I have been absent from this blog for a long time, basically since the LHC bumb went away, but not from physics. In the last days Sabinne Hossenfelder has been again doing posts against the idea of making new coliders as, for example Why a larger particle collider is not currently a good investment. I had already written a post, years ago, saying that it is necessary to make new coliders, and hope that Europeans and Chinese make one.

The topic of this entry is somewhat related to the LHC bumb. Let's remember the affair, an statistical in the LHC as around 3.5 $$ \sigma $$ was found and a lot of paper trying to find an explanation for it were published. A few mounts later the LHC published new data and the fluctuation had gone away. The usual interpretation was that the fluctuations were of pure statistical nature, which, of course, is the most rational interpretation, but here I wonder if beyond the standard model at least part of the new physic is a new kind of physic and that signal,, pointed that new kind of physic.

Before that fluctuations some others had been found, but with minor statistical significance, to later vanish.. Also, since them another ones had been found, and some have gone away also while a few others are waiting for new data to determine their fate.

Well, clearly statistical fluctuations are something that are very common, and it is not at all surprising to find them. In fact that is the reason of the $$ 5 \sigma$$ criteria to claim a discovery, and even this very high statistical significance could become not enough with the very large amount of data that the LHC is getting, according to some claims in Tomasso Dorigo's blog.

To understand why I use the expression "new kind of physic" we must wonder what we understand as a particle. In particle physics we have quantum fields, and they create particles. That particles created by the field have a definite mass, charge (under whatever gague group it is charged) and cross section of production but, could it be otherwise?

Well, if I wrote this entry is because I am considering that possibility. In that case maybe we have "erractic" particles that are defined by the possibility that their properties (mass, charge and coupling constants to standard model particles) can vary in time and/or space.

The idea of varying coupling constants is not new and goes back to Dirac, but usually it was considered that the considered coupling constants were the standard model ones, specially, the $$ \alpha$$ electromagnetic constant. My idea is that, someway, the standard model could be stable, but the should be new (low energy, in the sense of well bellow the planck energy, but still high energy for the earth colliders) physics could be unstable, with fluctuating characteristics. In his recent papers about the issue of whether string theory allows or not the existence of a deSitter vacua he mentions the possibility that the coupling constants of physic beyond the standard model could variate, but has not gone too far on it. .Lubos also has said in some posts that that variations should be related to moduli and they would have inconvenient statistical properties in cosmological data. That means that I am aware that there could be difficulties with the proposal, but still I think it is interesting to say a few things more about it.

To understand how this could happen we must go to string theory, but, in order to get it somewhat easier, we coould begin by the kaluza-klein scenario. As is well known there there we have a single extra dimension compactified in a circle. There a scalar field fit the relation

$$ p_\mu p^\mu - \frac{n^2}{R^2}=0 $$

That implies

$$ m_n=\frac{\mid n\mid }{R} $$

If the scalar field is a self interecting one, thorough a $$ \lambda\pi^4$$ (or maybe another power, it is not really relevant) the 4d coupling constant is related to the 5d one by

$$ \lambda_4=\frac{1}{R} \lambda_5 $$

Where V is the "volume" of the extra dimension, in this case $$ V= 2 \phi R $$

Well, now it comes the key ingredient, the way that we assign a value to R. It is a well known problem that was resolved assigning an scalar field to the geometrical moduli of the compactified space (in this simple case the radius of the cylinder) and a potential to the moduli so that the actual radius corresponds to the minimum of the potential of the moduli field. Still moduli stabilisation is a difficult issue. Usually the potential is generated by fluxes associated to the antisymmetric fields arising in string theory and sourced in branes. Still to get an stabilisation of all moduli is something that one put's by hand to avoid runaways, and to get defined values of the quantities but the question is that maybe we are prejudicing and that, in general, not all of them the are stabilised and that the physics beyond the standard model is not constant.

In a simple K-K scenario a variation of the radio doesn't give a grate variation of the masses and constants. If we change $$ R \mapsto R + \Delta R $$ the the mass changes as

$$ \Delta m_n= \frac{\mid n \mid }{R + \Delta R}=\mid n \mid \left ( - \frac{1}{R^2} \Delta R + o \left ( \Delta R^2 \right ) \right ) $$

In more general settings that Kaluza Klein the details are different in heterotic, F-theory, "plain braneworld" type II A, etc, but the dependence of the coupling constants on the inverse of the volume of the extra space remains true. In fact the mere KK mechanism is not too much the key ingredient determining the new physics and everything is more involved. One must first get a compactification that makes that the remaining supersymmetry is N=1 and later to chose some symmetry breaking mechanism, usually through a superpotential, and part of the characteristics of the low energy physics is fixed by that superpotential (others depend on the topology of the compatification space, and even in the metric, which is generally unknown, even in the simplests cases) The superpotential usually has perturbative and not perturbative contributions (instantons) but still depends in the geometry of the compactificated space.

I tried to do all the details in a generic scenario, the one described in a Dust and all paper from 2008 The LHC String Hunter's Companion but at last I considered that not being a payed investigator it didn't worth the effort.

But, in fact, the new physic searched by the LHC were not only superpartners or particles associated to new gague groups, usually some new U(1). Some star predictions for the LHC were micro black holes and Kaluza-Klein tower of the graviton in the Lisa Randall braneworlds scenario, and also that phsyic depends on the size of an extra dimension, which was assumed to have a fixed value. Still more interesting, in that scenarios the extra dimension was expected to have a size a lot bigger than the usual compactification scale. Previous to the braneworld there was the ADD scenarios, that is mathematically simplest.

In ADD the cross section to form a black hole in a collision of energy E is:

$$ \sigma (E) \sim \frac{1}{V_n M_*^{n+2} }\left ( \frac{E}{\sqrt{V_n M_*^{n+2} } } \right ) ^\alpha $$

Here $$ M_* $$ is true the (4+n) dimensional Planck scale (the 4d one would Mpl) The relation between them is different in ADD and in RS. In RS the relation is:

$$ M^2_{pl}= \frac{V_n}{M_*^{n+2}} $$

In ADD is:

$$ M^2_{pl} = \frac{M_*^3}{k}(1-e^{-2\pi k R} ) $$

The key ingredient is again present, the cross section depends on the size of the extra dimension. If this extra dimension size varies with time, the cross section of production of black holes is not constant. Even if it varies in space and not in time we have that as the earth moves in space it will move to zones with different value of the size, and them of the cross section.

I haven't searched in deep for the formula giving the dependence of the kaluza klein tower of the graviton but I am sure that it also depends on that size.

As I said at some point I am aware that maybe there are possible issues with the "erratic particles" scenario, and may be it is impossible in string theory anyway. For sure there are people that know string theory far better than me that could explain the issues in the improbable case that they consider that my proposal deserves their attention. And even if the scenario could be viable I have not made any estimation of what the variations of the sizes could be expected to be, nor the exact influence in a concrete realistic model of the variations of the measurable masses and coupling constants or whatever associated to the size variations.

One possible way to get that estimation could relay in cosmological considerations, but my knowledge of string cosmology is too bad to even try to purchase that objective. The only thing that I could conjecture is that maybe some properties of dark matter are fluctuating, and, perhaps, that could explain the very controversial claim of DAMA/LIBRA observation of an annual modulation of dark matter detection.

But, going very weird, we could think that physics beyond the standard model is not described by string theory but that, still, the properties of that new physic are not constant.

In any case the only really important thing is the LHC is searching for new physic in the conventional way, expecting that the characteristic of the new physic is as stable as the characteristics of the standard model physics, but may be that is a wrong starting point. Maybe there is something special in the standard model that makes it robust against fluctuations, but the same thing doesn't apply to the extra physics.

If true the people doing search for new physics should design a way to distinguish between statistical fluctuations of the type they are actually considering and another ones that are generated because the actual new physic is fluctuating. My knowledge of such issues is so null that I can even give a gimp of how they would do it.

And, to end my return to bloging, and the main reason for it: please, ignore Sabine campaign and build the FCC or whatever new collider you can!

P.S. I haven't mentioned it in the post, but there is a very important aspect in this scenario. The fluctuations in the new physic would not be random but there would be important correlations. One toy model is one in which there could be to possible particles A and Be to be produced in a collision. A would have certain value Qa of a certain charge (that would not be fluctuating) and a mass that maybe fluctuates among a central value $$ Ma= MCa \pm \Delta Ma$$ and the same for the particle B. The cross section of production would depend in many factors, but, mainly, in some coupling constant that would fluctuate. The key point would be that according to the coupling variations one would have periods where the production of A would be much more favourable, and another where the production of B would be the preferred one, and periods in between, giving some characteristic pattern. If there are more particles and coupling constants involved the pattern would be more complicated, but still there would be one, and maybe that kind of patterns would make the search simpler that if the fluctuations would be pure random.

P.S. 2 In the type of scenario I describe (see Mitchel comment for the suggestion of an slighthly different one) I think that string theory would gain because if correlated fluctuations are detected some aspects of the goemetry of the compatification could be inferred form low energy physics, something not quite possible in the standard wisdom.

## Friday, February 08, 2019

## Thursday, December 10, 2015

### Android aplication to edit mathematical equations in latex

I have been looking for a long time for a good android app that would allow me to edit equations, and get the latex code of it, but I haven´t found nothing really good, untill now ;-)

The first big step was my script smart note. It makes a great work in recognizing a handwriteen equation, and allos to export the latex code to a tex file.

But, in order to use it to write equations for a blog (or paper) I rather need to get the latex code writen to the wallpaper. For that purpose I begun writing an small app myself that imported the equations of that program. Althought my script makes a great work it doesn't allways recognizes things as you wan, so I decided to make a full equation's editor that can modify the imported equations, and ,of course, make an equation from zero. I have added symbols for most usual latex symbols, and also for the unsual. As far as I know the app is the most complete latex editor on the google play. It has a lot of symbols, and you have lot of flexibility to rearrange them to meet your preferences.

But I didn't stop there. In a typical use one needs to write some equations once and again, or variants of them, and it is pretty stupid to write them from scratch every time. To ride with that the app allows to save the latex code of the equations, together with additional information that allows to search them, in a database so one can construct it´s own equation's agend.

Once you have an equation you have buttons that copy the equation to the clipboard so you only need to paste them into the destination. It even wraps them with the tags corresponding to blogspot (if you use matjax), wordpress, or forums that are latex enabled (there is a button for every purpose).

If you have a note device (phablet, or even better, tablet) you can use the multiwindow features and use my app side by side with a document reader from wich you are actually copiying the equations, or a web navigator, or the blogspot/wordpress apps, or whatever you need, so you are far more productive.

I have made two versions of the app. The basic one, that is free (and withouth google adds by now) contains all the mentioned functionality. It is available at Google Play since now here

The free version is very complete, and powerfull, but there is a full version that has best quality graphs, and aditional features (export of registers, to share among devices, or other people, import the equations from a latex file in your device, etc) that will be available pretty soon ;-).

Beeing a blogger I have made it to make easy the most usual tasks of a scientific (mainly a physic) blogger, but It also makes a great work to write equations for a paper. As far as I know I honestly think that it is by far the best, and most complete equation editor in the android market, and, althought I don't know all the equations editors available, I think that even in windows there is nothing as complete and easy to use. But, of course, it may be because I have made it and I know how to use it, maybe others don't agree. In any case I hope you kike it, and find it usefull :).

The first big step was my script smart note. It makes a great work in recognizing a handwriteen equation, and allos to export the latex code to a tex file.

But, in order to use it to write equations for a blog (or paper) I rather need to get the latex code writen to the wallpaper. For that purpose I begun writing an small app myself that imported the equations of that program. Althought my script makes a great work it doesn't allways recognizes things as you wan, so I decided to make a full equation's editor that can modify the imported equations, and ,of course, make an equation from zero. I have added symbols for most usual latex symbols, and also for the unsual. As far as I know the app is the most complete latex editor on the google play. It has a lot of symbols, and you have lot of flexibility to rearrange them to meet your preferences.

But I didn't stop there. In a typical use one needs to write some equations once and again, or variants of them, and it is pretty stupid to write them from scratch every time. To ride with that the app allows to save the latex code of the equations, together with additional information that allows to search them, in a database so one can construct it´s own equation's agend.

Once you have an equation you have buttons that copy the equation to the clipboard so you only need to paste them into the destination. It even wraps them with the tags corresponding to blogspot (if you use matjax), wordpress, or forums that are latex enabled (there is a button for every purpose).

If you have a note device (phablet, or even better, tablet) you can use the multiwindow features and use my app side by side with a document reader from wich you are actually copiying the equations, or a web navigator, or the blogspot/wordpress apps, or whatever you need, so you are far more productive.

I have made two versions of the app. The basic one, that is free (and withouth google adds by now) contains all the mentioned functionality. It is available at Google Play since now here

The free version is very complete, and powerfull, but there is a full version that has best quality graphs, and aditional features (export of registers, to share among devices, or other people, import the equations from a latex file in your device, etc) that will be available pretty soon ;-).

Beeing a blogger I have made it to make easy the most usual tasks of a scientific (mainly a physic) blogger, but It also makes a great work to write equations for a paper. As far as I know I honestly think that it is by far the best, and most complete equation editor in the android market, and, althought I don't know all the equations editors available, I think that even in windows there is nothing as complete and easy to use. But, of course, it may be because I have made it and I know how to use it, maybe others don't agree. In any case I hope you kike it, and find it usefull :).

## Tuesday, July 28, 2015

### Stringy summer cinema

This summer, as usual, there has been two major string conferences. One of them is the strings 2015, celebrated in India, and has been announced on the Lubos and Woit blog. The most relevant page of the conference contains the talks and includes links to the slides and videos to most of the conferences. I have seen some of them and my idea is to see all of the most relevant ones.
Until now the one I have liked the more are the Maldacena one "Quantum mechanics and the geometry of spacetime"
where he talks about the basics of black holes quantum aspects and from there he goes to his conjecture, to the correspondence ER=EPR, to the fluid-gravity duality and to some still to develop "Entanglements is the fluid mechanics of gravity" or something like that, that sounds really intriguing.
Another very interesting one was the one by Ashoke Sen "Surviving in a metastable de Sitter space-time"
It studies how the actual mass of the Higgs, on the metastability zone, represents a treat for the survival of the universe because there could be a transition to a most stable one in a given point and that region would grow to the speed of light. That is well known, but the new thing is that an small cosmological constant makes our universe more safe against that problem, and also discusses how having redundant information in sufficiently separated points you can be well assured against that danger.
Ok, the stirngs 2015 has been announced in the blogs (although not as much as it usually was some years ago). But the thing is that recently in Madrid there was another very important string meeting String Phenomenology 2015. As with the strings 2015 there are links to the slides of the talks, and to the videos. The only difference is that instead of youtube the videos are located in another server and I am not sure for how long they will be available. The talks covers topics in string cosmology (mainly inflation). some of results and expectatives from the LHC lot of F-theory phenomenology and a few other mixed topics. Untill now I have just begun to watch the video of Eva Silverstein, that I find somewhat confuse so I can't say too much, but you can read more (in spanish) form the Francis blog El estado actual de las predicciones en teoría de cuerdas
Wll, it has been really a large time that I had not written here. I hope next entry would not take that long ;-)

## Saturday, October 04, 2014

### Why it is a good idea to build a new supercollider

Sabine Hossenfelder has posted an entry in it's blog claiming that maybe it is not a good idea to build a new supercollider: Is the next supercollider a good investment?.
I have nothing aagins Sabine, but I disagree with his arguments, and I have ansered her in her facebook. Because my answer has been long enought I have decided to copy it here in the form of a brief blog entry.
I seriously disagree with your viewpoint about this. Perhaps the most important reason is that you are neglecting the dark side (matter and energy). We have overhelming evidence that the standard model is not the whole history. In fact we know that it is only an small part of the mass and energy of the universe. The fact of not knowing from direct evidence nothing about the major cinstituent of the universe means that we have very few predictive power about it.
Of course we ca learn from the dark side by other type of experiments, but at the end of the day if we need detailled knowledge we need to product it in a collider (if we can).
On the practical size, well, untill know the most of aplications from the standard model (the specific part that only theorethical physics study of it) come from neutrinos. That is because they can go througnt matter quite easily. Well, dark matter also can do that, but at slow speeds. If at some time we know how to product it and how to detect it in a reasonable way we could learn a lot abut solid objects, as for example, the earth, that are not available by other means.
And you have also the possibility that dark matter is not simple -a single particle- and maybe you could do it for build something.
Another point is the Higgs. We know that it existes, and it's mass. But we also know that itself alone is somewhat exoteric, I mean, the problem of it's raunaway mass and the naturallnes question. We had expected that something very special should existe at near the higgs mass, and we have barely beguined to explore that energies. Maybe we could say that naturalness was not a good idea, but note before studiying that range of energies in deep.
And that drives me to another point. We even have not run the LHC at full energy. Maybe we could find a lot of things when it does. Even the mini black holes (and the associated braneworld scenaries)are not fully discarded at present for the next energies as far as I know. And that ,for sure, would be a great, great discover. I am almost sure that if there are mesoscopic extra dimensions and we can build miniblack holes there will be a lot of new aplied physics that will be build upon it.
And about SUSY. well, it is a pain in the ass, I agree. But the fact is that it's non existance would be somewhat tantalizing. My main point about it is that local supersymmetry brings a theory of gravity, and I would find a bizarre coincidence that we have a theory of gravity and a well motivated theory of particle physics that gives gravity, and that both are unrelated.
I am sure that I could elaborate a lot more on many of the subjects but I guess that anyone interested could do it by himself.

## Sunday, June 29, 2014

### Strings 2014

Long time since my last post. There is no particular reason, just a mix of circunstances.
I write now to advise, if someone is still not aware, about the annual convention about string theory.
The slides, and vídeos, are available online at talks online
Based on previous experience with similar cases I warm potential watchers about the fact that the videos usually are available online for a limited amount of timpe, typically one or two weeks, so harry up!

## Tuesday, December 31, 2013

### A prety good year ending in Arxiv

After a somewhat disappointing year in theorethical physics where the greater topic seems to have been the black holes firewall discussion (seemingly settled in a long - around 90 pages- paper) the end year in a promising way.
One one hand we have the second paper of the famous Amplituhedron construction (I told about it in my spanish blog when it appeared) that was announced by Nima Arkani-Hamed, but of which we only had seen a somewhat introductory paper. This is the second one: Into the Amplituhedron. I have not read it, neither I did it with the first. I still am reading, from time to time, a previous long paper about grassmanians and all that. I hope I'll give a reading to the first, and to the one of today, soon, but I guess that Lubos will write a fair better post about it that anything that I could do.
I must acknowledge to Lubos the pointing to another paper. Gravitation from Entanglement in Holographic CFTs. I had not paid attention to it because almost like a question of principles I don't pay attention to papers with the fancy names "holographic" or "entanglement" in the topic. But this time it looks pretty important because it looks like is they have made a great advance in a constructive derivation of general relativity from some thermodynamics consideration, improving the previous work by Ted Jacobson. But better read the entry in Lubos Blog Einstein's equations from first law of thermodynamics in AdS.
A very different type of article is Stars in M theory (made up of intersecting branes). The title is surprising and the subject certifies that we are in front of a very exotic paper.

*We study stars in M theory. First, we obtain the analog of Oppenheimer -- Volkoff equations in a suitably general set up. We obtain analytically the asymptotic solutions to these equations when the equations of state are linear. We study perturbations around such solutions in several examples and, following a standard method, use their behaviour to determine whether an instability is present or not. In this way, we obtain a generalisation of the corresponding results of Chavanis. We also find that stars in M theory have instabilities. Therefore, if sufficiently massive, such stars will collapse. We discuss the significance of these (in)stabilities within the context of Mathur's fuzz ball proposal.*The last paper I consider is this: Inflationary paradigm after Planck 2013. The first author is the very creator of the idea of Inflation, Alan guth, and at first sight it would seem a review of resoluts. In fact is somewhat of a reply to another paper, but still it is a good way to get an idea of how the Planck results have affected our view on Inflation. One last words for my previous entry. After a discussion of it in an spanish forum I was pointed to an old paper by Tholman - http://authors.library.caltech.edu/2596/1/TOLpr30d.pdf- and, seemengly, it could be considered in the kind of ideas discussed there. My example could be translated to emission of photons between bodies orbitating a central one. The gravitational blue/red shift would play the role of the expanding/contracting universe red/blue shift and that would accommodate into a stationary situation. If the difference of blue/red shift energy is greater than the temperature difference (boltzman factors mediated) then an inverse Clausius behaviour is possible. The key point is that the Clausius Law is not equivalent to the formulation of increase of entropy in a general relativistic setup and this last kind of law still holds, at least in static (probably stationary) gravitational fields, but I am not sure about how the same would still hold in a general spacetime.## Thursday, October 24, 2013

### Spacetime kills the second law of thermodynamics, real or paradox?

This summer, among many other things, have read the famous book of Susskind and Lindesay about black hole information paradox. Casually just after finishing ir's reading I have to teach in private tuition (hope I am saying it right ) statistical mechanics (a conventional introduction with the main average topics). In doing that I have revised the foundations of it from the viewpoint of what I had read in the Susskind, and also some other aspects that one must face when triying to apply statistical mechanic reasoning to non physical problems (for example ecology (I collaborated with a guy working in mathematical ecology for a while).

There are a few things that I am thinking about, but for the present entry I will concentrate in an easy mental experiment that, at least apparently, violates the second law of thermodynamics. The key of the violation is the lack of a proper definition of energy in general relativity, but for the present case I don't even need to go into mathematical details about it. My idea was clear, take a situation where that problem in the energy definition rises a paradox that violates the second law. I tried a few strategies that possible also work, but in the end I found a really easy one that I think is simple and representative.

I have found that the easier way of attack is to use the Clausius enunciate: No physical process can transfer heat from a cold body to a warm one. The way to circunvate the law is as follows: Take two "black bodies", for example the canonical ones consisting of a cavity with photons in equilibrium with the walls that are at different temperatures. Now in the warmer one open the also canonical small hole that allow a photon, or a few ones, to scape. This is made in an expanding universe, the photons that have exit form the body travel in that space time loosing energy. Them they arrive at the coldest body. The photons of the warmer body initially had more energy that the ones in the coldest body, but now, after travelling in the expanding space time, arrive to the second with less energy that the photons in the cold body. That means that the cold body becomes coldest after it gets in equilibrium with that photons. Now we make the reverse procedure, we send some photons from the cold (now coldest) body to the warmest one. But, as this is an imaginary experiment, choose to do so when the universe is contracting (for example we make the experiment in the edge of the time when a FRW goes for the expanding to the contracting phase). In the travel on this contracting universe the photons gain energy and when they arrive to the warmest body they can be, if we wait enough, be more energetic than the ones in equilibrium with the warm body and so they actually drive it hotter. As far as we can make the experiment as far as we want from anything else we are in a closed system and in that closed system we have effectively transferred heat from a cold body to a hot one, breaking the second law.

Of course we have not counted the entropy of spacetime, but how could we do so? In the Hawking laws of black holes we learned that classical gravity worked as entropy, the area of a black hole playing the role of entropy. And it is well known how the Hawking radiation, a semiclassical effect (so taking quantum mechanics into play) gave a further argument. String theory (and ulterior works using only geometry and CFT, the kerr/CFT correspondence) gave microscopic support to that correspondence among gravity and thermodinamics. And also are were known the, probably wrong, ideas about gravity as entropy, that have gained a rebirth with the paper of Verlinde about "gravity as en entropic force". But in this mind experiment I don't use nothing special in GR, something like an horizon, or quantum effects. Neither is any claim about saying that gravity is entropy. The whole point is that, if there is no mistake, if you don't know how to count the entropy of the spacetime, in this nonstationary case, you can violate the second law of thermodynamics, and that looks very unfunny, isn't it? ;)

The most similar situation that I know is the famous case of Hawkings telling once (and later felling shame about the idea) that in a contracting universe entropy would go in the opposite direction, and the worries of Sean Carrol and others about the thermodynamic arrow. But as far as I know none made such an explicit case as the one I am presenting here.

Just to avoid some trivial criticisms I clarify that in GR the energy of a body (or a system of bodies) is the time component of a cuadrivector, so it is not an invariant. As E=Q-W (first law) and \[ \Delta S= \Delta Q /T \] entropy also should be some time component of some cuadrivector and that makes it's precise definition somewhat tricky, but as far as I see this experiment could be suited in a single reference system so we don't need to care about that things.

Well, this is the idea, and probably I am making some very trivial mistake, or this simply has already been considered and discarded, but as I don't know for sure than that is the case I present the idea here so anyone can blame me if necessary ;).

There are a few things that I am thinking about, but for the present entry I will concentrate in an easy mental experiment that, at least apparently, violates the second law of thermodynamics. The key of the violation is the lack of a proper definition of energy in general relativity, but for the present case I don't even need to go into mathematical details about it. My idea was clear, take a situation where that problem in the energy definition rises a paradox that violates the second law. I tried a few strategies that possible also work, but in the end I found a really easy one that I think is simple and representative.

I have found that the easier way of attack is to use the Clausius enunciate: No physical process can transfer heat from a cold body to a warm one. The way to circunvate the law is as follows: Take two "black bodies", for example the canonical ones consisting of a cavity with photons in equilibrium with the walls that are at different temperatures. Now in the warmer one open the also canonical small hole that allow a photon, or a few ones, to scape. This is made in an expanding universe, the photons that have exit form the body travel in that space time loosing energy. Them they arrive at the coldest body. The photons of the warmer body initially had more energy that the ones in the coldest body, but now, after travelling in the expanding space time, arrive to the second with less energy that the photons in the cold body. That means that the cold body becomes coldest after it gets in equilibrium with that photons. Now we make the reverse procedure, we send some photons from the cold (now coldest) body to the warmest one. But, as this is an imaginary experiment, choose to do so when the universe is contracting (for example we make the experiment in the edge of the time when a FRW goes for the expanding to the contracting phase). In the travel on this contracting universe the photons gain energy and when they arrive to the warmest body they can be, if we wait enough, be more energetic than the ones in equilibrium with the warm body and so they actually drive it hotter. As far as we can make the experiment as far as we want from anything else we are in a closed system and in that closed system we have effectively transferred heat from a cold body to a hot one, breaking the second law.

Of course we have not counted the entropy of spacetime, but how could we do so? In the Hawking laws of black holes we learned that classical gravity worked as entropy, the area of a black hole playing the role of entropy. And it is well known how the Hawking radiation, a semiclassical effect (so taking quantum mechanics into play) gave a further argument. String theory (and ulterior works using only geometry and CFT, the kerr/CFT correspondence) gave microscopic support to that correspondence among gravity and thermodinamics. And also are were known the, probably wrong, ideas about gravity as entropy, that have gained a rebirth with the paper of Verlinde about "gravity as en entropic force". But in this mind experiment I don't use nothing special in GR, something like an horizon, or quantum effects. Neither is any claim about saying that gravity is entropy. The whole point is that, if there is no mistake, if you don't know how to count the entropy of the spacetime, in this nonstationary case, you can violate the second law of thermodynamics, and that looks very unfunny, isn't it? ;)

The most similar situation that I know is the famous case of Hawkings telling once (and later felling shame about the idea) that in a contracting universe entropy would go in the opposite direction, and the worries of Sean Carrol and others about the thermodynamic arrow. But as far as I know none made such an explicit case as the one I am presenting here.

Just to avoid some trivial criticisms I clarify that in GR the energy of a body (or a system of bodies) is the time component of a cuadrivector, so it is not an invariant. As E=Q-W (first law) and \[ \Delta S= \Delta Q /T \] entropy also should be some time component of some cuadrivector and that makes it's precise definition somewhat tricky, but as far as I see this experiment could be suited in a single reference system so we don't need to care about that things.

Well, this is the idea, and probably I am making some very trivial mistake, or this simply has already been considered and discarded, but as I don't know for sure than that is the case I present the idea here so anyone can blame me if necessary ;).

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