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.

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:

(2)   \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:

(3)   \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 ;)):

(4)   \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

(5)   \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:

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

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

(7)   \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

(8)   \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

(9)   \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:

(10)   \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:

(11)   \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*}


(12)   \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:

(13)   \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), the works by G.Y. Bogoslovsky are quite relevant. There, the metric itself becomes

(14)   \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:

(15)   \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

(16)   \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*}

5. 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:

(17)   \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,

(18)   \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.

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

(19)   \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

(20)   \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

(21)   \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

(22)   \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

(23)   \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

(24)   \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.

7. 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,

(25)   \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

(26)   \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

(27)   \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\leq \hbar and dt/d\omega\leq \hbar in order to have quantumness into the octonionic ultimate relativity.

8. 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.

9. 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.

10. 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

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


(29)   \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

(30)   \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:

(31)   \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:

(32)   \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

(33)   \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

(34)   \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

(35)   \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

(36)   \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:

(37)   \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.


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

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

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


(40)   \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

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


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

Case 2. Minimal acceleration only. Now, you write

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


(44)   \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*}


(45)   \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

(46)   \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

(47)   \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,

(48)   \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

(49)   \begin{align*} a_N=\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

(50)   \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*}


(51)   \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:

(52)   \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!

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