD, this is the Brans-Dicke coupling constant \omega. The higher the value of \omega, the more matter couples to the tensor part and the less to the scalar part of the scalar-tensor gravity. The value of \omega is not constrained by the theory: one has to look at the behaviour of the distribution of matter to constrain \omega.
In General Relativity, matter at a point can completely determine the Einstein curvature tensor at that point. In B-D, however, the scalar potential also has a role. So if the matter sources a solution in General Relativity, one needs an appropriate \omega to source a solution in B-D for a given value of the scalar potential.
The more we look at astrophysical phenomena, and the more we do direct tests in our solar system, the more strongly positive the lowest physically compatible value of \omega can be, or the closer to zero the magnitude of the scalar must be, or both.
If the value of the scalar is everywhere zero, it can simply be elided, and one gets General Relativity.
At \omega -> \infty BD has for all practical purposes identical behaviour to General Relativity for any value of the scalar. Parsimony in that case would favour GR: the scalar field, even at its |strongest|, would just not affect matter.
A goal of theoreticians trying to keep scalar-tensor theory alive and relevant as a physical possibility therefore is to have \omega very high everywhere except near compact objects such as (stellar!) black holes and neutron stars. Around them, they might hope, the strength of the self-interaction of the scalar field (like the self-interaction of the Einstein curvature tensor in General Relativity, and the tensor part of scalar-tensor) leads to interesting differences from General Relativity.
The result is a "cloud" of scalar self-interaction around compact objects caused by the very high but still not infinite coupling constant, and the presence of matter. This is called "spontaneous scalarization". In [paper ref [43]] the headline paper's second author explores whether there might be "spontaneous vectorization" if one substitutes a vector field for the scalar field. It does, and then generalizes to spontaneous "self-gravitating gravitational cloud" formation in any theory where matter couples to an additional gravitational component beyond the Einstein tensor.
Boiling this down: 2nd author Ramazanoğlu has been keen on finding some extra knobs to twiddle in strong gravity while ensuring the knob-twiddling produces no measurable effect in weak gravity, and he has a preference for vector-tensor rather than scalar-tensor gravity. In these, it's a dynamical vector field that couples to matter (only) gravitationally, so again one has a coupling constant to choose. Although the details of parametrizing the coupling differ from Brans-Dicke, the coupling must be weak in a universe with the sorts of structures that exist in our sky.
As he and his co-authors write in the headline paper, they are excited by the theoretical possbility that there may be '... new fundamental fields [which] lie "dormant" in weak-gravity environments and yet manage to have significant effects in strongly gravitating bodies and systems'.
Which is perfectly reasonable for a theorist, driven by theoretical considerations.
There is no astrophysical justification for such fields, however (and some evidence against them) and generically the effects of these fields have to die away around ordinary stars and large black holes, or collapsing systems (star and planet formation, galaxy and galaxy-cluster evolution) don't match what we see in the sky.
Worse, there are some further problems for the theorists: [a] there is evidence in multi-pulsar systems (Hulse-Taylor double and PSR J0337+1715 triple) for the strong equivalence principle (SEP), and one cannot faithfully reproduce the SEP when the magnitude of the vector component is large, as it must be near compact objects (like pulsars); [b] gradient instabilities where there are modes with negative propagation speed squared lead to an explosion of self-gravitating-gravity that is nothing like what we see in our universe, and is almost certainly incompatible with the hot big bang; [c] the presence of quantum ghosts, leading to an unstable vacuum state.
The [c] ghosts are the topic of the headline paper.
Start with a classical vacuum vector-tensor background, then equip it with a (matter) quantum field where we freeze every point in the quantum field at the vacuum expectation value. Now add a compact mass, and see what happens to the vector-scalar gravitational fields, and the matter quantum field as they interact according to the coupling. Do we get negative energy modes? If so, we must have https://en.wikipedia.org/wiki/Ghost_(physics)#Bad_ghosts whose causal cones are much wider than those of massless particles. Mechanisms that suppress negative energy modes break Lorentz invariance, and we have very very very good evidence that we have Lorentz invariance everywhere in our sky.
One might, with enormous care, contrive a globally hyperbolic (and thus strongly causal) toy universe even with these FTL ghosts, following some logic proposed by Bob Geroch (https://arxiv.org/abs/1005.1614). Geroch's argument is that one could not say a priori that such a toy universe would -- or even could -- develop structures like the ones in our universe. One would have to just go compute the evolution and find out.
At least two of the authors of the headline paper specialize in computational gravitational physics, so maybe that's on their todo list. (cf. their Acknowledgments section, final sentence).
In this paper they only went as far as finding out that (a) being haunted by bad ghosts seems pretty generic where one has a novel auxiliary gravitational field that is strong only in strong gravity and (b) there may be refuge in adapting the non-linear self-interactions. As a slogan, "gravity self-gravitates, but the vector and tensor parts self-interact differently". It's an interesting theoretical idea, but there won't be astrophysical evidence to support or refute it for at least many years, and only then if they commit to choosing a range of couplings (rather than adjusting the coupling constant to match new evidence if and as it appears, which would be awkward if the new evidence does not force them into closer accord with General Relativity, as has happened to Brans-Dicke). Indeed, this entire programme of "spontaneous scalarization" and "spontaneous vectorization" is because there is so much observational (and in-solar-system experimental) support for General Relativity, meaning they have to hide their differences from GR in less and less observationally-accessible settings. In particular, they explicitly aim for complete compatibility with General Relativity in the weak field limit (give or take some thoughts on gravitational waves), and differences only close to small compact objects, or deep within large black holes (and thus hidden from practical observation).
My takeaway:
* It's very strange that omega is not constant everywhere. I've read stranger things, but it's suspicious that it must change in a very specific way.
* This is not a mainline extension of GR. (Does this sentence even make sense? I mean, it's not like the supersymmetry extension of the standard model, that has a big chunk of the community behind it, but it's still far from confirmed.)
* When I read in the title "ghosts", I though it was a stupid press article translation of "perturbations" or something. I was confusing reading the paper. But now I understand it has a very specific technical meaning of a weird faster than light thing in weird objects (or they equations). If confirmed, are they real things, or just a math effect like in the phase vs group velocity problems?
The Brans-Dicke (BD) coupling constant \omega can be constant everywhere in a specifically-modelled universe, and usually is. What varies by point in spacetime is the strength of the scalar field. \omega just relates the strength of that scalar field to its gravitational effects on matter. A BD universe with a higher \omega needs a higher absolute scalar value to deviate from General Relativity (which has no scalar potential field, and no \omega parameter).
One can vary anything in a theory, so a f(\omega)+f(scalar) theory is certainly something you can write down and explore the material consequences of. Usually you get an obviously unphysical theory. Even if the theory is not obviously unphysical, if it forces one to add more parameters that one proceeds to counter-tunes to match the good physical theory without these parameters, what is being gained? Maybe a deeper understanding of the more physical of the theories?
No theory that has an auxiliary gravitational field is properly an extension of General Relativity, it is an alternative to GR. This seems like a fine distinction, but auxiliary fields tend to produce astonishingly different outcomes for matter compared to General Relativity. One has to do headstands to suppress these differences or one gets a pretty different set of orbits (visible in signal-timings between spaceships, or lasers bouncing off the moon, etc) in our solar system, a very different count and/or average shape of galaxies, or a very different "texture" to the cosmic microwave background.
To be a candidate for a physical theory of gravitation, the alternative theory must of course match observations at least as well as General Relativity does. GR is supported by a lot of observations, especially in the weak field limit, and especially where GR differs from Newtonian gravitation far from masses.
The authors show that when they do a headstand to make the effects of the vector auxiliary gravitational field vanish in our solar system but not around neutron stars or small black holes, then matter around the black holes can trigger a gravitational avalanche, making the small black hole bigger than is possible by throwing all the nearby matter into it. In fact, it can run away and grow without bound, with the central mass M exceeding all the matter in the modelled universe.
This is caused by ghosts appearing when one does three things: (1) make the auxiliary scalar or vector field dense around and within these compact massive objects but sparse at a distance, (2) make the coupling of matter to the auxiliary field relevant (rather than vanishingly weak) and (3) add a quantum mechanical matter field. Fluctuations in the quantum matter produce fluctuations in the scalar field, and those fluctuations can take a one-way trip across some notional zero: rather than fluctuating from e.g. + -> 0 -> - and then back again - -> 0 -> +, a ghost gets in the way and keeps the fluctuated mode always negative (which means stronger gravitation than one expects from the local distribution of quantum matter).
> ... very specific technical meaning ...
Ghosts are almost always unphysical -- their presence breaks symmetries of nature that are well-tested, such as the local conservation of the proton mass, or a proton's passive or active gravitational charge. The passive charge is how a proton responds to a large nearby mass, while the active charge is how the proton affects the large nearby mass. One well-tested symmetry is that passive gravitational charge = active gravitational charge = mass.
Breaking the symmetries of the Poincaré group (the symmetries of Special Relativity: invariance under translation, rotation, and boost) with respect to that relationship is usually a sign that one's alternative gravitational theory is unphysical. A proton's mass should be the same on the Earth, the moon, in the Cassini-Huygens probe, in the MESSENGER probe, or in the fusing areas of the sun. You should expect to prove an argument that one can allow the proton mass to differ in the most highly redshifted galaxies, or in cosmic rays ejected from them, or at the formation of the cosmic microwave background (or near neutron stars).
Here it might be instructive to look at a specific example of a different ghost, the https://en.wikipedia.org/wiki/Massive_gravity#The_Boulware%E... which haunts a large family of theories which limit the range of the gravitational interaction to be less than that of the electromagnetic interaction. This adapts the active gravitational charge of the proton, making it weaker at long range. The range-limitation is done by having gravitational waves obey the massive wave equation while light continues to obey the massless wave equation. A second-quantization of these waves produces a massive graviton. The mass is taken to be low, much less than neutrinos, but this still turns out to create problems matching everything that General Relativity does, and at observable large distance scales.
Auxiliary metric fields -- scalar, vector, and tensor -- were brought in to chase away the Boulware-Deser ghost, so that large gravitating structures (which are contain many many protons) match what we see in the sky, while still differing from General Relativity in an area of interest (inflation, accelerated expansion, the apparent MOND relation taken to the relativistic limit, and black hole singularities). But like the old woman who ate a fly, bringing in further mathematical objects to stabilize the relationship between curvature and matter appears to be a losing physical programme.
In the particular family of theories in the headline paper, which descends from this work with massive gravity, the BD ghost may be chased away, but a different ghost associated with an effective mass squared shows up and creates divergences in the field equations around black holes with perfectly reasonable spins, and at both the vacuum-atmosphere interface and shallowly below the surface of a neutron star.
[1] https://arxiv.org/abs/1308.6587v2 published in PRL.
I hope to see you in the next GR discussion.
Anyway, one last question:
Did you choose the mass of the proton in the example because it's special? (The mass of the proton is not just the sum of the mass of the quarks, moreover, the mass of the quarks is quite small.) Is it equivalent to use the mass of the electron? (That has no inner parts (almost).)
Chemistry and the absorption lines or astrophysical maser emissions of molecular clouds (of water, ammonia, and so on) works in all directions and at all scales, or we would very excitedly notice differences in the equivalents of the Lyman-alpha forest and wonder what was happening to protons in different parts of the universe. As far as we can tell, protons interact gravitationally, electromagnetically, and via weak interactions in identical fashion everywhere and everywhen.
We could certainly talk about electrons instead of protons. Focusing on baryons and discussing baryonic matter, rather than things like leptons, is more cultural than anything. For example: https://astronomy.swin.edu.au/cosmos/B/Baryonic+Matter (Electrons and neutrons are practically always dancing with protons; for the most part it's only protons that you find going solo in large numbers in ionized clouds. Likewise, where do you find most of the universe's quarks and gluons? In protons.)
Of course one could also say that as far as we can tell from observation, the Standard Model works well everywhere in the observed universe. There may be some small correcting terms yet to be added to make it work better or in regimes or regions not yet observed, but "small" is important there. (Personally, when I hear "SM" I hear relativistic QFT on curved spacetime, and start thinking of the behaviour of dozens of fields, and in that context electrons are already messy (cf. dressed electrons), and protons are almost incomprehensible: https://profmattstrassler.com/articles-and-posts/largehadron... just scratches the surface!)
At that early point, I realised that the article was either going to be bafflegab, or it was way over my head. Can anyone explain what a "tachyonic instability" is?
When you learn about quantum field theory you find out that particles can be modeled as linearized modes of fields. e.g. the photon is a mode of the electro-magnetic field. This is a more general theory than just treating particles as fundamental, because it makes predictions for creation and annihilation of particles. Again, squared mass appears in these formulea.
If you write down a field theory with a squared mass term that is _negative_, you might naively think that this gives you particles that travel faster than the speed of light. However, the modes of such a field would not oscillate - they would grow exponentially (at least until they get out of the linear regime).
This is what is meant by a "tachyonic instability".
Would it be reasonably plausible that such instabilities get covered up by appropriate event horizons? E.g. how do we know that such tachyonic instabilities aren’t a physical effect of some kind?