# 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:

https://thespectrumofriemannium.wordpress.com/2012/11/18/log054-barrow-units/

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:

$\boxed{c^2=\dfrac{1}{\sqrt{\varepsilon_0\mu_0}}}$

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:

$\boxed{c=299,792,458m/s}$

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

$\mathbf{P}^2c^2-E^2=-m^2c^4$

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
$hc=1.98644568\times 10^{-25}J\cdot m$
$hc=1.23984193 eV\cdot \mu m$

or

$\hbar c=0.1591549hc$ 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

$F_C=k_C\dfrac{Qq}{r^2}$

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

$k_B=1_{k_B}=1$

and thus the complete HEP system is defined if you set

$\boxed{c=\hbar=k_B=1}$

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

$\boxed{\left[L\right]=\left[T\right]=\left[E\right]^{-1}}$

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:

$\boxed{\left[E\right]^n}$

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

$\left[\partial_\mu\right]=\left[\Phi\right]=\left[A^\mu\right]=1$

$\left[\Psi_D\right]=\left[\Psi_M\right]=\left[\chi\right]=\left[\eta\right]=\dfrac{3}{2}$

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:

$\boxed{e=c=G_N=4\pi\varepsilon_0=1}$

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.,

$\boxed{G_N=c=\hbar=k_B=1}$

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.

Moreover:

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:

$\boxed{G_N=c=1}$

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

## Conversion Factors

This table from wikipedia is very useful:

where:

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]

$k_B/c^2$[kg/K]

Conversion from m, kg, C, K into s

$1/c$ [s/m]

$G_N/c^3$[s/kg]

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

$G_Nk_B/c^5$ [s/K]

Conversion from m, kg, s, K into C

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

$(G_N4\pi\varepsilon_0)^{1/2}$ [C/kg]

$c^3/(G_N/(4\pi\varepsilon_0))^{1/2}$[C/s]

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

Conversion from m, kg, s, C into K

$c^4/(G_Nk_B)$[K/m]

$c^2/k_B$ [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:

and

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:

http://arxiv.org/abs/physics/0110060

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#033. Electromagnetism in SR.

The Maxwell’s equations and the electromagnetism phenomena are one of the highest achievements and discoveries of the human kind. Thanks to it, we had radio waves, microwaves, electricity, the telephone, the telegraph, TV, electronics, computers, cell-phones, and internet. Electromagnetic waves are everywhere and everytime (as far as we know, with the permission of the dark matter and dark energy problems of Cosmology). Would you survive without electricity today?

The language used in the formulation of Maxwell equations has changed a lot since Maxwell treatise on Electromagnetis, in which he used the quaternions. You can see the evolution of the Mawell equations “portrait” with the above picture. Today, from the mid 20th centure, we can write Maxwell equations into a two single equations. However, it is less know that Maxwell equations can be written as a single equation $\nabla F=J$ using geometric algebra in Clifford spaces, with $\nabla =\nabla \cdot +\nabla\wedge$, or the so-called Kähler-Dirac-Clifford formalism in an analogue way.

Before entering into the details of electromagnetic fields, let me give some easy notions of tensor calculus. If $x^2=\mbox{invariant}$, how does $x^\mu$ transform under Lorentz transformations? Let me start with the tensor components in this way:

$x^\mu e_\mu=x^{\mu'}e_{\mu'}=\Lambda^{\mu'}_{\;\; \nu}x^\mu e_{\mu'}=\Lambda^{\mu'}_{\;\; \mu}x^\mu e_{\mu'}$

Then:

$e_\mu=\Lambda^{\mu'}_{\;\; \mu} e_{\mu'}\rightarrow e_{\mu'}=\left(\Lambda^{-1}\right)_{\;\; \mu'}^{\mu}e_\mu=\left[\left(\Lambda^{-1}\right)^T\right]^{\;\; \mu}_{\nu}e_\mu$

Note, we have used with caution:

1st. Einstein’s convention: sum over repeated subindices and superindices is understood, unless it is stated some exception.

2nd. Free indices can be labelled to the taste of the user segment.

3rd. Careful matrix type manipulations.

We define a contravariant vector (or tensor (1,0) ) as some object transforming in the next way:

$\boxed{a^{\mu'}=\Lambda^{\mu'}_{\;\; \nu}a^\nu}\leftrightarrow\boxed{a^{\mu'}=\left(\dfrac{\partial x^{\mu'}}{\partial x^\nu}\right)a^\nu}$

where $\left(\dfrac{\partial x^{\mu'}}{\partial x^\nu}\right)$ denotes the Jabobian matrix of the transformation.
In similar way, we can define a covariant vector ( or tensor (0,1) ) with the aid of the following equations

$\boxed{a_{\mu'}=\left[\left(\Lambda^{-1}\right)^{T}\right]_{\mu'}^{\:\;\; \nu}a_\nu}\leftrightarrow\boxed{a_{\mu'}=\left(\dfrac{\partial x^{\nu}}{\partial x^{\mu'}}\right)a_\nu}$

Note: $\left(\dfrac{\partial x^{\nu}}{\partial x^{\mu'}}\right)=\left(\dfrac{\partial x^{\mu'}}{\partial x^{\nu}}\right)^{-1}$

Contravariant tensors of second order ( tensors type (2,0)) are defined with the next equations:

$\boxed{b^{\mu'\nu'}=\Lambda^{\mu'}_{\;\; \lambda}\Lambda^{\nu'}_{\;\; \sigma}b^{\lambda\sigma}=\Lambda^{\mu'}_{\;\; \lambda}b^{\lambda\sigma}\Lambda^{T \;\; \nu'}_{\sigma}\leftrightarrow b^{\mu'\nu'}=\dfrac{\partial x^{\mu'}}{\partial x^\lambda}\dfrac{\partial x^{\nu'}}{\partial x^\sigma}b^{\lambda\sigma}}$

Covariant tensors of second order ( tensors type (0,2)) are defined similarly:

$\boxed{c_{\mu'\nu'}=\left(\left(\Lambda\right)^{-1}\right)^{T \;\;\lambda}_{\mu'}\left(\left(\Lambda\right)^{-1T}\right)^{\;\; \sigma}_{\nu'}c_{\lambda\sigma}=\left(\Lambda^{-1T}\right)^{\;\; \lambda}_{\mu'}c_{\lambda\sigma}\Lambda^{-1 \;\; \nu'}_{\sigma}\leftrightarrow c_{\mu'\nu'}=\dfrac{\partial x^{\lambda}}{\partial x^{\mu'}}\dfrac{\partial x^{\sigma}}{\partial x^{\nu'}}c_{\lambda\sigma}}$

Mixed tensors of second order (tensors type (1,1)) can be also made:

$\boxed{d^{\mu'}_{\;\; \nu'}=\Lambda^{\mu'}_{\;\; \lambda}\left(\left(\Lambda\right)^{-1T}\right)^{\;\;\;\; \sigma}_{\nu'}d^{\lambda}_{\;\;\sigma}=\Lambda^{\mu'}_{\;\; \lambda}d^{\lambda}_{\;\;\sigma}\left(\left(\Lambda\right)^{-1}\right)^{\sigma}_{\;\;\; \nu'}\leftrightarrow d^{\mu'}_{\;\; \nu'}=\dfrac{\partial x^{\mu'}}{\partial x^{\lambda}}\dfrac{\partial x^{\sigma}}{\partial x^{\nu'}}d^{\lambda}_{\;\; \sigma}}$

We can summarize these transformations rules in matrix notation making the transcript from the index notation easily:

1st. Contravariant vectors change of coordinates rule: $X'=\Lambda X$

2nd. Covariant vectors change of coordinates rule: $X'=\Lambda^{-1T} X$

3rd. (2,0)-tensors change of coordinates rule: $B'=\Lambda B \Lambda^T$

4rd. (0,2)-tensors change of coordinates rule: $C'=\Lambda^{-1T}C\Lambda^{-1}$

5th. (1,1)-tensors change of coordinates rule: $D'=\Lambda D \Lambda^{-1}$

Indeed, without taking care with subindices and superindices, and the issue of the inverse and transpose for transformation matrices, a general tensor type (r,s) is defined as follows:

$\boxed{T^{\mu'_1\mu'_2\ldots \mu'_r}_{\nu'_1\nu'_2\ldots \nu'_s}=L^{\nu_s}_{\nu'_s}\cdots L^{\nu_1}_{\nu'_1}L^{\mu'_r}_{\mu_r}\cdots L^{\mu'_1}_{\mu_1}T^{\mu_1\mu_2\ldots\mu_r}_{\nu_1\nu_2\ldots \nu_s}}$

We return to electromagnetism! The easiest examples of electromagnetic wave motion are plane waves:

$x=x_0\exp (iKX)=x_0\exp (ix^\mu p_\mu)$

where $\phi=XK=KX=X\cdot K=x^\mu p_\mu=\mathbf{k}\cdot\mathbf{r}-\omega t$

Indeed, the cuadrivector K can be “guessed” from the phase invariant ($\phi=\phi'$ since the phase is a dot product):

$K=\square \phi$

where $\square$ is the four dimensional nabla vector defined by

$\square=\left(\dfrac{\partial}{c\partial t},\dfrac{\partial}{\partial x},\dfrac{\partial}{\partial y},\dfrac{\partial}{\partial z}\right)$

and so

$K^\mu=(K^0,K^1,K^2,K^3)=(\omega/c,k_x,k_y,k_z)$

Now, let me discuss different notions of velocity when we are considering electromagnetic fields, beyond the usual notions of particle velocity and observer relative motion, we have the following notions of velocity in relativistic electromagnetism:

1st. The light speed c. It is the ultimate limit in vacuum and SR to the propagation of electromagnetic signals. Therefore, it is sometimes called energy transfer velocity in vacuum or vacuum speed of light.

2nd. Phase velocity $v_{ph}$. It is defined as the velocity of the modulated signal in a plane wave, if $\omega =\omega (k)=\sqrt{c^2\mathbf{k}-K^2}$, we have

$v_{ph}=\dfrac{\omega (\mathbf{k})}{k}$ where k is the modulus of $\mathbf{k}$. It measures how much fast the phase changes with the wavelength vector.

From the definition of cuadrivector wave length, we deduce:

$K^2=\omega^2\left(\dfrac{1}{v_p}-\dfrac{1}{c^2}\right)$

Then, we can rewrite the distinguish three cases according to the sign of the invariant $K^2$:

a) $K^2>0$. The separation is spacelike and we get $v_p.

b)$K^2=0$. The separation is lightlike or isotropic. We obtain $v_p=c$.

c)$K^2<0$. The separation is timelike. We deduce that $v_p>c$. This situation is not contradictory with special relativity since phase oscillations can not transport information.

3rd. Group velocity $v_g$. It is defined like the velocity that a “wave packet” or “pulse” has in its propagation. Therefore,

$v_g=\dfrac{dE}{dp}=\dfrac{d\omega}{dk}$

where we used the Planck relationships for photons $E=\hbar \omega$ and $p=\hbar k$, with $\hbar=\dfrac{h}{2\pi}$

4th. Particle velocity. It is defined in SR by the cuadrivector $U=\gamma (c,\mathbf{v})$

5th. Observer relative velocity, V. It is the velocity (constant) at which two inertial observes move.

There is a nice relationship between the group velocity, the phase velocity and the energy transfer, the lightspeed in vacuum. To see it, look at the invariant:

$K^2=\mathbf{k}^2-\omega^2/c^2$

Deriving this expression, we get $v_g=d\omega/dk=kc^2/\omega=c^2/v_{ph}$

so we have the very important equation

$\boxed{v_gv_{ph}=c^2}$

Other important concept in electromagnetism is “light intensity”. Light intensity can be thought like the “flux of light”, and you can imagine it both in the wave or particle (photon corpuscles) theory in a similar fashion. Mathematically speaking:

$\mbox{Light intensity=Flux of light}=\dfrac{\mbox{POWER}}{\mbox{Area}}\rightarrow I=\dfrac{\mathcal{P}}{A}=\dfrac{E/V}{tA/V}=\dfrac{uV}{tA}=uc$

so $I=uc$ where u is the energy density of the electromagnetic field and c is the light speed in vacuum. By Lorentz transformations, it can be showed that for electromagnetic waves, energy, wavelength, energy density and intensity change in the following way:

$E'=\sqrt{\dfrac{1+\beta}{1-\beta}}E$

$\lambda'=\sqrt{\dfrac{1-\beta}{1+\beta}}\lambda$

$u'= \dfrac{E'}{\lambda' NA}=\dfrac{1-\beta}{1+\beta}\dfrac{E}{N\lambda A}$

$I'=\dfrac{1-\beta}{1+\beta}I$

The relativistic momentum can be related to the wavelength cuadrivector using the Planck relation $P^\mu=\hbar K^\mu$. Under a Lorentz transformation, momenergy transforms $P'=\Lambda P$. Assign to the wave number vector $\mathbf{k}$ a direction in the S-frame:

$\vert \mathbf{k}\vert \left( \cos \theta, \sin \theta, 0 \right)=\dfrac{\omega}{c}\left(\cos\theta,\sin\theta,0\right)$

and then

$K^\mu=\dfrac{\omega}{c}\left(1,\cos\theta,\sin\theta,0\right)$

In matrix notation, the whole change is written as:

$\begin{pmatrix}\dfrac{\omega'}{c}\\ k'_x\\ k'_y\\k'_z\end{pmatrix}=\begin{pmatrix}\gamma & -\beta\gamma & 0 & 0\\ -\beta\gamma & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0& 1\end{pmatrix}\dfrac{\omega}{c}\begin{pmatrix}1\\ \cos\theta\\ \sin\theta\\ 0\end{pmatrix}$

so

$K'\begin{cases}\omega'=\gamma \omega(1-\beta\cos\theta)\\ \;\\ k'_x=\gamma\dfrac{\omega}{c}\\ \;\\ k'_y=\dfrac{\omega}{c}\sin\theta\\\;\\ k'_z=0\end{cases}$

Using the first two equations, we get:

$k'_x=\dfrac{\omega'}{c}\dfrac{\cos\theta-\beta}{1-\beta\cos\theta}$

Using the first and the third equation, we obtain:

$k'_y=\dfrac{\omega'}{c}\dfrac{\sin\theta}{\gamma\left(1-\beta\cos\theta\right)}$

Dividing the last two equations, we deduce:

$\dfrac{k'_x}{k'_y}=\dfrac{\sin\theta}{\gamma\left(\cos\theta-\beta\right)}=\dfrac{u'_y}{u'_x}=\tan\theta'$

This formula is the so-called stellar aberration formula, and we will dedicate it a post in the future.

If we write the first equation with the aid of frequency f (and $f_0$) instead of angular frequency,

$f=f_0\dfrac{1}{\gamma(1-\beta\cos\theta)}$

where we wrote the frequency of the source as $\omega'=2\pi f_0$ and the frequency of the receiver as $\omega=2\pi \nu$. This last formula is called the relativistic Doppler shift.

Now, we are going to introduce a very important object in electromagnetism: the electric charge and the electric current. We are going to make an analogy with the momenergy $\mathbb{P}=m\gamma\left(c,\mathbf{v}\right)$. The cuadrivector electric current is something very similar:

$\mathbb{J}=\rho_0\gamma\left(c,\mathbf{u}\right)=\rho\left(c,\mathbf{u}\right)=\left(\rho c,\mathbf{j}\right)$

where $\rho=\gamma \rho_0$ is the electric current density, and $\mathbf{u}$ is the charge velocity. Moreover, $\rho_0=nq$ and where $q$ is the electric charge and $n=N/V$ is the electric charge density number, i.e., the number of “elementary” charges in certain volume. Indeed, we can identify the components of such a cuadrivector:

$\mathbb{J}=\left(J^0,J^1,J^2,J^3\right)=\rho_0\left(c\gamma,\gamma\mathbf{v}\right)=\rho_0\gamma\left(c,\mathbf{u}\right)$. We can make some interesting observations. Suppose certain rest frame S where we have $\rho=\rho_++\rho_-=0$, i.e., a frame with equilibred charges $\rho_+=-\rho_-$, and suppose we move with the relative velocity of the electron (or negative charge) observer. Then $u=v(e)$ and $j_x=\rho_-v$, while the other components are $j_y=j_z=0$. Then, the charge density current transforms as follows:

$\begin{pmatrix}\rho'c\\ j'_x\\ j'_y\\j'_z\end{pmatrix}=\begin{pmatrix}\gamma & -\beta\gamma & 0 & 0\\ -\beta\gamma & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0& 1\end{pmatrix}\begin{pmatrix}0\\ \rho_- v\\ 0\\ 0\end{pmatrix}=\begin{pmatrix}-\gamma \beta \rho_- v\\ \gamma \rho_- v\\ 0\\ 0\end{pmatrix}$

and

$\rho'=-\gamma \beta^2\rho_-=\gamma\beta^2\rho_+$

$j'_x=\gamma\rho_- v=-\gamma \rho_+ v$

We conclude:

1st. Length contraction implies that the charge density increases by a gamma factor, i.e., $\rho_+\rightarrow \rho_+\gamma$.

2nd. The crystal lattice “hole” velocity $-v$ in the primed frame implies the existence in that frame of a current density $j'_x=-\gamma \rho_+ v$.

3rd. The existence of charges in motion when seen from an inertial frame (boosted from a rest reference S) implies that in a moving reference frame electric fields are not alone but with magnetic fields. From this perspective, magnetic fields are associated to the existence of moving charges. That is, electric fields and magnetic fields are intimately connected and they are caused by static and moving charges, as we do know from classical non-relativistic physics.

Remember now the general expression of the FORPOWER tetravector, or Power-Force tetravector, in SR:

$\mathcal{F}=\mathcal{F}^\mu e_\mu=\gamma\left(\dfrac{\mathbf{f}\cdot\mathbf{v}}{c},f_x,f_y,f_z\right)$

and using the metric, with the mainly plus convention, we get the covariant componets for the power-force tetravector:

$\mathcal{F}_\mu=\gamma\left(-\dfrac{\mathbf{f}\cdot\mathbf{v}}{c},f_x,f_y,f_z\right)$

We define the Lorentz force as the sum of the electric and magnetic forces

$\mathbf{f}_L=\mathbf{f}_e+\mathbf{f}_m=q\mathbf{E}+\mathbf{v}\times \mathbf{B}$

Noting that $(\mathbf{v}\times\mathbf{B})\cdot \mathbf{v}=0$, the Power-Force tetravector for the Lorentz electromagnetic force reads:

$\mathcal{F}_L=\mathcal{F}^\mu e_\mu=\gamma q\left(\dfrac{\mathbf{E}\cdot{\mathbf{v}}}{c},\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)$

And now, we realize that we can understand the electromagnetic force in terms of a tensor (1,1), i.e., a matrix, if we write:

$\mathcal{F}=\dfrac{q}{c}\mathbb{F}\mathbb{U}$

so

$\begin{pmatrix}\mathcal{F}^0\\ \mathcal{F}^1\\ \mathcal{F}^2\\ \mathcal{F}^3\end{pmatrix}=\dfrac{q}{c}\begin{pmatrix}0 &E_x & E_y & E_z\\ E_x & 0 & cB_z& -cB_y\\ E_y & -cB_z & 0 & cB_x\\ E_z & cB_y& -cB_x& 0\end{pmatrix}\begin{pmatrix}\gamma c\\ \gamma v_x\\ \gamma v_y\\ \gamma v_z\end{pmatrix}$

Therefore, $\mathcal{F}^\mu=\dfrac{q}{c}F^\mu_{\;\; \nu}U^\nu\leftrightarrow \mathcal{F}=\dfrac{q}{c}\mathbb{F}\mathbb{U}$

where the components of the (1,1) tensor can be read:

$\mathbb{F}=\mathbf{F}^\mu _{\;\; \nu}=\begin{pmatrix}0& E_x& E_y& E_z\\ E_x & 0 & cB_z& -cB_y\\ E_y& -cB_z& 0 & cB_x\\ E_z & cB_y& -cB_x& 0\end{pmatrix}$

We can lower the indices with the metric $\eta=diag(-1,1,1,1)$ in order to have a more “natural” equation and to read the symmetry of the electromagnetic tensor $F_{\mu\nu}$ (note that we can not study symmetries with indices covariant and contravariant),

$\mathbf{F}_{\mu\nu}=\eta_{\mu \alpha}\mathbf{F}^{\alpha}_{\;\; \nu}$

with

$\mathbf{F}_{\mu\nu}=\begin{pmatrix}0& -E_x& -E_y& -E_z\\ E_x & 0 & cB_z& -cB_y\\ E_y& -cB_z& 0 & cB_x\\ E_z & cB_y& -cB_x& 0\end{pmatrix}$

Similarly

$\mathbf{F}^{\mu\nu}=\mathbf{F}^{\alpha}_{\;\; \beta}\eta^{\beta \nu}=\begin{pmatrix}0& E_x& E_y& E_z\\ -E_x & 0 & cB_z& -cB_y\\ -E_y& -cB_z& 0 & cB_x\\ -E_z & cB_y& -cB_x& 0\end{pmatrix}$

Please, note that $F_{\mu\nu}=-F_{\nu\mu}$. Focusing on the components of the electromagnetic tensor as a tensor type (1,1), we have seen that under Lorentz transformations its components change as $F'=LFL^{-1}$ under a boost with $\mathbf{v}=\left(v,0,0\right)$ in such a case. So, we write:

$\boxed{F'=\begin{pmatrix}\gamma & -\beta\gamma & 0 & 0\\ -\beta\gamma & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0& 1\end{pmatrix}\begin{pmatrix}0& E_x& E_y& E_z\\ E_x & 0 & cB_z& -cB_y\\ E_y& -cB_z& 0 & cB_x\\ E_z & cB_y& -cB_x& 0\end{pmatrix}\begin{pmatrix}\gamma & \beta\gamma & 0 & 0\\ \beta\gamma & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0& 1\end{pmatrix}}$

$\boxed{F'=\mathbf{F}^{\mu'}_{\;\; \nu'}=\begin{pmatrix}0& E_x& \gamma (E_y-vB_z)& \gamma (E_z+vB_y)\\ E_x & 0 & c\gamma(B_z-\frac{v}{c^2}E_y) & -c\gamma (B_y+\frac{v}{c^2}E_z)\\ \gamma (E_y-vB_z)& -c\gamma (B_z-\frac{v}{c^2}E_y) & 0 & cB_x\\ \gamma (E_z+vB_y) & c\gamma (B_y+\frac{v}{c^2}E_z)& -cB_x& 0\end{pmatrix}}$

From this equation we deduce that:

$\mbox{EM fields after a boost}\begin{cases}E_{x'}=E_x,\; \; E_{y'}=\gamma \left( E_y-vB_z\right),\;\; E_{z'}=\gamma \left(E_z+vB_y\right)\\ B_{x'}=B_x,\;\; B_{y'}=\gamma \left(B_y+\frac{v}{c^2}E_z\right),\;\;B_{z'}=\gamma \left(B_z-\frac{v}{c^2}E_y\right)\end{cases}$

Example: In the S-frame we have the fields $E=(0,0,0)$ and $B=(0,B_y,0)$. The Coulomb force is $f_C=qE=(0,0,0)$ and the Lorentz force is $f_L=(0,0,qvB_y)$. How are these fields seen from the S’-frame? It is easy using the above transformations. We obtain that

$E'=(0,0,\gamma vB_y)$, $B'=(0,\gamma B_y,0)$, $f'_C=qE'=(0,0,\gamma qvB_y)$, $f'_L=(0,0,0)$

Surprinsingly, or not, the S’-observer sees a boosted electric field (non null!), a boosted magnetic field,  a boosted non-null Coulomb force and a null Lorentz force!

We can generalize the above transformations to the case of a general velocity in 3d-space $\mathbf{v}=(v_x,v_y,v_z)$

$\mathbf{E}_{\parallel'}=\mathbf{E}_\parallel$ $\mathbf{B}_{\parallel'}=\mathbf{B}_{\parallel}$

$\mathbf{E}_{\perp'}=\gamma \left[\mathbf{E}_\perp+(\mathbf{v}\times \mathbf{B})_\perp\right]=\gamma \left[\mathbf{E}_\perp+(\mathbf{v}\times \mathbf{B})\right]$

$\mathbf{B}_{\perp'}=\gamma \left[\mathbf{B}_\perp-\dfrac{1}{c^2}(\mathbf{v}\times \mathbf{E})_\perp\right]=\gamma \left[\mathbf{B}_\perp-\dfrac{1}{c^2}(\mathbf{v}\times \mathbf{E})\right]$

The last equal in the last two equations is due to the orthogonality of the position vector to the velocity in 3d space due to the cross product. From these equations, we easily obtain:

$E_\parallel=\dfrac{(v\cdot E)v}{v^2}=\dfrac{\beta\cdot E}{\beta^2}$

$E_\perp=E-E_\parallel=E-\dfrac{(v\cdot E)v}{v^2}$

and similarly with the magnetic field. The final tranformations we obtain are:

$\boxed{E'=E_{\parallel'}+E_{\perp'}=\dfrac{(v\cdot E)v}{v^2}+\gamma \left[ E-\dfrac{(v\cdot E)v}{v^2}+v\times B\right]}$

$\boxed{B'=B_{\parallel'}+B_{\perp'}=\dfrac{(v\cdot B)v}{v^2}+\gamma \left[ B-\dfrac{(v\cdot B)v}{v^2}-v\times E\right]}$

Equivalently

$\boxed{E'=\gamma \left(E+v\times B\right)-\left(\gamma-1\right)\dfrac{\left(v\cdot E\right) v}{v^2}}$

$\boxed{B'=\gamma \left(B-v\times \dfrac{E}{c^2}\right)-\left(\gamma-1\right)\dfrac{\left(v\cdot B\right) v}{v^2}}$

In the limit where $c\rightarrow \infty$ or $\dfrac{v}{c}\rightarrow 0$, we get that

$E'=E+v\times B$ $B'=B-\dfrac{v\times E}{c^2}$

There are two invariants for electromagnetic fields:

$I_1=\mathbf{E}\cdot\mathbf{B}$ and $I_2=\mathbf{E}^2-c^2\mathbf{B}^2$

It can be checked that

$\mathbf{E}\cdot\mathbf{B}=\mathbf{E}'\cdot\mathbf{B}'=invariant$

and

$\mathbf{E}^2-c^2\mathbf{B}^2=\mathbf{E'}^2-c^2\mathbf{B'}^2=invariant$  under Lorentz transformations. It is obvious since, up to a multiplicative constant,

$I_1=\dfrac{1}{4} F^\star_{\mu\nu}F^{\mu\nu}=\dfrac{1}{8}\epsilon_{\mu\nu\sigma \tau}F^{\sigma \tau}F^{\mu\nu}=\dfrac{1}{2}tr \left(F^{\star T}F\right)$

$I_2=\dfrac{1}{2}F_{\mu\nu}F^{\mu\nu}=tr\left(F^TF\right)$

and where we have defined the dual electromagnetic field as

$\star F=F^\star_{\mu\nu}=\dfrac{1}{2}\epsilon_{\mu\nu \sigma \tau}F^{\sigma \tau}$

or if we write it in components ( duality sends $\mathbf{E}$ to $\mathbf{B}$ and $\mathbf{B}$ to $-\mathbf{E}$)

$\star F=F^\star_{\mu\nu}=\begin{pmatrix}0& -B_x& -B_y& -B_z\\ B_x & 0 & -cE_z& cE_y\\ B_y& cE_z& 0 & -cE_x\\ B_z & -cE_y& cE_x& 0\end{pmatrix}$

We can guess some consequences for the electromagnetic invariants:

1st. If $E\perp B$, then $E\cdot B=0$ and thus $E_\perp B$ in every frame! This fact is important, since it shows that plane waves are orthogonal in any frame in SR. It is also the case of electromagnetic radiation.

2nd. As $E\cdot B=\vert E\vert \vert B\vert \cos \varphi$ can be in the non-orthogonal case either positive or negative. If $E\cdot B$ is positive, then it will be positive in any frame and similarly in the negative case. Morevoer, a transformation into a frame with $E=0$ (null electric field) and/or $B=0$ (null magnetic field) is impossible. That is, if a Lorentz transformation of the electric field or the magnetic field turns it to zero, it means that the electric field and magnetic field are orthogonal.

3rd. If E=cB, i.e., if $E^2-c^2B^2=0$, then it is valid in every frame.

4th. If there is a electric field but there is no magnetic field B in S, a Lorentz transformation to a pure B’ in S’ is impossible and viceversa.

5th. If the electric field is such that $E>cB$ or $E, then they can be turned in a pure electric or magnetic frame only if the electric field and the magnetic field are orthogonal.

6th. There is a trick to remember the two invariants. It is due to Riemann. We can build the hexadimensional vector( six-vector, or sixtor) and complex valued entity

$\boxed{\mathbf{F}=\mathbf{E}+ic\mathbf{B}}$

The two invariants are easily obtained squaring F:

$F^2=\mathbf{E}^2-c^2\mathbf{B}^2+2ic\mathbf{E}\cdot\mathbf{B}=invariant$

We can introduce now a vector potencial tetravector:

$\mathbb{A}=A^\mu e_\mu=\left( A^0,A^1,A^2,A^3\right)=\left(\dfrac{V}{c},\mathbf{A}\right)=\left(\dfrac{V}{c},A_x,A_y,A_z\right)$

This tetravector is also called gauge field. We can write the Maxwell tensor in terms of this field:

$F_{\mu\nu}=\partial_\mu A_\nu-\partial _\nu A_\mu$

It can be easily probed that, up to a multiplicative constant in front of the electric current tetravector, the first set of Maxwell equations are:

$\boxed{\partial_\mu F^{\mu\nu}=j^\nu \leftrightarrow \square \cdot \mathbf{F}=\mathbb{J}}$

The second set of Maxwell equations (sometimes called Bianchi identities) can be written as follows:

$\boxed{\partial_\mu F^{\star \mu \nu}=\dfrac{1}{2}\epsilon^{\nu\mu\alpha\beta}\partial_\mu F_{\alpha\beta}=0}$

The Maxwell equations are invariant under the gauge transformations in spacetime:

$\boxed{A^{\mu'}=A^\mu+e\partial^\mu \Psi}$

where the potential tetravector and the function $\Psi$ are arbitrary functions of the spacetime.

Some elections of gauge are common in the solution of electromagnetic problems:

A) Lorentz gauge: $\square \cdot A=\partial_\mu A^\mu=0$

B) Coulomb gauge: $\nabla \cdot \mathbf{A}=0$

C) Temporal gauge: $A^0=V/c=0$

If we use the Lorentz gauge, and the Maxwell equations without sources, we deduce that the vector potential components satisfy the wave equation, i.e.,

$\boxed{\square^2 A^\mu=0 \leftrightarrow \square^2 \mathbb{A}=0}$

Finally, let me point out an important thing about Maxwell equations. Specifically, about its invariance group. It is known that Maxwell equations are invariant under Lorentz transformations, and it was the guide Einstein used to extend galilean relativity to the case of electromagnetic fields, enlarging the mechanical concepts. But, the larger group leaving invariant the Maxwell equation’s invariant is not the Lorentz group but the conformal group. But it is another story unrelated to this post.