## Is it possible to get rid of exotic matter in warp drive?

12/05/2019

On 1994, Miguel Alcubierre proposed a solution of the Einstein equations (see here) describing a space-time bubble moving at arbitrary speed. It is important to notice that no violation of the light speed limit happens because is the space-time moving and inside the bubble everything goes as expected. This kind of solutions of the Einstein equations have a fundamental drawback: they violate Weak Energy Condition (WEC) and, in order to exist, some exotic matter with negative energy density must exist. Useless to say, nobody has ever seen such kind of matter. There seems to exist some clue in the way Casimir effect works but this just relies on the way one interprets quantum fields rather than an evidence of existence. Besides, since the initial proposal, a great number of studies have been published showing how pathological the Alcubierre’s solution can be, also recurring to quantum field theory (e.g. Hawking radiation). So, we have to turn to dream of a possible interstellar travel hoping that some smart guy will one day come out with a better solution.

Of course, Alcubierre’s solution is rather interesting from a physical point of view as it belongs to a number of older solutions, like wormholes, time machines and like that, yielded by very famous authors as Kip Thorne, that arise when one impose a solution and then check the conditions of its existence. This turns out to be a determination of the energy-momentum tensor and, unavoidably, is negative. Then, they violate whatever energy condition of the Einstein equations granting pathological behaviour. On the other side, they appear the most palatable for science fiction of possible futures of space and time travels. In these times where this kind of technologies are largely employed by the film industry, moving the fantasy of millions, we would hope that such futures should also be possible.

It is interesting to note the procedure to obtain these particular solutions. One engineers it on a desk and then substitute them into the Einstein equations to see when are really a solution. One fixes in this way the energy requirements. On the other side, it is difficult to come out from the blue with a solution of the Einstein equations that provides such a particular behaviour, moving the other way around. It is also possible that such solutions are not possible and imply always a violation of the energy conditions. Some theorems have been proved in the course of time that seem to prohibit them (e.g. see here). Of course, I am convinced that the energy conditions must be respected if we want to have the physics that describes our universe. They cannot be evaded.

So, turning at the question of the title, could we think of a possible warp drive solution of the Einstein equations without exotic matter? The answer can be yes of course provided we are able to recover the York time, or warp factor, in the way Alcubierre obtained it with its pathological solution. At first, this seems an impossible mission. But the space-time bubble we are considering is a very small perturbation and perturbation theory can come to rescue. Particularly, when this perturbation can be locally very strong. On 2005, I proposed such a solution (see here) together with a technique to solve the Einstein equations when the metric is strongly perturbed. My intent at that time was to give a proof of the BKL conjecture. A smart referee suggested to me to give an example of application of the method. The metric I have obtained in this way, perturbing a Schwarzaschild metric, yields a solution that has an identical York time (warp factor) as for the Alcubierre’s metric. Of course, I am respecting energy conditions as I am directly solving the Einstein equations that do.

The identity between the York times can be obtained provided the form factor proposed by Alcubierre is taken to be 1 but this is just the simplest case. Here is an animation of my warp factor.

It seen the bubble moving as expected along the x direction.

My personal hope is that this will go beyond a mathematical curiosity. On the other side, it should be understood how to provide such kind of perturbations to a given metric. I can think to the Einstein-Maxwell equations solved using perturbation theory. There is a lot of literature about and a lot of great contributions on this argument.

Finally, this could give a meaning to the following video by NASA.

## They did it!

11/02/2016

This is a great moment in history of physics: Gravitational waves were directly detected by the merging of two black holes by the LIGO Collaboration. This is a new world we arrived at and there will be a lot to be explored and understood. I do not know if it is for the direct proof of existence of gravitational waves or black holes that fixes this great moment forever in the memory of mankind. But by today we have both!

You can find an excellent recount here. This is the paper

Thank you for this great work!

## News on propulsion at NASA

01/11/2015

There has been a lot of rumor on measurements performed by Eagleworks labs at NASA this spring. After that, NASA imposed a veto on whatever information should coming out about the work of this group until peer-reviewed work should have appeared. Most of the problems come out from the question of the EmDrive. This is a presumed thruster obtained by pumping radio-frequency into a cavity shaped as a closed frustum. This device has been largely dismissed by the physicists’ community due to a blatant violation of conservation of momentum. Such an object should stand still aside from known physical effects as Lorentz force or thermal thrust arising from heating of the cavity in the air. The claimed effect is really tiny standing on measurements that has been done since now and so, mundane explanations remain the most credited. Notwithstanding this, people at NASA have kept on working. This is testified by the recent posts by Paul March at Nasaspaceflight forum. Paul march is a member of the NASA group working on new propulsion technologies and what he is claiming is really striking. I report this here

where he claims that, notwithstanding all the precautions, they keep on seeing a $100 \mu N$ thrust. They know perfectly that, for Maxwell theory, no thrust should be observed as stated by the following post

So, they see a thrust, after having removed all mundane effects, and the possible explanation for it is not classical electromagnetism as all said from the start. In particular, it is cited by the questioner the link to Greg Egan’s post explaining why there cannot be any thrust by known electromagnetism with this geometry (see Greg Egan’s post). I think that they will make their results known once the peer-review process will be concluded. I would like to remember that other NASA labs asked to concur to confirm their measurements.

This group also performed interference experiments on this cavity and observed an effect. If all this will be confirmed it will represent a breakthrough, not only from the technological side let me say, as a new physical effect will be proved at work with general relativity now to be experimentally managed on a tabletop device. Note that general relativity is always at work in this situation with a large density of electromagnetic energy as I also discussed here. It should be said that this has nothing to do with warp drive as conceived in Alcubierre metric and similar.

We hope to hear very soon from this group with more official channels. Surely, their results will provide a wealth of new avenues to pursue for research and technology.

Marco Frasca (2015). Einstein-Maxwell equations for asymmetric resonant cavities arXiv arXiv: 1505.06917v1

## NASA and warp drive: An update

25/04/2015

There is some excitement in the net about some news of Harold White’s experiment at NASA. I have uncovered it by chance at a forum. This is a well-frequented site with people at NASA posting on it and regularly updating about the work that they are carrying out. You can also have noticed some activity in the Wikipedia’s pages about it (see here at the section on EmDrive and here). Wikipedia’s section on EmDrive explains in a few lines what is going on. Running a laser inside the RF cavity of the device they observed an unusual effect. They do not know yet if this could be better explained by more mundane reasons like air heating inside the cavity itself. They will repeat the measurements in a vacuum chamber to exclude such a possibility. I present here some of the slides used by White to recount about this This is the current take by Dr. White as reported by one of his colleagues too prone to leak on nasaspaceflight forum:

…to be more careful in declaring we’ve observed the first lab based space-time warp signal and rather say we have observed another non-negative results in regards to the current still in-air WFI tests, even though they are the best signals we’ve seen to date. It appears that whenever we talk about warp-drives in our work in a positive way, the general populace and the press reads way too much into our technical disclosures and progress.

I would like to remember that White is not using exotic matter at all. Rather, he is working with strong RF fields to try to develop a warp bubble. This was stated here even if implicitly. Finally, an EmDrive device has been properly described here. Using strong external fields to modify locally a space-time has been described here. If this will be confirmed in the next few months, it will represent a major breakthrough in experimental general relativity since Eddington  confirmed the bending of light near the sun. Applications would follow if this idea will appear scalable but it will be a shocking result anyway. We look forward to hear from White very soon.

Marco Frasca (2005). Strong coupling expansion for general relativity Int.J.Mod.Phys.D15:1373-1386,2006 arXiv: hep-th/0508246v3

## What is going on at NASA?

09/01/2015

As a physicist I have been always interested about experiments that can corroborate theoretical findings. Most of these often become important applications for everyday life or change forever the course of the history of mankind. With this in view, I am currently following with great interest the efforts by the NASA group headed by Harold White. This work has arisen uproar in the web and in the media as it was come to envision the possibility to realize a warp drive, in the way Alcubierre devised it, and the stars were in the reach shortly. As it is well-known, Alcubierre drive implies exotic matter something that does not appear at hand neither in small nor in large quantity. On the other side, it was indirectly observed in the Casimir effect, a beautiful application of quantum field theory to real life. So, it is rather normal to link warp drive with exotic matter. It should be emphasized that nobody on Earth ever managed it in some way and it is not available at your nearest grocery store. The experiment carried out by Harold White and his group is realized with an interference device using lasers on an optical table. The idea is to observe a modification of space-time, a minuscule one, that would modify the paths of the laser beams. This would be comparable to the realization of the Chicago pile by Enrico Fermi that was the starting point for the Manhattan project. I would like to emphasize that such a laboratory small-scale manipulation of space-time would be a huge breakthrough in physics and would open up the way to a new kind of engineering, that of space-time. So, our hopes for a warp drive would be totally fulfilled.

There is an eager desire to obtain any possible information about the progress of White’s work but, of course, there are a couple of hurdles. The first one is that a scientist needs to be certain before to claim a result and we know very well why from some blatant examples in the last years. Extraordinary claims require extraordinary evidence. Last but not least, Harold White is employed at NASA and some restrictions could be required by the organization he is working with. So, it is really interesting a video appeared quite recently where White claims that the effect is there but further work is needed for confirmation. If you have a hour of your spare time, this video is worthwhile to be seen.

This video is interesting per se because Harold White is talking to his colleagues at NASA. But in the question time happens the interesting fact. A White’s colleague asks him “where is the exotic matter?”:

and here something interesting happens. White seems to avoid the question and admits that they talked before in the office. What is more interesting is what the White’s colleague is saying then unveiling some of the machinery behind the experiment. The colleague says that the experiment could be carried out in some strong coupling regime that makes the magic happen without any exotic matter. White denies and disagrees. We know that he is using strong electromagnetic fields in the interference zone. Indeed, the matter of the behaviour of the space-time in a strong perturbation was studied for cosmological aims by Belinski, Kalathnikov and Lifshitz, the BKL trio. This scenario was confirmed by numerical studies by David Garfinkle (see here). I was able to derive it by analysing the behaviour of the Einstein equations under a strong perturbation (see here) in analytical way. So, the chance to study such effects in a laboratory would be really striking and would mean an incredible breakthrough for people working in general relativity and related fields. What the exchange between White and his colleague implies is that this could be already at hand and without exotic matter. All the growing concerns about the work at NASA are then not applicable and a different kind of analysis would be needed. Particularly, Alcubierre drive should be devised in a different way. As a physicist, I am eager to learn more about this and to know the real answer, from the horse’s mouth, to the question “where is the exotic matter?”.

Miguel Alcubierre (2000). The warp drive: hyper-fast travel within general relativity Class.Quant.Grav.11:L73-L77,1994 arXiv: gr-qc/0009013v1

David Garfinkle (2003). Numerical simulations of generic singuarities Phys.Rev.Lett. 93 (2004) 161101 arXiv: gr-qc/0312117v4

Marco Frasca (2005). Strong coupling expansion for general relativity Int.J.Mod.Phys.D15:1373-1386,2006 arXiv: hep-th/0508246v3

## Standard Model at the horizon

08/12/2014

Hawking radiation is one of the most famous effects where quantum field theory combines successfully with general relativity. Since 1975 when Stephen Hawking uncovered it, this result has obtained a enormous consideration and has been derived in a lot of different ways. The idea is that, very near the horizon of a black hole, a pair of particles can be produced one of which falls into the hole and the other escapes to infinity and is seen as emitted radiation. The overall effect is to drain energy from the hole, as the pair is formed at its expenses, and its ultimate fate is to evaporate. The distribution of this radiation is practically thermal and a temperature and an entropy can be attached to the black hole. The entropy is proportional to the area of the black hole computed at the horizon, as also postulated by Jacob Bekenstein, and so, it can only increase. Thermodynamics applies to black holes as well. Since then, the quest to understand the microscopic origin of such an entropy has seen a huge literature production with the notable understanding coming from string theory and loop quantum gravity.

In all the derivations of this effect people generally assumes that the particles are free and there are very good reasons to do so. In this way the theory is easier to manage and quantum field theory on curved spaces yields definite results. The wave equation is separable and exactly solvable (see here and here). For a scalar field, if you had a self-interaction term you are immediately in trouble. Notwithstanding this, in  the ’80 Unruh and Leahy, considering the simplified case of two dimensions and Schwarzschild geometry, uncovered a peculiar effect: At the horizon of the black the interaction appears to be switched off (see here). This means that the original derivation by Hawking for free particles has indeed a general meaning but, the worst conclusion, all particles become non interacting and massless at the horizon when one considers the Standard Model! Cooper will have very bad times crossing Gargantua’s horizon.

Turning back from science fiction to reality, this problem stood forgotten for all this time and nobody studied this fact too much. The reason is that the vacuum in a curved space-time is not trivial, as firstly noted by Hawking, and mostly so when particles interact. Simply, people has increasing difficulties to manage the theory that is already complicated in its simplest form. Algebraic quantum field theory provides a rigorous approach to this (e.g. see here). These authors consider an interacting theory with a $\varphi^3$ term but do perturbation theory (small self-interaction) probably hiding in this way the Unruh-Leahy effect.

The situation can change radically if one has exact solutions. A $\varphi^4$ classical theory can be indeed solved exactly and one can make it manageable (see here). A full quantum field theory can be developed in the strong self-interaction limit (see here) and so, Unruh-Leahy effect can be accounted for. I did so and then, I have got the same conclusion for the Kerr black hole (the one of Interstellar) in four dimensions (see here). This can have devastating implications for the Standard Model of particle physics. The reason is that, if Higgs field is switched off at the horizon, all the particles will lose their masses and electroweak symmetry will be recovered. Besides, further analysis will be necessary also for Yang-Mills fields and I suspect that also in this case the same conclusion has to hold. So, the Unruh-Leahy effect seems to be on the same footing and importance of the Hawking radiation. A deep understanding of it would be needed starting from quantum gravity. It is a holy grail, the switch-off of all couplings, kind of.

Further analysis is needed to get a confirmation of it. But now, I am somewhat more scared to cross a horizon.

V. B. Bezerra, H. S. Vieira, & André A. Costa (2013). The Klein-Gordon equation in the spacetime of a charged and rotating black hole Class. Quantum Grav. 31 (2014) 045003 arXiv: 1312.4823v1

H. S. Vieira, V. B. Bezerra, & C. R. Muniz (2014). Exact solutions of the Klein-Gordon equation in the Kerr-Newman background and Hawking radiation Annals of Physics 350 (2014) 14-28 arXiv: 1401.5397v4

Leahy, D., & Unruh, W. (1983). Effects of a λΦ4 interaction on black-hole evaporation in two dimensions Physical Review D, 28 (4), 694-702 DOI: 10.1103/PhysRevD.28.694

Giovanni Collini, Valter Moretti, & Nicola Pinamonti (2013). Tunnelling black-hole radiation with $φ^3$ self-interaction: one-loop computation for Rindler Killing horizons Lett. Math. Phys. 104 (2014) 217-232 arXiv: 1302.5253v4

Marco Frasca (2009). Exact solutions of classical scalar field equations J.Nonlin.Math.Phys.18:291-297,2011 arXiv: 0907.4053v2

Marco Frasca (2013). Scalar field theory in the strong self-interaction limit Eur. Phys. J. C (2014) 74:2929 arXiv: 1306.6530v5

Marco Frasca (2014). Hawking radiation and interacting fields arXiv arXiv: 1412.1955v1

## Back to work

02/02/2014

I would like to have a lot more time to write on my blog. Indeed, time is something I have no often and also the connection is not so good as I would like in the places I spend most of it. So, I take this moment to give an update of what I have seen around in these days.

LHC has found no evidence of dark matter so far (see here). Dark matter appears even more difficult to see and theory is not able to help the search. This is also one of our major venues to go beyond the Standard Model. On the other side, ASACUSA experiment at CERN produced the first beam of antihydpogen atoms (see here, this article is free to read). We expect no relevant news about the very nature of Higgs until, on 2015, LHC will restart. It must be said that the data collected so far are saying to us that this particle is behaving very nearly as that postulated by Weinberg on 1967.

In these days there has been some fuss about the realization in laboratory of a Dirac magnetic monopole (see here).  Notwithstanding this is a really beautiful experiment, nobody has seen a magnetic monopole so far. It is a simulation performed with another physical system: A BEC. This is a successful technology that will permit us an even better understanding of physical systems that are difficult to observe. Studies are ongoing to realize a simulation of  Hawking radiation in such a system.  Even if this is the state of affairs, I have read in social networks and in the news that a magnetic monopole was seen in laboratory. Of course, this is not true.

The question of black holes is always at the top of the list of the main problems in physics. Mostly when a master of physics comes out with a new point of view. So, a lot of  fuss arose from this article in Nature involving a new idea from Stephen Hawking that the author published in a paper on arxiv (see here). Beyond the resounding title, Hawking is just proposing a way to avoid the concept of firewalls that was at the center of a hot debate in the last months. Again we recognize that a journalist is not making a good job but is generating a lot of noise around and noise can hide a signal very well.

Finally, we hope in a better year in science communication. The start was somewhat disappointing.

Kuroda N, Ulmer S, Murtagh DJ, Van Gorp S, Nagata Y, Diermaier M, Federmann S, Leali M, Malbrunot C, Mascagna V, Massiczek O, Michishio K, Mizutani T, Mohri A, Nagahama H, Ohtsuka M, Radics B, Sakurai S, Sauerzopf C, Suzuki K, Tajima M, Torii HA, Venturelli L, Wu Nschek B, Zmeskal J, Zurlo N, Higaki H, Kanai Y, Lodi Rizzini E, Nagashima Y, Matsuda Y, Widmann E, & Yamazaki Y (2014). A source of antihydrogen for in-flight hyperfine spectroscopy. Nature communications, 5 PMID: 24448273

M. W. Ray,, E. Ruokokoski,, S. Kandel,, M. Möttönen,, & D. S. Hall (2014). Observation of Dirac monopoles in a synthetic magnetic field Nature, 505, 657-660 DOI: 10.1038/nature12954

Zeeya Merali (2014). Stephen Hawking: ‘There are no black holes’ Nature DOI: 10.1038/nature.2014.14583

S. W. Hawking (2014). Information Preservation and Weather Forecasting for Black Holes arXiv arXiv: 1401.5761v1

## Some more news on warp drive

23/07/2013

Today, New York Times published an article with an interview to Harold “Sonny” White about NASA studies on warp drive (see here). This revamped the interest about what NASA is funding (with a really small budget being  just \$50,000) on this that have to be considered forefront research. For the readers that are not aware about what this research is aimed to, I invite them to read the very good article on Wikipedia about Alcubierre drive. As can be easily imagined, this article gets some new adding  each day and moves the curiosity of a myriad of people around the world. So, the activity of this NASA’s group is under a lot of attention by media and, with a lot of skepticism, by the scientific community. Alcubierre itself, the inventor of this idea, does not believe at all that is doable. The main reasons are well explained here (hat tip to Jennifer Ouellette) and one of these, the most important one maybe, is a lot of missing information as studies on this idea showed more its impossibility than else.

Anyhow,  we hope that Harold White will fill all the details at 2013 Starship Congress that he will attend giving a talk (see here). The schedule is here. He will speak on August 17th.

Miguel Alcubierre (2000). The warp drive: hyper-fast travel within general relativity Class.Quant.Grav.11:L73-L77,1994 arXiv: gr-qc/0009013v1

## Ashtekar and the BKL conjecture

18/02/2011

Abhay Ashtekar is a well-known Indian physicist working at Pennsylvania State University. He has produced a fundamental paper in general relativity that has been the cornerstone of all the field of research of loop quantum gravity. Beyond the possible value that loop quantum gravity may have, we will see in the future, this result of Ashtekar will stand as a fundamental contribution to general relativity. Today on arxiv he, Adam Henderson and David Sloan posted a beautiful paper where the Ashtekar’s approach is used to reformulate the Belinski-Khalatnikov-Lifshitz (BKL) conjecture.

Let me explain why this conjecture is important in general relativity. The question to be answered is the behavior of gravitational fields near singularities. About this, there exist some fundamental theorems due to Roger Penrose and Stephen Hawking. These theorems just prove that singularities are an unavoidable consequence of the Einstein equations but are not able to state the exact form of the solutions near such singularities. Vladimir Belinski, Isaak Markovich Khalatnikov and Evgeny Lifshitz put forward a conjecture that gave them the possibility to get the exact analytical behavior of the solutions of the Einstein equations near a singularity: When a gravitational field is strong enough, as near a singularity, the spatial derivatives in the Einstein equations can be safely neglected and only derivatives with respect to time should be retained. With this hypothesis, these authors were able to reduce the Einstein equations to a set of ordinary differential equations, that are generally more treatable, and to draw important conclusions about the gravitational field in these situations. As you may note, they postulated a gradient expansion in a regime of a strong perturbation!

Initially, this conjecture met with skepticism. People simply have no reason to believe to it and, apparently, there was no reason why spatial variations in a solution of a non-linear equation with a strong non-linearity should have to be neglected. I had the luck to meet Vladimir Belinski at the University of Rome “La Sapienza”. I was there to follow some courses after my Laurea and Vladimir was teaching a general relativity course that I took. The course showed the BKL approach and gravitational solitons (another great contribution of Vladimir to general relativity). Vladimir is also known to have written some parts of the second volume of the books of Landau and Lifshitz on theoretical physics. After the lesson on the BKL approach I talked to him about the fact that I was able to get their results as their approach was just the leading order of a strong coupling expansion. It was on 1992 and I had just obtained the gradient expansion for the Schroedinger equation, also known in literature as the Wigner-Kirkwood expansion, through my approach to strong coupling expansion. The publication of my proof happened just on 2006 (see here), 14 years after our colloquium.

Back to Ashtekar, Henderson and Sloan’s paper, this contribution is relevant for a couple of reasons that go beyond application to quantum gravity. Firstly, they give a short but insightful excursus on the current situation about this conjecture and how computer simulations are showing that it is right (a gradient expansion is a strong coupling expansion!). Secondly, they provide a sound formulation using Ashtekar variables of the Einstein equations that is better suited for its study. In my proof too I use a Hamiltonian formulation but through ADM formalism. These authors have in mind quantum gravity instead and so ADM formalism could not be the best for this aim. In any case, such a different approach could also reveal useful for numerical simulations.

Finally, all this matter is a strong support to my view started with my paper on 1992 on Physical Review A. Since then, I have produced a lot of work with a multitude of applications in almost all areas of physics. I hope that the current trend of confirmations of the goodness of my ideas about perturbation theory will keep on. As a researcher, it is a privilege to be part of this adventure of humankind.

Ashtekar, A. (1986). New Variables for Classical and Quantum Gravity Physical Review Letters, 57 (18), 2244-2247 DOI: 10.1103/PhysRevLett.57.2244

Abhay Ashtekar, Adam Henderson, & David Sloan (2011). A Hamiltonian Formulation of the BKL Conjecture arxiv arXiv: 1102.3474v1

Marco Frasca (2005). Strong coupling expansion for general relativity Int.J.Mod.Phys. D15 (2006) 1373-1386 arXiv: hep-th/0508246v3

Frasca, M. (1992). Strong-field approximation for the Schrödinger equation Physical Review A, 45 (1), 43-46 DOI: 10.1103/PhysRevA.45.43