LOG#194. On minimons, maximons and darkons.

If you believe in both, relativity AND quantum mechanics, something that I presume you do due to experimental support, you are driven to admit that the speed of light is the maximal velocity (at least, the maximal 1-speed, in single time theories), and the limit of the quantum of action to \hbar. The limit of quantum of action also implies that any EXTENDED structure with size \sim \lambda can not exceed not only the speed of light, but also an acceleration! Usually, this is not remarked or stressed but it is true, unless you presume certain class of fuzziness and non-locality. If

    \[v\leq c\rightarrow v^2\leq c^2\rightarrow \dfrac{v^2}{\lambda}\leq\dfrac{c^2}{\lambda}\]

and from simple dimensional analysis, you get

    \[a_c\leq \dfrac{c^2}{\lambda}\]

And, if you plug \lambda=\lambda_C=\hbar/m c you get

    \[a_c\leq \dfrac{mc^3}{\hbar}\]

This is the simplest argument I know to propose a maximal acceleration principle from the combined force of special relativity AND quantum mechanics. There are other arguments. If you followed my posts about Bohrlogy and fundamental challenges and issues in physmatics, you do know I believe there is a deep link we are missing in the search of quantum gravity and the ultimate theory. We are lacking a conducting principle. I am not claiming that maximal acceleration IS the principle, but only that it is pointing out towards it. And not too many people know this!

I have often highlighted too that quantum theory is not only a theory of numbers. It is also a theory of quantization of everything (via quanta of action!). Everything is the quantum of something. That is why number theory will matter in the final ultimate theory. Nature LOVES counting!  If you admit that there is a well defined vacuum, and that there is a lower bound to the energy, plus a higher limit on the number of microstates (atoms) of space-time in any volume (or more precisely, hypersurface, according to the holographic principle), you must agree that there is a minimum quantum (the minimon, the “vacua”) of everything and there is a maximal quantum that saturates the phase space-time, the maximon. Firstly, consider the following example. The Schwarzschild radius of general relativity has also a mysterious duality and connections to minimal length. Let me write:

    \[R_S=\dfrac{2G_NM}{c^2}=\dfrac{2L_p^2 c M}{\hbar}=2\dfrac{ML_p^2}{\lambda_C}\]

where we have used L_p^2=G\hbar/c^3 as definition of Planck length. Would you say the temporal part of the Schwarzschild metric scales like 1/r or 1/r^2? Suppose you make mass and r not constant but a fluctuating quantities due to the inclusion of quantum mechanics and the Heisenberg Uncertainty Principle (HUP). Then, \Delta E\Delta r\geq \hbar c/2. Now, saturating the bound, \Delta E=\hbar c/2\Delta r. Then, from

    \[2\dfrac{ML_p^2}{\lambda_C}\rightarrow 2\dfrac{G_NM}{rc^2}\rightarrow 2\dfrac{G_N\Delta M}{\Delta r c^2}=2\dfrac{L_p^2 c \Delta E}{\hbar \Delta r c^2}\]

and from the above argument

    \[2\dfrac{L_p^2 c \Delta E}{\hbar\Delta r c^2}=\dfrac{2L_p^2 \hbar  c^2}{2\hbar \Delta r^2}=\dfrac{L_p^2}{\Delta r^2}\]

Gravity plus quantum mechanics implies not only a minimal length, it also changes the way in which metric changes. Fluctuations of the metric are more sensible to squares of the Planck length. Note that you could be tempted to write g_{tt,q}\sim 1/r but it seems much more natural to say that the square of area is much more natural. You could be yourself puzzled like me, but some time ago, Jacob Bekenstein suggested something like this. Area quantization for the black hole (quantum!) spectrum. If you know loop quantum gravity, you also know that the area operator is much more natural than the volume or length operator. So, somehow, area, surface or its general ND generalization, the hypersurfaces are much more natural objects for quantum gravity that “points”, aren’t they? Who knows? Braners and stringers also know this, I am sure of it.

By the other hand, Caianiello’s papers about maximal acceleration had two big goals:

  1. Quantization as geometry in phase space-time.
  2. Quantum mechanics as geometry in some curved phase space-times.

Maximal acceleration can also be related to Sakharov limiting temperature, now more frequently quoted as Hagedorn temperature. It is very suggestive the symmetry you could get from any extended relativity theory with both, maximal 1-speed and maximal 1-acceleration:

    \[E=\dfrac{mc^2}{\sqrt{1-\dfrac{V^2}{c^2}}\sqrt{1-\dfrac{A^2}{a_M^2}}}\]

One comment is necessary with respect to maximal acceleration and the above arguments. \lambda is not, indeed, a fundamental length, so differente forces yield, in principle, different characteristic \lambda and different fundamental lengths! You can see like an issue. But, remember not every particle travels to the speed of light. Only massless bosons do! With respect to maximal acceleration, you can write:

    \[a_M=\dfrac{\mu c^2}{m\lambda}=\dfrac{c\hbar}{2m\lambda^2}=\dfrac{2\mu^2 c^3}{m\hbar}\]

where a new dual mass \mu beyond m has been introduced, so that

    \[\lambda \mu c=\dfrac{\hbar}{2}\]

Now, we are going to derive Sakharov maximal temperature (critical Hagedorn temperature), from the maximal acceleration principle. Take \mu=m and calculate the maximal acceleration in the center of mass system. Convince yourself that

    \[F_M=\dfrac{2m^2c^3}{\hbar}\]

Equate this to the gravitational force

    \[F_M=F_G=\dfrac{G_Nm^2}{R^2}\]

but note that in the center of mass you get ma/2=Gm^2R^2. From

    \[\dfrac{G_Nm^2}{R^2}=\dfrac{2m^2c^3}{\hbar}\]

    \[2R^2=\lambda^2=\dfrac{G_N\hbar}{c^3}=L_p^2\]

In the center of mass frame

    \[ma_M/2=\dfrac{m^2c^3}{\hbar}=G_N\dfrac{m^2}{R^2}\]

so R^2\sim \lambda^2\sim L_p^2, so the maximal acceleration is relevant at Planck scales. Via the Unruh effect:

    \[T_U=\dfrac{\hbar a}{2\pi c k_B}\]

any maximal limit to the acceleration implies a maximal temperature, as Sakharov suggested. Indeed:

    \[T_U=T_M=\dfrac{\hbar}{2\pi k_B c}a_M=\dfrac{\hbar}{2\pi k_B c}\dfrac{c^2}{\lambda}=\dfrac{1}{2\pi k_B}\sqrt{\dfrac{\hbar c^5}{G_N}}\]

Maximal temperature would be infinite if the speed of light were infinite, or the newtonian gravitational constant were zero. Exported to black hole physics, or even superstring/M-theories, maximal temperature seems to be a critical feature (critical point) that points out a phase transition to new degrees of freedom of (phase) space-time and conventional particle physics or geometry. Moreover, maximal acceleration can be seen as a feature in some generalized uncertainty principles (GUP). By duality, if extended to the full PHASE space-time, if there is a minimal length (the minimon), there is a maximal (cosmic?) length (the maximon). If there is a maximal velocity, there should be a minimal velocity (more on this in a nearby special post…). If there is a maximal acceleration, there is a minimal acceleration. So, we should have minimons and maximons of every stuff.

In 4D general relativity, you can get a natural maximal force c^4/4G_N, curiously the gravitational force in the event horizon of the Schwarzschild solution. It is natural. Furthermore, the Bronstein-Zelmanov-Okun cube also suggests a maximal power, c^5/4G_N. The recently observed first gravitational wave, GW150914 released about 4\cdot 10^{49}W<P_M, according to the aLIGO collaboration. It is sometimes discussed how much of the transient GW events turn into GW energy. One should expect that it were also emitted electromagnetic, neutrinos, and other forms of radiation. But, then, what is the origin of the minimal/maximal mass/energy or density? That is a deep unsolved problem, since the suggestion of minimons and maximons for any magnitude has not been discussed in a more general context. To my knowledge, only something similar is suggested by the Buchdahl inequalities and other black hole investigations. Quantum hypergraphs! What is going on then? In n space-time dimensions, Q\leq F/D^{n-3} are bounded. But there is also hints in superstrings. The string tension in natural units reads

    \[F_S=\dfrac{1}{2\pi\alpha^{'}}\]

If you support that F_g=c^4/4G_N is the maximal force, and equating to the above, you get a relation with the string tension. And from the Nambu-Goto action

    \[S_{NG}=F_S\int_\Sigma dxdt=\dfrac{1}{2\pi\alpha{'}}\int_\Sigma dA\]

So the maximal acceleration can be related to the maximal complexity and maximal action conjecture by Susskind et alii! Even more, this argument is much more general since you can apply it to any action, for instance, Born-Infeld theory, as follows

    \[S_{BI}=-F_S^2\int dvol\sqrt{-\det (g+2\pi\alpha^{'}F)}\]

Then, if you make a Taylor series from this, you get Maxwell theory plus corrections of order maximal tension, via relation with \alpha^{'}. The full non-linear theory implies non-perturbative states and Schwinger effect like excitations of vacuum. It is related to some critical fields, and thus, to maximal temperature as well! Of course, this could be broken at some level, OR, it could be something much more fundamental, if completely understood. In relation to all of this, there is a very curious surprising link (not always completely remarked) between Regge trajectories and Kerr-like solutions in black hole physics. Let me write 3 awesome bounds, stunningly similar. Firstly, the open relativistic string theory angular moment to mass connection:

    \[\boxed{J\leq \dfrac{\alpha^{'}}{c^3}M^2}\]

Secondly, the closed relativistic string theory angular moment to mass connection:

    \[\boxed{J\leq \dfrac{\alpha^{'}}{2c^3}M^2}\]

Thirdly, the Kerr black hole angular moment to mass bound to avoid naked singularities (and so, to keep the cosmic censorship true!):

    \[\boxed{J\leq \dfrac{G}{c}M^2}\]

If black holes are some type of rotating ring with tension, then we should expect

    \[\dfrac{G}{c}\sim \dfrac{\alpha^{'}}{c^3}\rightarrow \dfrac{1}{\alpha^{'}}\sim \dfrac{1}{Gc^2}\sim\dfrac{c^4}{G}\sim F_M\]

Therefore, Hagedorn temperature, Sakharov maximal temperature, Hawking temperature and the Unruh temperature, up to some constant, give a criticical value for those temperatures. It hints that gravity itself or fields are emergent concepts. Even space-time is effective. Any uniformly accelerated spacetime has a temperature with respect to inertial observers. By the other hand, we have found that something happens at some critical temperature. It implies a maximal acceleration and hidden dynamics there when the new degrees of freedom do appear. Duality is a key concept. Even more, duality on time dimensions are fun! Any quantum theory at non-zero temperature implies a link with TIME. Temperature and time are related via periodicity in IMAGINARY TIME, i.e., \beta=\hbar T=period. In any relativistic quantum theory, purely kinematical Unruh temperatures will naturally appear. Thus, we should expect that, in any theory with minimal length, maximal temperature and maximal acceleration are key. In fact, as I tried to say to a professor as undergraduated (even before knowing all of this stuff!), maximal acceleration (temperature) can NOT emerge from classical general relativity or classical string theory since they do NOT include it in any obvious formulation. I have no knowledge of such a formulation of classical general relativity or string/M-theory including a maximal acceleration principle from the beginning, and its effect to them. Maximal acceleration (and minimal acceleration), maximal length (minimal length) could be related to some unknown dynamics of string theory and quantum gravity. Perhaps, I can not explain myself clear. I do not know. But I am becoming more and more confident that a general min-max. principle is operating in effective theories of quantum gravity. And they are completely general. In fact, black hole physics is critical field theory in a sense. When you equate the black hole Hawking temperature to the Unruh temperature, or more interestingly, to the Schwinger temperature, you get maximal acceleration (maximal field!). If you do it yourself you will get a_M=c^4/(4G_NM), g_M=2Ec/\hbar, A_M=c^6/(4G_NE). Please, note the duality transformation in energy, and the equivalence, if you make 2c/\hbar=c^6/4G_N.

A more formal derivation of maximal acceleration can be tracked to Caianiello himself, using a more general HUP, and remarking that it implies some sort of process doubling the degrees of freedom (coordinates) in space-time to phase space-time. Let me write the HUP:

    \[\Delta E\Delta g\geq \dfrac{\hbar}{2}\vert \dfrac{dg}{dt}\vert \]

For g=v it yields

    \[\Delta E\Delta v\geq \dfrac{\hbar}{2}\vert \dfrac{dv}{dt}\vert\]

or

    \[a=\vert \dfrac{dv}{dt}\vert\leq \dfrac{2\Delta E \Delta v}{\hbar}\]

For some fluctuations \Delta E=E=Mc^2 (quantum of energy!), and we do know that \Delta v\leq c, by special relativity, so we have at last

    \[\boxed{a\leq \dfrac{2Mc^3}{\hbar}}\]

Q.E.D.

You can get this result, even if you do not believe in quanta of time (chronons or choraons?), in the following way

    \[\Delta v=v-\langle v\rangle=v'(0)\Delta t+v''(0)\Delta t^2+\mathcal{O}(\Delta t^3)\]

From this,

    \[\Delta v=\vert a(0)\vert \Delta t=\langle a\rangle \Delta t\]

and from classical HUP

    \[\Delta t\geq \dfrac{\hbar}{2E}\]

    \[\dfrac{\Delta v}{\langle v\rangle}\geq \dfrac{\hbar}{2E}\]

    \[\langle a\rangle\leq \dfrac{2E\Delta v}{\hbar}=\dfrac{2Ec}{\hbar}\]

and therefore again

    \[\boxed{a_M\leq \dfrac{2Mc^3}{\hbar}=\dfrac{2Ec}{\hbar}}\]

Why is this stuff important? Why should it? Well, Penrose, here https://arxiv.org/abs/1707.04169 , introduced erebons (planckons, maximons,…) as dark matter particles. Erebons (or darkons) as the quanta of dark matter or dark energy naturally arise from minimal/maximal acceleration. Even more, you could guess a MONDian (MOdified Newtonian Dynamics) associated to them. Indeed, I gave a talk about this topic about a year ago…Let me begin with MINIMAL acceleration a_0. Suppose you have a mass bounded by gravity in which there are some type of background and minimal acceleration a_0, Newton’s second law provides:

    \[\dfrac{v^2}{R}=\dfrac{G_NM}{R^2}+a_0\]

or

    \[v^2=\dfrac{G_NM}{R}+a_0R\]

Square it, to obtain

    \[\boxed{v^4=\dfrac{G_N^2M^2}{R^2}+a_0^2R^2+2G_NMa_0}\]

If you assume that G_N^2<<1, a_0^2<<1, you obtain the MONDian law

    \[\boxed{v^4=2G_NMa_0}\]

Remark: You could fit the minimal acceleration to the cosmological constant, via a_0=c\sqrt{\Lambda}. Is there any other physically appealing definition?

Suppose now you include a Hooke law term as well. Of course it is related to the cosmological constant term (but it has opposite sign to normal Hooke law from springs!). You could think the cosmic hookean term as certain class of maximal length term. By duality, some type of maximal tension/minimal tension. Mimicking the above arguments:

    \[\dfrac{v^2}{R}=\dfrac{G_NM}{R^2}+a_0+\Lambda R\]

    \[v^2=\dfrac{G_NM}{R}+a_0R+\Lambda R^2\]

Square it…

    \[\boxed{v^4=\dfrac{G_N^2M^2}{R^2}+a_0^2R^2+\Lambda^2 R^4+2G_NMa_0+2G_NMR\Lambda+2a_0\Lambda R^3}\]

I claim that in the limit where you neglect G^2_N, a_0^2, \Lambda^2<<1, you get a correction to MOND. Interestingly, this correction to MOND are two terms proportional to the cosmological constant, that become more and more important as R grows. This generalized MOND must be provided by some quanta of spacetime…The darkons, because they are associated to dark matter and dark energy. Of course, this is the same reason why Penrose suggested the name erebons for his test of cyclic cosmology. The origin of a_0, \Lambda is the issue. And of course, the G_N and the relative strength of the different terms in the velocity-radius curve. I believe this law suggest certain dynamics between a_0, \Lambda, and G_N, \Lambda that could explain why MOND fails in some particular systems. Or not!  Note that you can get maximal/minimal length or acceleration whenever you write c(c/L)^n.

As final off-topic, let me talk about the differences between the generation of electromagnetic (EW) and gravitational waves (GW).  EMs require charged dipoles as generators, and they cause atoms to decay (up to ground stable states). GWs require massive bodies and quadrupole generators (since mass-energy is conserved). GWs require fast bodies. GW, in principle, travel to the speed of light, or don’t they? Some people argue now that EW do not move at the speed “of light”, they move at the maximal speed that allows the space-time itself, that would be the GW speed or c! GW have also polarizations, as EW theirselves. GW are distortions of space-time when it wobblies. Timey wimey stuff? Some GW pocket formulae for you today as bonus! Define two circular orbiting bodies a some distance r, with masses M_1, M_2. Define \mu=M_1M_2/M_1+M_2, M=M_1+M_2, and remember the Kepler 3rd law, that says

    \[\Omega^2=GM/r^3\]

For circular orbits, the GW frequency is f_{GW}=2f_{orb}, \omega_{GW}=2\Omega, and then, the binary system radiates so that, in the quadrupole approximation,

    \[\boxed{P=L=\dfrac{dE}{dt}=\dfrac{32G_N\Omega^6r^4\mu^2}{5c^5}=\dfrac{32G_N^4(M_1+M_2)M_1^2M_2^2}{5c^5r^5}}\]

and

    \[\boxed{\dfrac{d\Omega}{dt}=\dfrac{96M_2M_2G^{5/3}(M_1+M_2)^{-1/3}\Omega^{11/3}}{5c^5}}\]

This equation can be integrated to get

    \[\Omega^{-8/3}=\Omega_0^{-8/3}-\dfrac{8}{3}K(t-t_0)\]

where

    \[K\equiv \dfrac{96M_1M_2G^{5/3}(M_1+M_2)^{-1/3}}{c^5}\]

See you in another wonderful blog post!

P.S.: Epitachyons, particles with a>a_M, could be existing entities. Indeed, as you can see from extended relativity, superluminal superaccelerated epitachyons have real energy.

P.S.(II): It seems that the SAME idea (maximons and minimons) have been discussed in the past by M. A. Markov and others. Markovian minimons are neutrinos under a Planckian seesaw we do know it can not work with current data. m_\nu=M_D^2/M_X can be explained by a dimension 5 five operator (Weinberg’s operator) containing heavy Majorana mass neutrinos (right-handed). A neutrino is kicked by a Higgs, turning it into a very heavy Majorana neutrino (unobserved), then it is kicked again and turned into a light neutrino (type I seesaw).

P. S. (III): The maximon is a mayan deity as well!

P. S. (IV): Haven’t you got enough continuous time to read this? Take a chronon or choraeon (discrete quantum of time!). In Caldirola’s theory

    \[\theta_0=\dfrac{1}{6\pi\varepsilon_0}\dfrac{e^2}{mc^3}\]

or equivalently

    \[\theta_0=\dfrac{2}{3}\dfrac{K_Ce^2}{mc^3}\simeq \alpha\dfrac{\hbar}{mc^2}\sim\dfrac{\hbar}{\Gamma}\]

where \Gamma is the decay width.

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