LOG#209. A cosmic link with a non-trivial zeta zero?

Some weeks ago, the Riemann zeta function and the Riemann hypothesis were again in the news. Sir Michael Atiyah proposed a (wrong) failed proof of the Riemann hypothesis. My blog has an history of being controversial from time to time. So I am going to reveal another of my craziest ideas here. I can be wrong and it is just a curious coincidence I noted long ago but as it is short, I decided to postpone it to a transitional post like this. What is this thing about? Let me explain me better…

The cosmological constant that Einstein introduced like a property of spacetime is related to a possible new length scale (usually tied to the Hubble length in Cosmology), L_\Lambda. It is also related to de Sitter spacetimes and the so-called Garidi mass in de Sitter group representations. Taking \Lambda as fundamental constant, together the triple (h,c,G_N), you get the quadruple set of parameters (h,c,G_N,\Lambda). With this quadruple set, you can build up 8at least) two mass/length scales:

1st. M_W=\dfrac{h}{c}\sqrt{\dfrac{\Lambda}{3}}\sim 10^{-68}kg. This is known from Wesson or Garidi in de Sitter spacetime representation theory.

2nd. M_U=\dfrac{c^2}{G_N}\sqrt{\dfrac{3}{\Lambda}}\sim 10^{53}kg.

We recall that we can define the Planck length l_p^2=\dfrac{Gh}{c^3} and the cosmological length l_\Lambda^2=\dfrac{3}{\Lambda}, and you can easily prove (I have done this before in this blog) that

(1)   \begin{equation*} \dfrac{M_W}{M_U}=\dfrac{l_p^2}{l_\Lambda^2}=\dfrac{Gh\Lambda}{3c^3}=\alpha_C\approx 6.51\cdot 10^{-122} \end{equation*}

is a sort of cosmic fine structure constant \alpha_C^{-1}\approx 1.54\cdot 10^{121}. What has all this to do with my post and the Riemann zeta function? In principle nothing!!!! Just a weird numerical coincidence I noted some years ago and I did not publish because is completely crazy. We have as you probably know two different determinations of \Lambda, we can denote them by \Lambda_{th} and \Lambda_{exp}. They differ about just the above cosmic fine structure constant factor. It is huge. Theoretical physicists are just depressed when they got a prediction that it is about 121 orders of magnitude different from observational data! Now, a curious coincidence (I like to compare it to the moonshine conjecture since it involves large numbers, like those also used in Dirac-Eddington hypothesis):

(2)   \begin{equation*} \boxed{\Lambda_{th}=\sqrt{\exp\left(4\pi^2\cdot  t_1\right)}\Lambda_{exp}\simeq 1.5\cdot 10^{121}} \end{equation*}


(3)   \begin{equation*} \boxed{\Lambda_{th}=\sqrt[8]{\exp\left(16\pi^2\cdot  t_1\right)}\Lambda_{exp}\simeq 1.5\cdot 10^{121}} \end{equation*}

and where t_1 is the first non-trivial Riemann zeta function zero


Of course, I am not pretending to say the above relation is exact or even true. It is just another moonshine-like coincidence I wanted to share with my readers and all over the world…Why the hell  the first Riemann zeta function non-trivial zero has to do with the ratio between the Planck length and the cosmological constant (Hubble) length? Nothing, unless the Riemann zeta function associated system, the riemannium, has something to do with quantum gravity and cosmology. We already know that zeroes behave like the spectrum of some “heavy” system. The nature of that system is presently just unknown. A heavy nuclei? A black hole? A complex system? A quantum quasicrystal? Maybe some n-body classically chaotic system? We don’t know.

See you in a new blog bost!

P.S.: Do you want to win one million dollars? Prove the Riemann hypothesis. Do you want to live forever? Prove the Riemann hypothesis. Fortune and glory? Prove the Riemann hypothesis!

LOG#208. Monsters and LISA.

Black holes and other astrophysical objects are true monsters. Some interesting tools from web pages to learn about these free catalogues: WATCHDOG and BLACKCAT. Also, the blackhole.org page and the sounds of spacetime for gravitational waves are suitable for you. Furthermore, you can enjoy the BH stardate online encyclopedia.

Two of the biggest and coolest (literally) black holes are quasars. Their names: OJ 287 and 3C 273. Persistent BH sources do emit X-rays, and they are powerful sources of X-rays in our galaxy (and beyond!). Transient BH events are also studied. Beyond X-rays, you study BH accretion and merging, likely BH growing if you are “lucky” or a true believer.

A list of “monsters” (not necessarily BH):

  1. Moon’s magnetic field.
  2. Asteroid with rings. 10199 Chariklo its name.
  3. Sixtail asteroids (not steroids!)
  4. Red storm (Jupiter’s famous biggest spot!).
  5. Hot Jupiter messes.
  6. HD 106906b.
  7. Uranus storms.
  8. KIC2856960.
  9. UV underproduction sources.
  10. Shape of dark matter (DM).
  11. Galaxies with age of billions of years (10 billions or more!).
  12. Iciness Saturn rings.
  13. GR bursts and fast radio bursts.
  14. Cataclysmic variable stars.
  15. Universe smoothness.
  16. Strange structures in the Universe (Multiverse?).

The new born gravitational wave astronomy is going to outer space. LISA (Laser Interferometer Space Antenna) will study a completely different set of sources beyond the ground based gravitational wave observatories. What are they?

  1. WD (White dwarf) binaries.
  2. NS (Neutron Star) binaries.
  3. BH+NS/WD systems (binaries).
  4. BH+BH mergers (supermassive or even intermediate; EMRI and IMRI are expected to be observed). EMRI=Extreme Mass-Ratio Inspirals. IMRI=Intermediate Mass-Ratio Inspirals.

LISA science targets in its timelife (2-10 years):

  1. Hubble constant (after 2 years, by binary inspirals, with accuracy of a few per cent).
  2. Equation of state of dark energy.
  3. EMRI/IMRI sources/observations.
  4. Tests of fundamental physics with gravitational waves.
  5. Ultracompact binaries.
  6. Surprises we have not expected or thought about ;).

By the other hand, beyond GW astronomy, some cool experiments are the HAWC water telescope, Cherenkov telescopes (CTA!), and the AMON network.

Primordial BH are interesting objects. In the window 10^{-16}-10^{-7}M_\odot they offer the option to be ALL the dark matter.

Gravitational wave luminosity (power!) is given for a binary (non-eccentric) system by:

    \[L_{GW}=-\dfrac{dE}{dt}=\left(\dfrac{32}{5}\right)G^{7/3}\left(M_c\pi f_{GW}\right)^{10/3}\]

where f_{GW}=2f_{orb}, M_c=\mu^{3/5}M^{2/5} is the chirp mass, and the reduced mass \mu=M_1M_2/(M_1+M_2), with M=M_1+M_2. Sometimes, the symmetric mass ratio \eta=\mu/M is used. Non-zero binary eccentricity formula do exist but it will not be considered in this post. GW higher harmonics are useful in GW astronomy. However, for circular orbits, you have


For the strain amplitude, averaging, you get

    \[h=1.5\cdot 10^{-21}\left(\dfrac{f_{GW}}{10^{-3}Hz}\right)^{2/3}\left(\dfrac{M_c}{M_\odot}\right)^{5/3}\left(\dfrac{D}{kpc}\right)^{-1}\]

Remember: the gravitational wave frequency evolution for a binary system from t_0 up to coalescence is twice the orbital frequency:

(1)   \begin{equation*} f_{GW}=2f_{orb}=\dfrac{1}{\pi}\left(\dfrac{GM_c}{c^3}\right)^{-5/8}\left(\dfrac{5}{256(t_0-t)}\right)^{3/8} \end{equation*}

where the chirp mass reads off

(2)   \begin{equation*} M_c=\sqrt[5]{\dfrac{(M_1M_2)^3}{(M_1+M_2)}} \end{equation*}

and the power of the gravitational wave is equal to

(3)   \begin{equation*} P_{GW}=\dfrac{32G}{5c^5}\mu^2\omega^6 r^4 \end{equation*}

with \omega=2\pi f_{GW} and \mu=M_1M_2/M, M=M_1+M_2.

By the way, the maximal frequency on Earth that we can detect from binary black hole mergers is related to the innermost  stable circular orbit (ISCO). Roughly, this orbit corresponds to a radius or separation from the center of mass equal to r=3r_s/2=6GM/c^2. For this orbit, the frequency should be:

(4)   \begin{equation*} f_{max,c}=f_{isco}=\dfrac{c^3}{6^{3/2}\pi GM}\approx 4.4\dfrac{M}{M_\odot} kHz \end{equation*}

Thus, ground bases observatories could catch on mergers of intermediate black holes if sensitive enough. However, they will catch more easily mergers of tens or hundreds of solar masses. What are the biggest monsters? The new created black holes species with more than millions of solar masses: the ultramassive black holes (with more than 10^{10} solar masses). Example: IC 1101, TON 618, NGC 4889, NGC 6166, NGC 1270, and others you can read off from the this wiki-list.

We will learm more about the sounds of spacetime in forthcoming entries! See you in other wonderful blog post…

P.S.: From Orosz et al. and other sources from the arxiv, you get something like this list (several versions included)

LOG#207. Beck, Zeldovich and maximal acceleration: the vacuum.

Some time ago, Zeldovich derived the following expression for the vacuum energy density:

(1)   \begin{equation*} \rho_{V}=\dfrac{Gm^6c^2}{\hbar^4} \end{equation*}

or equivalently, with a link to Caianiello’s maximal acceleration formula,

(2)   \begin{equation*} \rho_V=G_N\left(\dfrac{mc^3}{\hbar}\right)^6\left(\dfrac{\hbar}{c^8}\right)^2 \end{equation*}

Remark: the above vacuum energy density is related to the cosmological constant via the mathematical formula

(3)   \begin{equation*} \rho_V=\dfrac{\Lambda c^4}{8\pi G}=\rho_{CC} \end{equation*}

Now to far away, Christian Beck also proposed a formula for the measured cosmological constant and he derived it from pure informational axioms. It reads:

(4)   \begin{equation*} \rho_{CC}^{Beck}=\left(\dfrac{c}{\hbar}\right)^4\left(\dfrac{G_N}{8\pi}\right)\left(\dfrac{m_e}{\alpha_{em}}\right)^6 \end{equation*}

and where \alpha_{em}=K_Ce^e/\hbar c is the known fine structure constant. Why the cosmological constant is so small when our current theories based on standard Quantum Field Theories predice it should be HUGE is a mystery. We have some ideas based on supersymmetry and non-perturbative damping due to Schwinger effects that could work, but no one has managed a clear explanation. Some people believe we need a better theory. I agree partly, we need also experiments. Are the Beck formula or Zeldovich proposal right? Can we test them somehow? It is the work of future physicists to enlighten the dark issue of the vacuum energy and its radical mismatch between microscales and macroscales. What is vacuum or vacuum energy after all?


LOG#206. Multitemporal theories.

Newton’s gravity reads:


In extra dimensions, D=d+1, d-1=2+n, it reads


For extra dimensions, if their size are much smaller than considered distances, R>>R_{XD}, then by matching



So, gravitational is weak in our scale because it gets diluted through extra dimensions. Real Planck scale gravity is much stronger. In terms of mass scales (large ADD scenario):


More generally, we can substitute R^n by a volumen V_n:


    \[\boxed{M^2_D V_n=M^2_P}\]

What if you get extra time-like dimensions. Let N=n+1+d the number of dimensions. Then,


The proof is also straightforward:

    \[F^{xt}=G_D\dfrac{\varepsilon_i M^{ij}\varepsilon_j}{R^d}\]

with \varepsilon_i the time vectors, such that

    \[\cos\theta=\varepsilon_i\cdot \varepsilon_j/\vert\varepsilon_i\vert\vert\varepsilon_j\vert\]

    \[F^{xt}=G^{(ij)}\dfrac{M^i_{\; k}M^{\; k j}}{R^d}\]


    \[\boxed{F^{xt}_{ij}=G_{(ij)}\dfrac{M^i_{\; k}M^{\; k j}}{R^d}\cos^2\theta}\]

where, with d=2+s, N=n+1+d=n+1+2+s=n+2+s. Therefore,


and gravity is “small” due to the almost orthogonality of time vectors. Equivalently,


We can indeed combine the extradimensional space-like case with the time-like case:


    \[\boxed{M_P^{2 (ij)}=M_D^{2 (ij)} V^s \cos^2\theta}\]


  1. What are more interesting, extra time-like or extra space-like dimensions?
  2. Why extra time-like dimensions are IMPORTANT despite being generally neglected by theorists, excepting a few excepcional cases?

LOG#205. Ether wind, SR and light clocks.

From the Michelson-Morley experiment, some clever experimentalist tried to derive the light speed through the “ether wind”. It is very similar to a river being rowing by sailors in a boat. The times for journey “upstream” and “downstream” are:

    \[cT_\parallel (1)=L+vT_\parallel (1)\]

    \[cT_\parallel (2)=L-vT_\parallel (2)\]

Journey across the ether wind uses the following pythagorean theorem:



    \[T_\parallel=T_\parallel (1)+T_\parallel (2)=\dfrac{L}{c-v}+\dfrac{L}{c+v}\]




you get the time difference

    \[\Delta T=T_\parallel-T_\perp=\dfrac{2Lc}{c^2-v^2}-\dfrac{2L}{\sqrt{c^2-v^2}}\approx \dfrac{L}{c}\left(\dfrac{V}{c}\right)^2\]

The Michelson-Morley experiment had enough resolution to detect fringes caused by the above time difference. However, there were no time difference. There is no shift in light ways through the heavens. However, the ether hypothesis was kept until Einstein times…

Einstein derived the relativity of time with the tool of light clocks. Suppose a rest frame and a moving rocket with constant speed v.. Inside the rocket a light beam perpendicularly, to measure time in its frame. Supposing it travels d upside and d upside down, the time it yields is


That is the lab light clock time. From the outside, the rocket light clock time is different, since it follows an oblique trajectory. The distance of one side is D, so the time in this frame will be


By trigonometry, the base the rocket travels is x=vt_{RC}, so the semibase reads x/2=vt_{RC}/2. Let \Delta t the time interval between events in lab frame, and \Delta t' the time interval the lab sees or the rocket clock measures between some events. Again, pythagorean theorem rocks:




and then

    \[\Delta t'=\Delta t\sqrt{1-v^2/c^2}\]

Time moves “slower” for rocket clocks seen from outside, and measured by the lab outside. Similar arguments work out for lenght and we have length contraction! If L_0 is the length of a rod measured with a light beam in the rocket frame, and L is the length of the rod as measured in the LAB frame OUTSIDE. Front of rod crosses a point P at time t'_1 in the rocket and t_1 in the lab. The back of rod crosses a the point P at time t'_2 in the rocket and t_2 in the lab. Since




    \[L=v\Delta t=v\Delta t'\sqrt{1-v^2/c^2}=L'_0\sqrt{1-v^2/c^2}\]



Therefore, the movin rod in the LAB frame outside appears (to the lab observers) length contracted L<<L_0. The rod would be normal from the rocket inside observers. There is an invariant interval of spacetime, as it was shown in my notes on special relativity here:

    \[\Delta\tau^2=c^2\Delta t^2-\Delta x^2\]

That number is the same in all frames moving at constant speed with respect to each other. Simultaneity is also relative, as space and time measurements as well.

What happens with energy and momentum? In the lab frame, particle has at time t position x. In the particle frame (rocket frame), we have t', x'=0. Thus,


Then, we form the invariant


provided the transverse momentum


and the canonical momentum





    \[p_T=\sqrt{p^2c^2+m^2c^4}\approx mc^2+\dfrac{p^2}{2m}\]

Note that p_T=E_{total}=Mc^2. However, p=0 yields E_0=mc^2 are rest energy.

Particles of light, from the classical side, are radiation. Wave light phenomena are classical electromagnetic waves. Usually, accelerated point-like particles of matter emit electromagnetic waves. Waves are also associated to the Maxwell field described by the pair E, B. In the quantum world, things are a little different. However, we see (yet!) phenomena like interference at the classical level!

    \[\vert A\vert^2=I=\vert A_1+A_2\vert^2\neq I_1+I_2\]

The Heisenberg uncertainty principle provides \Delta x\Delta p\geq \hbar/2. Quantum physics says that probability is related to \vert A_1+A_2\vert^2. Hydrogen atoms are quantized by Bohr rules, via L=n\hbar =hh/2\pi. The interaction of light with matters surprised people again when we found out that wave physics could NOT explain the photoelectric effect! A linear relation between kinetic energy and the frequency of light is NOT expected from wave light theory! Exercise: use what you know from the harmonic oscillator or waves to prove this fact. However, quantum light theory, as Einstein taught us, solves the issue of the photoelectric effect giving us the right theory with


Photons are quanta of light, with E=hf=\hbar\omega. Classically, beyond a different dependency of kinetic energy and frequency of light, we would obtain f=f'. However, interaction with atoms or matter quanta changes this naive idea. The total momentum and energy of light and atoms are conserved. Take p=h/\lambda for light, and E=pc. You invest some of the light momentum for make electrons free of the bounding forces at the matter surface. p=hf/\lambda changes, but the total momentum and energy is conserved before and after the photon hitting the electron and metal surface in the photoelectric effect! Interactions of light and matter are quantum in nature. Quantum interactions are more complicated due to fluctuations. However, in general, energy, momentum and angular momentum are conserved. Left-handed and right-handed electrons interact in the same way. Compton scattering is another interesting phenomenon. It can be seen as a consequence of gauge U(1) invariance associated to charge conservation. Antimatter interacts in a parallel way, only changing the sign of charge and we have also the CPT theorem in any local special relativistic framework. Annihilation of matter and antimatter becomes possible at quantum level. Radiation arises from high energy physics. Particle colliders use these facts to create particles. Quantum Field Theory (QFT) is a misnomer for a quantum mechanical special relativistic theory that allows to the particle number to vary! Number of particles changes in any QFT. Particle creation/destruction phenomena is the ABC of QFT. For instante, in Q.E.D., the QFT theory for light and matter, any gauge (electromagnetic) compesating field is A_\mu(x,t), it has a potential \varphi(x,t), and matter fields are \Psi(x,t). The Heisenberg principle applies, to yield:

    \[\Delta p\Delta x\sim h\]

    \[\Delta E\geq \dfrac{\hbar c}{L}\]

    \[\Delta p\geq \dfrac{\hbar}{L}\]

    \[\Delta E>E\]


Beyond light, beyond photons…What happens to quanta of MATTER? The question is complex. A complete theory for quanta of matter required time, 15-20 years, in the first third of the 20th century. Using the de Broglie relation, p=h/\lambda, just as we have a wave-like equation for “light”


given suitable \varphi, the wave-like theory for electrons is much more complex because it implies the square root of the wave equation to understand that. Dirac derived the next equation in 1928:





Matter fields follow Pauli exclusion principle (PEP), they have negative energy states and they imply the existence of antimatter. Light is its own antiparticle and photons are bosons. Electrons and other matter field are FERMIONS. Under rotations, fermions are described by spinors, they need 4\pi radians or twists to become the same object. If not, their wavefunction changes by a minus sign! Dirac equation predicts antimatter. Positrons were discovered a years after Dirac wrote its equation (a Clifford algebra structure is behind it, to be discussed here in the near future!).

See you in another blog post!

LOG#204. Mechanics and light.

Gravitational or electrical forces are inverse squared laws:



Strikingly similar, they are both also conservative forces. For gravity:




for the electrical force. Defining the potentials V_g=U_g/m, and V_e=U_e/q, you get the gravitational and electrical potentials



Conservative fields are defined from these potentials

    \[E=-\nabla V_e\]

    \[g=-\nabla V_g\]

In general, for any field \Psi, if conservative, \Psi=-\nabla V. The gravitational field reads, from newtonian gravity (module a sign)


and you would get E=-K_CQ/R^2 in the coulombian field case! Focusing on the gravitational case (a similar field could be done with the electrical field)…The momentum


is conserved under any vertical (radial) gravitational field. Imagine you do a traslation


The momentum in the x component reads p_x=m\dfrac{dx}{dt}=m\dfrac{dx'}{dt}! Note the momentum in the y or vertical component would not be conserved due to F_g! Thus, symmetry is important. Imagine a spring holding from the upper horizontal surface. Then

    \[x(t)=A\sin(\omega t)\]

where A=constant and

    \[\dot{x}=A\omega \cos (\omega t)\]

with m\omega^2=k, then



    \[E=T+U=\dfrac{m}{2}\dot{x}^2+\dfrac{k}{2}x^2=A^2\dfrac{k}{2}=\dfrac{m\omega^2 A^2}{2}=constant\]

Since the spring force is conservative, F=-kx=-kd(x^2/2)/dx, the total energy is conserved. Note the symmetry that says E does not depend on the time and it is constant!

Going 3D is important here. We will use components to avoid vector arrows for convenience. Newton’s second law is

    \[\sum_i F_i=ma_i\]




If F_i=0, then p_i=constant!

Kinetic energy for non-relativistic particles read


If p_i is conserved, then the kinetic energy is also conserved. This is valid for the free particle. In the case of conservative forces, the potential energy reads


and it yields a uniform motion with solution





The first term is precisely:

    \[W(0\rightarrow f)=\Delta T=\dfrac{1}{2}\Delta v^2\]

And thus,

    \[W(i\rightarrow f)=-\Delta E_p=-\Delta U\]


    \[U=-F\cdot \Delta x\]



i. e., f=-\nabla U, Q.E.D. for any conservative force. E=T+U holds for conservative forces with certain properties in the potential energy (depending on coordinates in a homogeneus way). For the harmonic oscillator:



and the solution

    \[x(t)=A\sin (\omega t)+B\cos (\omega t)\]

with x(0)=x_0 at t=t_0. t=t_0=0 in general, so

    \[x(t)=x_0\cos (\omega t)\]

or any other sinusoidal waveform as well. The velocity

    \[v(t)=\dot{x}=-x_0 \omega \sin (\omega t)\]

    \[a(t)=-x_0\omega^2\cos^2(\omega t)\]

and then


as before!

Light can NOT be described with classical NEWTONIAN mechanics. It took several decades an roughly speaking several centuries to code electromagnetic laws into a single set of equations. Maxwell wrote the synthesis of our current electromagnetic knowledge of light:

  1. Gauss law for the electric field: \nabla \cdot E=\div E=\rho/\varepsilon_0=4\pi K_C\rho. Equivalently, \phi=\oint_S E\cdot dS=4\pi K_CQ=Q/\varepsilon_0. For point particles, this law provides E_i=K_CQx_i/r_i^3=Qu_i/4\pi\varepsilon_0 r_i^2. Moreover, F_i=qE_i, and the gauge field E_i=-\nabla_i\varphi. \varphi=V=Q/4\pi\varepsilon_0 r_i is the potential.
  2. Faraday’s law: \nabla\times E=-\partial_t B, or equivalently \oint_\Gamma E\cdot dl=-\partial_t\oint_C B\cdot dS.
  3. Gauss law for the magnetic field (no magnetic monopoles in standard electromagnetism): \nabla\cdot B=0, or \oint_SB\cdot dS=\phi_B=0.
  4. Ampere’s law: \nabla\times B=j/\varepsilon_0c^2. This original Ampere’s law does not conserve electric charge, so Maxwell added an extra term, the displacement current, yielding

        \[\nabla\times B=j/\varepsilon_0c^2+(\partial_t E)/c^2\]


The combination of the 4 equations above produces wave-like equations for E, B:

    \[\dfrac{1}{c^2}\partial_t^2 E_i-\nabla^2 E_i=\left(\dfrac{1}{c^2}\partial_t^2-\nabla^2\right) E_i=0\]

    \[\dfrac{1}{c^2}\partial_t^2 B_i-\nabla^2 B_i=\left(\dfrac{1}{c^2}\partial_t^2-\nabla^2\right) B_i\]

Plane wave solutions are allowed, with \sin(\omega t-k_i x^i) or generally

    \[E_i\sim c B_i\sim e^{\omega t-k_i x^i}\]

Wave speed is given by


so Maxwell cleverly pointed out that light should be an electromagnetic wave! Furthermore, E\perp B\perp v in general. Light can also be polarized. Polarization or fluctuations in the directions of (E, B) is due to the transverse character of the electromagnetic waves. Maxwell’s equations unify E,B into a single framework. All the electromagnetic phenomena from a common dynamics. Special relativity allows to condense Maxwell equations into \partial_\mu F^{\mu\nu}=j^\nu and \varepsilon_{\mu\nu\sigma\tau}\partial^\nu F^{\sigma\tau}=0. Clifford algebra simplify these equations into a single \partial F=j. Differential forms also allows for such a simplification. Maxwell equations have a new invariance beyond galilean invariance: Lorentz invariance. Essentially, Maxwell equations imply Special Relativity.

In presence of matter, Maxwell equation must be completed with constitutive relations for the electromagnetic fields, plus

    \[\nabla \cdot D=\rho_{free}\]

    \[\nabla \cdot B=0\]

    \[\nabla \times E=-\partial_t B\]

    \[\nabla \times H=j_{free}+\partial_t D\]

and the boundary conditions for

    \[D=\varepsilon_0 E+P\;\;\; P=P(E)\]

    \[H=\dfrac{B}{\mu_0}+M\;\;\; M=M(E)\]

where P, M are the polarization and the magnetization for the (D, H) pair.

Remark: Beyond mechanics and light, today we care about entropic forces,

    \[F_i^{ent}=T\dfrac{\partial S}{\partial q^i}\]

Entropic forces and conservative forces can be added

    \[F_t=F_i^{ent}+F_i^{cons}=T\dfrac{\partial S}{\partial q^i}-\dfrac{\partial U}{\partial q^i}\]

Only in the zero temperature limit, we get the usual conservative terms. The above force can be obtained from


i.e., the Helmholtz free energy.

LOG#203. Action gym.

A challenge post! 😉

Quantum physics becomes important when the magnitude called action is of order of Planck constant (an action itself!). Action is quantized. It is a much more essential quantization than that of energy or angular momentum (action itself the latter!). Whenever S\sim 10^{-34}J\cdot s you have quantum effects.

Exercise 1. Compute the action from an antenna with power 1kW and frequency f=1MHz. Is it quantum?

Exercise 2. A pocket clock. It uses a device of size 10^{-4}m and mass 10^{-4}kg to get times with precision of 1s. Compute its action. Is it quantum?

Exercise 3. Compute the action for a single atomic nucleus. Typical energies are about 1 MeV and distances about 1 fm (10^{-15}m). Binding energy per nucleon is about 10^{-12}J, mass of the nucleus can be taken as the proton mass. Compute the action. Is it quantum?

Exercise 4. Compute the action for an electron in the 1s shell of the hydrogen atom. Is it quantum?

Exercise 5. Compute the so-called Fermi energy in the case of potassium (metal). The atomic weight is about 39 g/mol, and its atom density 0.86 g per cubic centimeter. Suppose a single electron per atom. Compute its action. Is it quantum?


    \[\left[\hbar\right]=ML^2T^{-1}=\mbox{Energy}\cdot \mbox{Time}=\mbox{Momentum x Length}=\mbox{angular momentum x angle}\]

LOG#202. Harmonic oscillator.

Some clever physicists say that everything is an harmonic oscillator, and that every hard problem is just solvable in terms of a suitable set of harmonic oscillators (even true with string theory!):

In classical mechanics (CM) you have a the following standard harmonic oscillator lagrangian:

    \[L_{HO}=\dfrac{1}{2}m\dot{q}^2-\dfrac{1}{2}kq^2=T-U=\mbox{Kinetic Energy-Potential Energy}\]

The first order lagrangian given above depends upon the generalized velocities in the kinetic energy part. It provides the following Euler-Lagrange equations (EL):

    \[E(L)\equiv \dfrac{\partial L}{\partial q}-\dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{q}}\right)=0\]

For the L_{HO} given above, you obtain

    \[ \dfrac{\partial L}{\partial q}=-kq\]

    \[\dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{q}}\right)=\dfrac{dp}{dt}=m\ddot{q}\]

where p=m\dot{q} is the generalized momentum. Putting the two terms together, you get

    \[-m\ddot{q}-kq=0\leftrightarrow m\ddot{q}+kq=0\leftrightarrow \ddot{q}+\dfrac{k}{m}=0\leftrightarrow \ddot{q}+\omega^2 q=0\]

Indeed, you recognize this equation as the classical harmonic oscillator solution, that of course you can also get from Newton’s second for a Hooke’s law F=-kq. Moreover, you can also be general, and from the prescription:


derive the Newton’s law from this energetic approach, since EL applied to it implies

    \[\dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{q}}\right)=\dfrac{dp}{dt}\]

if you define the generalized momentum as

    \[p\equiv \dfrac{\partial L}{\partial \dot{q}}\]

and, by the other hand, for the potential energy depending ONLY on the generalized coordinates you get

    \[\dfrac{\partial L}{\partial q}=-\dfrac{\partial U}{\partial q}\]

Note that the last term is only the prescription for a conservative force F=-\nabla U.


  1. What if U=U(q,\dot{q}) or U(q,\dot{q},t). Nothing changes, unless the potential energy depends explicitly on time, what renders issues to the problem. Energy could be not conserved. And generally it is not conserved, unless time is not present explicitly in the lagrangian.
  2. What about non-conservative forces? Well, there are some issues too. Several methods have been developed to account for in into the lagrangian method. E.g.: Rayleigh dissipative function, fractional calculus techniques and others.
  3. What if lagrangians act on fractional derivatives?
  4. Riewe’s mechanism using dynamics is remarkable.
  5. What about the field theory extension of the harmonic oscillator?

Furthermore, consider the canonical HO lagrangian

    \[L_{HO}\equiv L_1=\dfrac{1}{2}m\dot{q}^2-\dfrac{1}{2}kq^2\]

Next, consider the change of the lagrangian by a piece (extra langrangian)


A change or variation in the lagrangian of the form

    \[L\rightarrow L+\dfrac{d}{dt}f(q,\dot{q})\]

is generally called gauge invariance for L. The addition of the total time derivative to the lagrangian does not change the equations of motion (EOM). In a field theory, the addition of a divergence (total derivative with respect all the spacetime indices) does not change the EOM. Then,


This trick, however, has a caveat here, since I used a function f(q,\dot{q}). The lagrangian L_3 depends on the generalized accelerations and you will have to use a second order EL equations to do the job. The proof is simple:



    \[\boxed{L_3\equiv L_{HO}^{HO}=-\dfrac{mq\ddot{q}}{2}-\dfrac{1}{2}kq^2}\]

is the higher order harmonic oscillator (HOHO) lagragian! If you dislike the minus signs, you can even define L_4=-L_3 and to continue the next steps below. The second order (higher order) EL equations read as follows:

    \[E(L)\equiv \dfrac{\partial L}{\partial q}-\dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{q}}\right)+\dfrac{d^2}{dt^2}\left(\dfrac{\partial L}{\partial \ddot{q}}\right)=0\]

Now, you can recover the HO equation from this higher (second) order lagrangian. Proof:

    \[\dfrac{\partial L}{\partial q}=-\dfrac{m\ddot{q}}{2}-kq\]

    \[\dfrac{\partial L}{\partial \dot{q}}=0\rightarrow \dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{q}}\right)=0\]

    \[\dfrac{\partial L}{\partial \ddot{q}}=-\dfrac{mq}{2}\rightarrow \dfrac{d^2}{dt^2}\left(\dfrac{\partial L}{\partial \ddot{q}}\right)=-\dfrac{m\ddot{q}}{2}\]

And so, from E(L)=0, adding the above last equations, you also get in the second order formalism





Thus, the theories with L_1 and L_3, equivalently, L_{HO} and L_{HO}^{HO}, are completely equivalent at the level of the EOM, since they are related by a gauge transformation, or they differ by total time derivative. It is also related to (non) canonical transformations in phase space. But this will be treated in a future classical mechanics thread…

In conclusion:


is a higher (second) order lagrangian giving us the EOM of a single harmonic oscillator, and it is related to the canonical standard HO lagrangian via a gauge transformation (a time derivative of a function f(q,\dot{q})).

Remark (I): You can generalize this to field theory as well.

Remark (II): The theory of equivalent lagrangians and higher mechanics is subtle, but it exists.

LOG#201. Generalized absement.

Absement is generally defined as

(1)   \begin{equation*} \vec{A}=\int \vec{r}(t) dt \end{equation*}

We can even integrate the absement (p-1)-times (position integrated n-times) to get

(2)   \begin{equation*} \vec{A}_n=\int \vec{r}(t) dt^n=\int \vec{A}(t) dt^{p-1}=\int \cdots \int\vec{r}(t) (dt)\underbrace{\cdots}_{n-times} dt \end{equation*}


(3)   \begin{equation*} \vec{A}_n=\int \cdots \int\vec{A}(t) (dt)\underbrace{\cdots}_{(p-1)-times} dt \end{equation*}

with dimensions of LT^n and units of ms^n. However, if time is multidimensional, and the time vector is written as

(4)   \begin{equation*} \vec{t}=\left(t_1,\ldots, t_r\right) \end{equation*}

We can indeed define some interesting generalizations of the absement.

Firstly, the scalar absement:

(5)   \begin{equation*} A_s=\int \vec{r}\left(t_1,\ldots,t_r\right)\cdot d\vec{t} \end{equation*}

Second generalization: the pseudovector (bivector) absement (valid in when the cross product is avalaible):

(6)   \begin{equation*} \vec{A}_B=\int \vec{r}\left(t_1,\ldots,t_r\right)\times d\vec{t} \end{equation*}

Thirdly, the exterior (scalar, vector or tensor) absement (valid in arbitrary dimensions!)

(7)   \begin{equation*} A_E=\int X\left(t_1,\ldots,t_r\right)\wedge  dt_1 \wedge \cdots \wedge dt_r \end{equation*}

(8)   \begin{equation*} A_V=\int X^\mu\left(t_1,\ldots,t_r\right)\wedge  dt_1 \wedge \cdots \wedge dt_r \end{equation*}

(9)   \begin{equation*} A_T=\int X^{\mu_1\cdots \mu_s}_{\;\;\;\;\;\;\;\;\;\;\nu_1\cdots \nu_p}\left(t_1,\ldots,t_r\right)\wedge  dt_1 \wedge \cdots \wedge dt_r \end{equation*}

Of course, you can easily include the spatial variables as well in the above definitions, but it would drive us to think absement in terms of field theory, what it is indeed “natural”…Note that separability of variables is a subtle issue in quantum mechanics due to entanglement! Something similar happens in relativity when two tensors are decomposable into pieces with tensor product or when you have an antisymmetric tensor decomposable into exterior products (decomposable p-forms).

Moreover, if X is a 1-vector or 1-form (by duality), but we can even generalize this to an arbitrary antisymmetric q-tensor (q-form, by duality). Thus, in that case, the fourth generalized absement is a (q+r)-form (or (q+r)-vector):

(10)   \begin{equation*} A=\int X_q \wedge  dt_1 \wedge \cdots \wedge dt_r=\int X^{\mu_1\cdots \mu_q} d^r T \end{equation*}

Indeed, we can even go further, by going to the full-relativistic case, defining the speed of light vector (1-form dual)

(11)   \begin{equation*} \vec{C}=(c_1,\ldots,c_r) \end{equation*}

With this definition, in 1T physics, absement has units of L^2\sim LT, as dimension of length is the same of that of time, and hinting that area or absement are MORE fundamental

(12)   \begin{equation*} \vec{A}=\int \vec{r}(t) d(ct) \end{equation*}

Integrating the absement (p-1)-times (the position n-times) to get

(13)   \begin{equation*} \vec{A}_n=\int \vec{r}(t) dt^p=\int \cdots \int\vec{r}(t) (cdt)\underbrace{\cdots}_{n-times} (cdt) \end{equation*}

we get something with units LT^{n}, or using the speed of light (p-1)-times, L^{p} units. If the q-form is separable or not is irrelevant. The generalized absement will have dimensions (after speed of light conversion) of L^{p}. Again, only a fundamental length is important and write:

(14)   \begin{equation*} \vec{A}_B=\int \vec{r}\left(t_1,\ldots,t_r\right)\times d(\vec{ct}) \end{equation*}

(15)   \begin{equation*} A_E=\int X\left(t_1,\ldots,t_r\right)\wedge  d(c_1t_1) \wedge \cdots \wedge d(c_rt_r) \end{equation*}

(16)   \begin{equation*} A=\int X_q \wedge  dt_1 \wedge \cdots \wedge dt_r=\int X^{\mu_1\cdots \mu_q} d^r (CT) \end{equation*}

Finally, from C-space (Clifford spaces) or super-spaces (and even hyper-superspaces!), you can create the most general absement-like object as follows. From the Clifford product AB=A\cdot B+A\wedge B, and

(17)   \begin{equation*} \mathcal{A}=\int X dT=\int X\cdot dT+ X\wedge dT \end{equation*}


(18)   \begin{equation*} \mathcal{A}=\int X d(CT)=\int X\cdot (CdT)+ X\wedge d(CT) \end{equation*}

and more generally

(19)   \begin{equation*} \mathcal{A}=\int \langle X d(CT)\rangle_{w} \end{equation*}

where we take the order w-vector from the Clifford product (or just a sum of some grades up to order w). This global generalization for any multivector and polyvector has a very interesting feature. It simplifies the relationship between area and absement. Formally,

(20)   \begin{equation*} \mathcal{A}=\int XdX=\dfrac{X^2}{2}\sim X^2 \end{equation*}

where we have assumed that X is a classical commuting c-number, so absement is like an area or reciprocally, spacetime is the SQUARE root of absement! If X were a grassmannian absement (Oh, God! I love this variation!) field, it would provide absement of order one, since integration acts as derivative for grassmannian variables according to the Berezinian prescriptions. Is out there any generalized absement useful for physmatics? Is there a reason why action is more fundamental than power or energy or are we just biased? If action is fundamental, could absement play a more fundamental role in future physmatics? Surely it could be…After all, action, power or energy are only three possible variables between many of them! Maybe, the privileged role of actergy, a.k.a., action is a consequence of our human perception and we could chose any other kinematical and dynamical variable for describing motion. Absement based dynamics is something it could be even more than possible in the future.

By the way, generalized absement is also related to generalized zilches and helicities of higher dimensional stuff. Ah, yeah…What the hell are zilches and helicities? That is another future blog story!

See you in another blog post!!!!

LOG#200. TSOR: The Spectrum Of Relativity.

Happy New Year 2018, dear followers! Today is a special blog post! My 200th post! After a really terrible 2017 from the viewpoint of family, health and productivity, I wish this one will be a great year for me and for my readers!

My first gift to the blogsphere is to continue with my traditional special posts. I decided long ago to make special posts every 50 blog posts. 50 seems to be special. Well, what is going on in this special one? Well…I faced a singular problem to decide the name of my domain and site when I was beginning my blogger experience. I decided to be the spectrum of something cool…And well, there were two natural candidates. As many of you do know, I decided that TSOR should become The Spectrum Of Riemannium, but the alternative was the also wonderful The Spectrum Of Relativity. Indeed, this is the topic today…What it could be, or what it can be or what it can become the other name of my blog and why it is also cool. Because, there are MANY theories of relativity (Theory Of Relativity=THOR?), even if you don’t know them all, I am going to introduce some of these fascinating and forgotten (by many) non-trivial extensions of the theory of relativity.The bunch of theories of relativity is what I call The Spectrum Of Relativity (also TSOR, for short!).


For a theoretical physicist like me (also astrophysicist, also astronomer or cosmologist from certain viewpoint!), relativity is a misnomer and a name of at least three theories. Yes, you heard OK, 3 theories:

  1. Galilean Relativity. Relativity did not begin with Einstein himself. It was Galileo Galilei, the first modern scientist, who stated the first Theory of Relativity (THOR, for short…Yes, I like this acronym). Galileo realized that the laws of physics were the same for observers who are in uniform velocity motion with respect to each other. That is what we know as galilean relativity. Motion is relative but the laws of physics are THE SAME, if you are in rest or in relativity motion with uniform speed with respect to any other reference frame. Of course, Galileo did not know the laws of electromagnetism, so galilean invariance holds only for Mechanics, as Newton himself would state with his infinitesimal calculus (or Leibniz’s version, in current fashion).
  2. Special relativity. The first breakdown or “extension” of relativity was to include the universality of physical laws for electromagnetic fields. After Faraday and Maxwell synthesis of the equations of electromagnetism, it became clear that something was “wrong” in comparison with Mechanics and the relativity principle. It seems that relativity could not be saved in the frame of the ether…The negative searches for the absolute speed of light with respect to the ether was wrong and the logical consequence waited for Einstein explanation: there is no ether as we thought. Relativity is saved with a caveat: the transformation laws are not the galilean transformations but the so-called Lorentz transformations that make invariant the Maxwell equations of electromagnetism! At velocities close to the speed of light, the laws of physics and the measures of space and time change with respect to those we knew from galilean physics. Time and space are unified into space-time, there is no absolute time or space, only spacetime interval measurements. Invariant quantities change in “special relativity” so that energy and mass show to be equivalent, just as space or time are two sides of the same objet, or momentum has an extra component we can interpret as energy. Moreover, simultaneity is also a relative concept…Thus, Newton notions of absolute time and space, and some other ideas discussed by 19th philosophers are wrong! What a time for doing physics the 20th century was. Of course, this thing also implied a terrible consequence: the law of newtonian gravity should be modified to accomplish for the non-instantaneous propagation of physical signals that special relativity proposed. Therefore, the 2nd relativity drove us to the third relativity revolution…
  3. General relativity. Gravity is (at least locally) equivalent to uniformly accelerated systems. This observation, or something like it, also noted as well by Galileo himself but dubbed as “anecdote”, was the key Rosetta stone to build a relativistic version of gravity. The equivalence principle has several variations (I am not going to go into details today) but if inertial mass is equivalent to gravitational charge, then gravity is a pseudoforce. Indeed, gravity itself is a manifestation of the curvature of spacetime! If you also postulate that the laws of physics in “general coordinate systems” are (at least those who are locally free falling, i.e., dropping with constant acceleration) the same (something we could call general covariance), you get a theory that we name General Relativity. Einstein’s himself did not like the name…Maybe he preferred theory of invariants or, in current time, we should named it as the theory of gravitation locally special, since in every point of the spacetime continuum special relativity holds, but the name General Relativity is conventional and it has remained until our days. Of course, this relativity does many other things special relativity alone does not. It predicts several effects that we have confirmed (not a full list): the precession of the perihelion in the Mercury orbit, the deflection of light near a massive body (or more generally the gravitational lensing effect), the gravitational time delay, the expansion of the Universe, the possible existence of a cosmological constant, the existence of black holes (highly curved regions of spacetime, where light can not classically escape), gravitomagnetism, or even the more spectacular discovery of the existence of gravitational waves from ripples of spacetime itself! LIGO discovery is going to be remembered as one of the brightest discoveries of human Science. It also opens a new window to observe the Universe in a way we could have never imagine in Newton’s time. Newton’s theory of gravity does not predict gravitational waves, general relativity does. There are of course alternative theories of relativity that also predict gravitational waves, but general relativity is the simplest of these theories. Einstein did know very well what he was doing after all. Didn’t he?

Standard theories are everywhere. You can read them in school, high school or the university and post-graduate schools too! Standard theories are very successful! For instance, standard thermodynamics, from the principles (now four, but maybe five?) and framework using the more advanced ideas of thermodynamic potentials…Today we even know that statistical mechanics of atoms and molecules are the hidden basis of conventional thermodynamics of solids and conventional matter, but quantum theory (standard, of course) is necessary in order to understand some quantum phases at very low and very high temperatures! Ensembles! Yes, we need ensembles! And well, sometimes equipartition of energy is democratically good…Or not! Standard special relativity and quantum mechanics are good to predict the distribution of thermal bosons and fermions, or even go beyond with anyons, q-onic and (q,p)-onic statistics! Special relavitiy is simple. It is based in one or two postulates (a single one if you agree that Lorentz invariance encodes both, the special relativity principle and the constancy of the speed of light in vacuum!). Special relativity also rises some “fake” paradoxes, and it rises some questions about the nature of space and time that are not yet solved. Quantum Mechanics is another Monster. The real Monster, I may say. It began from the results of lots of experiments, and it is now a set of rules (axioms) that pervades the microscopic fundaments of Nature, excepting gravity, who resist any trivial or easy way to be quantized. Quantum Mechanics is an real “monster”. It destroys determinism at the level of postulates, even when the theory does have a deterministic evolution law through unitarity, measurements are mysterious and terrible. They collapse the vector state. And the vector state is itself meaningless, as it is a complex number. Only the amplitude, understood as a modulus squared is meaningful and the terrible (for many) truth is that it is probabilistic at origin. But fortunately for us, probabilities are conserved…That is one of the quantum beasts. There are two more things: firstly quantum entanglement. You can take two have states of a system that are not separable or independent, and if you carry those parts far, far way from each other…Well, you can know what are the state in the another even if relativity itself would imply a faster than light thing that of course is not possible. This kind of non-locality is real. And it is one of the secret powers of quantum mechanics. Quantum entanglement and quantum mechanics open the possible creation of quantum computers,…Quantum statistical mechanics and quantum information theory are solid at current time. We can not deny the experimental evidence that favor quantum mechanics even if you have philosophical issues with its axioms! People who are not happy with quantum mechanics think that maybe some subquantum theory will supersede it in the future. However, there are many powerful no-go theorems about how those subquantum or hidden-variables theories could be. Many of them are even weirder than the Quantum Monster! For instance, any viable hidden variable theory reproducing quantum mechanics has to be CONTEXTUAL in order to mimic quantum mechanics results (e.g., the violation of Bell inequalities and every experiment that quantum mechanics or quantum experiments with spin show to be…True). Wait! What happens with relativity at this level? Well, indeed relativity and gravity push the idea of QM to be generalized (but not to be completely wrong!). Modern views, like that General Relativity is in fact equivalent to Quantum Mechanics, not only that ER=EPR, is being discussed by Leonard Susskind and that entanglement DOES MAKE spacetime is an idea on the air right now. We do not know the whole details, though!

Gravity is the missing piece into the general quantum scheme! Standard gravity, understood as general relativity now, is hard to quantize. It is not impossible, but it rises “infinite issues”. Literally. Quantum Field Theory handles infinities with a very special technique giving up infinities (with the aid of renormalization and regularization tools), but gravity is a CLASSICAL monster. The three equivalence principles (the weak, the strong and the Einstein’s) can be tracked to times of Ernst Mach and its principle of “motion”. What is motion after all? Riemann’s legacy, its habilitation thesis on the foundations of geometry itself, was the basis of relativity. And well, you can now know that this is also a reason for my dual name TSOR. Einstein’s finally realized that the metric field and the curvature of the space-time manifold were the origin of the gravitational (pseudo)-force. However, nothing forbids us to go beyond metric theories of space-time, or to restrict ourselves to quadratic metric functions, something that Riemann stated in his thesis as well. However, the quadratic form metric is the simplest theory, and it yields the simplest theory of geometrical space-time theory of gravity. It is called general relativity (GR). GR has lots of known and confirmed results: gravitational time delays (GPS confirmed!), the precession of Mercury’s perihelion, gravitational lensing or now the famous detection of gravitational waves (GW). GR also has cosmological consequences. Using some extra simple principles, you can even derive the LCDM model of the whole Universe, describing the Universe at big scales from almost the beginning of time (well, it is a little more tricky than all this, but I don’t want to be precise at this moment). However, GR predicts its own breakdown. Black holes and space-time singularities are the issues it faces. BUT, anyway, you can use something called black hole thermodynamics, and some tools of quantum field theory (QFT) in curved spacetime to derive effects like those by Hawking or Unruh! Black holes do evaporate due to quantum effects and they have temperature even if you use the approximation of space-time being a classical thing. Any uniformly accelerating (Rindler) observer sees a different VACUUM that a rest observer. Amazing! Your absence of faith in quantum gravity or unification, is it disturbing enough? Gauge theories describing the Standard Model of particles and fields (handling everything excepting gravity) can only describe…The 5% of the Universe. The remaining 95% is “dark” stuff waiting for a new explanation. You can use the power of the lagrangians and hamiltonians in both classical and quantum sense, but you can only explain, at current time, a poor 5% of the whole Universe. Scary? No! Leptons and quarks, the electroweak and the strong forces, are now confirmed by experiments. Even the Higgs mechanism is being tested right now, and we do know there is a Higgs-like particle permeating the space and time…125GeV/c² is its mass (more precisely, the mass of the Higgs field). Of course, gauge fixing any theory is non-trivial, you need Fadeev-Popov ghosts and some extra pieces in the lagrangian beast the SM is…But it fits experiments. It rocks. It is solid. During the last 50 years we were and we are yet searching for a break in the SM…Nothing happened. We are yet searching for supersymmetry(SUSY), superstrings and branes, preons, micro-BH, sphalerons, magnetic monopoles, axions, and other weirdos…The issue with the Higgs field is its mass as well, as the Yukawa interactions coming from nowhere everywhere with the aid of the Higgs vev, not coming from any SM symmetry. Crazy. The Higgs field is not protected by any SM symmetry, so it could be in principle mucho more heavy. It does not! The Large Hadron Collider is not enough to test the Higgs elf-coupling or to study some specific new physics searches. It is too nasty to shock protons with protons. Maybe we are wrong with the new physics scale. It could be anywhere between the 100GeV or the GUT-Planck scales, but the Higgs-like particle is too light. However, hopes to find something new are banishing as I write these article. Take care, I am not saying it is impossible, but it is unlikely the LHC will find something new but I would wish to be wrong! The known running of coupling constants in QED and QCD supports the Standard Model (SM) and a crazy theory called asymptotically safe gravity could indeed say that the SM plus gravity is OK until the Planck scale, with the only addition of superheavy right-handed neutrinos (indeed, something strange happens with high energy neutrinos, as seen by Ice-Cube! Something is happening with PeV neutrinos…).

The mixing of quarks and neutrinos are very different but “complementary”. No one has an idea of why. The mixing of the CKM quark matrix is almost minimal (the matrix is almost the identity). The mixing of neutrinos is almost maximal (the PMNS is close to be a rotation giving us maximal CP maximal violation). Of course, the details are subtle. The search for CP violation in the neutrino sector is the following stage. We do know CP is violated in the weak/electroweak sector, and a new generation of experiments is searching for CP violation in pure neutrino sectos. It is important…However, the strong force is different. The absence of a CP violating term in the QCD sector is something called the strong CP problem. The simplest solution, a Peccei-Quinn symmetry involving that detergent…Oh, sorry, that axion like particle…It should be a Nobel Prize search if found. Axions are also a candidate for dark matter, so we expect axions are out there…In fact, excepting sterile neutrinos or right-handed neutrinos, axions are a cheap alternative to make up dark matter without involving too much new physics we have not found. We will need axion telescopes!

Before going further into the real topic of this special blog post, let me add some extra points:

  1. Classical geometry should have a quantum geometry dictionary.
  2. Classical geometry can be somehow thought as symplectic or multisymplectic mechanics, quantum geometry is projective geometry.
  3. Standard theories should be classical standard theories based on classical groups, quantum geometries should be something else when quantum space-time is involved. After all, the thermodynamics is classically a very well-understood example of standard statistical mechanics where information is important and, though as probability, conserved due to unitarity properties.
  4. What we do know we do not know is key! The origin of mass, space-time or gauge theories theirselves. In the modern view, they should be emergent (non-fundamental!) from another concept or from other “degrees of freedom”. Even the all-mighty superstring theory/M-theory is poorly understood due the Hagedorn phase transitions where the theory itself can not be defined or where the perturbation theory is meaningless. What is the high-T phase of superstring theory? What is M-theory? 10^{500} vacua in M-theory, or more…About 10²⁷²⁰⁰⁰ in F-theory are certainly disturbing, specially if you want a single Universe, not if you play with the Multiverse idea also expected from the inflation thing…But that is another story.
  5. Beyond Standard Thermodynamics or Statistical Physics you could play with non-equilibrium thermodynamics (I. Prigogine would be happy, I think…) or you could involve non-extensive thermodynamics or some kind of superstatistics. Are you ready for all of them? The chaosmos approaches you now…

At last, we face with the real challenge of this post…However, you will need a little bit more more for the next part. I decided to be easy with the introduction, motivating…A bit silly I should say, but this post deserved something easy for non-mathematical readers as well. So I wanted to write something for them as well. Let me begin :).

A map for Theories of Relativity & Gravity

How many relativities do you know? Of course, you will answer that special and general relativity. You should also include galilean relativity, something Einstein himself knew before his struggle with the apparent tension between electromagnetism and classical mechanics. The incredible thing was to discover that electromagnetism itself has an enlarged notion (or special) of relativity that guided Einstein all his life, even to realize that newtonian gravity was not consistent with it and thus a generalized theory of relativity, a.k.a., general relativity (really, a locally special relativistic theory of gravity) was built. But, returning to the question…How many relativities out there? Well, let me list them:

Type A. Enlarged multidimensional relativities. Extensions of usual space-time to more dimensions.

1. Special relativity in D=s+t (s space-like, t time-like) dimensions. However, due to experimental bias and/or evidence, we select D=3+1 spacetime. String theory/superstring theory and M-theory work with more than 3+1 dimensions, as old Kaluza-Klein theories (KK). KK theories focus on space-like dimensions despite we could also include the option of time-like dimensions.

2. General relativity in D=s+t dimensions. Again, due to experimental evidence, we select general relativity in D=3+1 spacetime. Curiously, an old solution to the famous cosmological constant problem was given by a general theory of relativity with an extra time-like dimension. Don’t worry, small enough to avoid closed time-like curves and malicious time machines.

Special relativity or general relativity with more than one time dimensions, even infinite time dimensions, were advocated by several authors, remarkly the multitemporal papers by Kalitzin, and Barashenkov, the latter with only 3 dimensions of time and special focus on mechanics and electromagnetism. However, in principle, they are only versions of 1 and 2. Also, EAB Cole (as soon as 1982, I was a kid there), studied special relativity with 3 times as well.

3. Phase space-time relativity or Born reciprocal relativities. Extend the relativity doubling coordinates to include momentum. That was done by Born in 1938. He wanted to unify relativity and the quantum theory. Thus, he realized that the notion of duality or relativity in electromagnetism could be extended to the space-time metric, and by reciprocity between position and momentum variables, he doubled space-time into phase space-time:

(1)   \begin{equation*} dS^2_B=dX^MdX_M=dx^\mu dx_\mu+dp^\mu dp_\mu \end{equation*}

Born reciprocal relativity pushed to the limit, would allow to the whole space-time to addopt a crystal-like, quasicrystal-like or even poly(quasi)crystal-like behavior. Even more, it could imply certain cellular space-time structure, recently advocated by Petr Jizba with his approach of the world space-time crystal, but previously tried by A. Das and E.A.B. Cole. Also, there is a jet-bundle quantum version of this relativity:

(2)   \begin{equation*} -d\Sigma^2=dX_\mu dX^\mu+\dfrac{\hbar^2}{M^2c^2}d\dot{X}_\mu d\dot{X}^\mu+\kappa\dfrac{\hbar^4}{M^4c^4}d\ddot{X}_\mu d\ddot{X}^\mu+\ldots \end{equation*}

Maybe could we try as well the gravitational maximal force relativity

(3)   \begin{align*} -d\Sigma^2_{ext}=dX_\mu dX^\mu+\dfrac{\hbar^2}{M^2c^2}d\dot{X}_\mu d\dot{X}^\mu+\kappa_Q\dfrac{\hbar^4}{M^4c^4}d\ddot{X}_\mu d\ddot{X}^\mu+\ldots\\+\dfrac{1}{F_m^2}dP_\mu dP^\mu+\dfrac{1}{F_m^4}d\dot{P}_\mu d\dot{P}^\mu+\kappa_G\dfrac{1}{F_m^6}d\ddot{P}_\mu d\ddot{P}^\mu\ldots \end{align*}

and where F_m=c^4/4G or a multiple of it.

4. Fractal spacetime and scale relativity. Considered by some scientists postdictional or even crackpottery, Laurent Nottale proposed a theory in which the microscale and macroscale get merged due to scale properties. Scale relativity forces us to consider the possibility of a fractal spacetime in which the “state of scale” is relative as length or time measurements. Interestingly, it provides a way to understand the quantum world in terms of non-differentiable calculus in a way not too much have appreaciated until now. A short and fun reference (quite hidden) in the BBC conference and later book, Superstrings: a theory of everything, was included when referring to the djinn variable that Nottale created. Also, an extra transformation for the state of scale are usually included. But the theory is not mainstream even know.

Type B. Deformed relativities.

What is a deformed relativity? Well, there is no universal notion of that. I understand deformed relativities in a very broad sense. I will define a deformed relativity to that theory of relativity including a new dispersion relationship E=E(p,m) or even satisfying the same relativistic dispersion equation, they have a DIFFERENT or extended gamma factor. Under this definition, even some multitemporal relativities belong to both, type A and type B relativities. But I was aware of that non-null intersection.

1. Multitemporal deformed relativity. Kalitzian relativity include the generalized gamma factor:

(4)   \begin{equation*} \Gamma^{-1} =\sqrt{1-\beta_s^2+\sum_{k\neq s}\left(\dfrac{c_kdt_k}{c_sd\tau_s}\right)^2}=\sqrt{1-\dfrac{v_{(s)}^2}{c_s^2}+\sum_{k\neq s}\left(\dfrac{c_kdt_k}{c_sd\tau_s}\right)^2} \end{equation*}

Multitemporal theories are not quantum-friendly in general, it seems, despite some interesting ideas. However, twistor like ideas certainly contain some “vectorial time” quantities embedded into the spinorial structure of spacetime. Of course, this is not the usual way to do or understand spinors. However, hypertwistors and some not to old ideas tried to recover the idea that time could be higher dimensional, despite we observe, apparently, a single time direction (perhaps a projection? An illusion?).

2. Relativity in de-Sitter spaces. Kalitzian multitemporal relativity was also studied by G. Arcidiacono, but he focused instead on de-Sitter like properties (not multitemporal stuff excepting some short reviews of kalitzian relativity), he rediscovered Fantappie’s final relativity, and discovered the more general ultraspherical (highly symmetrical) relativity (dS-like), and the deformed gamma factor showed to be:

(5)   \begin{equation*} \Gamma=\dfrac{1+\alpha^2-\gamma^2}{\sqrt{\left(1+\alpha^2-\gamma^2\right)\left(1-\beta^2\right)+\left(\vec{\alpha}\times\vec{\gamma}-\gamma\right)^2}} \end{equation*}

where \alpha=\vec{x}/r, \beta=\vec{v}/c,\gamma=\vec{t}/t_0, and H=1/t_0=c/r=c/R_U. This projective relativity imply, e.g., a generalized Doppler projective effect (can we erase the need for dark energy with this? Try it yourself ;)):

(6)   \begin{equation*} \omega_p=\sqrt{\left(\dfrac{1-\beta}{1+\beta}\right)\left(1+\alpha^2\dfrac{1+\beta}{1-\beta}\right)} \end{equation*}

and the dS-Arcidiacono-Fantappie momentum reads

(7)   \begin{equation*} \pi_i=P_i+\dfrac{X_iX_s}{r^2}P_s=\left(1+\dfrac{x^2}{r^2}\right)P_i+\dfrac{x^s}{r^2}L_{is} \end{equation*}

where x^2=x_sx^s. The above momentum equation is indeed typical from dS relativities where the cosmological constant appears like a kinematical invariant quantity to be taken into account. This thing is known since times of Bacry and Levy-Leblond, who studied the maximal possible kinematical groups under standard degrees of freedom. Let me to put this in context a bit more. Deformed relativities introduce MODREs( MOdified Dispersion RElationships) like:

(8)   \begin{equation*} E^2=p^2+m^2 \end{equation*}

where we set c=1 for simplicity. KK theories generalize that to:

(9)   \begin{equation*} E^2=p^2+m^2+ \sum_{i=1}^s\dfrac{N_i^ 2}{R_i^2}-\sum_{j=1}^t\dfrac{n_j^2}{r_j^2} \end{equation*}

Conformal relativities have a MODRE

(10)   \begin{equation*} E^2=k^2+M^2 \end{equation*}

And for dS relativities, where you have something like g=\sqrt{3}L_P/R and g^2=G\hbar \Lambda/3c^3\sim 10^{-122} provide a deformed gamma

(11)   \begin{equation*} \Gamma^{-1}_{dS}=\sqrt{\left(1-\eta_{ij}\dfrac{X^iX^j}{R^2}\right)\left(1+\eta_{ij}\dfrac{\dot{X}^i\dot{X}^{j}}{c^2}\right)+\dfrac{2t\eta_{ij}X^i\dot{X}^j-\eta_{ij}\dot{X}^i\dot{X}^jt^2}{R^2}+\dfrac{(\eta_{ij}X^i\dot{X}^j)^2}{c^2R^2}} \end{equation*}

Also, you can find out the dS gamma written as follows:

(12)   \begin{equation*} E^2_{dS}=m^2+P_{cR}^2+\dfrac{c^2}{R^2}(L^2-R^2) \end{equation*}

Chiatti gives for Arcidiacono-Fantappie dS what we could call Arcidiacono-Fantappie-Chiatti gamma:

(13)   \begin{equation*} \Gamma^{-1}=\left[1-\beta^2+(\alpha-\beta\gamma)^2-(\alpha\wedge\beta)^2\right]^{1/2}(1-\alpha^2-\gamma^2) \end{equation*}


(14)   \begin{equation*} E^2_{dS}=p^2+m^2A^4+\dfrac{M_{ik}M^{ik}}{2r^2} \end{equation*}

with \alpha=d/R, \beta=v/c, \gamma=t/t_0, A^2=1+\alpha^2-\gamma^2.

3. Discrete crystal-like models of spacetime. Pioneered maybe by the noncommutative Snyder-spacetime, A. Das and E.A.B. Cole proposed models of spacetime that resemble cells or discrete structures. The obvious bad thing is that they break down Lorentz invariance at scales where the fundamental length appears, but that could be good for renormalization understtanding in the future. After all, no one knows why renormalization works yet! We could call these approaches cellular or (quasi)(poly)crystalline relativities.

4. Finslerian relativity. After B. Riemann seminal thesis on non-euclidean geometries, everyone geometrized physics or tried it. Indeed, Grassmann and Clifford ideas were present in the superb work of Riemann’s habilitation thesis. Gauss was pleased to listen and read it. And even more, Riemann himself realized that, in principle, there are many notions of non-euclidean geometry. He focused on quadratic forms, but he pointed out that other geometries with cubic and higher order forms could be possible. Giving up the quadratic constraint to the metric, you get something called Finsler (Weyl-Finsler is sometimes used) geometry. It includes the effect of higher curvatures and accelerations, an thus, the effect on velocity, acceleration and higher derivatives in the metric form. The generalized geometry of this type has an interesting story, not to be written today, and a Japanese fan: Kawaguchi. Finsler-Kawaguchi geometries have received only recent attention by the string theory community, but the areolar invariants they seem to provide should certainly interest to people with holographic development ideas. The simplest realization of finslerian relativity, at kinematical special level (not the general relativistic level) is to include a maximal acceleration a_m deformed gamma, so finslerian maximal acceleration theories include the generalized or deformed gamma:

(15)   \begin{equation*} \gamma_{v,a}=\dfrac{1}{\sqrt{1-v^2/c^2}\sqrt{1-a^2/a_m^2}} \end{equation*}

Caianiello tried to explain quantum mechanics with the aid of finslerian relativity and advocated a maximal acceleration principle. Did you know about tachyons? What about epitachyons with a>a_m? Well, tachyonic epitachyons have real \gamma_{v,a}…I am not sure of what that is the mean of those entities. Furthermore, you can extend finslerian relativity to even higher derivatives or accelerations. Maximal jerk? Maximal jounce? Maximal pop? Oh, yeah…You have never ever heard of those.

Finslerian relativity can be related to maximal acceleration special relativities like that of Caianiello, or in general relativistic settings (generalized special relativistic too).

5. Anisotropic special and general relativity. The works by G.Y. Bogoslovsky are quite relevant. There, the metric itself becomes

(16)   \begin{equation*} ds^2_B=(dx_0^2-dx^2)\left[\dfrac{(dx_0-\nu\cdot dx)^2}{dx_0^2-dx^2}\right]^r \end{equation*}

and more generally, the line element, in any (generally anisotropic) spacetime dimension becomes:

(17)   \begin{equation*} ds=\left[\dfrac{(\nu_i dx^i)^2}{g_{ik}dx^i dx^k}\right]^{r/2}\sqrt{g_{ik}dx^i dx^k} \end{equation*}

In addition to this metric, you get

(18)   \begin{align*} (p_0^2-p^2)=mc^2\left(1-r\right)^{(1-r)}\left(1+r\right)^{(1+r)}\left[\dfrac{(p_0-p\cdot \nu)^2}{p_0^2-p^2}\right]^r\\ E\equiv p_0c=\dfrac{mc^2}{\sqrt{1-\dfrac{v^2}{c^2}}} \left(\dfrac{1-v\cdot\nu/c}{\sqrt{1-\dfrac{v^2}{c^2}}}\right)^r \left[1-r+r\dfrac{1-v^2/c^2}{1-v\cdot\nu/c}\right]\\ p=\dfrac{mc^2}{\sqrt{1-\dfrac{v^2}{c^2}}} \left(\dfrac{1-v\cdot\nu/c}{\sqrt{1-\dfrac{v^2}{c^2}}}\right)^r \left[(1-r)v/c+r\nu\dfrac{1-v^2/c^2}{1-v\cdot\nu/c}\right]\\ L_{aSR}=-mc^2\left(\dfrac{1-v\cdot\nu/c}{\sqrt{1-\dfrac{v^2}{c^2}}}\right)^r\sqrt{1-\dfrac{v^2}{c^2}}\\ L_{aSR}^{nonrel}\approx -mc^2-mc r(v\cdot\nu)+(1-r)m\dfrac{v^2}{2}+r(1-r)\dfrac{m(v\cdot\nu)^2}{2}\\ \Gamma^{-1}_{aSR}=\left(\dfrac{1-v\cdot\nu/c}{\sqrt{1-\dfrac{v^2}{c^2}}}\right)^r\sqrt{1-\dfrac{v^2}{c^2}}\\ M_{\alpha\beta}=m(1-r)\left(\delta_{\alpha\beta}+r\nu_\alpha\nu_\beta\right) \end{align*}

6. Minimal velocity relativity (minimal acceleration and so on). Some years ago, I was reading and studying about every possible generalization of special and general relativity and I read a quite cute paper by a brazilian physicist, Cláudio Nassiff. He introduced a version of what we know now as doubly special relativity (DSR) based on a minimal velocity of quantum origin (due to the uncertainty principle, motion can not be stopped due to inevitable quantum fluctuations of the fields). This is a bit tricky, since any ultra-referential as he proposed has to respect the known SR world in which we live. However, the existence of such a minimal velocity can not be neglected at the quantum level. In fact, it introduces a new generalized gamma factor:

(19)   \begin{equation*} \lambda=\dfrac{h}{mc^2}\sqrt{\dfrac{1-V^2/c^2}{1-c_0^2/V^2}} \end{equation*}

and of course,

(20)   \begin{equation*} E=mc^2\sqrt{\dfrac{1-c_0^2/V^2}{1-V^2/c^2}} \end{equation*}

and a new deformed version of the addition of velocities taking into account the existence of the new ultra-referential, or cosmic doubly special invariant frame. Triply special relativities include a big combination of these doubly special relativity and dS relativity. It is mathematically complex, but it is not the place where I am going to develop it.

7. Ketsaris’ relativity in 7D and 10D. Ketsaris studied the kinematical relativity in which:

(21)   \begin{equation*} ds^2=dx^2_1+dx_2^2+dx_3^2-c^2dt^2+R^2d\theta_1^2+R^2d\theta_2^2+R^2d\theta_3^2 \end{equation*}

7D arises under a rotational internal relativity, and 10D chrono-rotational invariant

(22)   \begin{equation*} ds^2=dx^2_1+dx_2^2+dx_3^2-c^2dt^2+R^2d\theta_1^2+R^2d\theta_2^2+R^2d\theta_3^2+T^2dv_1^2+T^2dv_2^2+T^2v_3^2 \end{equation*}

and he also introduced a generalized gamma parameter to be

(23)   \begin{equation*} \gamma_{v,a,\omega}^{-1}=\sqrt{1-\dfrac{v^2}{c^2}-\dfrac{a^2}{A^2}+\dfrac{\omega^2}{\Omega^2}} \end{equation*}

and with

(24)   \begin{equation*} E(P,M)^2=m^2+p^2c^2+c^2f^2T^2-\dfrac{c^2l^2}{R^2} \end{equation*}

where f=\gamma ma, p=\gamma mv, l=\gamma J\Omega are the 10d relativistic force, momentum and angular momentum. Ketsaris, predating and advanced the emergence of Clifford spacetime relativity (see next) introduced the generalized spacetime

(25)   \begin{equation*} dX^2=g_{ij}dx^{i}dx^{j}+a_{a_1a_2a_3a_4}dx^{a_1a_2}dx^{a_3a_4} \end{equation*}

and more generally the kinematical order invariant geometries

(26)   \begin{equation*} dX^2=g_{i_1k_1}dx^{i_1}dx^{k_1}+g_{i_1i_2,k_1k_2}dx^{i_1i_2}dx^{k_1k_2}+\ldots+g_{i_1\ldots i_n,k_1\ldots k_n}dx^{i_1\ldots i_n}dx^{k_1\ldots k_n} \end{equation*}

In this way, Ketsaris could envision a generalized spacetime dS_K^2=G_{AB}dX^AdX^B for multi-indices that is now common in “extended” relativities. However, he forgot to say what happens with odd indices, i.e., odd metrics, very electromagnetic (or Yang-Mills) like. He could have added too the extra fields, and the scalar field (the zero order metric) is absent in his theory.

8. Gogberashvili octonionic relativity. Following the octonionic ultimate religion, the octonions represent the ultimate layer of physical reality and physical signals are built from them. M. Gogberarashvili took this credo a step beyond the usual coincidental 3,4,6,10 (1,2,4,8) division algebra knowledge and build in a new relativity, that we would also classify into the DSR type, allowing to the spacetime itself to acquire octonionic (really split octonionic) reality. Thus,

(27)   \begin{equation*} s^2=c^2dt^2-x_nx^n+\hbar^2\lambda_n\lambda^n-c^2\hbar^2\omega^2 \end{equation*}

with octonionic signal

(28)   \begin{equation*} s=c(t+\hbar I\omega)+J_n(x^n+\hbar I\lambda^n) \end{equation*}

The invariance of octonionic norm provides the relationship, quantum-like gamma factor

(29)   \begin{equation*} \dfrac{d\tau}{dt}=\gamma^{-1}=\sqrt{\left[1-\hbar^2\left(\dfrac{d\omega}{dt}\right)^2\right]-\dfrac{v^2}{c^2}\left[1-\hbar^2\left(\dfrac{d\lambda_n}{dx^n}\right)^2\right]} \end{equation*}

where v^2=\dfrac{dx_n}{dt}\dfrac{dx^n}{dt} and the DSR octonionic model provides the bounds v^2\leq c^2, dx_n/d\lambda^n\geq \hbar and dt/d\omega\geq \hbar in order to have quantumness into the octonionic ultimate relativity.

9. Zi-hua Weng sedenionic (and beyond) relativity. The Chinese Zi-huan Weng introduced a sedenionic and beyond version of relativity. You can read his papers and understand why they are hard. Zi-hua Weng has worked out some predictions in gravitomagnetism of his theory and some nice additional effects I urge you to check. It is falsiable easily as every other extension of relativity.

As you have seen, standard relativity theories have very well known and not well-known (non-standard) generalizations. Many of these relativities, even if uncorrect, should be named the forgotten or “lost” relativities (or the others, for short). Even, the DSR community would not recognize some of the relativities I mentioned and reviewed above of these lines. However, weird people have worked out these non-standard theories. The same happens with gravitational theories, but all that ins much more known. If you are a theoretical physicist or working mathematician, I am sure you have heard about theories “beyond” general relativity. For example, you know string theory and supergravity in 4 and 11 dimensions (maximal supergravity and M-theory live in 11d). Gauss-Bonet theories, and higher order gravitational theories, even non-local with and infinite number of derivatives are in the air again into the theoretical community. However, from the observational viewpoint, General Relativity rocks as much as the Standard Model, and the LCDM model rules over the data and the current phenomenology. MONDians and MOGians tried to find out a theory going beyond the galactic scales and cluster scales or even beyond to explain data, but the controversy and doubts on these theories rise when you plug the simulations and different experiments are contrasted.

10. Total relativity. The genius and Nobel Prize F. Wilczek dubbed total relativity to the theory in which, realizing the Mach principle one and for all motion, you could describe every motion in an invariant proper way. Not too much about the mathematical framework of this principle of total theory was proposed by him, but this blog post is going into that idea of total, final or projective relativity that many others have tried to develop to describe the quantum universe.

Type C. Extended (or enlarged) relativities. They are formulated into an enlarge spacetime with new degrees of freedom and include, in principle, every higher order correction from a jet-bundle viewpoint.

11. Super-relativity. SR is based on the notion of invariance. In particular, invariance under a pseudo-orthogonal group SO(s,t), more generally an orthogonal group if we give up the distinction between space and time coordinates due to the speed of light turning space into space and vice-versa. Supersymmetry is an extension of space-time allowing for anti-commutative c-numbers, or Grassmann numbers (indeed, spinor-like variables). The space-time invariant interval can be extended from the orthogonal group O(D) (or the O(D,D) in phase-spacetime approaches) to the supersymmetric version of it, that is

(30)   \begin{equation*} S^2=\sum_{i=1}^D (X^i)^2= X^iX_i \end{equation*}


(31)   \begin{equation*} S^2_{SUSY}=\sum_{i=1}^D (X^i)^2+\theta^a C_{ab} \theta^b \end{equation*}

where C_{ab} is an antisymmetric matrix. In general, two natural choices for the super-relativity theory is to consider one of the infinite families of groups allowing for a real representation of the generalized length, namely, the orthosymplectic group OSp(N,M) group based on the above invariant or the unitarysymplectic group SU(N,M) leaving invariant

(32)   \begin{equation*} S^2_{SUSY}=(X^i)^*X^i+ g_{ab}(\theta^a)^*\theta^b \end{equation*}

and where g_{ab}=\pm \delta_{ab}.

The ER: Extended Relativity

Carlos Castro and Matej Pavŝiĉ worked out the elements of an extended theory of relativity going beyond the usual notions of spacetime, and including new degrees of freedom…Even more interestingly, they were able to build a supersymmetric version of this extension, since the Clifford degrees of freedom allows us to introduce very naturally supercoordinates whenever we specify the Clifford spacetime in N=2^D dimensions:

(33)   \begin{equation*} {\bf X } ~ = ~ s~ {\bf 1 } ~+ ~ x^\mu ~ \gamma_\mu ~+~ x^{\mu_1\mu_2} ~ \gamma_{\mu_1} \wedge \gamma_{\mu_2} ~+ ~ x^{\mu_1 \mu_2\mu_3} ~ \gamma_{\mu_1}\wedge \gamma_{\mu_2} \wedge \gamma_{\mu_3}~ + \ldots~ +~x^{\mu_1\cdots \mu_D} ~ \gamma_{\mu_1} \wedge \cdots \wedge \gamma_{\mu_D} \end{equation*}

Remark: just when you need a speed to unify space and time in special relativity, in order to unify the multivector degrees of freedom into the Clifford polyvector you need a new dimensional quantity, a fundamental length, in order to keep terms on equal footing. We set that fundamental length (Planck length, string scale, …?) to the unit for convenience.

You will now observe why I mentioned about C-space previously when talking about Ketsaris relativity. Ketsaris relativity is certain realization of more general C-spaces:

(34)   \begin{equation*} ds^2=G_{AB}dX^AdX^B=d\sigma^2+g_{\mu_1\mu_2}dx^{\mu_1}dx^{\mu_2}+\cdots+g_{\mu_1\cdots \mu_D\nu_1\cdots\nu_D}dX^{\mu_1\cdots \mu_D}dX^{\nu_1\cdots\nu_D} \end{equation*}

or equivalently

(35)   \begin{equation*} ds^2=d\sigma^2+dx^\mu dx_\mu+ dx^{\mu\nu}dx_{\mu\nu}+\cdots+dx^{\mu_1\mu_2\cdots\mu_D}dx_{\mu_1\mu_2\cdots\mu_D} \end{equation*}

and you can also have any generalized electromagnetic-YM field into a polyvector

(36)   \begin{equation*} A=\sigma 1+A_\mu \gamma^\mu+A_{\mu_1\mu_2} \gamma^{\mu_1\mu_2}+\cdots +A_{\mu_1\cdots \mu_D}\gamma^{\mu_1\cdots \mu_D} \end{equation*}

Note the resemblance of the field strength to the position expansion in C-space. Really there are a complete symmetry between position and field in this theory! An old friend for our basic classical field theory courses now generalized into C-spaces. You can extend these polyvectors to a phase C-spacetime and dually invariant doubly geometry Born-like, as reciprocity hinted to M. Born 80 years ago. Even more, you can add supersymmetry. And with care enough, you can easily read off the higher spin dictionary between objects with higher indices and much more.

The UR: Ultimate Relativity

Two years ago, and about 4 months, I was invited to give a talk in Slovenia. The topic was Clifford space relativity, so I decided to punch with one idea I have not developed too much since then (due to some health and “life” issues). What if we could go further into the total or final relativity ideas of Wilczek, Fantappie-Arcidiacono and those ideas about the Mach principle (which one of the different machian principles?). I present the idea of Clifford spacetime (C-spacetime) to make easy for M. Pavŝiĉ give his talk about membranes and p-branes and I wanted to be original so, I introduced an additional hypothesis I called Ultimate Relativity (UR, from the german word, prime or primitive and original): special relativity pushed to the requirements of a minimal an maximal length, by the principle of duality, should be enlarged in order to include effects of maximal/minimal velocities, accelerations, jerks,…Even more, you could imagine a theory in which you can formulate the dynamics in terms of integrals, like the absement, and higher integrals as the dynamical variables. Also, you could complicate things with fractional order and complex-like derivatives, but I am not going deeper today here. With a minimal an maximal velocity, aceleration, and so on, you should understand that those can be made of some quantities we already know, like the speed of light. For instance, you get

(37)   \begin{equation*} \mbox{Max}\left(\dfrac{d^{n+1}X}{dt^{n+1}}\right)\leq c\left(\dfrac{c}{L_p}\right)^n \end{equation*}

for nth maximal 1st, 2nd, and so on n-th derivative. For minimal velocity and so on, the dual relationship should be

(38)   \begin{equation*} \mbox{Min}\left(\dfrac{d^{n+1}X}{dt^{n+1}}\right)\geq c_0\left(\dfrac{c_0}{L_\Lambda}\right)^n \end{equation*}

Then, also a generalized gamma factor should arise:

(39)   \begin{equation*} \Gamma (X,V,A,\ldots)=\dfrac{\sqrt{1-L_p^2/X^2}\sqrt{1-V_0^2/V^2}\sqrt{1-A_0^2/A^2}\cdots}{\sqrt{1-X^2/L_\Lambda^2}\sqrt{1-V^2/c^2}\sqrt{1-A^2/A_M^2}\cdots} \end{equation*}

This is a first step into the Ultima Relativity, the Ultimate Relativity, in which the basic object should be a modular invariant, and mostly, to understand the whole reality or realities in terms of all the true degrees of freedom, that are those of numbers and pure informational. To test Ultimate Relativity and the gamma factor above, we need more experiments and likely a new view of the quantum theory (a new view and understanding, not a new interpretation, it is wrong to neglect what quantum experiments say about our reality; reality is highly dependent on the variables you use or measure, just when you try to understand what is space or time separated in SR, it is not the right way to understand electromagnetism or SR phenomena without taking the merger of space and time seriously).

It’s OK. What is this useful for? Well, some pieces I have were in fact suggested in my 2016 talk at IARD, but with other equivalent calculations, is that MOND equals to certain minimal/maximal acceleration (length):

Case 1. Maximal acceleration only.


(40)   \begin{equation*} a_N=\dfrac{a}{\sqrt{1-a^2/A^2}} \end{equation*}

From the above equation, you get easiy with the aid of simple algebra that

(41)   \begin{equation*} a_N^2=\dfrac{a^2}{1-a^2/A^2} \end{equation*}


(42)   \begin{align*} a_N^2(1-\dfrac{a^2}{A^2})=a^2\\ a_N^2-\dfrac{a_N^2a^2}{A^2}=a^2\\ a^2+a^2\dfrac{a_N^2}{A^2}=a_N^2\\ a^2 \left(1+\dfrac{a_N^2}{A^2}\right)=a_N^2 \end{align*}

and then

(43)   \begin{equation*} \boxed{a=\dfrac{a_N}{\sqrt{1+a_N^2/A^2}}} \end{equation*}


(44)   \begin{equation*} \boxed{a=A\dfrac{1}{\sqrt{1+A^2/a_N^2}}} \end{equation*}

Case 2. Minimal acceleration only. Now, you write

(45)   \begin{equation*} \sqrt{1-a_0^2/a^ 2}a=a_N \end{equation*}


(46)   \begin{align*} a_N^2=a^2(1-a_0^2/a^2)\\ a_N^2=a^2-a_0^2\\ a^2=a_N^2+a_0^2\\ a=\sqrt{a_N^2+a_0^2}\\ a=a_0\sqrt{1+a_N^2/a_0^2} \end{align*}


(47)   \begin{equation*} \boxed{a=a_0\sqrt{1+\dfrac{a_N^2}{a_0^2}}} \end{equation*}

And the rest acceleration can be defined as a_{r}=a-a_0, or

(48)   \begin{equation*} a_r=a_0\left[\sqrt{1+\dfrac{a_N^2}{a_0^2}}-1\right] \end{equation*}

Note that, whenever a\sim a_0, then a_N/a_0<<1 (newtonian approximation is no longer valid) and you get the series

(49)   \begin{equation*} a_0\sqrt{1+(a_N/a_0)^2}\approx a_0\left(1+\dfrac{1}{2}\left(\dfrac{a_N}{a_0}\right)^2\right)=a_0+\dfrac{a_N^2}{a_0} \end{equation*}

Therefore, you get a minimal acceleration (dark energy?) plus a dark matter MONDIAN effect with the hypothesis of a minimal acceleration only. Remarkly, as you know,

(50)   \begin{equation*} \dfrac{(v^2/r)^2}{2}=G\dfrac{M}{r^2} \end{equation*}

and you get the know constant flat v^4=2GM law for galaxies.

Case 3. Maximal+minimal acceleration.

A more complicated case with maximal and minimal acceleration can be taken into account from the previous two cases. You can write now

(51)   \begin{align*} a_N=a\dfrac{\sqrt{1-\left(\dfrac{a_0}{a}\right)^2}}{\sqrt{1-\left(\dfrac{a}{A}\right)^2}}\\ a_N^2=a^2\dfrac{1-\dfrac{a_0^2}{a^2}}{1-\dfrac{a^2}{A^2}}\\ a_N^2(1-\dfrac{a^2}{A^2})=a^2(1-\dfrac{a_0^2}{a^2})\\ a^2=\dfrac{a_N^2+a_0^2}{1+\dfrac{a_N^2}{A^2}}\\ a=\sqrt{\dfrac{a_N^2+a_0^2}{1+\left(\dfrac{a_N}{A}\right)^2}}\\ a_r=a-a_0 \end{align*}

and therefore we get

(52)   \begin{equation*} \boxed{a=\sqrt{\dfrac{a_N^2+a_0^2}{1+\left(\frac{a_N}{A}\right)^2}}=A\sqrt{\dfrac{1+\frac{a_0^2}{a_N^2}}{1+\frac{A^2}{a_N^2}}}=a_0\sqrt{\dfrac{1+\frac{a_N^2}{a_0^2}}{1+\frac{a_N^2}{A^2}}}} \end{equation*}


(53)   \begin{equation*} a_r=a-a_0=a_0\left(\sqrt{\dfrac{1+\left(\frac{a_N}{a_0}\right)^2}{1+\left(\frac{a_N}{A}\right)^2}}-1\right) \end{equation*}

TSOR: The Spectrum Of Relativity

TSOR, The Spectrum Of Riemannium, is also The Spectrum Of Relativity by now (indeed I was considering that name as well for my blog at the beginning). Indeed, this is an adventure of how we can go beyond the quantum and the relativity theories we all know to the most general and generalized theory and the field theory concept itself. An incredible journey through different types of knowledge and ideas that were written before but not completely realized. If the quantum theory and the relativistic theory of fields we know are yet not complete, they must be completed or extended (enlarged).

Beyond the Quantum and Relativity?

Of course, there is a big new symmetry principle from which total relativity, final relativity and/or ultimate relativity (UR) should arise. A link between the continuous and the discrete worlds and the world of the different representations and dualities between theories from the last decades. A third revolution in string theory, loop quantum gravity and unification is yet to come. New algebraic tools and the full power of the number theory and special functions are waiting for it. The concept of number and time or space will never be the same. If the space-time is entanglement and a code, it can be decoded, and most importantly, if quantum, entanglement is a key concept, just as superposition. If gravity can be reduced or explained fully with entanglement of something, other forces as well. Or…Have you achieved any ultra-instinct for ultra-relativity and ultra-quantum theories? Oh, right…I am crazy enough to go for a Dragon Ball Super to Ultra joke here…

A guess of why we could not see extended dimensions of time, anisotropies or C-space degrees of freedom? The first motive is that they are all wrong, but dark energy and dark matter fit in these new degrees of freedom we can not see (yet!). Usually, we assume that enlarged theories reduce to previous theories with a tiny parameter and we get the new theory in natural units. There is another alternative. Well, tropicalization is an interesting idea: we drop out extra terms due to max/min semirings or tropical algebra/arithmetics…But I can not go further here today. However, from the mathematical viewpoint, we can not have any other option. The ultimate theory should approach a non-archimedean or ultra-metric (ultras everywhere? Is that bad? It depends on the intelligence of the ultrametricity you get).

Hints? Much more:

Conclusions and outlook

I have reviewed (shallowly, I may say), some of the forgotten and lost relativities, and, as special post number 200 (after six years of blogging), and I tried to show you details of some theories not too understood or foreseen in the path to the final unification of physics.You have read that we have many theories of special relativity, just as we have many theories of gravity beyond general relativity (Lanczos-Lovelock, Gauss-Bonnet, supergravities, superstring theories, M-theory, Horndeski theories, tensor-scalar theories, extended gravities in C-spaces,…).

Sometimes, I admit I am controversial, just like multiple dimensions of time (by the way, if you take into account all the possible observers, and not a single observer, you need multiple dimensions of time to describe the different realities these observers measure) but I believe that knowledge has to be shared in order to avoid its destruction. It is an issue of being a survivor. After all, we are all doomed…Aren’t we? The knowledge of fundamental physics is fundamental to ensure our survival in the Cosmos, irrespectively of any other thing. The only way to become an interstellar or intergalactic species is to master quantum gravity and the fabric of quantum spacetime, whatever it be. Unless human being were genetically modified and/or we could modify locally the force of gravity, every plan or effort to return to a permanent lunar base or settle colonies on Mars and other places in the Solar System will be in vane. There are edges of spacetime and frontiers in physics we have to solve in order to go to outer space with guarantees. Otherwise, it would be unrealistic to consider to travel out there.

Appendix A: Future at TSOR

I don’t know what is my future. And a I have several interesting projects for TSOR right now, my issue is time. I have been ill (maybe I am yet, I am not sure) and with difficulties. I considered to give up of writing this blog, but I thought I wanted to share pieces of what I have learned and I know in my timelife. Every person in this world has issues. I have been fighting against myself and my own life destiny during the last 3 years. I plan to renew the blog appearance in the near future. Maybe even to make a podcast or videologs. I don’t know how to record properly and to edit video or record sound for science. So, at the moment, I will try to post the unfinished posts I had (this blog post is being latexed in order to be turned a review paper of forgotten relativities). I am also to do thematic posts relative to some of these relativities very soon. If time (I work as High School teacher right now- yet, unhappy of me) allows me, I am finishing interesting blog posts about different topics I wanted to write about.

Appendix B: List of future topics

A non-exhaustive list is the following:

  1. Generalized absement.
  2. A C-space  relativity thread.
  3. The Noether theorems.
  4. Nima’s works about the amplituhedra, associahedra and the eft-hedra.
  5. Nambu dynamics.
  6. Magnetic monopoles.
  7. More about Bohrlogy.
  8. More about higher dimensional stuff. Indeed, I gave a seminar past year to my students about higher dimensional mathematics and physics.
  9. Gravitational wave physics.
  10. Black hole physics and wormholes.
  11. Time travel.
  12. Space (interstellar, interdimensional and intergalactic) travel theory.

This article is dedicated to the memory of the man who discovered the radiation of black holes, a great man who left us past Mars and that became inmortal with the discovery of the formulae:

(54)   \begin{align*} S=\dfrac{1}{4}k_B\dfrac{A}{L_p^2}\\ T_{BH}=\dfrac{\hbar c^3}{8\pi G_NMk_B}\\ t_{ev}=\dfrac{5120\pi G^2_NM_0^3}{\hbar c^4} \end{align*}

R.I.P. S. W. Hawking, you will never know how many times I read your thesis and main papers the last months, your equations will be with us forever. We will surely miss you…Your Brief Story of Time will continue to attract people into Science in the future, as long as study and intelligence be valued by our current species.

See you in a forthcoming blog post!!!!!!!!!!!

P.S.: Some pics from older works (not by me)

P.S.(II): If you like what I post, consider make a donation or support/spread my posts in social media! Thank you in advance!