LOG#100. Crystalline relativity.



CENTENARY BLOG POST! And dedicatories…

1. Serendipitous thoughts about my 100th blog post

2. The search for unification and higher dimensional theories

3. Final relativity

4. Kalitzin’s metric: multitemporal relativity

5. Spacetime crystals and crystalline relativity: concepts and results

6. Enhanced galilean relativity

7. Conformal two-time relativity and gravitation

8. Hyperspherical electromagnetism and multitemporal relativity

9. Conclusions

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!

2. The search for unification and higher dimensional theories

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


\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?

3. Final relativity

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!}}



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


and the hypersphere


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/


riemannsphere2 riemannsphere4riemannsphere5riemannsphere6

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”:


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


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}}


\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




But we know that

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



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


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!

4. Kalitzin’s metric: multitemporal relativity

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





\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


\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}

5. Spacetime crystals and crystalline relativity: concepts and results

 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


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


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

Then, we have




\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}


\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


then we can easily obtain that


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!

6. Enhanced galilean relativity

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



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




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



The second simpler example is two time enhaced galilean relativity:


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

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


and then


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



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


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


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


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…

7. Conformal two-time relativity and gravitation

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


with a conformal time


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




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


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.

8. Hyperspherical electromagnetism and multitemporal relativity

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



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




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



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


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


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



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



and where

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

(2) Fluid indices for




(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


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.

9. Conclusions

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<<c_2<<c_3<<\ldots 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?

LOG#070. Natural Units.


Happy New Year 2013 to everyone and everywhere!

Let me apologize, first of all, by my absence… I have been busy, trying to find my path and way in my field, and I am busy yet, but finally I could not resist without a new blog boost… After all, you should know the fact I have enough materials to write many new things.

So, what’s next? I will dedicate some blog posts to discuss a nice topic I began before, talking about a classic paper on the subject here:


The topic is going to be pretty simple: natural units in Physics.

First of all, let me point out that the election of any system of units is, a priori, totally conventional. You are free to choose any kind of units for physical magnitudes. Of course, that is not very clever if you have to report data, so everyone can realize what you do and report. Scientists have some definitions and popular systems of units that make the process pretty simpler than in the daily life. Then, we need some general conventions about “units”. Indeed, the traditional wisdom is to use the international system of units, or S (Iabbreviated SI from French language: Le Système international d’unités). There, you can find seven fundamental magnitudes and seven fundamental (or “natural”) units:

1) Space: \left[ L\right]=\mbox{meter}=m

2) Time: \left[ T\right]=\mbox{second}=s

3) Mass: \left[ M\right]=\mbox{kilogram}=kg

4) Temperature: \left[ t\right]=\mbox{Kelvin degree}= K

5) Electric intensity: \left[ I\right]=\mbox{ampere}=A

6) Luminous intensity: \left[ I_L\right]=\mbox{candela}=cd

7) Amount of substance: \left[ n\right]=\mbox{mole}=mol(e)

The dependence between these 7 great units and even their definitions can be found here http://en.wikipedia.org/wiki/International_System_of_Units and references therein. I can not resist to show you the beautiful graph of the 7 wonderful units that this wikipedia article shows you about their “interdependence”:


In Physics, when you build a radical new theory, generally it has the power to introduce a relevant scale or system of units. Specially, the Special Theory of Relativity, and the Quantum Mechanics are such theories. General Relativity and Statistical Physics (Statistical Mechanics) have also intrinsic “universal constants”, or, likely, to be more precise, they allow the introduction of some “more convenient” system of units than those you have ever heard ( metric system, SI, MKS, cgs, …). When I spoke about Barrow units (see previous comment above) in this blog, we realized that dimensionality (both mathematical and “physical”), and fundamental theories are bound to the election of some “simpler” units. Those “simpler” units are what we usually call “natural units”. I am not a big fan of such terminology. It is confusing a little bit. Maybe, it would be more interesting and appropiate to call them “addapted X units” or “scaled X units”, where X denotes “relativistic, quantum,…”. Anyway, the name “natural” is popular and it is likely impossible to change the habits.

In fact, we have to distinguish several “kinds” of natural units. First of all, let me list “fundamental and universal” constants in different theories accepted at current time:

1. Boltzmann constant: k_B.

Essential in Statistical Mechanics, both classical and quantum. It measures “entropy”/”information”. The fundamental equation is:

\boxed{S=k_B\ln \Omega}

It provides a link between the microphysics and the macrophysics ( it is the code behind the equation above). It can be understood somehow as a measure of the “energetic content” of an individual particle or state at a given temperature. Common values for this constant are:

k_B=1.3806488(13)\times 10^{-23}J/K = 8.6173324(78)\times 10^{-5}eV/K

k_B=1.3806488(13)\times 10^{-16}erg/K

Statistical Physics states that there is a minimum unit of entropy or a minimal value of energy at any given temperature. Physical dimensions of this constant are thus entropy, or since E=TS, \left[ k_B\right] =E/t=J/K, where t denotes here dimension of temperature.

2. Speed of light.  c.

From classical electromagnetism:


The speed of light, according to the postulates of special relativity, is a universal constant. It is frame INDEPENDENT. This fact is at the root of many of the surprising results of special relativity, and it took time to be understood. Moreover, it also connects space and time in a powerful unified formalism, so space and time merge into spacetime, as we do know and we have studied long ago in this blog. The spacetime interval in a D=3+1 dimensional space and two arbitrary events reads:

\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-c^2\Delta t^2

In fact, you can observe that “c” is the conversion factor between time-like and space-like coordinates.  How big the speed of light is? Well, it is a relatively large number from our common and ordinary perception. It is exactly:


although you often take it as c\approx 3\cdot 10^{8}m/s=3\cdot 10^{10}cm/s.  However, it is the speed of electromagnetic waves in vacuum, no matter where you are in this Universe/Polyverse. At least, experiments are consistent with such an statement. Moreover, it shows that c is also the conversion factor between energy and momentum, since


and c^2 is the conversion factor between rest mass and pure energy, because, as everybody knows,  E=mc^2! According to the special theory of relativity, normal matter can never exceed the speed of light. Therefore, the speed of light is the maximum velocity in Nature, at least if specially relativity holds. Physical dimensions of c are \left[c\right]=LT^{-1}, where L denotes length dimension and T denotes time dimension (please, don’t confuse it with temperature despite the capital same letter for both symbols).

3. Planck’s constant. h or generally rationalized \hbar=h/2\pi.

Planck’s constant (or its rationalized version), is the fundamental universal constant in Quantum Physics (Quantum Mechanics, Quantum Field Theory). It gives

\boxed{E=h\nu=\hbar \omega}

Indeed, quanta are the minimal units of energy. That is, you can not divide further a quantum of light, since it is indivisible by definition! Furthermore, the de Broglie relationship relates momentum and wavelength for any particle, and it emerges from the combination of special relativity and the quantum hypothesis:

\lambda=\dfrac{h}{p}\leftrightarrow \bar{\lambda}=\dfrac{\hbar}{p}

In the case of massive particles, it yields

\lambda=\dfrac{h}{Mv}\leftrightarrow \bar{\lambda}=\dfrac{\hbar}{Mv}

In the case of massless particles (photons, gluons, gravitons,…)

\lambda=\dfrac{hc}{E} or \bar{\lambda}=\dfrac{\hbar c}{E}

Planck’s constant also appears to be essential to the uncertainty principle of Heisenberg:

\boxed{\Delta x \Delta p\geq \hbar/2}

\boxed{\Delta E \Delta t\geq \hbar/2}

\boxed{\Delta A\Delta B\geq \hbar/2}

Some particularly important values of this constant are:

h=6.62606957(29)\times 10^{-34} J\cdot s
h=4.135667516(91)\times 10^{-15}eV\cdot s
h=6.62606957(29)\times 10^{-27} erg\cdot s
\hbar =1.054571726(47)\times 10^{-34} J\cdot s
\hbar =6.58211928(15)\times 10^{-16} eV\cdot s
\hbar= 1.054571726(47)\times 10^{-27}erg\cdot s

It is also useful to know that
hc=1.98644568\times 10^{-25}J\cdot m
hc=1.23984193 eV\cdot \mu m


\hbar c=0.1591549hc or \hbar c=197.327 eV\cdot nm

Planck constant has dimension of \mbox{Energy}\times \mbox{Time}=\mbox{position}\times \mbox{momentum}=ML^2T^{-1}. Physical dimensions of this constant coincide also with angular momentum (spin), i.e., with L=mvr.

4. Gravitational constant. G_N.

Apparently, it is not like the others but it can also define some particular scale when combined with Special Relativity. Without entering into further details (since I have not discussed General Relativity yet in this blog), we can calculate the escape velocity of a body moving at the speed of light

\dfrac{1}{2}mv^2-G_N\dfrac{Mm}{R}=0 with v=c implies a new length scale where gravitational relativistic effects do appear, the so-called Schwarzschild radius R_S:

\boxed{R_S=\dfrac{2G_NM}{c^2}=\dfrac{2G_NM_{\odot}}{c^2}\left(\dfrac{M}{M_{\odot}}\right)\approx 2.95\left(\dfrac{M}{M_{\odot}}\right)km}

5. Electric fundamental charge. e.

It is generally chosen as fundamental charge the electric charge of the positron (positive charged “electron”). Its value is:

e=1.602176565(35)\times 10^{-19}C

where C denotes Coulomb. Of course, if you know about quarks with a fraction of this charge, you could ask why we prefer this one. Really, it is only a question of hystory of Science, since electrons were discovered first (and positrons). Quarks, with one third or two thirds of this amount of elementary charge, were discovered later, but you could define the fundamental unit of charge as multiple or entire fraction of this charge. Moreover, as far as we know, electrons are “elementary”/”fundamental” entities, so, we can use this charge as unit and we can define quark charges in terms of it too. Electric charge is not a fundamental unit in the SI system of units. Charge flow, or electric current, is.

An amazing property of the above 5 constants is that they are “universal”. And, for instance, energy is related with other magnitudes in theories where the above constants are present in a really wonderful and unified manner:

\boxed{E=N\dfrac{k_BT}{2}=Mc^2=TS=Pc=N\dfrac{h\nu}{2}=N\dfrac{\hbar \omega}{2}=\dfrac{R_Sc^4}{2G_N}=\hbar c k=\dfrac{hc}{\lambda}}

Caution: k is not the Boltzmann constant but the wave number.

There is a sixth “fundamental” constant related to electromagnetism, but it is also related to the speed of light, the electric charge and the Planck’s constant in a very sutble way. Let me introduce you it too…

6. Coulomb constant. k_C.

This is a second constant related to classical electromagnetism, like the speed of light in vacuum. Coulomb’s constant, the electric force constant, or the electrostatic constant (denoted k_C) is a proportionality factor that takes part in equations relating electric force between  point charges, and indirectly it also appears (depending on your system of units) in expressions for electric fields of charge distributions. Coulomb’s law reads


Its experimental value is

k_C=\dfrac{1}{4\pi \varepsilon_0}=\dfrac{c^2\mu_0}{4\pi}=c^2\cdot 10^{-7}H\cdot m^{-1}= 8.9875517873681764\cdot 10^9 Nm^2/C^2

Generally, the Coulomb constant is dropped out and it is usually preferred to express everything using the electric permitivity of vacuum \varepsilon_0 and/or numerical factors depending on the pi number \pi if you choose the gaussian system of units  (read this wikipedia article http://en.wikipedia.org/wiki/Gaussian_system_of_units ), the CGS system, or some hybrid units based on them.

H.E.P. units

High Energy Physicists use to employ units in which the velocity is measured in fractions of the speed of light in vacuum, and the action/angular momentum is some multiple of the Planck’s constant. These conditions are equivalent to set

\boxed{c=1_c=1} \boxed{\hbar=1_\hbar=1}

Complementarily, or not, depending on your tastes and preferences, you can also set the Boltzmann’s constant to the unit as well


and thus the complete HEP system is defined if you set


This “natural” system of units is lacking yet a scale of energy. Then, it is generally added the electron-volt eV as auxiliary quantity defining the reference energy scale. Despite the fact that this is not a “natural unit” in the proper sense because it is defined by a natural property, the electric charge,  and the anthropogenic unit of electric potential, the volt. The SI prefixes multiples of eV are used as well: keV, MeV, GeV, etc. Here, the eV is used as reference energy quantity, and with the above election of “elementary/natural units” (or any other auxiliary unit of energy), any quantity can be expressed. For example, a distance of 1 m can be expressed in terms of eV, in natural units, as

1m=\dfrac{1m}{\hbar c}\approx 510eV^{-1}

This system of units have remarkable conversion factors

A) 1 eV^{-1} of length is equal to 1.97\cdot 10^{-7}m =(1\text{eV}^{-1})\hbar c

B) 1 eV of mass is equal to 1.78\cdot 10^{-36}kg=1\times \dfrac{eV}{c^2}

C) 1 eV^{-1} of time is equal to 6.58\cdot 10^{-16}s=(1\text{eV}^{-1})\hbar

D) 1 eV of temperature is equal to 1.16\cdot 10^4K=1eV/k_B

E) 1 unit of electric charge in the Lorentz-Heaviside system of units is equal to 5.29\cdot 10^{-19}C=e/\sqrt{4\pi\alpha}

F) 1 unit of electric charge in the Gaussian system of units is equal to 1.88\cdot 10^{-18}C=e/\sqrt{\alpha}

This system of units, therefore, leaves free only the energy scale (generally it is chosen the electron-volt) and the electric measure of fundamentl charge. Every other unit can be related to energy/charge. It is truly remarkable than doing this (turning invisible the above three constants) you can “unify” different magnitudes due to the fact these conventions make them equivalent. For instance, with natural units:

1) Length=Time=1/Energy=1/Mass.

It is due to x=ct, E=Mc^2 and E=hc/\lambda equations. Setting c and h or \hbar provides

x=t, E=M and E=1/\lambda.

Note that natural units turn invisible the units we set to the unit! That is the key of the procedure. It simplifies equations and expressions. Of course, you must be careful when you reintroduce constants!

2) Energy=Mass=Momemntum=Temperature.

It is due to E=k_BT, E=Pc and E=Mc^2 again.

One extra bonus for theoretical physicists is that natural units allow to build and write proper lagrangians and hamiltonians (certain mathematical operators containing the dynamics of the system enconded in them), or equivalently the action functional, with only the energy or “mass” dimension as “free parameter”. Let me show how it works.

Natural units in HEP identify length and time dimensions. Thus \left[L\right]=\left[T\right]. Planck’s constant allows us to identify those 2 dimensions with 1/Energy (reciprocals of energy) physical dimensions. Therefore, in HEP units, we have


The speed of light identifies energy and mass, and thus, we can often heard about “mass-dimension” of a lagrangian in the following sense. HEP units can be thought as defining “everything” in terms of energy, from the pure dimensional ground. That is, every physical dimension is (in HEP units) defined by a power of energy:


Thus, we can refer to any magnitude simply saying the power of such physical dimension (or you can think logarithmically to understand it easier if you wish). With this convention, and recalling that energy dimension is mass dimension, we have that

\left[L\right]=\left[T\right]=-1 and \left[E\right]=\left[M\right]=1

Using these arguments, the action functional is a pure dimensionless quantity, and thus, in D=4 spacetime dimensions, lagrangian densities must have dimension 4 ( or dimension D is a general spacetime).

\displaystyle{S=\int d^4x \mathcal{L}\rightarrow \left[\mathcal{L}\right]=4}

\displaystyle{S=\int d^Dx \mathcal{L}\rightarrow \left[\mathcal{L}\right]=D}

In D=4 spacetime dimensions, it can be easily showed that



where \Phi is a scalar field, A^\mu is a vector field (like the electromagnetic or non-abelian vector gauge fields), and \Psi_D, \Psi_M, \chi, \eta are a Dirac spinor, a Majorana spinor, and \chi, \eta are Weyl spinors (of different chiralities). Supersymmetry (or SUSY) allows for anticommuting c-numbers (or Grassmann numbers) and it forces to introduce auxiliary parameters with mass dimension -1/2. They are the so-called SUSY transformation parameters \zeta_{SUSY}=\epsilon. There are some speculative spinors called ELKO fields that could be non-standandard spinor fields with mass dimension one! But it is an advanced topic I am not going to discuss here today. In general D spacetime dimensions a scalar (or vector) field would have mass dimension (D-2)/2, and a spinor/fermionic field in D dimensions has generally (D-1)/2 mass dimension (excepting the auxiliary SUSY grassmanian fields and the exotic idea of ELKO fields).  This dimensional analysis is very useful when theoretical physicists build up interacting lagrangians, since we can guess the structure of interaction looking at purely dimensional arguments every possible operator entering into the action/lagrangian density! In summary, therefore, for any D:

\boxed{\left[\Phi\right]=\left[A_\mu\right]=\dfrac{D-2}{2}\equiv E^{\frac{D-2}{2}}=M^{\frac{D-2}{2}}}

\boxed{\left[\Psi\right]=\dfrac{D-1}{2}\equiv E^{\frac{D-1}{2}}=M^{\frac{D-1}{2}}}

Remark (for QFT experts only): Don’t confuse mass dimension with the final transverse polarization degrees or “degrees of freedom” of a particular field, i.e., “components” minus “gauge constraints”. E.g.: a gauge vector field has D-2 degrees of freedom in D dimensions. They are different concepts (although both closely related to the spacetime dimension where the field “lives”).

In summary:

i) HEP units are based on QM (Quantum Mechanics), SR (Special Relativity) and Statistical Mechanics (Entropy and Thermodynamics).

ii) HEP units need to introduce a free energy scale, and it generally drives us to use the eV or electron-volt as auxiliary energy scale.

iii) HEP units are useful to dimensional analysis of lagrangians (and hamiltonians) up to “mass dimension”.

Stoney Units

In Physics, the Stoney units form a alternative set of natural units named after the Irish physicist George Johnstone Stoney, who first introduced them as we know it today in 1881. However, he presented the idea in a lecture entitled “On the Physical Units of Nature” delivered to the British Association before that date, in 1874. They are the first historical example of natural units and “unification scale” somehow. Stoney units are rarely used in modern physics for calculations, but they are of historical interest but some people like Wilczek has written about them (see, e.g., http://arxiv.org/abs/0708.4361). These units of measurement were designed so that certain fundamental physical constants are taken as reference basis without the Planck scale being explicit, quite a remarkable fact! The set of constants that Stoney used as base units is the following:

A) Electric charge, e=1_e.

B) Speed of light in vacuum, c=1_c.

C) Gravitational constant, G_N=1_{G_N}.

D) The Reciprocal of Coulomb constant, 1/k_C=4\pi \varepsilon_0=1_{k_C^{-1}}=1_{4\pi \varepsilon_0}.

Stony units are built when you set these four constants to the unit, i.e., equivalently, the Stoney System of Units (S) is determined by the assignments:


Interestingly, in this system of units, the Planck constant is not equal to the unit and it is not “fundamental” (Wilczek remarked this fact here ) but:

\hbar=\dfrac{1}{\alpha}\approx 137.035999679

Today, Planck units are more popular Planck than Stoney units in modern physics, and even there are many physicists who don’t know about the Stoney Units! In fact, Stoney was one of the first scientists to understand that electric charge was quantized!; from this quantization he deduced the units that are now named after him.

The Stoney length and the Stoney energy are collectively called the Stoney scale, and they are not far from the Planck length and the Planck energy, the Planck scale. The Stoney scale and the Planck scale are the length and energy scales at which quantum processes and gravity occur together. At these scales, a unified theory of physics is thus likely required. The only notable attempt to construct such a theory from the Stoney scale was that of H. Weyl, who associated a gravitational unit of charge with the Stoney length and who appears to have inspired Dirac’s fascination with the large number hypothesis. Since then, the Stoney scale has been largely neglected in the development of modern physics, although it is occasionally discussed to this day. Wilczek likes to point out that, in Stoney Units, QM would be an emergent phenomenon/theory, since the Planck constant wouldn’t be present directly but as a combination of different constants. By the other hand, the Planck scale is valid for all known interactions, and does not give prominence to the electromagnetic interaction, as the Stoney scale does. That is, in Stoney Units, both gravitation and electromagnetism are on equal footing, unlike the Planck units, where only the speed of light is used and there is no more connections to electromagnetism, at least, in a clean way like the Stoney Units do. Be aware, sometimes, rarely though, Planck units are referred to as Planck-Stoney units.

What are the most interesting Stoney system values? Here you are the most remarkable results:

1) Stoney Length, L_S.

\boxed{L_S=\sqrt{\dfrac{G_Ne^2}{(4\pi\varepsilon)c^4}}\approx 1.38\cdot 10^{-36}m}

2) Stoney Mass, M_S.

\boxed{M_S=\sqrt{\dfrac{e^2}{G_N(4\pi\varepsilon_0)}}\approx 1.86\cdot 10^{-9}kg}

3) Stoney Energy, E_S.

\boxed{E_S=M_Sc^2=\sqrt{\dfrac{e^2c^4}{G_N(4\pi\varepsilon_0)}}\approx 1.67\cdot 10^8 J=1.04\cdot 10^{18}GeV}

4) Stoney Time, t_S.

\boxed{t_S=\sqrt{\dfrac{G_Ne^2}{c^6(4\pi\varepsilon_0)}}\approx 4.61\cdot 10^{-45}s}

5) Stoney Charge, Q_S.

\boxed{Q_S=e\approx 1.60\cdot 10^{-19}C}

6) Stoney Temperature, T_S.

\boxed{T_S=E_S/k_B=\sqrt{\dfrac{e^2c^4}{G_Nk_B^2(4\pi\varepsilon_0)}}\approx 1.21\cdot 10^{31}K}

Planck Units

The reference constants to this natural system of units (generally denoted by P) are the following 4 constants:

1) Gravitational constant. G_N

2) Speed of light. c.

3) Planck constant or rationalized Planck constant. \hbar.

4) Boltzmann constant. k_B.

The Planck units are got when you set these 4 constants to the unit, i.e.,


It is often said that Planck units are a system of natural units that is not defined in terms of properties of any prototype, physical object, or even features of any fundamental particle. They only refer to the basic structure of the laws of physics: c and G are part of the structure of classical spacetime in the relativistic theory of gravitation, also known as general relativity, and ℏ captures the relationship between energy and frequency which is at the foundation of elementary quantum mechanics. This is the reason why Planck units particularly useful and common in theories of quantum gravity, including string theory or loop quantum gravity.

This system defines some limit magnitudes, as follows:

1) Planck Length, L_P.

\boxed{L_P=\sqrt{\dfrac{G_N\hbar}{c^3}}\approx 1.616\cdot 10^{-35}s}

2) Planck Time, t_P.

\boxed{t_P=L_P/c=\sqrt{\dfrac{G_N\hbar}{c^5}}\approx 5.391\cdot 10^{-44}s}

3) Planck Mass, M_P.

\boxed{M_P=\sqrt{\dfrac{\hbar c}{G_N}}\approx 2.176\cdot 10^{-8}kg}

4) Planck Energy, E_P.

\boxed{E_P=M_Pc^2=\sqrt{\dfrac{\hbar c^5}{G_N}}\approx 1.96\cdot 10^9J=1.22\cdot 10^{19}GeV}

5) Planck charge, Q_P.

In Lorentz-Heaviside electromagnetic units

\boxed{Q_P=\sqrt{\hbar c \varepsilon_0}=\dfrac{e}{\sqrt{4\pi\alpha}}\approx 5.291\cdot 10^{-19}C}

In Gaussian electromagnetic units

\boxed{Q_P=\sqrt{\hbar c (4\pi\varepsilon_0)}=\dfrac{e}{\sqrt{\alpha}}\approx 1.876\cdot 10^{-18}C}

6) Planck temperature, T_P.

\boxed{T_P=E_P/k_B=\sqrt{\dfrac{\hbar c^5}{G_Nk_B^2}}\approx 1.417\cdot 10^{32}K}

From these “fundamental” magnitudes we can build many derived quantities in the Planck System:

1) Planck area.

A_P=L_P^2=\dfrac{\hbar G_N}{c^3}\approx 2.612\cdot 10^{-70}m^2

2) Planck volume.

V_P=L_P^3=\left(\dfrac{\hbar G_N}{c^3}\right)^{3/2}\approx 4.22\cdot 10^{-105}m^3

3) Planck momentum.

P_P=M_Pc=\sqrt{\dfrac{\hbar c^3}{G_N}}\approx 6.52485 kgm/s

A relatively “small” momentum!

4) Planck force.

F_P=E_P/L_P=\dfrac{c^4}{G_N }\approx 1.21\cdot 10^{44}N

It is independent from Planck constant! Moreover, the Planck acceleration is

a_P=F_P/M_P=\sqrt{\dfrac{c^7}{G_N\hbar}}\approx 5.561\cdot 10^{51}m/s^2

5) Planck Power.

\mathcal{P}_P=\dfrac{c^5}{G_N}\approx 3.628\cdot 10^{52}W

6) Planck density.

\rho_P=\dfrac{c^5}{\hbar G_N^2}\approx 5.155\cdot 10^{96}kg/m^3

Planck density energy would be equal to

\rho_P c^2=\dfrac{c^7}{\hbar G_N^2}\approx 4.6331\cdot 10^{113}J/m^3

7) Planck angular frequency.

\omega_P=\sqrt{\dfrac{c^5}{\hbar G_N}}\approx 1.85487\cdot 10^{43}Hz

8) Planck pressure.

p_P=\dfrac{F_P}{A_P}=\dfrac{c^7}{G_N^2\hbar}=\rho_P c^2\approx 4.6331\cdot 10^{113}Pa

Note that Planck pressure IS the Planck density energy!

9) Planck current.

I_P=Q_P/t_P=\sqrt{\dfrac{4\pi\varepsilon_0 c^6}{G_N}}\approx 3.4789\cdot 10^{25}A

10) Planck voltage.

v_P=E_P/Q_P=\sqrt{\dfrac{c^4}{4\pi\varepsilon_0 G_N}}\approx 1.04295\cdot 10^{27}V

11) Planck impedance.

Z_P=v_P/I_P=\dfrac{\hbar^2}{Q_P}=\dfrac{1}{4\pi \varepsilon_0 c}\approx 29.979\Omega

A relatively small impedance!

12) Planck capacitor.

C_P=Q_P/v_P=4\pi\varepsilon_0\sqrt{\dfrac{\hbar G_N}{ c^3}} \approx 1.798\cdot 10^{-45}F

Interestingly, it depends on the gravitational constant!

Some Planck units are suitable for measuring quantities that are familiar from daily experience. In particular:

1 Planck mass is about 22 micrograms.

1 Planck momentum is about 6.5 kg m/s

1 Planck energy is about 500kWh.

1 Planck charge is about 11 elementary (electronic) charges.

1 Planck impendance is almost 30 ohms.


i) A speed of 1 Planck length per Planck time is the speed of light, the maximum possible speed in special relativity.

ii) To understand the Planck Era and “before” (if it has sense), supposing QM holds yet there, we need a quantum theory of gravity to be available there. There is no such a theory though, right now. Therefore, we have to wait if these ideas are right or not.

iii) It is believed that at Planck temperature, the whole symmetry of the Universe was “perfect” in the sense the four fundamental foces were “unified” somehow. We have only some vague notios about how that theory of everything (TOE) would be.

The physical dimensions of the known Universe in terms of Planck units are “dramatic”:

i) Age of the Universe is about t_U=8.0\cdot 10^{60} t_P.

ii) Diameter of the observable Universe is about d_U=5.4\cdot 10^{61}L_P

iii) Current temperature of the Universe is about 1.9 \cdot 10^{-32}T_P

iv) The observed cosmological constant is about 5.6\cdot 10^{-122}t_P^{-2}

v) The mass of the Universe is about 10^{60}m_p.

vi) The Hubble constant is 71km/s/Mpc\approx 1.23\cdot 10^{-61}t_P^{-1}

Schrödinger Units

The Schrödinger Units do not obviously contain the term c, the speed of light in a vacuum. However, within the term of the Permittivity of Free Space [i.e., electric constant or vacuum permittivity], and the speed of light plays a part in that particular computation. The vacuum permittivity results from the reciprocal of the speed of light squared times the magnetic constant. So, even though the speed of light is not apparent in the Schrödinger equations it does exist buried within its terms and therefore influences the decimal placement issue within square roots. The essence of Schrödinger units are the following constants:

A) Gravitational constant G_N.

B) Planck constant \hbar.

C) Boltzmann constant k_B.

D) Coulomb constant or equivalently the electric permitivity of free space/vacuum k_C=1/4\pi\varepsilon_0.

E) The electric charge of the positron e.

In this sistem \psi we have

\boxed{G_N=\hbar =k_B =k_C =1}

1) Schrödinger Length L_{Sch}.

L_\psi=\sqrt{\dfrac{\hbar^4 G_N(4\pi\varepsilon_0)^3}{e^6}}\approx 2.593\cdot 10^{-32}m

2) Schrödinger time t_{Sch}.

t_\psi=\sqrt{\dfrac{\hbar^6 G_N(4\pi\varepsilon_0)^5}{e^{10}}}\approx 1.185\cdot 10^{-38}s

3) Schrödinger mass M_{Sch}.

M_\psi=\sqrt{\dfrac{e^2}{G_N(4\pi\varepsilon_0)}}\approx 1.859\cdot 10^{-9}kg

4) Schrödinger energy E_{Sch}.

E_\psi=\sqrt{\dfrac{e^{10}}{\hbar^4(4\pi\varepsilon_0)^5G_N}}\approx 8890 J=5.55\cdot 10^{13}GeV

5) Schrödinger charge Q_{Sch}.

Q_\psi =e=1.602\cdot 10^{-19}C

6) Schrödinger temperature T_{Sch}.

T_\psi=E_\psi/k_B=\sqrt{\dfrac{e^{10}}{\hbar^4(4\pi\varepsilon_0)^5G_Nk_B^2}}\approx 6.445\cdot 10^{26}K

Atomic Units

There are two alternative systems of atomic units, closely related:

1) Hartree atomic units: 

\boxed{e=m_e=\hbar=k_B=1} and \boxed{c=\alpha^{-1}}

2) Rydberg atomic units:

\boxed{\dfrac{e}{\sqrt{2}}=2m_e=\hbar=k_B=1} and \boxed{c=2\alpha^{-1}}

There, m_e is the electron mass and \alpha is the electromagnetic fine structure constant. These units are designed to simplify atomic and molecular physics and chemistry, especially the quantities related to the hydrogen atom, and they are widely used in these fields. The Hartree units were first proposed by Doublas Hartree, and they are more common than the Rydberg units.

The units are adapted to characterize the behavior of an electron in the ground state of a hydrogen atom. For example, using the Hartree convention, in the Böhr model of the hydrogen atom, an electron in the ground state has orbital velocity = 1, orbital radius = 1, angular momentum = 1, ionization energy equal to 1/2, and so on.

Some quantities in the Hartree system of units are:

1) Atomic Length (also called Böhr radius):

L_A=a_0=\dfrac{\hbar^2 (4\pi\varepsilon_0)}{m_ee^2}\approx 5.292\cdot 10^{-11}m=0.5292\AA

2) Atomic Time:

t_A=\dfrac{\hbar^3(4\pi\varepsilon_0)^2}{m_ee^4}\approx 2.419\cdot 10^{-17}s

3) Atomic Mass:

M_A=m_e\approx 9.109\cdot 10^{-31}kg

4) Atomic Energy:

E_A=m_ec^2=\dfrac{m_ee^4}{\hbar^2(4\pi\varepsilon_0)^2} \approx 4.36\cdot 10^{ -18}J=27.2eV=2\times(13.6)eV=2Ry

5) Atomic electric Charge:

Q_A=q_e=e\approx 1.602\cdot 10^{-19}C

6) Atomic temperature:

T_A=E_A/k_B=\dfrac{m_ee^4}{\hbar^2(4\pi\varepsilon_0)^2k_B}\approx 3.158\cdot 10^5K

The fundamental unit of energy is called the Hartree energy in the Hartree system and the Rydberg energy in the Rydberg system. They differ by a factor of 2. The speed of light is relatively large in atomic units (137 in Hartree or 274 in Rydberg), which comes from the fact that an electron in hydrogen tends to move much slower than the speed of light. The gravitational constant  is extremely small in atomic units (about 10−45), which comes from the fact that the gravitational force between two electrons is far weaker than the Coulomb force . The unit length, LA, is the so-called and well known Böhr radius, a0.

The values of c and e shown above imply that e=\sqrt{\alpha \hbar c}, as in Gaussian units, not Lorentz-Heaviside units. However, hybrids of the Gaussian and Lorentz–Heaviside units are sometimes used, leading to inconsistent conventions for magnetism-related units. Be aware of these issues!

QCD Units

In the framework of Quantum Chromodynamics, a quantum field theory (QFT) we know as QCD, we can define the QCD system of units based on:

1) QCD Length L_{QCD}.

L_{QCD}=\dfrac{\hbar}{m_pc}\approx 2.103\cdot 10^{-16}m

and where m_p is the proton mass (please, don’t confuse it with the Planck mass M_P).

2) QCD Time t_{QCD}.

t_{QCD}=\dfrac{\hbar}{m_pc^2}\approx 7.015\cdot 10^{-25}s

3) QCD Mass M_{QCD}.

M_{QCD}=m_p\approx 1.673\cdot 10^{-27}kg

4) QCD Energy E_{QCD}.

E_{QCD}=M_{QCD}c^2=m_pc^2\approx 1.504\cdot 10^{-10}J=938.6MeV=0.9386GeV

Thus, QCD energy is about 1 GeV!

5) QCD Temperature T_{QCD}.

T_{QCD}=E_{QCD}/k_B=\dfrac{m_pc^2}{k_B}\approx 1.089\cdot 10^{13}K

6) QCD Charge Q_{QCD}.

In Heaviside-Lorent units:

Q_{QCD}=\dfrac{1}{\sqrt{4\pi\alpha}}e\approx 5.292\cdot 10^{-19}C

In Gaussian units:

Q_{QCD}=\dfrac{1}{\sqrt{\alpha}}e\approx 1.876\cdot 10^{-18}C

Geometrized Units

The geometrized unit system, used in general relativity, is not a completely defined system. In this system, the base physical units are chosen so that the speed of light and the gravitational constant are set equal to unity. Other units may be treated however desired. By normalizing appropriate other units, geometrized units become identical to Planck units. That is, we set:


and the remaining constants are set to the unit according to your needs and tastes.

Conversion Factors

This table from wikipedia is very useful:



i) \alpha is the fine-structure constant, approximately 0.007297.

ii) \alpha_G=\dfrac{m_e^2}{M_P^2}\approx 1.752\cdot 10^{-45} is the gravitational fine-structure constant.

Some conversion factors for geometrized units are also available:

Conversion from kg, s, C, K into m:

G_N/c^2  [m/kg]

c [m/s]

\sqrt{G_N/(4\pi\varepsilon_0)}/c^2 [m/C]

G_Nk_B/c^2 [m/K]

Conversion from m, s, C, K into kg:

c^2/G_N [kg/m]

c^3/G_N [kg/s]

1/\sqrt{G_N4\pi\varepsilon_0} [kg/C]


Conversion from m, kg, C, K into s

1/c [s/m]


\sqrt{\dfrac{G_N}{4\pi\varepsilon_0}}/c^3 [s/C]

G_Nk_B/c^5 [s/K]

Conversion from m, kg, s, K into C


(G_N4\pi\varepsilon_0)^{1/2} [C/kg]


k_B\sqrt{G_N4\pi\varepsilon_0}/c^2   [C/K]

Conversion from m, kg, s, C into K


c^2/k_B [K/kg]

c^5/(G_Nk_B) [K/s]

c^2/(k_B\sqrt{G_N4\pi\varepsilon_0}) [K/C]

Or you can read off factors from this table as well:




Advantages and Disadvantages of Natural Units

Natural units have some advantages (“Pro”):

1) Equations and mathematical expressions are simpler in Natural Units.

2) Natural units allow for the match between apparently different physical magnitudes.

3) Some natural units are independent from “prototypes” or “external patterns” beyond some clever and trivial conventions.

4) They can help to unify different physical concetps.

However, natural units have also some disadvantages (“Cons”):

1) They generally provide less precise measurements or quantities.

2) They can be ill-defined/redundant and own some ambiguity. It is also caused by the fact that some natural units differ by numerical factors of pi and/or pure numbers, so they can not help us to understand the origin of some pure numbers (adimensional prefactors) in general.

Moreover, you must not forget that natural units are “human” in the sense you can addapt them to your own needs, and indeed,you can create your own particular system of natural units! However, said this, you can understand the main key point: fundamental theories are who finally hint what “numbers”/”magnitudes” determine a system of “natural units”.

Remark: the smart designer of a system of natural unit systems must choose a few of these constants to normalize (set equal to 1). It is not possible to normalize just any set of constants. For example, the mass of a proton and the mass of an electron cannot both be normalized: if the mass of an electron is defined to be 1, then the mass of a proton has to be \approx 6\pi^5\approx 1936. In a less trivial example, the fine-structure constant, α≈1/137, cannot be set to 1, because it is a dimensionless number. The fine-structure constant is related to other fundamental constants through a very known equation:

\alpha=\dfrac{k_Ce^2}{\hbar c}

where k_C is the Coulomb constant, e is the positron electric charge (elementary charge), ℏ is the reduced Planck constant, and c is the again the speed of light in vaccuum. It is believed that in a normal theory is not possible to simultaneously normalize all four of the constants c, ℏ, e, and kC.

Fritzsch-Xing  plot

Fritzsch and Xing have developed a very beautiful plot of the fundamental constants in Nature (those coming from gravitation and the Standard Model). I can not avoid to include it here in the 2 versions I have seen it. The first one is “serious”, with 29 “fundamental constants”:


However, I prefer the “fun version” of this plot. This second version is very cool and it includes 28 “fundamental constants”:


The Okun Cube

Long ago, L.B. Okun provided a very interesting way to think about the Planck units and their meaning, at least from current knowledge of physics! He imagined a cube in 3d in which we have 3 different axis. Planck units are defined as we have seen above by 3 constants c, \hbar, G_N plus the Boltzmann constant. Imagine we arrange one axis for c-Units, one axis for \hbar-units and one more for G_N-units. The result is a wonderful cube:


Or equivalently, sometimes it is seen as an equivalent sketch ( note the Planck constant is NOT rationalized in the next cube, but it does not matter for this graphical representation):


Classical physics (CP) corresponds to the vanishing of the 3 constants, i.e., to the origin (0,0,0).

Newtonian mechanics (NM) , or more precisely newtonian gravity plus classical mechanics, corresponds to the “point” (0,0,G_N).

Special relativity (SR) corresponds to the point (0,1/c,0), i.e., to “points” where relativistic effects are important due to velocities close to the speed of light.

Quantum mechanics (QM) corresponds to the point (h,0,0), i.e., to “points” where the action/angular momentum fundamental unit is important, like the photoelectric effect or the blackbody radiation.

Quantum Field Theory (QFT) corresponds to the point (h,1/c,0), i.e, to “points” where both, SR and QM are important, that is, to situations where you can create/annihilate pairs, the “particle” number is not conserved (but the particle-antiparticle number IS), and subatomic particles manifest theirselves simultaneously with quantum and relativistic features.

Quantum Gravity (QG) would correspond to the point (h,0,G_N) where gravity is quantum itself. We have no theory of quantum gravity yet, but some speculative trials are effective versions of (super)-string theory/M-theory, loop quantum gravity (LQG) and some others.

Finally, the Theory Of Everything (TOE) would be the theory in the last free corner, that arising in the vertex (h,1/c,G_N). Superstring theories/M-theory are the only serious canditate to TOE so far. LQG does not generally introduce matter fields (some recent trials are pushing into that direction, though) so it is not a TOE candidate right now.

Some final remarks and questions

1) Are fundamental “constants” really constant? Do they vary with energy or time?

2) How many fundamental constants are there? This questions has provided lots of discussions. One of the most famous was this one:


The trialogue (or dialogue if you are precise with words) above discussed the opinions by 3 eminent physicists about the number of fundamental constants: Michael Duff suggested zero, Gabriel Veneziano argued that there are only 2 fundamental constants while L.B. Okun defended there are 3 fundamental constants

3) Should the cosmological constant be included as a new fundamental constant? The cosmological constant behaves as a constant from current cosmological measurements and cosmological data fits, but is it truly constant? It seems to be…But we are not sure. Quintessence models (some of them related to inflationary Universes) suggest that it could vary on cosmological scales very slowly. However, the data strongly suggest that

P_\Lambda=-\rho c^2

It is simple, but it is not understood the ultimate nature of such a “fluid” because we don’t know what kind of “stuff” (either particles or fields) can make the cosmological constant be so tiny and so abundant (about the 72% of the Universe is “dark energy”/cosmological constant) as it seems to be. We do know it can not be “known particles”. Dark energy behaves as a repulsive force, some kind of pressure/antigravitation on cosmological scales. We suspect it could be some kind of scalar field but there are many other alternatives that “mimic” a cosmological constant. If we identify the cosmological constant with the vacuum energy we obtain about 122 orders of magnitude of mismatch between theory and observations. A really bad “prediction”, one of the worst predictions in the history of physics!

Be natural and stay tuned!

LOG#053. Derivatives of position.

Position or displacement and its various derivatives define an ordered hierarchy of meaningful concepts. There are special names for the derivatives of position (first derivative is called velocity, second derivative is called acceleration, and some other derivatives with proper name), up to the eighth derivative and down to the -9th derivative (ninth integral).

We are going to study the derivatives of position and their corresponding names and special meaning in Physmatics.

0th derivative is position

In Physics, displacement or position is the vector that specifies the change in position of a point, particle, or object. The position vector directs from the reference point to the present position.

A sensor is said to be displacement-sensitive when it responds to absolute position.

For example, whereas a dynamic microphone is a velocity receiver (responds to the derivative of sound pressure or position), a carbon microphone is a displacement receiver in the sense that it responds to sound pressure or diaphragm position itself. The physical dimension of position vector or the distance is length, i.e., \left[\mathbf{x}\right]=\left[ d\right]=L

1st derivative is velocity

Velocity is defined as the rate of change of position or the rate of displacement. It is a vector physical quantity, both speed and direction are required to define it. In the SI(metric)  system, it is measured in meters per second (m/s).

The scalar absolute value (magnitude)  of velocity is called speed. For example, “5 metres per second” is a speed and not a vector, whereas “5 metres per second east” is a vector. The average velocity (v) of an object moving through a displacement \Delta x in a straight line during a time interval \Delta t is described by the formula:

\mathbf{v}_m=\dfrac{\Delta \mathbf{x}}{\Delta t}

Therefore,  velocity is change in position per unit of time. If the change is made “infinitesimally”, i.e., taking two very close points in time, we can define the instantanous velocity ( a.k.a, the derivative) as the limit of the average speed or two very close points when the time interval tends to zero:

\displaystyle{\mathbf{v}=\lim_{\Delta t\rightarrow 0}\dfrac{\Delta \mathbf{x}}{\Delta t}\equiv \dfrac{d\mathbf{x}(t)}{dt}}

Most piano-style music keyboards are approximately velocity-sensitive, within a certain specific, though limited range of key travel, i.e. to a first-order approximation, a note is made louder by hitting a key faster. Most electronic music keyboards are also velocity sensitive, and measure the time interval between switch contact closures at two different positions of key travel on each key.

The physical dimensions of velocity are  \left[\mathbf{v}\right]=LT^{-1}

2nd derivative is acceleration

Acceleration is defined as the rate of change of velocity. It is thus a vector quantity with dimension LT^{-2}. We can define average aceleration and instantaneous acceleration in the same way we did with the velocity:

\mathbf{a}_m=\dfrac{\Delta \mathbf{v}}{\Delta t}

\displaystyle{\mathbf{a}=\lim_{\Delta t\rightarrow 0}\dfrac{\Delta \mathbf{v}}{\Delta t}\equiv \dfrac{d\mathbf{v}(t)}{dt}}

In SI units acceleration is measured in m/s^2. The term “acceleration” generally refers to the change in instantaneous velocity. Average acceleration can also be defined with the above formula.

The physical dimensions of acceleration are \left[\mathbf{a}\right]=LT^{-2}.

3rd derivative is jerk

Jerk (sometimes called jolt in British English, but less commonly so, due to possible confusion with use of the word to also mean electric shock), surge or lurch, is the rate of change of acceleration; more precisely, the derivative of acceleration with respect to time, the second derivative of velocity or the third derivative of displacement. Jerk is described by the following equations:



1) \mathbf{a} is the acceleration.

2) \mathbf{v} is the velocity.

3) \mathbf{x} is the position or displacement.

4) t is the time parameter.

Physical dimensions of jerk are \left[\mathbf{j}\right]=LT^{-3}.

4th derivative is jounce

Jounce (also known as snap) is the fourth derivative of the position vector with respect to time, with the first, second, and third derivatives being velocity, acceleration, and jerk, respectively; in other words, jounce is the rate of change of the jerk with respect to time.


Physical dimensions of snap are \left[\mathbf{s}\right]=LT^{-4}

5th and beyond: Higher-order derivatives

Following jounce (snap), the fifth and sixth derivatives of the displacement vector are sometimes referred to as crackle and pop, respectively. Dork has also been suggested for the sixth derivative. Although the reasons given were less than entirely sincere, dork does have an appealing ring to it, specially for geeks, freaks and dorks. The seventh and eighth derivatives of the displacement vector are sometimes referred to as lock and drop. Their respective formulae can be obtained in a simple way from the previous formalism.

In general, physical dimensions of higher order derivatives of position are defined to be quantities with \left[\mathbf{Q}\right]=LT^{-r}, for any integer number r greater or equal than zero.

-1st derivative (integral) of position is absement

Absement (or absition) refers to the -1th time-derivative of displacement (or position), i.e. the integral of position over time. Mathematically speaking:

\displaystyle{\mathbf{A}=\int \mathbf{x}dt}

The rate of change of absement is position. Absement is a quantity with dimension LT. In SI units, absement is measured in ms or metre seconds.

One meter-second corresponds to being absent from an origin or other reference point 1 meter away for a duration of one second. This amount of absement is equal to being two metres away from the origin for one half second, or being one half a metre from the origin for two seconds, or a 1mm absence for 1000 seconds, a 1km absence for 1 millisecond, and so on.

The word “absement” is a blend of the words absence and displacement.

The physical dimensions of absement are \left[\mathbf{A}\right]=LT.

Useful applications of absement

Whereas most musical keyboard instruments, such as the piano, and many electronic keyboards, respond to velocity at which keys are struck, and some such as the tracker-organ, respond to displacement (how far down a key is pressed), flow-based musical instruments such as the hydraulophone, respond to the integral of displacement, i.e. to a time-distance product. Thus “pressing” a key (water jet) on a hydraulophone down for a longer period of time will result in a buildup of the sound level, as fluid (water) begins to fill the sounding mechanism (reservoir), up to a certain maximum filling point beyond which the sound levels off (along with a slow decay). Hydraulophone reservoirs have an approximate integrating effect on the distance or displacement applied by the musician’s fingers to the “keys” (water jets). Whereas the piano provides more articulation and enunciation of individual note-onsets than the organ, the hydraulophone provides a more continuously fluidly varying sound than either the organ or piano.

Of course all these models are approximate: hydraulophones are approximately presement-responsive, pianos are approximately velocity-responsive, etc..

The concepts of absement and presement originated in regards to flow-based musical instruments like hydraulophones, but may be applied to any area of physics, as they exist along the hierarchy of the derivatives of displacement.

A very slow-responding pipe-organ with tracker-action can often exhibit an effect similar to that of a hydraulophone, when it takes time for the wind and sound levels to build up, so that the sound level is approximately the product of how far down a key is pressed and how long it is held down for.

The concept of absement may also be applied to communications theory. For example, the difficulty in maintaining a communications channel (wired or wireless) increases with distance as well as with the time for which the channel must be kept active.

As a crude but simple example, absement may be used, very approximately, to model the cost of a long-distance phone call as the product of distance and time. A short-duration call over a long distance might, for example, represent the same quantity of absement as a long-duration call over a shorter distance.

Absement may also be used in sociological studies, i.e. we might express loneliness or homesickness as a product of distance from home and time away from home. Simply put, the old aphorism “absence makes the heart grow fonder” has been expressed as “absement makes the heart grow fonder”, to suggest that it matters both how absent one is (i.e. how far), as well as for how long one is absent.

Absement versus presement

Absement refers to the time-distance product (or more precisely the integral of displacement) away from a reference point, whereas the integral of reciprocal position, called presement, refers to the closeness, compounded over time.

The word “presement” is a portmanteau constructed from the words presence and displacement.

Placement (scalar quantity, nearness) is defined as the reciprocal of the position’s magnitude ( i.e., the reciprocal of the distance, an scalar quantity), and presement refers to the time-integral of placement. Most notably, with some high-pressure hydraulophones, it is physically impossible to fully obstruct a water jet, so position can never reach zero, and thus placement remains finite, as does its time integral, presement.

\mbox{Placement}\equiv \dfrac{1}{d}

\displaystyle{\mbox{Presement}=\int dt \dfrac{1}{d}}

and where d is the distance d=\sqrt{x^2+y^2+z^2}, with the origin fixed to the zero vector. Simply put, absement is the time-integral of farness, and presement is the time-integral of nearness, to a given point (e.g. farness or nearness of a musicians finger to/from the exit port of a water jet in a hydraulophone).

Physical dimensions of placement are \left[\mbox{Placement}\right]=L^{-1} while the physical dimensions of presement are \left[\mbox{Presement}\right]=L^{-1}T

Lower-order derivatives (higher-order integrals)

Some hydraulophones, such as the North Nessie (the hydraulophone on the North side of hydraulophone circle) at the Ontario Science Centre consist of cascaded hydraulophonic mechanisms, resulting in a double-integrating effect. In particular, the hydraulophone is linked indirectly to the North pipes, such that the water in direct physical contact with the fingers of the musician is not the same water in the organ pipes. As a result of this indirection, the instrument itself responds to presement/absement, the first integral of position whereas the pipes respond absemently to the action in the instrument, i.e. to the second integral of position of the player’s fingers. The time-integral of the time-integral of position is called absity/presity.

Absity is a portmanteau formed from the words absement (or absence) and velocity.

Following this pattern, higher time integrals of displacement may be named as follows:

1) Absement or absition is the integral of displacement.

2) Absity is the double integral of displacement.

3) Abseleration is the triple integral of displacement.

4) Abserk is the fourth integral of displacement.

5) Absounce is the fifth integral of displacement.

Likewise, presement, presity, preseleration, and similar words, are the integrals of reciprocal displacement (nearness).

Although there are no three-stage hydraulophones currently being manufactured as products, there are a number of three-stage (and some with higher numbers of stages) hydraulophone prototypes, in which some elements of the sound production respond to absity/presity, abseleration/preseleration, etc.

Derivatives of momentum

In Physics, momentum is defined as the product of mass and velocity, i.e.,


or mathematically speaking


Moreover, we define the concept of “force” as the rate of change of momentum with respect to time, i.e.,


It mass does not depend on the time, we get \mathbf{F}=m\mathbf{a}

Can we define names for the next derivatives of momentum with respect to time? Of course, we can. It is only a nominal issue. There is a famous “poem” about this:

Momentum equals mass times velocity. Force equals mass times acceleration. Yank equals mass times jerk. Tug equals mass times snap. Snatch equals mass times crackle. Shake equals mass times pop.

If mass is not constant, the common definitions of higher derivatives of momentum are as follows ( the last equality is obtained supposing the mass is constant with time):

0th time derivative of momentum is of course The Momentum itself ( I am sorry, Mom-entum is not related with your Mom).

\mathbf{p}=m\mathbf{v}=\dfrac{d^0\mathbf {p}}{dt^0}.

1st time derivative of momentum is The Force ( I am sorry. It is a Star Wars joke).


2nd time derivative of momentum is The Yank ( I am sorry, it is not a tank or a yankie from USA).


3rd time derivative of momentum is The Tug ( I am sorry. It is not a bug in the deepest part of The Matrix).


4th time derivative of momentum is The Snatch ( I am sorry, it is not the golden Snitch).


5th time derivative of momentum is The Shake ( I am sorry, it is not the japanese sake or a sweet tropical milk-shake).


Notations for derivatives/integrals

Lebiniz operational notation: f(x) has a derivative with respect to x written as \dfrac{df}{dx}. Then, the derivative is denoted as the operator D=\dfrac{d}{dx}. Higher order derivatives and integrals can be defined recursively:

D^2=\left(\dfrac{d}{dx}\right)^2\equiv \dfrac{d}{dx}\left(\dfrac{d}{dx}\right)=\dfrac{d^2}{dx^2}

D^r=\left(\dfrac{d}{dx}\right)^r\equiv \underbrace{\dfrac{d}{dx}\cdots\left(\dfrac{d}{dx}\right)}_\text{r-times}=\dfrac{d^r}{dx^r}, \;\; \forall r\geq 0

\displaystyle{D^{-1}=\int dx}

\displaystyle{D^{-2}=\int d^2x=\int (dx)^2=\int dx dx'}

\displaystyle{D^{-r}=\int d^rx=\int (dx)^r=\int dx\cdots dx^{(r)}=\int \underbrace{dx\cdots}_\text{r-times}}

Newton dot notation: Derivatives are marked as dotted functions, e.g.,

\dot{f}=\dfrac{df}{dx} \ddot{f}=\dfrac{d^2f}{dx^2} \dddot{f}=\dfrac{d^3f}{dx^3} and so on. Integrals are written in the usual form we do today.

Modern primed notation: Derivatives are marked as primed functions, e.g.,

f'=\dfrac{df}{dx} f''=\dfrac{d^2f}{dx^2} f'''=\dfrac{d^3f}{dx^3} and so on. Integrals are written in the usual form we do today.

Modern sublabel notation: Derivatives are marked with a subindex label denoting the variable with respect to we are making the derivative. Integrals are represented in the usual form. Thus,

f_x=\dfrac{df}{dx} f_{xx}=\dfrac{d^2f}{dx^2} f_{xxx}=\dfrac{d^3f}{dx^3} and so on.

These notations have their own advantanges and disadvantanges, but if we use them carefully, any of them can be very powerful.

Remarkable relationships

Physicists like to relate physical quantities in Mechanics/Dynamics to 4 main variables: force, power, action and energy. We can even dedude some interesting relationships between them and displacement, time, momentum, absement, placement, and presement.

1) Equations relating force and other magnitudes. Force dimensions are MLT^{-2}. Then, we have the identities:



2) Equations relating power and other magnitudes. Power dimensions are ML^2T^{-3}. We easily get:



3) Equations relating action and other magnitudes. Action dimensions are ML^2T^{-1}. We obtain in this case:

\mbox{Action}=\mbox{Energy}\times \mbox{Time}=\mbox{Displacement}\times\mbox{Momentum}=\mbox{Power}\times\mbox{(Time)}^2

\mbox{Action}=\mbox{Force}\times \mbox{Absement}=\dfrac{\mbox{Momentum}}{\mbox{Placement}}=\mbox{Mass}\times\begin{pmatrix}\mbox{Areolar}\\ \mbox{Velocity}\end{pmatrix}

4) Equations relating energy and other magnitudes. Energy dimensions are ML^2T^{-2}. We deduce from this last case


\mbox{Energy}=\mbox{Momentum}\times \mbox{Velocity}=\mbox{Power}\times\mbox{Time}


In the same way, we can also deduce more fascinating identities:



since we easily get


\mbox{Absement}=\mbox{Presement}\times \mbox{(Displacement)}^2=L^{-1}TL^2=LT


and of course


Moreover, we also have





and the next interesting result as well:

\boxed{(\mbox{Placement})(\mbox{Presement})=\begin{pmatrix}\mbox{Areolar}\\ \mbox{Velocity}\end{pmatrix}^{-1}=L^{-2}T}

or equivalently

\boxed{\begin{pmatrix}\mbox{Areolar}\\ \mbox{Velocity}\end{pmatrix}=v_A=\dfrac{1}{\mbox{Placement}\times \mbox{Presement}}=L^2T^{-1}}

Music, elements and Physics

The inspiring guide to the new names and variables was the theory of hydraulophones and music. In fact, there is a recent proposal to classify every musical instrument according to its physical origin instead of the classical element. It also makes sense to present the four states-of-matter in increasing order of energy: Earth/Solid first, Water/Liquid second, Air/Gas third, and Fire/Plasma fourth. At absolute zero, if it were possible, everything is a solid. then as things heat up they melt, then they evaporate, and finally, with enough energy, would become a ball of plasma, thus establishing a natural physical ordering as follows:

1) Earth/Solid played instruments. Geolophones. They produce sound pulsing the matter (“Earth”) of some object (string, membrane,…). Ordered in increasing dimension, from 1d to 3d, they can be: I) Chordophones (Played strings, streched objects with cross-section negligible respect to their length), II) Membranophones (Played membranes with thickness negligible respect to their area), III) Idiophones/Bulkphones (played 3d tensionless branes or higher).

2) Water/Liquid played instruments. Hydraulophones. These instruments produce vibrating sound pulsing jets of liquids (“Water”).

3) Air/Gas played instruments. Aerophones. These instuments produce vibrations and sound touching the flux of gases (“Air”).

4) Fire/Plasma played instruments. Ionophones. These instruments produce sonic waves playing the flux of plasma (“Fire”).

5) Quintessence/Idea/Information/Informatics played instruments.  These instruments produce “sound”  by computational means, whether optical, mechanical, electrical, or otherwise. We could name these instruments with some cool word. Akashaphones (from the sanskrit word/prefix “akasha”, meaning “aether, ether” or as Western tradition would say, “quintessence, fifth element”) will be the names of such instruments.

This classification matches the range of acoustic transducers that exist today (excepting the quintessencial transducer, of course) as well: 1) Geophone, 2) Hydrophone, 3) Microphone or speaker, and 4) Ionophone. In the same way I have never known a term for the akashaphones before, for the fifth transducer we should use a new term. Loakashaphone, from the same sanskrit origin than akashaphone, would be the analogue 5th transducer.


The following list is a summary of the derivatives of displacement/position:

A) Time integrals of position/displacement.

Order -9. Absrop. SI units ms^9.  Time integral of absock. Dimensions: LT^9.

Order -8. Absock. SI units ms^8. Time integral of absop. Dimensions: LT^8.

Order -7. Absop. SI units ms^7. Time integral of absrackle. Dimensions: LT^7.

Order -6.  Absrackle. SI units ms^6. Time integral of absounce. Dimensions: LT^6.

Order -5. Absounce. SI units ms^5. Time integral of abserk. Dimensions: LT^5.

Order -4. Abserk. SI units ms^4. Time integral of abseleration. Dimensions: LT^4.

Order -3. Abseleration. SI units ms^3. Time integral of absity. Dimensions: LT^3.

Order -2. Absity. SI units ms^2. Time integral of absement. Dimensions: LT^2.

Order -1. Absement. SI units ms. Time integral of position. Dimensions: LT.

Order 0. Position/Displacement. SI units m. Dimensions: L.

Remark: Integrals with respect to time of position measure “farness”.

B) Time derivatives of position/displacement.

Order 0. Position/Displacement. SI units m. Dimensions: L.

Order 1. Velocity. SI units m/s. Rate of change of position. Dimensions: LT^{-1}.

Order 2. Acceleration. SI units m/s^2. Rate of change of velocity. Dimensions: LT^{-2}.

Order 3. Jerk/jolt/surge/lurch. SI units m/s^3. Rate of change of acceleration. Dimensions: LT^{-3}.

Order 4. Jounce/snap. SI units m/s^4. Rate of change of jerk. Dimensions: LT^{-4}.

Order 5. Crackle. SI units m/s^5. Rate of change of jounce. Dimensions: LT^{-5}.

Order 6. Pop. SI units m/s^6. Rate of change of crackle. Dork has also been suggested for the sixth derivative. Although the reasons given were less than entirely sincere, dork does have an appealing ring to it. Dimensions: LT^{-6}.

Order 7. Lock. SI units m/s^7. Rate of change of pop. Dimensions: LT^{-7}.

Order 8. Drop. SI units m/s^8. Rate of change of lock. Dimensions: LT^{-8}.

Remark: Derivatives of position with respect to time measure “swiftness”.

C) Reciprocals of position/displacement and their time integrals.

Order 0. Placement. SI units m^{-1}. Placement (scalar quantity, nearness) is the reciprocal of position (scalar quantity distance), i.e., 1/x. Dimensions: L^{-1}.

Order -1. Presement. SI units m^{-1}s. Time integral of placement. Dimensions: L^{-1}T.

Order -2. Presity. SI units m^{-1}s^2. Time integral of presement. Dimensions: L^{-1}T^2.

Order -3. Preseleration. SI units m^{-1}s^3. Time integral of presity. Dimensions: L^{-1}T^3.

Order -4. Preserk. SI units m^{-1}s^4. Time integral of preseleration. Dimensions: L^{-1}T^4.

Order -5. Presounce. SI units m^{-1}s^5. Time integral of preserk. Dimensions: L^{-1}T^5.

Order -6. Presackle. SI units m^{-1}s^6. Time integral of presounce. Dimensions: L^{-1}T^6.

Order -7. Presop. SI units m^{-1}s^7. Time integral of presackle. Dimensions: L^{-1}T^7.

Order -8. Presock. SI units m^{-1}s^8. Time integral of presop. Dimensions: L^{-1}T^8.

Order -9. Presrop. SI units m^{-1}s^9. Time integral of presock. Dimensions: L^{-1}T^9.

Remark: Integrals of reciprocal displacement with respect to time measure “nearness”.

D) Time derivatives of momentum.

Order 0. Momentum. \mathbf{p}. SI units kgms^{-1}. Momentum equals mass times velocity. Dimensions: MLT^{-1}, where M denotes mass dimension.

Order 1. Force. \mathbf{F}. SI units are newtons. N=kg\cdot ms^{-2}. Time derivative of momentum, or rate of change of momentum with respect to time. Dimensions: MLT^{-2}.

Order 2. Yank. \mathbf{Y}. SI units N\cdot s^{-1}=kgms^{-3}. Time integral of presement. Rate of change of force with respect to time. Dimensions: MLT^{-3}.

Order 3. Tug. \mathbf{T}. SI units N\cdot s^{-2}=kgms^{-4}. Rate of change of yank with respect to time. Dimensions: MLT^{-4}.

Order 4. Snatch. \mathbf{S}. SI units N\cdot s^{-3}=kgms^{-5}. Rate of change of tug with respect to time. Dimensions: MLT^{-5}.

Order 5. Shake. \mathbf{Sh}. SI units N\cdot s^{-4}=kgms^{-6}. Rate of change of snatch with respect to time. Dimensions: MLT^{-6}.

Remark: Derivatives of momentum with respect to time measure “strengthness” or “forceness”.

So we have to remember 4 fascinating ideas,

i) Time integrals  of position measure “farness”.

ii) Time derivatives of position measure “swiftness”.

iii) Time integrals of reciprocal position measure “nearness”.

iv) Time derivatives of momentum measure “forceness”.

And a fifth further great idea… Physics, Mathematics or more generally Physmatics own an inner “Harmony” or “Music” in their deepest principles and theories.

Some additional questions can be asked further:

0th. What about “infinite” order derivatives and integrals?

1st. What if time is not a continuous function?

2nd. What if time is not a scalar quantity?

3rd. What about fractional order/irrational order/complex order derivatives/X-order derivatives?

4th. What if (space) time/displacement does not exist?

5th. Can Mechanics/Dynamics of particles/fields/strings/branes/… be formulated in terms of integrals/reciprocals of “position” and “momentum” variables, i.e., as the power of negative and/or higher/lower derivatives? Would such a formulation of Mechanics/Dynamics be useful/meaningful for something deeper? That is, what are the right variables to study in Dynamics if some classical/quantum concepts are absent?

We could answer to some of these questions. For instance, the answer to the 0th question is interesting but it requires to know about jet spaces and/or path integrals. Moreover, the solution to the 3rd question would require the introduction of the fractional/fractal calculus. But that is another long story/log-entry to be told in a forthcoming future post!

Stay tuned!