# LOG#105. Einstein’s equations.

In 1905,  one of Einstein’s achievements was to establish the theory of Special Relativity from 2 single postulates and correctly deduce their physical consequences (some of them time later).  The essence of Special Relativity, as we have seen, is that  all the inertial observers must agree on the speed of light “in vacuum”, and that the physical laws (those from Mechanics and Electromagnetism) are the same for all of them.  Different observers will measure (and then they see) different wavelengths and frequencies, but the product of wavelength with the frequency is the same.  The wavelength and frequency are thus Lorentz covariant, meaning that they change for different observers according some fixed mathematical prescription depending on its tensorial character (scalar, vector, tensor,…) respect to Lorentz transformations.  The speed of light is Lorentz invariant.

By the other hand, Newton’s law of gravity describes the motion of planets and terrrestrial bodies.  It is all that we need in contemporary rocket ships unless those devices also carry atomic clocks or other tools of exceptional accuracy.  Here is Newton’s law in potential form:

$4\pi G\rho = \nabla ^2 \phi$

In the special relativity framework, this equation has a terrible problem: if there is a change in the mass density $\rho$, then it must propagate everywhere instantaneously.  If you believe in the Special Relativity rules and in the speed of light invariance, it is impossible. Therefore, “Houston, we have a problem”.

Einstein was aware of it and he tried to solve this inconsistency.  The final solution took him ten years .

The apparent silly and easy problem is to develop and describe all physics in the the same way irrespectively one is accelerating or not. However, it is not easy or silly at all. It requires deep physical insight and a high-end mathematical language.  Indeed,  what is the most difficult part are  the details of Riemann geometry and tensor calculus on manifolds.  Einstein got  private aid from a friend called  Marcel Grossmann. In fact, Einstein knew that SR was not compatible with Newton’s law of gravity. He (re)discovered the equivalence principle, stated by Galileo himself much before than him, but he interpreted deeper and seeked the proper language to incorporante that principle in such a way it were compatible (at least locally) with special relativity! His  “journey” from 1907 to 1915 was a hard job and a continuous struggle with tensorial methods…

Today, we are going to derive the Einstein field equations for gravity, a set of equations for the “metric field” $g_{\mu \nu}(x)$. Hilbert in fact arrived at Einstein’s field equations with the use of the variational method we are going to use here, but Einstein’s methods were more physical and based on physical intuitions. They are in fact “complementary” approaches. I urge you to read “The meaning of Relativity” by A.Einstein in order to read a summary of his discoveries.

We now proceed to derive Einstein’s Field Equations (EFE) for General Relativity (more properly, a relativistic theory of gravity):

Step 1. Let us begin with the so-called Einstein-Hilbert action (an ansatz).

$S = \int d^4x \sqrt{-g} \left( \dfrac{c^4}{16 \pi G} R + \mathcal{L}_{\mathcal{M}} \right)$

Be aware of  the square root of the determinant of the metric as part of the volume element.  It is important since the volume element has to be invariant in curved spacetime (i.e.,in the presence of a metric).  It also plays a critical role in the derivation.

Step 2. We perform the variational variation with respect to the metric field $g^{\mu \nu}$:

$\delta S = \int d^4 x \left( \dfrac{c^4}{16 \pi G} \dfrac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} + \dfrac{\delta (\sqrt{-g}\mathcal{L}_{\mathcal{M}})}{\delta g^{\mu \nu}} \right) \delta g^{\mu \nu}$

Step 3. Extract out  the square root of the metric as a common factor and use the product rule on the term with the Ricci scalar R:

$\delta S = \int d^4 x \sqrt{-g} \left( \dfrac{c^4}{16 \pi G} \left ( \dfrac{\delta R}{\delta g^{\mu \nu}} +\dfrac{R}{\sqrt{-g}}\dfrac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} \right) +\dfrac{1}{\sqrt{-g}}\dfrac{\delta ( \sqrt{-g}\mathcal{L}_{\mathcal{M}})}{\delta g^{\mu\nu}}\right) \delta g^{\mu \nu}$

Step 4.  Use the definition of a Ricci scalar as a contraction of the Ricci tensor to calculate the first term:

$\dfrac{\delta R}{\delta g^{\mu \nu}} = \dfrac{\delta (g^{\mu \nu}R_{\mu \nu})}{\delta g^{\mu \nu} }= R_{\mu\nu} + g^{\mu \nu}\dfrac{\delta R_{\mu \nu}}{\delta g^{\mu \nu}} = R_{\mu \nu} + \mbox{total derivative}$

A total derivative does not make a contribution to the variation of the action principle, so can be neglected to find the extremal point.  Indeed, this is the Stokes theorem in action. To show that the variation in the Ricci tensor is a total derivative, in case you don’t believe this fact, we can proceed as follows:

Check 1. Write  the Riemann curvature tensor:

$R^{\rho}_{\, \sigma \mu \nu} = \partial _{\mu} \Gamma ^{\rho}_{\, \sigma \nu} - \partial_{\nu} \Gamma^{\rho}_{\, \sigma \mu}+ \Gamma^{\rho}_{\, \lambda \mu} \Gamma^{\lambda}_{\, \sigma \nu} - \Gamma^{\rho}_{\, \lambda \nu} \Gamma^{\lambda}_{\, \sigma \mu}$

Note the striking resemblance with the non-abelian YM field strength curvature two-form

$F=dA+A \wedge A = \partial _{\mu} A_{\nu} - \partial _{\nu} A_{\mu} + k \left[ A_\mu , A_{\nu} \right]$.

There are many terms with indices in the Riemann tensor calculation, but we can simplify stuff.

Check 2. We have to calculate the variation of the Riemann curvature tensor with respect to the metric tensor:

$\delta R^{\rho}_{\, \sigma \mu \nu} = \partial _{\mu} \delta \Gamma^{\rho}_{\, \sigma \nu} - \partial_\nu \delta \Gamma^{\rho}_{\, \sigma \mu} + \delta \Gamma ^{\rho}_{\, \lambda \mu} \Gamma^{\lambda}_{\, \sigma \nu} - \delta \Gamma^{\rho}_{\lambda \nu}\Gamma^{\lambda}_{\, \sigma \mu} + \Gamma^{\rho}_{\, \lambda \mu}\delta \Gamma^{\lambda}_{\sigma \nu} - \Gamma^{\rho}_{\lambda \nu} \delta \Gamma^{\lambda}_{\, \sigma \mu}$

One cannot calculate the covariant derivative of a connection since it does not transform like a tensor.  However, the difference of two connections does transform like a tensor.

Check 3. Calculate the covariant derivative of the variation of the connection:

$\nabla_{\mu} ( \delta \Gamma^{\rho}_{\sigma \nu}) = \partial _{\mu} (\delta \Gamma^{\rho}_{\, \sigma \nu}) + \Gamma^{\rho}_{\, \lambda \mu} \delta \Gamma^{\lambda}_{\, \sigma \nu} - \delta \Gamma^{\rho}_{\, \lambda \sigma}\Gamma^{\lambda}_{\mu \nu} - \delta \Gamma^{\rho}_{\, \lambda \nu}\Gamma^{\lambda}_{\, \sigma \mu}$

$\nabla_{\nu} ( \delta \Gamma^{\rho}_{\sigma \mu}) = \partial _\nu (\delta \Gamma^{\rho}_{\, \sigma \mu}) + \Gamma^{\rho}_{\, \lambda \nu} \delta \Gamma^{\lambda}_{\, \sigma \mu} - \delta \Gamma^{\rho}_{\, \lambda \sigma}\Gamma^{\lambda}_{\mu \nu} - \delta \Gamma^{\rho}_{\, \lambda \mu}\Gamma^{\lambda}_{\, \sigma \nu}$

Check 4. Rewrite the variation of the Riemann curvature tensor as the difference of two covariant derivatives of the variation of the connection written in Check 3, that is, substract the previous two terms in check 3.

$\delta R^{\rho}_{\, \sigma \mu \nu} = \nabla_{\mu} \left( \delta \Gamma^{\rho}_{\, \sigma \nu}\right) - \nabla _{\nu} \left(\delta \Gamma^{\rho}_{\, \sigma \mu}\right)$

Check 5. Contract the result of Check 4.

$\delta R^{\rho}_{\, \mu \rho \nu} = \delta R_{\mu \nu} = \nabla_{\rho} \left( \delta \Gamma^{\rho}_{\, \mu \nu}\right) - \nabla _{\nu} \left(\delta \Gamma^{\rho}_{\, \rho \mu}\right)$

Check 6. Contract the result of Check 5:

$g^{\mu \nu}\delta R_{\mu \nu} = \nabla_\rho (g^{\mu \nu} \delta \Gamma^{\rho}_{\mu\nu})-\nabla_\nu (g^{\mu \nu}\delta \Gamma^{\rho}_{\rho \mu}) = \nabla _\sigma (g^{\mu \nu}\delta \Gamma^{\sigma}_{\mu \nu}) - \nabla_\sigma (g^{\mu \sigma}\delta \Gamma ^{\rho}_{\rho \mu})$

Therefore, we have

$g^{\mu \nu}\delta R_{\mu \nu} = \nabla_\sigma (g^{\mu \nu}\delta \Gamma^{\sigma}_{\mu\nu}- g^{\mu \sigma}\delta \Gamma^{\rho}_{\rho\mu})=\nabla_\sigma K^\sigma$

Q.E.D.

Step 5. The variation of the second term in the action is the next step.  Transform the coordinate system to one where the metric is diagonal and use the product rule:

$\dfrac{R}{\sqrt{-g}} \dfrac{\delta \sqrt{-g}}{\delta g^{\mu \nu}}=\dfrac{R}{\sqrt{-g}} \dfrac{-1}{2 \sqrt{-g}}(-1) g g_{\mu \nu}\dfrac{\delta g^{\mu \nu}}{\delta g^{\mu \nu}} =- \dfrac{1}{2}g_{\mu \nu} R$

The reason of the last equalities is that $g^{\alpha\mu}g_{\mu \beta}=\delta^{\alpha}_{\; \beta}$, and then its variation is

$\delta (g^{\alpha\mu}g_{\mu \nu}) = (\delta g^{\alpha\mu}) g_{\mu \nu} + g^{\alpha\mu}(\delta g_{\mu \nu}) = 0$

Thus, multiplication by the inverse metric $g^{\beta \nu}$ produces

$\delta g^{\alpha \beta} = - g^{\alpha \mu}g^{\beta \nu}\delta g_{\mu \nu}$

that is,

$\dfrac{\delta g^{\alpha \beta}}{\delta g_{\mu \nu}}= -g^{\alpha \mu} g^{\beta \nu}$

By the other hand, using the theorem for the derivation of a determinant we get that:

$\delta g = \delta g_{\mu \nu} g g^{\mu \nu}$

since

$\dfrac{\delta g}{\delta g^{\alpha \beta}}= g g^{\alpha \beta}$

because of the classical identity

$g^{\alpha \beta}=(g_{\alpha \beta})^{-1}=\left( \det g \right)^{-1} Cof (g)$

Indeed

$Cof (g) = \dfrac{\delta g}{\delta g^{\alpha \beta}}$

and moreover

$\delta \sqrt{-g}=-\dfrac{\delta g}{2 \sqrt{-g}}= -g\dfrac{ \delta g_{\mu \nu} g^{\mu \nu}}{2 \sqrt{-g}}$

so

$\delta \sqrt{-g}=\dfrac{1}{2}\sqrt{-g}g^{\mu \nu}\delta g_{\mu \nu}=\dfrac{1}{2}\sqrt{-g}g_{\mu \nu}\delta g^{\mu \nu}$

Q.E.D.

Step 6. Define the stress energy-momentum tensor as the third term in the action (that coming from the matter lagrangian):

$T_{\mu \nu} = - \dfrac{2}{\sqrt{-g}}\dfrac{(\sqrt{-g} \mathcal{L}_{\mathcal{M}})}{\delta g^{\mu \nu}}$

or equivalently

$-\dfrac{1}{2}T_{\mu \nu} = \dfrac{1}{\sqrt{-g}}\dfrac{(\sqrt{-g} \mathcal{L}_{\mathcal{M}})}{\delta g^{\mu \nu}}$

Step 7. The extremal principle. The variation of the Hilbert action will be  an extremum when the integrand is equal to zero:

$\dfrac{c^4}{16\pi G}\left( R_{\mu \nu} - \dfrac{1}{2} g_{\mu \nu}R\right) - \dfrac{1}{2} T_{\mu \nu} = 0$

i.e.,

$\boxed{R_{\mu \nu} - \dfrac{1}{2}g_{\mu \nu} R = \dfrac{8\pi G}{c^4}T_{\mu\nu}}$

Usually this is recasted and simplified using the Einstein’s tensor

$G_{\mu \nu}= R_{\mu \nu} - \dfrac{1}{2}g_{\mu \nu} R$

as

$\boxed{G_{\mu\nu}=\dfrac{8\pi G}{c^4}T_{\mu\nu}}$

This deduction has been mathematical. But there is a deep physical picture behind it. Moreover,  there are a huge number of physics issues one could go into. For instance, these equations bind to particles with integral spin which is good for bosons, but there are matter fermions that also participate in gravity coupling to it. Gravity is universal.  To include those fermion fields, one can consider the metric and the connection to be independent of each other.  That is the so-called Palatini approach.

Final remark: you can add to the EFE above a “constant” times the metric tensor, since its “covariant derivative” vanishes. This constant is the cosmological constant (a.k.a. dark energy in conteporary physics). The, the most general form of EFE is:

$\boxed{G_{\mu\nu}+\Lambda g_{\mu\nu}=\dfrac{8\pi G}{c^4}T_{\mu\nu}}$

Einstein’s additional term was added in order to make the Universe “static”. After Hubble’s discovery of the expansion of the Universe, Einstein blamed himself about the introduction of such a term, since it avoided to predict the expanding Universe. However, perhaps irocanilly, in 1998 we discovered that the Universe was accelerating instead of being decelerating due to gravity, and the most simple way to understand that phenomenon is with a positive cosmological constant domining the current era in the Universe. Fascinating, and more and more due to the WMAP/Planck data. The cosmological constant/dark energy and the dark matter we seem to “observe” can not be explained with the fields of the Standard Model, and therefore…They hint to new physics. The character of this  new physics is challenging, and much work is being done in order to find some particle of model in which dark matter and dark energy fit. However, it is not easy at all!

May the Einstein’s Field Equations be with you!

# LOG#100. Crystalline relativity.

## Centenary blog post and dedicatories

My blog is 100 posts “old”. I decided that I wanted a special topic and subject for it, so I have thinking during several days if I should talk about Physmatics, tropical mathematics or polylogarithms, but these topics deserve longer entries, or a full thread to discuss them with details I consider very important, so finally I changed my original mind and I took a different path.

This blog entry is dedicated specially to my friends out there. They are everywhere in the world. And specially to Carlos Castro, M. Pavsic (inventors of C-space, M-space relativity in Clifford spaces and the brane M-space approach to relativity with Clifford Algebras, respectively), my dear friend S.Lukic (now working hard in biomathematics and mathematical approaches to genetics), A. Zinos (a promising Sci-Fi writer), J. Naranja (my best friend, photographer and eclectic man) and all my (reduced) Spanish friends (yes, you know who are you, aren’t you?). I dedicated this special blog entry to my family (even if they don’t know what I am doing with this stuff, likely they have no idea at all…) and those special people who keep me up and make me feel alive (from time to time, when they write me, in russian worlds), even when the thunder sounds and the storm arises and I switch off from almost all the real world. And finally, it is also dedicated to all my unbiased followers around the world… Wherever you are… It is also dedicated to all of you…

Well, firstly I should eat a virtual take, don’t you think so?

1. Serendipitous thoughts about my 100th blog post

Here, in my 100th post, I am going to write about some old fashioned idea/s, likely “crackpot” to some current standards, but it also shares interesting ideas with Sci-Fi and real scientific topics like the recently introduced “time crystals” by Wilczek. The topic today is: a forgotten (likely wrong) multitemporal theory of relativity!

Why did I choose such a crazy topic? Firstly, it is an uncommon topic. Multitemporal theories or theories with extra time-like dimensions are generally given up or neglected by the physics community. The reasons seem to be broad: causality issues (closed time-like curves “are bad”), the loss of experimental evidence (time seems to be 1D, doesn’t it?), vacuum instabilities induced/triggered by QM with extra time-like dimensions and many others (some of them based on phislophical prejudices, I would say). From the pure mathematical viewpoint, extra time-like dimensions are posible and we can handle them in a similar way to space-like dimensions, but some differences arise. Let me remark that there is a complete branch of mathematics (sometimes called semi-riemannian geometry) that faces with spaces with multiple temporal dimensions (spaces with more than one temporal coordinate, generally more than minus, or plus-dependind on your sign convention).

The second reason is that I am very interested in any theory beyond the Standard Model, and particularly, any extension of Special Relativity that has been invented and in any extension that could be built from first principles. Extended theories of relativity beyond Special Relativiy do exist. The first theory Beyond Standard Special Relativity, to my knowledge, was metarelativity, namely: extended special relativity allowing “tachyons”. It was pioneered by Recami, Sudarshan, Pavsic and some other people, to quote only some of the people I have in mind right now. Perhaps, the next (known) trial was Snyder Non-Commutative spacetime. It extends relativity beyond the realm of commutative spacetime coordinates. After these “common” extended relativities, we also have (today): deformed special relativities like Doubly or Triply Special Relativities and q-deformed versions like kappa-Minkovski spacetime and some other models like the de Sitter (dS) relativity. These theories are “non mainstream” today,  but they certainly have some followers (I am one of them) and there are clever people involved in their development. Let me note that Special Relativity seems to hold yet in any High Energy experiment, so extended relativities have to explain the data in such a way that their deformation parameters should approach the Minkonvskian geometry in certain limits. Even the Kaluza-Klein approach to extra spacelike dimensions is “a deformed” version of Special Relativity somehow!

Some more modern versions of extended relativities are the theory of relativity in Clifford spaces ( pioneered by Carlos Castro Perelman and Matej Pavsic, and some other relatively unknown researchers), a theory based in relativity in (generalized) phase spaces with a (generalized) Finsler geometry or the very special relativity.  In fact, Finsler geometries triggered another extension of special relativity long ago. People call this extension VERY SPECIAL relativity (or Born reciprocal relativity in phase space, a Finsler spacetime), and other people anisotropic spacetime relativity (specially some people from Russia and Eastern Europe). Perhaps, there are some subtle details, but they share similar principles and I consider very special relativity and finslerian relativity as “equivalent” models (some precision should be done here from a mathematical perspective, but I prefer an intuitive approach in this post). Remember: all these extensions are out there, irrespectively you believe in them or not, such theories do exist. A different issue IS if Nature obeys them or not closer or not, they can be built and either you neglect them due to some conservative tastes you have (Occam’s razor: you keep Minkovskian/General Relativity since they can fit every observation at a minimum ” theoretical cost”) or you find some experimental fact that can falsify them (note that they can fix their deformation parameters in order you avoid the experimental bounds we have at current time).

My third reason to consider this weird and zenzizenzizenzic post is to be an open mind. String theory or loop quantum gravity have not been “proved” right in the experiments. However, they are great mathematical and theoretical frameworks. Nobody denies that, not me at least. But no new evidences from the alledged predictions of string theory/Loop Quantum Gravity have been confirmed so far. Therefore, we should consider new ideas or reconsider old fashioned ideas in order to be unbiased. Feynman used to say that the most dangerous thing in physics was that everyone were working on the same ideas/theories. Of course, we can coincide in some general ideas or principles, but theory and experiment advances are both necessary. With only one theory or idea in the city, everything is boring. Again, the ultimate theory, if it exists, could be a boring theory, something like SM plus gravity (asymptotically safe) until and even beyond the Planck scale, but some people think otherwise. There are many “dark” and unglued pieces yet in Physmatics…

The final reason I will provide you is that…I like strange ideas! Does it convert me in a crackpot? I wish you think otherwise! I wouldn’t be who I am if I enjoyed dogmatic ideas only. I use to distinguish crackpottery from “non-standard” models, so maybe, a more precise definition or rule should be provided to know what is the difference between them (crackpottery and non-stardardness) but I believe that it is quite “frame dependent” at the end. So…Let me begin now with a historial overview!

The unification of fundamental forces in a single theory or unified field theory was Einstein’s biggest dream. After the discovery that there was a pseudoeuclidean 4D geometry and a hidden symmetry in the Maxwell’s equations, Einstein’s quest was to express gravity in way that were consistent with the Minkovskian geometry in certain limit. Maxwell’s equations in 4D can be written as follows in tensor form:

$\partial^\mu F_{\mu\nu}=\mbox{Div} F_{\mu\nu}=J_\nu$

and

$\mbox{Rot}F_{\mu\nu}=\dfrac{1}{2}\epsilon_{\mu\nu\sigma\tau}\partial^\nu F^{\sigma\tau}=0$

where $J_\nu=(-c\rho,\vec{j})$ is the electromagnetic four-current. The symmetry group of these classical electromagnetic equations is the Poincare group, or to be more precise, the conformal group since we are neglecting the quantum corrections that break down that classical symmetre. I have not talked about the conformal group in my group theory thread but nobody is perfect! Eintein’s field equations for gravity are the following equations (they are “common knowledge” in general relativity courses):

$G_{\mu\nu}=\kappa T_{\mu\nu}$

The invariance group of (classical or standard) general relativity is something called the diffeomorphism group (due to general covariance). The diffeomorphism group invariace tells us that every (inertial or not) frame is a valid reference frame to every physical laws. Gravity can be “locally given away” if you use a “free fall” reference frame. The fact that you can “locally” forget about gravity is the content of the Einstein’s equivalence principle. I will discuss more the different classes of existing equivalence principles in a forthcoming thread of General Relativity, but this issue is not important today.

What else? Well, 4D theories seem not to be good enough to explain everything! Einstein’s himself devoted the last years of his life to find the unified theory of electromagnetism and gravity, ignoring the nuclear (quantum) interactions. It was his most famous failure beyond his struggles against the probabilistic interpretation of the  “new” Quantum Mechanics. Eintein’s unification dreams was tried by many others: Weyl, Kaluza, Klein, Eddington, Dirac himself, Heisenberg,…Remember that Faraday himself tried to find out a relation between gravity and electromagnetism! And those dreams continue alive today! In fact, quantum field theory “unifies” electromagnetism and weak nuclear forces with the electroweak theory inside the Standard Model. It is believed  that a Grand Unified Theory(GUT) should unify the electroweak force and the strong (nuclear) interaction at certain energy scale $E_X$. X is called the GUT scale, and it is generally believed that it arises at about $latez 10^{15}$ GeV. Unification with gravity is thought to be “relevant” at Planck scale $E_P$, or about $10^{19}$ GeV. Therefore, we can observe that there are two main “approaches” to the complete unification of the known “fundamental interactions”:

1st. The Particle Physics path. It began with the unification of electricity and magnetism. Then we discovered the nuclear interactions. Electromagnetism and weak interactions were unified in the 70s of the past 20th century. Then, it was conjectured that GUT unification would happen at high energy with Quantum Chromodynamics (the gauge theory of strong nuclear forces), and finally, the unification with gravity at Planck energy. Diagramatically speaking:

$\mbox{EM}\longrightarrow \mbox{Nuclear Forces}\longrightarrow \mbox{EW theory}+\mbox{QCD}\longrightarrow \mbox{EW+QCD}+\mbox{Gravity}$

2nd. The Faraday-Einstein unification path. It begins with the unification of gravity and electromagnetism first! Today, it can be said that the entropic gravity/force approach by Verlinde is a revival of this second path. It is also the classical road followed by Kaluza-Klein theories: gauge fields are higher dimensional components of a “big metric tensor” which becomes “quantized” somehow. Diagramatically:

$\mbox{EM}\longrightarrow \mbox{Gravity}\longrightarrow \mbox{EM theory}+\mbox{Gravity}\longrightarrow \mbox{EM+Gravity}+\mbox{nuclear forces}$

An interesting question is if these two paths are related and how we bring out together the best ideas of both of them. From a purely historical reason, the first path has been favoured and it has succeeded somehow. The classical “second” path is believed to be “wrong” since it neglects Quantum Mechanics and generally it finds issues to explain what Quantum Field Theories do explain. Is it a proof? Of course, it is NOT, but Physics and Physmatics have experimental foundations we can not avoid. It is not only a question of “pure thought” to invent a “good theory”. You have to test it. It has to explain everything you do know until now. That is how the Occam’s razor works in Science. You have experiments to do and observations to explain…You can not come with a new theory if it is in contradiction with well-tested theories. The new theory has to include the previous theories in some limit. Otherwise, you have a completely nonsense theory.

The second path to unification has lots of “hidden” stories and “strange theories”. Einstein’s works about teleparallelism and non-symmetrical metric tensor theories were induced by this road to unification. Has someone else followed this path?

Answer to the last question: Yes! I am going to explain you the generally unknown theory of projective relativity! It was originally created by the italian physicist Fantappie, and it was studied and extended to multiple time-like dimensions via a bulgarian guy called Kalitzin and an italian physicist known as G. Arcidiacono. Perhaps it shares some points with the current five-dimensional theory advocated by P.Wesson, but it is a completely different (parallel likely) history.

Fantappie (1901-1956) built a “projective” version of special relativity the he called “final relativity”. Today, it is known as de Sitter-relativity or de Sitter projective relativity, and according to Levy-Leblond, is one of the maximal deformations of kinematical groups available in classical physics! In fact, we can “see” the Fantappie’s final (projective) relativity as an anticipation of the cosmological constant as a physical reality. The cosmological constant IS a physical parameter in final relativity associated to the radius of the Universe. If you take this statement as “true”, you are driven to think that the cosmological constant is out there as a true “thing”. Giving up the mismatch between our current QFT calculations of vacuum energy, de Sitter relativity/final projective relativity does imply the existence of the cosmological constant! Of course, you should explain why our QFT are wrong in the way they are…But that is a different story. At current time, WMAP/Planck have proved that Dark Energy, a.k.a. the cosmological constant, is real. So, we should rethink about the way in which it enters in physics. Should we include a new symmetry in QFT (de Sitter symmetry) in order to solve the cosmological constant problem? It is a challenge! Usually, QFT are formulated in Minkovski space. But QFT calculations in Minkovski spacetime give no explanation of its cosmological value. Maybe, we should formulate QFT taking into accont the cosmological constant value. As far as I know, QFT defined on de Sitter spaces are much less developed that anti de Sitter spaces, since these ones are popular because of the adS/CFT correspondence. There are some interestings works about QFT in dS spaces in the arxiv. There are issues, though, e.g., the vacuum definition and QFT calculations in dS background are “harder” than the equivalent Minkovskian counterparts! But I believe it is a path to be explored further!

Fantappie had also a hierarchical “vision” on higher dimensional spaces. He defined “hyperspherical” universes $S_n$ contained in rotational groups $R_{n+1}$ with $(n+1)$ euclidean dimensions and $n(n+1)/2$ group parameters. He conjectured that the hierarchy of hyperspherical universes $S_3, S_4, \ldots, S_n$ provided a generalization of Maxwell equations, and with the known connection between $S_n$ and $R_{n+1}$, Fantappie tried the construction of a unified theory with extra dimensions (a cosmological theory, indeed), with the aid of his projective relativity principle. He claimed to be able to generalize Einstein’s gravitational field equations to electromagnetism, following then the second path to unification that I explained above. I don’t know why Fantappie final projective relativity (or de Sitter relativity) is not more known. I am not expert in the History of Physics, but some people and ideas remain buried or get new names (de Sitter relativity is “equivalent” to final relativity) without an apparent reason at first sight. Was Fantappie a crackpot? Something tells me that Fantappie was a weird italian scientist like Majorana but he was not so brilliant. After all, Fermi, Pauli and other conteporary physicists don’t quote his works.

From projective relativity to multitemporal relativity

What about “projective relativity”? It is based on projective geometry. And today we do know that projective geometry is related and used in Quantum Mechanics! In fact, if we take the $r=R\longrightarrow \infty$ limit of “projective” geometry, we end with “classical geometry”, and then $S_n$ becomes $E_n$, the euclidean spaces, when the projective radius tends to “infinity”. Curiously, this idea of projective geometry and projective relativity remained hidden during several decades after Fantappie’s death (it seems so). Only G. Arcidiacono and N. Kalitzin from a completely different multitemporal approach worked in such “absolutely crazy” idea. My next exposition is a personal revision of the Arcidiacono-Kalitzin multitemporal projective relativity. Suppose you are given, beyond the 3 standard spatial dimensions $(n-3)$ new parameters. They are ALL time-like, i.e., you have a $(n-3)$ time vector

$\vec{t}=\left( t_1,t_2,\ldots,t_{n-3}\right)$

We have $(n-3)$ timelike coordinates and $(n-3)$ “proper times” $\tau_s$, with $s=1,2,\ldots,n-3$. Therefore, we will also have $(n-3)$ different notions or “directions” of “velocity” that we can choose mutually orthogonal and normalized. Multitemporal (projective) relativity arise in this $n$ dimensional setting. Moreover, we can introduce $(n-3)$ “different” ( a priori) universal constants/speeds of light $c_s$ and a projective radius of the Universe, R. Kalitzin himself worked with complex temporal dimensions and even he took the limit of $\infty$ temporal dimensions, but we will not follow this path here for simplicity. Furthermore, Kalitzin gave no physical interpretation of those extra timelike dimensions/paramenters/numbers. By the other hand, G. Arcidiacono suggested the following “extension” of Galilean transformations:

$\displaystyle{\overline{X}=f(X)=\sum_{n=0}^\infty \dfrac{X^{(n)}(0)t^n}{n!}}$

$\overline{X}=X(0)+X'(0)t+X''(0)\dfrac{t^2}{2}+\ldots=X(0)+V^{(1)}t+V^{(2)}t^2/2+\ldots$

$\overline{X}=x+Vt+At^2/2+\ldots$

These transformations are nonlinear, but they can be linearized in a standard way. Introduce $(n-3)$ normalized “times” in such a way:

$t_1=t, t_2=t^2/2,\ldots, t_s=t^{s}/s!$

Remark: To be dimensionally correct, one should introduce here some kind of “elementary unit of time” to match the different powers of time.

Remark(II): Arcidiacono claimed that with 2 temporal dimensions $(t,t')$, and $n=5$, one gets “conformal relativity” and 3 universal constants $(R,c,c')$. In 1946, Corben introduced gravity in such a way he related the two speeds of light (and the temporal dimensions) so you get $R=c^2/c'$ when you consider gravity. Corben speculated that $R=c^2/c'$ could be related to the Planck’s legth $L_p$. Corben’s article is titled A classical theory of electromagnetism and gravity, Phys. Rev. 69, 225 (1946).

Arcidiacono’s interpretation of Fantappie’s hyperspherical universes is as follows: the Fantappie’s hyperspheres represent spherical surfaces in n dimensions, and these surfaces are embedded in certain euclidean space with (n+1) dimensions. Thus, we can introduce (n+1) parameters or coordinates

$(\xi_1,\xi_2,\ldots,\xi_n,\xi_0)$

and the hypersphere

$\xi_0^2+\xi_1^2+\ldots+\xi_n^2=r^2=R^2$

Define transformations

$\xi'_A=\alpha_{AB}\xi_B$ with $A,B=0,1,2,\ldots,n$

where $\alpha_{AB}$ are orthogonal $(n+1)\times (n+1)$ matrices with $\det \alpha_{AB}=+1$ for proper rotations. Then, $R_{n+1}\supset R_n$ and rotations in the $(\xi_A,\xi_B)$ plane are determined by $n(n+1)/2$ rotation angles. Moreover, you can introduce (n+1) projective coordinates $(\overline{x}_0,\overline{x}_1,\ldots,\overline{x}_n)$ such as the euclidean coordinates $(x_1,x_2,\ldots,x_n)$ are related with projective coordinates in the following way

$\boxed{x_i=\dfrac{r\overline{x}_i}{\overline{x}_0}}\;\; \forall i=1,2,\ldots,n$

Projective coordinates are generally visualized with the aid of the Beltrami-Reimann sphere, sometimes referred as Bloch or Poincarè sphere in Optics. The Riemann sphere is used in complex analysis. For instance:

This sphere is also used in Quantum Mechanics! In fact, projective geometry is the natural geometry for states in Quantum Physics. It is also useful in the Majorana representation of spin, also called star representation of spin by some authors, and riemann spheres are also the fundamental complex projective objects in Penrose’s twistor theory! To remark these statements, let me use some nice slides I found here http://users.ox.ac.uk/~tweb/00006/

Note: I am not going to explain twistor theory or Clifford algebra today, but I urge you to read the 2 wonderful books by Penrose on Spinors and Spacetime, or, in the case you are mathematically traumatized, you can always read his popular books or his legacy for everyone: The Road to Reality.

Projective coordinates are “normalized” in the sense

$\overline{x}_0^2+\ldots+\overline{x}_n^2=r^2$ or $\overline{x}_A\overline{x}_A=r^2$ $\forall A=0,1,\ldots,n$

It suggests us to introduce a pythagorean (“euclidean-like” ) projective “metric”:

$ds^2=d\overline{x}_Ad\overline{x}_A$

It is sometimes called the Beltrami metric. You can rewrite this metric in the following equivalent notation

$A^4ds^2=A^2(dx^idx^i)-(\alpha_i dx^i)^2$

with

$A^2=1+\alpha_s\alpha_s$ and $\alpha_s=x_s/r$

Some algebraic manipulations provide the fundamental tensor of projective relativity:

$\boxed{A^4 g_{ik}=A^2\delta_{ik}-\dfrac{x_ix_k}{r^2}}$

Here

$\vert g_{ik}\vert =g=A^{-2(n+1)}$ so

$\boxed{g^{ik}=(g_{ik})^{-1}=A^2\left( \delta_{ik}+\dfrac{x_ix_k}{r^2}\right)}$

The D’Alembertian operator is defined to be in this projective space

$\boxed{\square^2 \varphi =\dfrac{1}{\sqrt{g}}\partial_i\left(\sqrt{g}g^{ik}\partial_k \varphi\right)=0}$

Using projective “natural” coordinates with $r=1$ to be simpler in our analysis, we get

$A^{n+1}\partial_i\left[A^{-n-1}A^2(\delta_{ik}+x_ix_k)\partial_k\varphi\right]=0$

or

$A^{n+1}\left[(\partial_iA^{1-n})\partial_i\varphi+(A^{1-n})\partial_i(x_ix_k\partial_k\varphi)\right]=0$

But we know that

$\partial_iA^{1- n}=(1-n)A^{-1-n}x_i$

$\partial_ix_i=n$

$x_sx_s=A^2-1$

And then, if $r\neq 1$, we have the projective D’Alembertian operator

$\boxed{r^2\square^2=A^2\left(r^2\partial_i\partial_i\varphi +x_ix_k\partial_i\partial_k\varphi+2x_k\partial_k\varphi\right)=0}$

Here, $R_{n+1}$ is the tangent space (a projective space) with $\overline{x}'_A=\alpha_{AB}\overline{x}_B$, and where $A,B=0,1,\ldots,n$. We can return to “normal” unprojective relativistic framework choosing

$x'_i=\dfrac{\alpha_{ik}x_k+\alpha_{i0}}{\alpha_{00}+\left(\dfrac{\alpha_{i0}x_i}{r}\right)}$

with $x_i=0$ and $A=1$, and $\overline{x}_A=(r,0,\ldots,0)$. That is, in summary, we have that in projective relativity, using a proper relativistic reference frame, the position vector has NULL components excepting the 0th component $x_0=r=R$! And so, $\overline{x}_A=(r,0,\ldots,0)$ is a “special” reference frame in projective relativity. This phenomenon does not happen in euclidean or pseudoeuclidean relativity, but there is a “similar” phenomenon in group theory when you reduce the de Sitter group to the Poincaré group using a tool named “Inönü-Wigner” group contraction. I will not discuss this topic here!

It should be clear enough now that from $(x_1,\ldots,x_n)$, via $\overline{x}_i=x_i/A$ and $\overline{x}_0=r/A$, in the limit of infinite radius $R\longrightarrow \infty$, it reduces to the cartesian euclidean spaces $E_3,E_4,\ldots,E_n$. Nicola Kalitzin (1918-1970), to my knowledge, was one of the few (crackpot?) physicists that have studied multitemporal theories during the 20th century. He argued/claimed that the world is truly higher-dimensional, but ALL the extra dimensions are TIME-like! It is quite a claim, specially from a phenomenological aside! As far as I know he wrote a book/thesis, see here http://www.getcited.org/pub/101913498 but I have not been able to read a copy. I learned about his works thanks to some papers in the arxiv and a bulgarian guy (Z.Andonov) who writes about him in his blog e.g. here http://www.space.bas.bg/SENS2008/6-A.pdf

Arcidiacono has a nice review of Kalitzin multitemporal relativity (in the case of finite $n$ temporal dimensions), but I will modify it a litte bit to addapt the introduction to modern times. I define the Kalitzin metric as the following semiriemannian metric

$\boxed{\displaystyle{ds^2_{KAL}=dx_1^2+dx_2^2+dx_3^2-c_1^2dt_1^2-c_2^2dt_2^2-\ldots -c_{n-3}^2dt_{n-3}^2=\sum_{i=1}^3dx_i^2-\sum_{j=1}^{n-3}c_j^2dt_j^2}}$

Remark (I): It is evident that the above metric reduce to the classical euclidean metric or the Minkovski spacetime metric in the limites where we write $c_j=0$ and $c_1=c, c_{j+1}=0\forall j=1,2,\ldots,n-3$. There is ANOTHER way to recover these limits, but it involves some trickery I am not going to discuss it today. After all, new mathematics requires a suitable presentation! And for all practical purposes, the previous reduction makes the job (at least today).

Remark (II): Just an interesting crazy connection with algebraic “stuff” ( I am sure John C. Baez can enjoy this if he reads it)…

i) If $n-3=0$, then we have $n=3+0$ or 3D “real” (euclidean) space, with 0 temporal dimensions in the metric.

ii) If $n-3=1$, then we have $n=3+1$ or 4D pseudoeuclidean (semiriemannian?) spacetime, or equivalently, the (oldfashioned?) $x_4=ict$ relativity with ONE imaginary time, i.e. with 1 temporal dimension and 1 “imaginary unit” related to time!

iii) If $n-3=2$, then we have $n=3+2=5$ or 5D semiriemannian spacetime, a theory with 2 temporal imaginary dimensions, or 1 complex number (after complexification, we can take one real plus one imaginary unit), maybe related to projective dS/adS relativity in 5D, with $-i_0^2=-1=i_1^2$?

iv) If $n-3=3$, then we have $n=3+3=6$ or 6D semiriemannian spacetime, a theory with 3 temporal dimensions and 3 “imaginary units” related to …Imaginary quaternions $i^2=j^2=k^2=-1$?

v) If $n-3=7$, then we have $n=3+7=10$ or 10D semiriemannian spacetime, a theory with 3 temporal dimensions and 7 “imaginary units” related to …Imaginary octonions $i_1^2=i_2^2=\ldots =i_7^2=-1$?

vi) If $n-3=8$, then we have $n=3+7=11$ or 11D semiriemannian spacetime, a theory with 3 temporal dimensions and 8 “units” related to …Octonions $-i_0^2=i_1^2=i_2^2=\ldots =i_7^2=-1$?

Remark (III): The hidden division algebra connection  with the temporal dimensions of higher dimensional relativities and, in general, multitemporal relativities can be “seen” from the following algebraic facts

$n-3=0\leftrightarrow n=3=3+0\leftrightarrow t\in\mathbb{R}$

$n-3=1\leftrightarrow n=3=3+1\leftrightarrow t\in\mbox{Im}\mathbb{C}$

$n-3=2\leftrightarrow n=5=3+2\leftrightarrow t\in\mathbb{C}$

$n-3=3\leftrightarrow n=6=3+3\leftrightarrow t\in\mbox{Im}\mathbb{H}$

$n-3=4\leftrightarrow n=7=3+4\leftrightarrow t\in\mathbb{H}$

$n-3=7\leftrightarrow n=10=3+7\leftrightarrow t\in \mbox{Im}\mathbb{O}$

$n-3=8\leftrightarrow n=11=3+8\leftrightarrow t\in\mathbb{O}$

Remark (IV): Was the last remark suggestive? I think it is, but the main problem is how do we understand “additional temporal dimensions”? Are they real? Do they exist? Are they a joke as Feynman said when he derived electromagnetism from a non-associative “octonionic-like” multitemporal argument? I know, all this is absolutely crazy!

Remark (V): What about $(n-3)\longrightarrow \infty$ temporal dimensions. In fact, Kalitzin multitemporal relativity and Kalitzin works speculate about having $\infty$ temporal dimensions! I know, it sounds absolutely crazy, it is ridiculous! Specially due to the constants it would seem that there are convergence issues and some other weird stuff, but it can be avoided if you are “clever and sophisticated enough”.

Kalitzin metric introduces $(n-3)$ (a priori) “different” lightspeed species! If you faced problems understanding “light” in 4D minkovskian relativity, how do you feel about $\vec{C}=(c_1,\ldots,c_{n-3})$? Therefore, we can introduce $(n-3)$ proper times ( note that as far as I know at current time, N. Kalitzin introduces only a single proper time; I can not be sure since I have no access to his papers at the moment, but I will in future, I wish!):

$\boxed{-c_s^2d\tau_s^2=dx_1^2+dx_2^2+dx_3^2-c_1^2dt_1^2-\ldots-c_{n-3}^2dt_{n-3}^2}\;\forall s=1,\ldots,n-3$

Therefore, we can define generalized the generalized $\beta_s$ and $\Gamma_s$ parameters, the multitemporal analogues of $\beta$ and $\gamma$ in the following way. Fix some $s$ and $c_s, \tau_s$. Then, we have

$c_s^2d\tau_s^2=-dx_1^2-dx_2^2-dx_3^2+c_1^2dt_1^2+\ldots+c_s^2dt_s^2+\ldots+c_{n-3}^2dt_{n-3}^2$

$c_s^2\dfrac{d\tau_s^2}{dt_s^2}=-\dfrac{(d\vec{x})^2}{(dt_s)^2}+c_1^2\dfrac{dt_1^2}{dt_s^2}+\ldots+c_{n-3}^2\dfrac{dt_{n-3}^2}{dt_s^2}$

$c_s^2\dfrac{d\tau_s^2}{dt_s^2}=-\dfrac{(d\vec{x})^2}{(dt_s)^2}+c_1^2dt_1^2+\ldots+c_s^2+\ldots+c_{n-3}^2dt_{n-3}^2$

$\dfrac{d\tau_s^2}{dt_s^2}=-\dfrac{(d\vec{x})^2}{c_s^2(dt_s)^2}+c_1^2dt_1^2+\ldots+1+\ldots+c_{n-3}^2dt_{n-3}^2$

$\displaystyle{\dfrac{d\tau_s^2}{dt_s^2}=1-\dfrac{(d\vec{x})^2}{c_s^2(dt_s)^2}+\sum_{k\neq s}\dfrac{c_k^2dt_k^2}{c_s^2dt_s^2}}$

Define $B_s= v_{(s)}/c_s$ and $B_s= 1/\Gamma_s$ (be aware with that last notation), where $\Gamma_s, B_s$ are defined via the next equation:

$\boxed{\displaystyle{B_s= \dfrac{1}{\Gamma_s}=\sqrt{1-\beta_s^2+\sum_{k\neq s}\left(\dfrac{c_kdt_k}{c_sd\tau_s}\right)^2}=\sqrt{1-\dfrac{v_{(s)}^2}{c_s^2}+\sum_{k\neq s}\left(\dfrac{c_kdt_k}{c_sd\tau_s}\right)^2}}}$

and where

$\overrightarrow{V}_{(s)}=\vec v_s=\dfrac{d\vec{x}_\alpha}{dt_s}\;\;\forall \alpha=1,2,3$

Then

$\boxed{d\tau_s=B_sd\tau_s}$ or $\boxed{dt_s=\Gamma_s d\tau_s}$

Therefore, we can define $(n-3)$ different notions of “proper” velocity:

$\boxed{u_i^{(s)}=V^{(s)}=\dfrac{dx_i}{d\tau_s}=\dfrac{1}{B_s}\dfrac{dx_i}{dt_s}=\Gamma_s\dfrac{dx_i}{dt_s}=\Gamma_s \vec v_s}$

In the reference frame where $x_i=0$ AND/IFF $B_s=1$, then $u_i=0$ for all $i=1,2,3$ BUT there are $(s+3)$ “imaginary” components! That is, we have in that particular frame

$\boxed{u_{s+3}^{s}=ic_s} \;\;\forall s$

and thus

$\boxed{u_k^{(r)}u_k^{(s)}=-c_s^2\delta_{rs}}$

This (very important) last equation is strikingly similar to the relationship of reciprocal vectors in solid state physics but extended to the whole spacetime (in temporal dimensions!)! This is what I call “spacetime crystals” or “crystalline (multitemporal) relativity”. Relativity with extra temporal dimensions allows us to define some kind of “relativity” in which the different proper velocities define some kind of (relativistic) lattice. Wilczek came to the idea of “time crystal” in order to search for “periodicities” in the time dimension. With only one timelike dimension, the possible “lattices” are quite trivial. Perhaps the only solution to avoid that would be consider 1D quasicrystals coming from “projections” from higher dimensional “crystals” (quasicrystals in lower dimensions can be thought as crystals in higher dimensions). However, if we extend the notion of unidimensional time, and we study several time-like dimensions, new possibilities arise to build “time crystals”. Of course, the detection of extra timelike dimensions is an experimental challenge and a theoretical one, but, if we give up or solve the problems associated to multiple temporal dimensions, it becomes clear that the “time crystals” in D>1 are interesting objects in their own! Could elementary particles be “phonons” in a space-time (quasi)crystal? Is crystalline (multitemporal) relativity realized in Nature? Our common experience would suggest to the contrary, but it could be interesting to pursue this research line a little bit! What would it be the experimental consequence of the existence of spacetime crystals/crystalline relativity? If you have followed the previous discussion: spacetime crystals are related to different notions of proper velocity (the analogue of reciprocal vectors in solid state physics) and to the existence of “new” limit velocities or “speeds of light”. We only understand the 5% of the universe, according to WMAP/Planck, so I believe that this idea could be interesting in the near future, but at the moment I can not imagine some kind of experiment to search for these “crystals”. Where are they?

Remark: In Kalitzinian metrics, “hyperphotons” or “photons” are defined in the usual way, i.e., $ds_{KAL}^2=0$, so

$\mbox{Hyperphotons}: ds_{KAL}^2=0\leftrightarrow dx_1^2+dx_2^2+dx_3^2=c_1^2dt_1^2+\ldots+c_{n-3}^2dt_{n-3}^2$

Remark(II): In multitemporal or crystalline relativities, we have to be careful with the notion of “point” at local level, since we have different notions of “velocity” and “proper velocity”. Somehow, in every point, we have a “fuzzy” fluctuation along certain directions of time (of course we can neglect them if we take the limit of zero/infinity lightspeed along some temporal direction/time vectors). Then, past, present and future are “fuzzy” notions in every spacetime when we consider a multitemporal approach! In the theory of relativity in Clifford spaces, something similar happens when you consider every possible “grade” and multivector components for a suitable cliffor/polyvector. The notion of “point” becomes meaningless since you attach to the point new “degrees of freedom”. In fact, relativity in Clifford spaces is “more crystalline” than multitemporal relativity since it includes not only vectors but bivectors, trivectors,… See this paper for a nice review: http://vixra.org/pdf/0908.0084v1.pdf

Remark (III):  Define the “big lightspeeds” in the following way

$\boxed{C_s^2=v_s^2=\dfrac{(dx_i)^2}{(dt_s)^2}}\;\;\forall s=1,2,\ldots,n-3$

or

$\boxed{C_s=v_s=\dfrac{dx_i}{dt_s}}\;\;\forall s=1,2,\ldots,n-3$

Then, we have

$C_s^2=\dfrac{c_1^2dt_1^2}{dt_s^2}+\ldots+\dfrac{c_{n-3}^2dt_{n-3}^2}{dt_s^2}$

$C_s^2=c_s^2\dfrac{c_1^2dt_1^2}{c_s^2dt_s^2}+\ldots+c_s^2+\ldots+c_s^2\dfrac{c_{n-3}^2dt_{n-3}^2}{c_s^2dt_s^2}$

$C_s^2=c_s^2\left(\dfrac{c_1^2dt_1^2}{c_s^2dt_s^2}+\ldots+1+\ldots+\dfrac{c_{n-3}^2dt_{n-3}^2}{c_s^2dt_s^2}\right)$

$\displaystyle{C_s^2=c_s^2\left(1+\sum_{k\neq s}^{n-3}\left(\dfrac{c_kdt_k}{c_sdt_s}\right)^2\right)}$

where we note that

$\boxed{\displaystyle{C_s^2=c_s^2\left(1+\sum_{k\neq s}^{n-3}\left(\dfrac{c_kdt_k}{c_sdt_s}\right)^2\right)}\geq c_s^2}$

or

$\boxed{\displaystyle{C_s=c_s\sqrt{\left(1+\sum_{k\neq s}^{n-3}\left(\dfrac{c_kdt_k}{c_sdt_s}\right)^2\right)}}\geq c_s}$

The bound is saturated whenever we have $c_s\longrightarrow\infty$ or $c_k=0$. Such conditions, or the hypothesis of unidimensional time, leave us with the speed of light barrier, but IT IS NO LONGER A BARRIER IN A MULTITEMPORAL SET-UP!

Remark (I): Just for fun…Sci-Fi writers are wrong when they use the “hyperspace” to skip out the lightspeed barrier. What allows to give up such a barrier is MULTITEMPORAL TIME or the hypertime. Of course, if they mean “hyperspacetime”, it would not be so wrong. It is trivial to observe that if you include extra SPACE-LIKE dimensions, and you keep Lorentz Invariance in higher-dimensions, you can NOT scape from the speed of light limit in a classical “way”. Of course, you could use wormholes, Alcubierre drives or quantum “engines”, but they belong to a different theoretical domain I am not going to explain here. Not now.

Remark (II): If we suppose that every speed of light is constant (homogeneity in extradimensional time) and if we suppose, in addition to it, that they are all equal to the same number, say the known $c$, i.e., if we write

$c_1=c_2=\ldots=c_{n-3}=c$

then we can easily obtain that

$\boxed{C_s=c_s\sqrt{1+(n-4)}=c_s\sqrt{n-3}}$

And then, we have

1) n=3 (0 timelike dimensions) implies that $C_s=c_s=0$

2) n=4 (1 timelike dimension) implies that $C_s=c_s=c$

3) n=5 (2 timelike dimensions) implies that $C_s=\sqrt{2}c_s\approx 1.4c$

3) n=6 (3 timelike dimensions) implies that $C_s=\sqrt{3}c_s\approx 1.7c$

4) n=7 (4 timelike dimensions) implies that $C_s=\sqrt{4}c_s=2c_s$

5) n=8 (5 timelike dimensions) implies that $C_s=\sqrt{5}c_s\approx 2.2c$

6) n=9 (6 timelike dimensions) implies that $C_s=\sqrt{6}c_s\approx 2.4c$

7) n=10 (7 timelike dimensions) implies that $C_s=\sqrt{7}c_s\approx 2.6c$

8) n=11 (8 timelike dimensions) implies that $C_s=\sqrt{8}c_s\approx 2.8c$

9) n=12 (9 timelike dimensions) implies that $C_s=\sqrt{9}c_s=3c$

10) $n=\infty$ ($\infty -3=\infty$  timelike dimensions) implies that $C_s=\infty$, and you can travel to virtually any velocity !!!!!!But of course, it seems this is not real, infinite timelike dimensions sound like a completely crazy stuff!!!!! I should go to the doctor…

Remark(III): The main lesson you should learn from this is that spacelike dimensions can not change the speed of light barrier. By the contrary, the true power of extra timelike dimensions is understood when you realize that “higher dimensional” excitations of “temporal dimensions” provide a way to surpass the speed of light. I have no idea of how to manage this, I am only explaining you what are the consequences of the previous stuff.

Remark (IV): Just for fun (or not). I am a big fan of Asimov’s books. Specially the Foundation series and the Robot stories. When I discovered these facts, long ago, I wondered myself if Isaac Asimov met Kalitzin/Arcidiacono (I think he could not meet Fantappie or Fantappie’s works about projective relativity but I am sure he knew a little bit about hyperspace and hypertime, despite the fact he, as many others at current time, confused the idea of hyperspace and hypertime, but sometimes he seemed to know more than he was explaining. I am not sure. I am not a Sci-fi writer…But I suppose he knew “something”…But not exactly these facts). I think to remember a quote from one of his books in which a character said something like “(…)One of the biggest mistakes of theoretical physicists is to confuse the hyperspace unlimited C with the bounded velocity c in usual relativity(…)”. I think these are not the exact words, but I remember I read something like that in some of his books. I can not remember what and I have no time to search for it right now, so I leave this activity to you…To find out where Asimov wrote something very close to it. Remember my words are not quite exact, I presume…I have not read a “normal” Sci-Fi book since years ago!

Arcidiacono worked out a simple example of multitemporal theory. He formulated the enhacen galilean group in the following way

$x'=x+V_1t+V_2t^2/2+\ldots+V_{n-3}t^{n-3}/(n-3)!$

$t'=t$

with $V_1$ the velocity, $V_2$ the acceleration, $V_3$ the jerk,…$V_{n-3}$ the (n-3)th order velocity. He linearized that nonlinear group using the transformations

$t_s=t^s/s!$ $\forall s=1,2,\ldots,n-3$

and it gives

$x'=x+V_1t_1+V_2t_2+\ldots+V_{n-3}t_{n-3}$

$t'_1=t_1$

$t'_2=t_2$$t'_{n-3}=t_{n-3}$

So we have a group matrix

$G=\begin{pmatrix}1 & V_1 & \cdots & V_{n-3}\\ 0 & 1 & \cdots & 0\\ \cdots & \cdots & \cdots & \cdots\\ \cdots & \cdots & \cdots & 1\end{pmatrix}$

The simplest case is usual galilean relavity

$x'=x+Vt$

$t'=t$

The second simpler example is two time enhaced galilean relativity:

$x'=x+Vt_1+V_2t_2$

$t'_1=t_1$ $t'_2=t'_2$

If we use that $V_1=V$ and $t_s=t^s/s!$, then we have

$\dfrac{dx}{dt_s}=\dfrac{s-1}{t^{s-1}}\dfrac{dx}{dt}$

and then

$V_s=(s-1)t^{1-s}V$

With 2 times, we have $V_2=V/t$, and moreover, the free point particle referred to $t_s$ satisfies (according to Arcidiacono)

$\dfrac{d^2x}{dt_s^2}=0\leftrightarrow \dfrac{d^2x}{dt^2}-\left(\dfrac{s-1}{t}\right)\dfrac{dx}{dt}=0$

Let us work out this case with more details

$X'=X+Vt+At^2/2$

$T=t$

where we have 3 spatial coordinates (x,y,z) and two times (t,t’). Performing the above transformations

$X'=x+Vt+At'$

$T=t$ $T'=t'$

with velocities

$V=\dfrac{dx}{dt}$ and $V'=\dfrac{dx}{dt'}$, and with $V'=V/t$. If $V=At$, then $V'=A$, so a second order velocity becomes the constant acceleration in that frame. Furthermore

$\dfrac{d^2x}{dt'^2}=0$

implies that

$\dfrac{dV}{dt}=\dfrac{V}{t}$ and $x=At^2/2$

That is, invariant mechanics under uniformly accelerated motion with “multiple” velocities is possible! In fact, in this framework, uniformly accelerated motion seems to be “purely inertial”, or equivalently, it seems to be “fully machian”!!!!

If uniformly accelerated gravitational field is applied to the point particle, then, in this framework, it seems to suggest that it “changes” the time scale a quantity

$t'=t^2/2$

and it becomes a uniform motion! If a body moves unofrmorly, changing the scale of time, in multitemporal relativity, ib becomes uniformaly accelerated! I don’t understand this claim well enough, but it seems totally crazy or completely …Suggestive of a purely machian relativity? Wilczek called it “total relativity” long ago…

A conformal relativity with two time dimensions and two time dimensions was also studied by Arcidiacono (quite naively, I believe). He studied also a metric

$ds^2=dx^2+dy^2+dz^2-c^2dt^2-c'^2dt'^2$

with a conformal time

$t'=\dfrac{c^2t^2-x^2}{2c^2}$

Note that $c\longrightarrow \infty$ implies that $t'=t^2/2$. It implies some kind of hyperbolic motion

$V=\dfrac{At}{\sqrt{1+\dfrac{A^2t^2}{c^2}}}$

and

$x=\dfrac{c^2}{A^2}\left[-1+\sqrt{1+\dfrac{A^2t^2}{c^2}}\right]$

Remark: $Ax^2+2c^2x-Ac^2t^2=0\leftrightarrow x=\dfrac{A}{2c^2}\left(c^2t^2-x^2\right)$. Introductin a second time $x=At'$, then $V'=A$, where

$V'=\dfrac{V}{t-\dfrac{Vx}{c^2}}$

and again $V'=A$ produces the “classical relativity”.

Remark(II): Projective special relativity should produce some kind of “projective general relativity” (Arcidiacono claimed). This is quite a statement, since the diffeomorphism group in general relativity contains “general coordinate transformations”. I am not sure what he meant with that. Anyway, a projective version of “general relavity” is provided by twistor theory or similar theories, due to the use of complex projective spaces and generalizations of them. Conformal special relativity should imply some class of conformal general relativity. However, physical laws are not (apparently) invariant under conformal transformations in general. What about de Sitter/anti de Sitter spaces? I have to learn more about that and tell you about it in the future. Classical electromagnetism and even pure Yang-Mills theories at classical level can be made invariant under conformal transformations only with special care. Quantum Mechanics seems  to break that symmetry due to the presence of mass terms that spoil the gauge invariance of the theory, not only the conformal symmetry. Only the Higgs mechanism and “topological” terms allow us to introduce “mass terms” in a gauge invariant way! Any way, remember that Classical Mechanics is based on symplectic geometry, very similar to projective geometry in some circumstances, and Classical Field Theories also contain fiber fundles and some special classes of field theories, like Conformal Field Theories or even String Theory, have some elements of projective geometry in their own manner. Moreover, conformal symmetries are also an alternative approach to new physics. For instante, Georgi created the notion of a “hidden conformal sector” BSM theory, something that he called “unparticles”. People generalized the concept and you can read about “ungravity” as well. Unparticles, ungravity, unforces…Really weird stuff!!! Did you think multiple temporal dimensions were the only uncommon “ugly ducks” in the city? No, they weren’t…Crazy ideas are everywhere in theoretical physics. The real point is to find them applications and/or to find them in real experiments! It happened with this Higgs-like particle about 127GeV/c². And I think Higgs et alii will deserve a Nobel Prize this year due to it.

Remark (III): Final relativity, in the sense of Fantappie’s ideas, has to own a different type of Cosmology… In fact it has. It has a dS relativity Cosmology! The Stantard Cosmological Model fits the vacuum energy (more precisely we “fit” $\Omega_\Lambda$). It is important to understand what $\Lambda$ is. The Standard Cosmological Model does not explain it at all. We should explore the kinematical and cosmological models induced by the de Sitter group, and its associated QFT. However, QFT on dS spaces are not fully developed. So, that is an important research line for the future.

Arcidiacono generalizes electromagnetism to multitemporal dimensions (naively he “wrongly” thought he had unified electromagnetism and hydrodynamics) with the followin equations

$\mbox{Rot}H_{AB}=J_{ABC}$

$\mbox{Div}H_{AB}=I_A$

where $A,B=0,1,\ldots, n$. The tensor $H_{AB }$ have $n(n +1)/2$ components. The integrability conditions are

$\mbox{Rot}J_{ABC}=0$

and

$\mbox{Div}I_A=0$

We can build some potentials $U_A$, and $V_{ABC}$, so

$\mbox{Div}U_A=0$

$\mbox{Rot}V_{ABC}=0$

with $H_{AB}=\mbox{Div}V_{ABC}+\mbox{Rot}U_A$

we have

$\square^2V_{ABC}=J_{ABC}$ and $\square^2 U_A=I_A$

A generalized electromagnetic force is introduce

$2f_A=J_{ABC}H_{BC}-2I_BH_{AB}$

If $f_A=\mbox{Div}T_{AB}$, then the energy-momentum tensor will be

$T_{AB}=H_{AS}H_{SB}+\dfrac{1}{4}H_{RS}H_{RS}\delta_{AB}$

For position vectors $\overline{x}_A$, we have $(n-3)$ projectie velocities \$late \overline{u}_A^s, such as

$\overline{u}^s_A=\dfrac{d\overline{x}_A}{d\tau_s}$

$\boxed{\overline{u}_A^{(r)}\overline{u}_A^{(s)}=-c_s^2\delta_{rs}}$

where $\overline{x}_A\overline{x}_A=r^2$ and $\overline{x}_A\overline{u}_A^s=0$. From $H_{AB}$ we get

(1) $c_A$ hydrodynamics vector plus (n-3) magnetic vectors $h_A^s$ such as

$c_A=H_{AB}x_B$

$h_A^s=H_{AB}u^s_B=H_{AB}u^s_B$

and where

$c_Ax_A=0$ and $h_A^su^s_A=0$.

(2) Fluid indices for

$f^s=H_{AB}x_Au_B^s$

$f^{rs}=H_{AB}u_A^ru_B^s$

with

$(n-3)+\begin{pmatrix}n-3\\ 2\end{pmatrix}=\begin{pmatrix}n-2\\ 2\end{pmatrix}=\dfrac{(n-2)(n-3)}{2}$ total components. Note that if you introduce n=4 you get only 1 single independent component.

(3) The dual tensor $\star H_{ABC\ldots D}$ to $H_{AB}$ has (n-1) undices, so we can make

$K_{AB}=\star H_{ABC\ldots D}u_A^1u_B^2\ldots u_C^{n-3}$ and then $K_{AB}u_B^s=0$. The generalized electric field reads

$e=K_{AB}x_B$

so $e_Ax_A=e_Au_A^s=0$

Note that in this last equation, projective relativity means a total equivalence in a transformation changing position and multitemporal velocities, i.e., invariance under $x_A\leftrightarrow u_A^s$ is present in the last equation for electric fields in a multitemporal setting.

1) Multitemporal theories of relativity do exist. In fact, Dirac himself and De Donder studied this types of theories. However, they did not publish too much papers about this crazy subject.

2) Fantappie’s final relativity is an old idea that today can be seen as de Sitter Relativity. The contraction of the de Sitter group provides the Lorentz groupo. Final relativity/de Sitter relativity is based on “projective geometry” somehow.

3) Kalitzin’s and Arcidiacono’s ideas, likely quite naive and likely wrong, does not mean that multitemporal dimensions don’t exist. The only problem is to explain why the world is 3+1 if they exist or, equivalently, just as the space-like dimensions, the perception of multiple temporal dimensions is an experimental issue.

4) The main issues for extra timelike dimensions are: closed time-like curves, causality and vacuum instabilities (“imposible” processes) when Quantum Mechanics is taken into account in multi-time setting.

5) Beyond multi-time theories, there are interesting extensions of special relativity, e.g., C-space relativity.

6) Multiple temporal dimensions make the notion of point and event a little “fuzzy”.

7) Multiple time-like dimensions are what make possible to overpass the invariant speed of light. I am not going to prove it here, in the case of $c_k=c\forall k$ the maximum invariant velocity is equal to $\sqrt{n-3}c$. When the speeds of light are “different” the invariant velocity is a harder formula, but it does exist. From this viewpoint, it is hypertime dimensions and not hyperspace dimensions what make possible the faster than light travel (Giving up CTC, causality issues and vacuum instabilities triggered by quantum theories).

8) Hyperphotons are the equivalent concept of photons in multitemporal relativities and they are not tachyons, but they have a different invariant speed.

9) Philosophers have discussed the role of multitemporal dimensions. For instance, I read about Bennett 3d time, with 3 components he called time, hyparxis and eternity long ago, see here http://en.wikipedia.org/wiki/John_G._Bennett.

10) Isaac Asimov stories, beyond the imagination and intuition Asimov had, match the theory of relavity with extra time-like and space-like dimensions. I don’t know if he met Kalitzin, Dirac or some other physicist working on this field, but it is quite remarkable from the purely layman approach!

11) Theories with extra temporal dimensions have been studied by both mathematicians and physicists. At current time, maybe I can point out that F-theory has two timelike dimensions, Itzhak Bars has papers about two-time physics, semiriemannian (multitemporal) metrics are being studied by the balkan and russian schools and likely many others.

12) The so-called problem of time is even more radical when you deal with multi-time theories because the relation of multitemporal coordinates with the physical time is obscure. We don’t understand time.

13) We can formulate theories in a multi-time setting, but it requires a harder framework than in normal relativity: velocity becomes “a matrix”, there are different notions of accelerations, energy becomes a vector, “mass” is a “tensor”, multi-time electrodynamics becomes more difficult and many other issues arise with a multi-time setting. You have to study: jet theory, Finsler spaces, nonlinear connections, and some more sophisticated machinery in order to understand it.

14) Are multi-time theories important? Maybe…The answer is that we don’t know for sure, despite the fact that they are “controversial” and “problematic”. However, if you think multi-time theories are “dark”, maybe you should thing about that “dark stuff” forming the 95% of the Universe. However, Irina Aref’eva and other authors have studied the physical consequences of multi-time theores. Aref’eva herself, in collaboration with other russian physicists, proved that an additional timelike dimension can solve the cosmological constant problem (giving up any issue that an additional time dimension produces).

15) The idea of “time crystals” is boring in 1d time. It becomes more interesting when you thing about multi-time crystals as some of the ingredients of certain “crystalline relativity”. In fact, a similar idea has been coined by P. Jizba et alii, and it is known as “World Crystal”.

16) Final questions:

i) Can multi-time relativity be used by Nature? The answer can only be answered from an experimental viewpoint!

ii) Do we live in an anisotropic spacetime (quasi)crystal? I have no idea! But particles theirselves could be seen as (quantum) excitations of the spacetime crystal. In fact, I am wondering if the strange spectrum of the Standard Model could be some kind of 3d+1 time quasicrystal. If it is so, it could be that in certain higher dimensions, the spectrum of the SM could be more “simple”. Of course, it is the idea of extra dimensions, but I have not read any paper or article studying the SM particle spectrum from a quasicrystal viewpoint. It could be an interesting project to make some investigations about this idea.

iii) How many lightspeeds are there in the Universe? We can put by hand that every “lightspeed” species is equal to the common speed of light, but is it right? Could exist new lightspeed species out there? Note that if we considered those “higher lightspeeds” very large numbers, they could be unnoticed by us if the “electromagnetism” in the extra temporal dimensions were far different than the known electromagnetism. That is, it could be that $c=c_1< or that some of them were very small constants…In both cases, normal relativity could be some kind of “group” reduction.

iv) Could the time be secretly infinite-dimensional? Experiments show that the only invariant speed is c, but could it be an illusion?

v) Can we avoid the main problems of multi-time theories? I mean causality, Closed Timelike Curves (CTC), and vacuum instabilities as the most important of all of them.

vi) Is the problem of time related to the the multitemporality of the world?

Imagine that an idealised bug of negligible dimensions is hiding at the end of a hole of length L. A rivet has a shaft length of $a.

Clearly the bug is “safe” when the rivet head is flush to the (very resiliente) surface. The problem arises as follows. Consider what happens when the rivet slams into the surface at a speed of $v=\beta c$, where c is the speed of light and $0<\beta<1$. One of the essences of the special theory of relativity is that objects moving relative to our frame of reference are shortened in the direction of motion by a factor $\gamma^{-1}=\sqrt{1-\beta^2}$, where $\gamma$ is generally called the Lorentz dilation factor, as readers of this blog already know. However, from the point of view (frame of reference) of the bug, the rivet shaft is even shorter and therefore the bug should continue to be safe, and thus fast the rivet is moving.

Apparently, we have:

$a_{app}=\dfrac{a}{\gamma}=a\sqrt{1-\beta^2}$

Remark: this idea assumes that both objects are ideally rigid! We will return to this “fact” later.

From the frame of reference of the rivet, the rivet is stationary and unchanged, but the hole is moving fast and is shortened by the Lorentz contraction to

$L_{app}=\dfrac{L}{\gamma}=L\sqrt{1-\beta^2}$

If the approach speed is fast enough, so that $L_{app}, then the end of the hole slams into the tip of the rivet before the surface
can reach the head of the rivet. The bug is squashed! This is the “paradox”: is the bug squashed or not?

There are many good sources for this paradox (a relative of the pole-barn paradox), such as:

2) A nice animation can be found here  http://math.ucr.edu/~jdp/Relativity/Bug_Rivet.html

In this blog post we are going to solve this “paradox” in the framework of special relativity.

SOLUTION

One of the consequences of special relativity is that two events that are simultaneous in one frame of reference are no longer simultaneous in other frames of reference. Perfectly rigid objects are impossible.

In the frame of reference of the bug, the entire rivet cannot come to a complete stop all at the same instant. Information
cannot travel faster than the speed of light. It takes time for knowledge that the rivet head has slammed into the surface to
travel down the shaft of the rivet. Until each part of the shaft receives the information that the rivet head has stopped, that part keeps going at speed $v=\beta c$. The information proceeds down the shaft at speed c while the tip continues to move at speed $v=\beta c$.

The tip cannot stop until a time

$T_1=\dfrac{\dfrac{a}{\gamma}}{c-\beta c}=\dfrac{a}{\gamma c (1-\beta)}$

after the head has stopped. During that time the tip travels a distance $D_1=vT_1$. The bug will be squashed if

$vT_1>L-\dfrac{a}{\gamma}$

This implies that

$\dfrac{\beta c a}{\gamma c (1-\beta)}>L-\dfrac{a}{\gamma}\leftrightarrow \dfrac{a}{\gamma}\left(\dfrac{\beta}{1-\beta}+1\right) >L\leftrightarrow \dfrac{a}{\gamma}\left(\dfrac{\beta+1-\beta}{1-\beta}\right) >L$

From $\gamma^{-1}=\sqrt{1-\beta^2}$ we can calculate that

$\dfrac{1}{\gamma (1-\beta)}=\dfrac{\sqrt{1-\beta^2}}{1-\beta}=\dfrac{\sqrt{(1+\beta)(1-\beta)}}{1-\beta}=\sqrt{\dfrac{1+\beta}{1-\beta}}$

The bug will be squashed if the following condition holds

$a\sqrt{\dfrac{1+\beta}{1-\beta}}>L\leftrightarrow \dfrac{a}{L}>\sqrt{\dfrac{1-\beta}{1+\beta}}\leftrightarrow \left(\dfrac{a}{L}\right)^2> \dfrac{1-\beta}{1+\beta}$

or equivalently, after some algebraic manipulations, the bug will be squashed if:

$\beta>\dfrac{1-\left(\dfrac{a}{L}\right)^2}{1+\left(\dfrac{a}{L}\right)^2}$

Conclusion (in bug’s reference frame): the bug will be definitively squashed when $v_{min}=\beta_{min}c$ such as

$\boxed{\beta_{min}=\dfrac{1-\left(\dfrac{a}{L}\right)^2}{1+\left(\dfrac{a}{L}\right)^2}}$

Check: It can be verified that the limits $\displaystyle{\lim_{a\rightarrow 0^+}\beta_{min}=1^{-}}$ and $\displaystyle{\lim_{a\rightarrow L^-}\beta_{min}=0^{+}}$ are valid and physically meaningful.

Note that the impact of the rivet head always happens before the bug is squashed.

In the frame of reference of the rivet, the bug is definitively squashed whenever $\dfrac{L}{\gamma}.

Then,

$L\sqrt{1-\beta^2}

or equivalently

$\beta>\sqrt{1-\left(\dfrac{a}{L}\right)^2}$

or

$\beta>\beta_{min2}$ where $\boxed{\beta_{min2}=\sqrt{1-\left(\dfrac{a}{L}\right)^2}}$

The bug is squashed before the impact of the surface on the rivet head. This last equation (and thus $\beta_{min2}$) is a velocity higher than $\beta_{min}$.

Conclusion (in rivet’s reference frame): The entire surface cannot come to an abrupt stop at the same instant. It takes time for the information about the impact of the rivet tip on the end of the hole to reach the surface that is rushing towards the rivet head. Let us now examine the case where the speed is not high enough for the Lorentz-contracted hole to be shorter than the rivet shaft in the frame of reference of the rivet. Now the observers agree that the impact of the rivet head happens first. When the surface slams into contact with the head of the rivet, it takes time for information about that impact to travel down to the end of the hole. During this time the hole continues to move towards the tip of the rivet.

The time it takes for the propagating information to reach the tip of the stationary rivet is

$T_2=\dfrac{a}{c}$

during which time the bug moves a distance $D_2=vT_2=\dfrac{\beta c a}{c}=\beta a$

In the rivet’s reference frame, therefore, The bug is squashed if the following condition holds

$vT_2>\dfrac{L}{\gamma}-a\leftrightarrow \beta a>\dfrac{L}{\gamma}-a\leftrightarrow (1+\beta)a>\dfrac{L}{\gamma}\leftrightarrow \dfrac{a}{L}>\dfrac{1}{1+\beta}\sqrt{1-\beta^2}$

and then

$\dfrac{\sqrt{(1+\beta)(1-\beta)}}{1+\beta}<\dfrac{a}{L}\leftrightarrow \sqrt{\dfrac{1-\beta}{1+\beta}}<\dfrac{a}{L}$

and from this equation, we get same minimum speed that guarantees the squashing of the bug as was the case in the frame of reference of the bug! That is:

$\boxed{\beta_{min}=\dfrac{1-\left(\dfrac{a}{L}\right)^2}{1+\left(\dfrac{a}{L}\right)^2}}$

Note that observers travelling with each of the two frames of reference (bug and rivet) agree that the bug is squashed IF $\beta>\beta_{min}$, and that resolves the “paradox”. They also agree that the impact of rivet head on surface happens before the bug is squashed, provided that the following condition is satisfied:

$\beta_{min}<\beta<\beta_{min2}$

i.e., they agree if the impact of rivet head on surface happens before the bug is squashed

$\boxed{\dfrac{1-\left(\dfrac{a}{L}\right)^2}{1+\left(\dfrac{a}{L}\right)^2}<\beta<\sqrt{1-\left(\dfrac{a}{L}\right)^2}}$

Otherwise, they disagree on which event happens first.  For instance, if

$\beta>\beta_{min2}=\sqrt{1-\left(\dfrac{a}{L}\right)^2}$

For speeds this high, the observer in the bug’s frame of reference still deduces that the rivet-head impact happens first, but the other observer deduces that the bug is squashed first. This is consistent with the relativity of simultaneity! At the critical speed, when $\beta=\beta_c=\beta_{min2}$ the two events are simultaneous in the frame of the rivet, (the river fits perfectly in the shortened hole), but they are not simultaneous in the other frame of reference.

See you in the next blog post!

The Batmobile “fake paradox” helps us to understand Special Relativity a little bit. This problem consists in the next experiment:

There are two observers. Alfred, the external observer, and Batman moving with his Batmobile.

Now, we will suppose that the Batmobile is moving at a very fast constant speed with respect to the garage. Let us suppose that $v=0.866c=\dfrac{\sqrt{3}}{2}c$. Then, we have the following situation from the external observer:

However, with respect to the Batmobile reference frame, we have:

The question is. Who is right? Alfred or Batman? The surprinsig answer from Special Relativity is that Both are correct. Alfred and Batman are right! Let’s see why it is true. For Alfred, there is a time during which the Batmobile is completely inside the garage with both doors closed:

By the other hand, for Batman, the front and rear doors are not closed simultaneously! So there is never a time during which the Batmobile is completely inside the garage with both doors closed.

So, there is no paradox at all, if you are aware about the notion of simultaneity and its relativity!

# LOG#056. Gravitational alpha(s).

The topic today is to review a beautiful paper and to discuss its relevance for theoretical physics. The paper is: Comment on the cosmological constant and a gravitational alpha by R.J.Adler. You can read it here: http://arxiv.org/abs/1110.3358

One of the most intriguing and mysterious numbers in Physics is the electromagnetic fine structure constant $\alpha_{EM}$. Its value is given by

$\alpha_{EM}=7.30\cdot 10^{-3}$

or equivalenty

$\alpha_{EM}^{-1}=\dfrac{1}{\alpha_{EM}}=137$

Of course, I am assuming that the coupling constant is measured at ordinary energies, since we know that the coupling constants are not really constant but they vary slowly with energy. However, I am not going to talk about the renormalization (semi)group in this post.

Why is the fine structure constant important? Well, we can undertand it if we insert the values of the constants that made the electromagnetic alpha constant:

$\alpha_{EM}=\dfrac{e^2}{\hbar c}$

with $e$ being the electron elemental charge, $\hbar$ the Planck’s constant divided by two pi, c is the speed of light and where we are using units with $K_C=\dfrac{1}{4\pi \varepsilon_0}=1$. Here $K_C$ is the Coulomb constant, generally with a value $9\cdot 10^9Nm^2/C^2$, but we rescale units in order it has a value equal to the unit. We will discuss more about frequently used system of units soon.

As the electromagnetic alpha constant depends on the electric charge, the Coulomb’s electromagnetic constant ( rescaled to one in some “clever” units), the Planck’s constant ( rationalized by $2\pi$ since $\hbar=h/2\pi$) and the speed of light, it codes some deep information of the Universe inside of it. The electromagnetic alpha $\alpha_{EM}$ is quantum and relativistic itself, and it also is related to elemental charges. Why alpha has the value it has is a complete mystery. Many people has tried to elucidate why it has the value it has today, but there is no reason of why it should have the value it has. Of course, it happens as well with some other constants but this one is particularly important since it is involved in some important numbers in atomic physics and the most elemental atom, the hydrogen atom.

In atomic physics, there are two common and “natural” scales of length. The first scale of length is given by the Compton’s wavelength of electrons. Usint the de Broglie equation, we get that the Compton’s wavelength is the wavelength of a photon whose energy is the same as the rest mass of the particle, or mathematically speaking:

$\boxed{\lambda=\dfrac{h}{p}=\dfrac{h}{mc}}$

Usually, physicists employ the “reduced” or “rationalized” Compton’s wavelength. Plugging the electron mass, we get the electron reduced Compton’s wavelength:

$\boxed{\lambda_C=\dfrac{\lambda}{2\pi}=\dfrac{\hbar}{m_ec}=\dfrac{\hbar}{m_ec}=3.86\cdot 10^{-13}m}$

The second natural scale of length in atomic physics is the so-called Böhr radius. It is given by the formula:

$\boxed{a_B=\dfrac{\hbar^2}{m_e e^2}=5.29\cdot 10^{-11}m}$

Therefore, there is a natural mass ratio between those two length scales, and it shows that it is precisely the electromagnetic fine structure constant alpha $\alpha_{EM}$:

$\boxed{R_\alpha=\dfrac{\mbox{Reduced Compton's wavelength}}{\mbox{B\"{o}hr radius}}=\dfrac{\lambda_C}{a_B}=\dfrac{\left(\hbar/m_e c\right)}{\left(\hbar^2/m_ee^2\right)}=\dfrac{e^2}{\hbar c}=\alpha_{EM}=7.30\cdot 10^{-3}}$

Furthermore, we can show that the electromagnetic alpha also is related to the mass ration between the electron energy in the fundamental orbit of the hydrogen atom and the electron rest energy. These two scales of energy are given by:

1) Rydberg’s energy ( electron ground minimal energy in the fundamental orbit/orbital for the hydrogen atom):

$\boxed{E_H=\dfrac{m_ee^4}{2\hbar^2}=13.6eV}$

2) Electron rest energy:

$\boxed{E_0=m_ec^2}$

Then, the ratio of those two “natural” energies in atomic physics reads:

$\boxed{R'_E=\dfrac{\mbox{Rydberg's energy}}{\mbox{Electron rest energy}}=\dfrac{m_ee^4/2\hbar^2}{m_ec^2}=\dfrac{1}{2}\left(\dfrac{e^2}{\hbar c}\right)^2=\dfrac{\alpha_{EM}^2}{2}=2.66\cdot 10^{-5}}$

or equivalently

$\boxed{\dfrac{1}{R'_E}=37600=3.76\cdot 10^4}$

R.J.Adler’s paper remarks that there is a cosmological/microscopic analogue of the above two ratios, and they involve the infamous Einstein’s cosmological constant. In Cosmology, we have two natural (ultimate?) length scales:

1st. The (ultra)microscopic and ultrahigh energy (“ultraviolet” UV regulator) relevant Planck’s length $L_P$, or equivalently the squared value $L_P^2$. Its value is given by:

$\boxed{L_P^2=\dfrac{G\hbar}{c^3}\leftrightarrow L_P=\sqrt{\dfrac{G\hbar}{c^3}}=1.62\cdot 10^{35}m}$

This natural length can NOT be related to any “classical” theory of gravity since it involves and uses the Planck’s constant $\hbar$.

2nd. The (ultra)macroscopic and ultra-low-energy (“infrared” IR regulator) relevant cosmological constant/deSitter radius. They are usualy represented/denoted by $\Lambda$ and $R_{dS}$ respectively, and they are related to each other in a simple way. The dimensions of the cosmological constant are given by

$\boxed{\left[\Lambda \right]=\left[ L^{-2}\right]=(\mbox{Length})^{-2}}$

The de Sitter radius and the cosmological constant are related through a simple equation:

$\boxed{R_{dS}=\sqrt{\dfrac{3}{\Lambda}}\leftrightarrow R^2_{dS}=\dfrac{3}{\Lambda}\leftrightarrow \Lambda =\dfrac{3}{R^2_{dS}}}$

The de Sitter radius is obtained from cosmological measurements thanks to the so called Hubble’s parameter ( or Hubble’s “constant”, although we do know that Hubble’s “constant” is not such a “constant”, but sometimes it is heard as a language abuse) H. From cosmological data we obtain ( we use the paper’s value without loss of generality):

$H=\dfrac{73km/s}{Mpc}$

This measured value allows us to derive the Hubble’s length paremeter

$L_H=\dfrac{c}{H}=1.27\cdot 10^{26}m$

Moreover, the data also imply some density energy associated to the cosmological “constant”, and it is generally called Dark Energy. This density energy from data is written as:

$\Omega_\Lambda =\Omega^{data}_{\Lambda}$

and from this, it can be also proved that

$R_{dS}=\dfrac{L_H}{\sqrt{\Omega_\Lambda}}=1.46\cdot 10^{26}m$

where we have introduced the experimentally deduced value $\Omega_\Lambda\approx 0.76$ from the cosmological parameter global fits. In fact, the cosmological constant helps us to define the beautiful and elegant formula that we can call the gravitational alpha/gravitational cosmological fine structure constant $\alpha_G$:

$\boxed{\alpha_G\equiv \dfrac{\mbox{Planck's length}}{\mbox{normalized de Sitter radius}}=\dfrac{L_P}{\dfrac{R_{dS}}{\sqrt{3}}}=\dfrac{\sqrt{\dfrac{G\hbar}{c^3}}}{\sqrt{\dfrac{1}{\Lambda}}}=\sqrt{\dfrac{G\hbar\Lambda}{c^3}}}$

or equivalently, defining the cosmological length associated to the cosmological constant as

$L^2_\Lambda=\dfrac{1}{\Lambda}=\dfrac{R^2_{dS}}{3}\leftrightarrow L_\Lambda=\sqrt{\dfrac{1}{\Lambda}}=\dfrac{R_{dS}}{\sqrt{3}}$

$\boxed{\alpha_G\equiv \dfrac{\mbox{Planck's length}}{\mbox{Cosmological length}}=\dfrac{L_P}{L_\Lambda}=\dfrac{\sqrt{\dfrac{G\hbar}{c^3}}}{\sqrt{\dfrac{1}{\Lambda}}}=\sqrt{\dfrac{G\hbar\Lambda}{c^3}}=L_P\sqrt{\Lambda}=L_P\dfrac{R_{dS}}{\sqrt{3}}}$

If we introduce the numbers of the constants, we easily obtaint the gravitational cosmological alpha value and its inverse:

$\boxed{\alpha_G=1.91\cdot 10^{-61}\leftrightarrow \alpha_G^{-1}=\dfrac{1}{\alpha_G}=5.24\cdot 10^{60}}$

They are really small and large numbers! Following the the atomic analogy, we can also create a ratio between two cosmologically relevant density energies:

1st. The Planck’s density energy.

Planck’s energy is defined as

$\boxed{E_P=\dfrac{\hbar c}{L_P}=\sqrt{\dfrac{\hbar c^5}{G}}=1.22\cdot 10^{19}GeV=1.22\cdot 10^{16}TeV}$

The Planck energy density $\rho_P$ is defined as the energy density of Planck’s energy inside a Planck’s cube or side $L_P$, i.e., it is the energy density of Planck’s energy concentrated inside a cube with volume $V=L_P^3$. Mathematically speaking, it is

$\boxed{\rho_P=\dfrac{E_P}{L_P^3}=\dfrac{c^7}{\hbar G^2}=2.89\cdot 10^{123}\dfrac{GeV}{m^3}}$

It is an huge density energy!

Remark: Energy density is equivalent to pressure in special relativity hydrodynamics. That is,

$\mathcal{P}_P=\rho_P=\tilde{\rho}_P c^2=4.63\cdot 10^{113}Pa$

wiht Pa denoting pascals ($1Pa=1N/m^2$) and where $\tilde{\rho}_P$ represents here matter (not energy) density ( with units in $kg/m^3$). Of course, turning matter density into energy density requires a multiplication by $c^2$. This equivalence between vacuum pressure and energy density is one of the reasons because some astrophysicists, cosmologists and theoretical physicists call “vacuum pressure” to the “dark energy/cosmological constant” term in the study of the cosmic components derived from the total energy density $\Omega$.

2nd. The cosmological constant density energy.

Using the Einstein’s field equations, it can be shown that the cosmological constant gives a contribution to the stress-energy-momentum tensor. The component $T^{0}_{\;\; 0}$ is related to the dark energy ( a.k.a. the cosmological constant) and allow us to define the energy density

$\boxed{\rho_\Lambda =T^{0}_{\;\; 0}=\dfrac{\Lambda c^4}{8\pi G}}$

Using the previous equations for G as a function of Planck’s length, the Planck’s constant and the speed of light, and the definitions of Planck’s energy and de Sitter radius, we can rewrite the above energy density as follows:

$\boxed{\rho_\Lambda=\dfrac{3}{8\pi}\left(\dfrac{E_P}{L_PR^2_{dS}}\right)=4.21 \dfrac{GeV}{m^3}}$

Thus, we can evaluate the ration between these two energy densities! It provides

$\boxed{R_\rho =\dfrac{\mbox{Planck's energy density}}{\mbox{CC energy density}}=\dfrac{\rho_P}{\rho_\Lambda}=\left( \dfrac{3}{8\pi}\right)\left(\dfrac{L_P}{R_{dS}}\right)^2=\left(\dfrac{1}{8\pi}\right)\alpha_G^2=1.45\cdot 10^{-123}}$

and the inverse ratio will be

$\boxed{\dfrac{1}{R_\rho}=6.90\cdot 10^{122}}$

So, we have obtained two additional really tiny and huge values for $R_\rho$ and its inverse, respectively. Note that the power appearing in the ratios of cosmological lengths and cosmological energy densities match the same scaling property that the atomic case with the electromagnetic alpha! In the electromagnetic case, we obtained $R\sim \alpha_{EM}$ and $R_E\sim \alpha_{EM}^2$. The gravitational/cosmological analogue ratios follow the same rule $R\sim \alpha_G$ and $R_\rho\sim \alpha_G^2$ but the surprise comes from the values of the gravitational alpha values and ratios. Some comments are straightforward:

1) Understanding atomic physics involved the discovery of Planck’s constant and the quantities associated to it at fundamental quantum level ( Böhr radius, the Rydberg’s constant,…). Understanding the Cosmological Constant value and the mismatch or stunning ratios between the equivalent relevant quantities, likely, require that $\Lambda$ can be viewed as a new “fundamental constant” or/and it can play a dynamical role somehow ( e.g., varying in some unknown way with energy or local position).

2) Currently, the cosmological parameters and fits suggest that $\Lambda$ is “constant”, but we can not be totally sure it has not varied slowly with time. And there is a related idea called quintessence, in which the cosmological “constant” is related to some dynamical field and/or to inflation. However, present data say that the cosmological constant IS truly constant. How can it be so? We are not sure, since our physical theories can hardly explain the cosmological constant, its value, and why it is current density energy is radically different from the vacuum energy estimates coming from Quantum Field Theories.

3) The mysterious value

$\boxed{\alpha_G=\sqrt{\dfrac{G\hbar\Lambda}{c^3}}=1.91\cdot 10^{-61}}$

is an equivalent way to express the biggest issue in theoretical physics. A naturalness problem called the cosmological constant problem.

In the literature, there have been alternative definitions of “gravitational fine structure constants”, unrelated with the above gravitational (cosmological) fine structure constant or gravitational alpha. Let me write some of these alternative gravitational alphas:

1) Gravitational alpha prime. It is defined as the ratio between the electron rest mass and the Planck’s mass squared:

$\boxed{\alpha'_G=\dfrac{Gm_e^2}{\hbar c}=\left(\dfrac{m_e}{m_P}\right)^2=1.75\cdot 10^{-45}}$

$\boxed{\alpha_G^{'-1}=\dfrac{1}{\alpha_G^{'}}=5.71\cdot 10^{44}}$

Note that $m_e=0.511MeV$. Since $m_{proton}=1836m_e$, we can also use the proton rest mass instead of the electron mass to get a new gravitational alpha.

2) Gravitational alpha double prime. It is defined as the ratio between the proton rest mass and the Planck’s mass squared:

$\boxed{\alpha''_G=\dfrac{Gm_{prot}^2}{\hbar c}=\left(\dfrac{m_{prot}}{m_P}\right)^2=5.90\cdot 10^{-39}}$

and the inverse value

$\boxed{\alpha_G^{''-1}=\dfrac{1}{\alpha_G^{''}}=1.69\cdot 10^{38}}$

Finally, we could guess an intermediate gravitational alpha, mixing the electron and proton mass.

3) Gravitational alpha triple prime. It is defined as the ration between the product of the electron and proton rest masses with the Planck’s mass squared:

$\boxed{\alpha'''_G=\dfrac{Gm_{prot}m_e}{\hbar c}=\dfrac{m_{prot}m_e}{m_P^2}=3.22\cdot 10^{-42}}$

and the inverse value

$\boxed{\alpha_G^{'''-1}=\dfrac{1}{\alpha^{'''}_G}=3.11\cdot 10^{41}}$

We can compare the 4 gravitational alphas and their inverse values, and additionally compare them with $\alpha_{EM}$. We get

$\alpha_G <\alpha_G^{'} <\alpha_G^{'''} < \alpha_G^{''}<\alpha_{EM}$

$\alpha_{EM}^{-1}<\alpha^{''-1}_G <\alpha^{'''-1}_G <\alpha^{'-1}_G < \alpha^{-1}_G$

These inequations mean that the electromagnetic fine structure constant $\alpha_{EM}$ is (at ordinary energies) 42 orders of magnitude bigger than $\alpha_G^{'}$, 39 orders of magnitude bigger than $\alpha_G^{'''}$, 36 orders of magnitude bigger than $\alpha_G^{''}$ and, of course, 58 orders of magnitude bigger than $\alpha_G$. Indeed, we could extend this analysis to include the “fine structure constant” of Quantum Chromodynamics (QCD) as well. It would be given by:

$\boxed{\alpha_s=\dfrac{g_s^2}{\hbar c}=1}$

since generally we define $g_s=1$. We note that $\alpha_s >\alpha_{EM}$ by 3 orders of magnitude. However, as strong nuclear forces are short range interactions, they only matter in the atomic nuclei, where confinement, and color forces dominate on every other fundamental interaction. Interestingly, at high energies, QCD coupling constant has a property called asymptotic freedom. But it is another story not to be discussed here! If we take the alpha strong coupling into account the full hierarchy of alphas is given by:

$\alpha_G <\alpha_G^{'} <\alpha_G^{'''} < \alpha_G^{''}<\alpha_{EM}<\alpha_s$

$\alpha_s^{-1}<\alpha_{EM}^{-1}<\alpha^{''-1}_G <\alpha^{'''-1}_G <\alpha^{'-1}_G < \alpha^{-1}_G$

Fascinating! Isn’t it? Stay tuned!!!

ADDENDUM: After I finished this post, I discovered a striking (and interesting itself) connection between $\alpha_{EM}$ and $\alpha_{G}$. The relation or coincidence is the following relationship

$\dfrac{1}{\alpha_{EM}}\approx \ln \left( \dfrac {1}{16\alpha_G}\right)$

Is this relationship fundamental or accidental? The answer is unknown. However, since the electric charge (via electromagnetic alpha) is not related a priori with the gravitational constant or Planck mass ( or the cosmological constant via the above gravitational alpha) in any known way I find particularly stunning such a coincidence up to 5 significant digits! Any way, there are many unexplained numerical coincidences that are completely accidental and meaningless, and then, it is not clear why this numeral result should be relevant for the connection between electromagnetism and gravity/cosmology, but it is interesting at least as a curiosity and “joke” of Nature.

Some quotes about the electromagnetic alpha from wikipedia http://en.wikipedia.org/wiki/Fine-structure_constant

“(…)There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won’t recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we don’t know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly! (…)”. R.P.Feynman, QED: The Strange Theory of Light and Matter, Princeton University Press, p.129.

“(…) If alpha [the fine-structure constant] were bigger than it really is, we should not be able to distinguish matter from ether [the vacuum, nothingness], and our task to disentangle the natural laws would be hopelessly difficult. The fact however that alpha has just its value 1/137 is certainly no chance but itself a law of nature. It is clear that the explanation of this number must be the central problem of natural philosophy.(…)” Max Born, in A.I. Miller’s book Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung. p. 253. Publisher W.W. Norton & Co.(2009).

“(…)The mystery about α is actually a double mystery. The first mystery – the origin of its numerical value α ≈ 1/137 has been recognized and discussed for decades. The second mystery – the range of its domain – is generally unrecognized.(…)” Malcolm H. Mac Gregor, M.H. MacGregor (2007). The Power of Alpha.

# LOG#052. Chewbacca’s exam.

I found this fun (Spanish) exam about Special Relativity at a Spanish website:

Solutions:

1) $v=25/29 c$

2) $1.836 \times 10^{12} m = 12 A.U.$

3) t=13.6 months = 13 months and 18 days.

Calculations:

1) We use the relativistic addition of velocities rule. That is,

$V=(u-v)/(1-(uv/c^2))$

where u=Millenium Falcon velocity, v=imperial cruiser velocity= c/5, y V=relative speed=4c/5.

Using units with c=1:

4/5=(v-1/5)/(1-v/5)

4/5(1-v/5)=v-1/5

4/5-4/25v=v-1/5

29/25 v=1

v=25/29

Then, $v=25/29 c$ reinserting units.

2) This part is solved with the length contraction formula and the velocity calculated in the previous part (1). Moreover, we obtain:

$\Delta x'=\Delta x/\gamma$

Using the result we got from (1), and plugging that velocity v and the fact that $\Delta t'$ is equal to one hour, then es

$\Delta x'=v\Delta t'=\Delta x/\gamma$ , and from this

$\Delta x=\gamma v\Delta t'$

Substituting the numerical values, we obtain the given solution easily.

$\Delta x =1.97 ( 25/29 c )1hour =1.7 hc=1.836 \times 10^{12} =12A.U.$

3) Simple application of time dilation formula provides:

$\Delta t'=\gamma \Delta t$

Inserting, in this case, our given velocity, we obtain the solution we wrote above:

$\Delta t' = 1.97 ( 9 months) = 13.6 months = 13 months 18 days$.