LOG#070. Natural Units.

NaturalCup

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”:

SI_base_unit.svg

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:

ConversionTableNatUnits

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:

GeosystemConvFactors1

and

GeomSystemEMfactors

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”:

Fritsch-XingPlotFundConstants

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

fritz-xingPlotFromTalk

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:

OkunCube2

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):

Okuncube

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!

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LOG#041. Muons and relativity.

QUESTION: Is the time dilation real or is it an artifact of our current theories?

There are solid arguments why time dilation is not an apparent effect but a macroscopic measurable effect. Today, we are going to discuss the “reality” of time dilation with a well known result:

Muon detection experiments!

Muons are enigmatic elementary particles from the second generation of the Standard Model with the following properties:

1st. They are created in upper atmosphere at altitudes of about 9000 m, when cosmic rays hit the Earth and they are a common secondary product in the showers created by those mysterious yet cosmic rays.
2nd. The average life span is 2\times 10^{-6}s\approx 2ms
3rd. Typical speed is 0.998c or very close to the speed of light.
So we would expect that they could only travel at most d=0.998c\times 2 \times 10^{-6}\approx 600m
However, surprisingly at first sight, they can be observed at ground level! SR provides a beautiful explanation of this fact. In the rest frame S of the Earth, the lifespan of a traveling muon experiences time dilation. Let us define

A) t= half-life of muon with respect to Earth.

B) t’=half-life of muon of the moving muon (in his rest frame S’ in motion with respect to Earth).

C) According to SR, the time dilation means that t=\gamma t', since the S’ frame is moving with respect to the ground, so its ticks are “longer” than those on Earth.

A typical dilation factor \gamma for the muon is about 15-100, although the value it is quite variable from the observed muons. For instance, if the muon has v=0.998c then \gamma \approx 15. Thus, in the Earth’s reference frame, a typical muon lives about 2×15=30ms, and it travels respect to Earth a distance

d'=0.998c\times 30ms\approx 9000m.

If the gamma factor is bigger, the distance d’ grows and so, we can detect muons on the ground, as we do observe indeed!

Remark:  In the traveling muon’s reference frame, it is at rest and the Earth is rushing up to meet it at 0.998c. The distance between it and the Earth thus is shorter than 9000m by length contraction. With respect to the muon, this distance is therefore 9000m/15 = 600m.

An alternative calculation, with approximate numbers:

Suppose muons decay into other particles with half-life of about 0.000001sec. Cosmic ray muons have speed now about v = 0.99995 c.
Without special relativity, muon would travel

d= 0.99995 \times 300000 km/s\times 0.00000156s=0.47 km only!

Few would reach earth’s surface in that case. It we use special relativity, then plugging the corresponding gamma for v=0.99995c, i.e.,  \gamma =100, then muons’ “tics” run slower and muons live 100 times longer. Then, the traveled distance becomes

d'=100\times 0.9995\times 300000000 m/s\times 0.000001s= 30000m

Conclusion: a lot of muons reach the earth’s surface. And we can detect them! For instance, with the detectors on colliders, the cosmic rays detectors, and some other simpler tools.


LOG#034. Stellar aberration.

In this entry, we are going to study a relativistic effect known as “stellar aberration”.

From the known Lorentz transformations of velocities (inverse case), we get:

v_x=\dfrac{v'_x+V}{1+\dfrac{v'_xV}{c^2}}

v_y=\dfrac{v'_y\sqrt{1-\beta^2}}{1+\dfrac{v'xV}{c^2}}

v_z=\dfrac{v'_z\sqrt{1-\beta^2}}{1+\dfrac{v'zV}{c^2}}

The classical result (galilean addition of velocities) is recovered in the limit of low velocities V\approx 0 or sending the light speed get the value “infinite” c\rightarrow \infty. Then,

v_x=v'_x+V v_y=v'_y v'_z=v_z

Let us define

\theta =\mbox{angle formed by x}\; \mbox{and}\; v_x

\theta' =\mbox{angle formed by x'}\; \mbox{and}\; v'_x

Thus, we get the component decomposition into the xy and x’y’ planes:

v_x=v\cos\theta v_y=v\sin\theta

v'_x=v'\cos\theta' v'_y=v'\sin\theta'

From this equations, we get

\tan \theta=\dfrac{v'\sin\theta'\sqrt{1-\beta^2}}{v'\cos\theta'+V}

If v=v'=c

\boxed{\tan \theta=\dfrac{\sin\theta'\sqrt{1-\beta_V^2}}{\cos\theta'+\beta_V}}

and then

\boxed{\cos\theta=\dfrac{\beta_V+\cos\theta'}{1+\beta_V\cos\theta'}}

\boxed{\sin\theta=\dfrac{\sin\theta'\sqrt{1-\beta_V^2}}{1+\beta_V\cos\theta'}}

From the last equation, we get

\sin\theta'\sqrt{1-\beta_V^2}=\sin\theta\left(1+\beta_V\cos\theta'\right)

From this equation, if V<<c, i.e., if \beta_V<<1 and \theta'=\theta+\Delta\theta with \Delta \theta<<1, we obtain the result

\Delta \theta=\theta'-\theta=\beta_V\sin\theta

By these formaulae, the angle of a light beam propagating in space depends on the velocity of the source respect to the observer. We can observe this relativistic effect every night (supposing a good approximation that Earth’s velocity is non-relativistic, as it shows). The physical interpretation of the above aberration formulae (for the stars we watch during a skynight) is as follows: due to the Earth’s motion, a star in the zenith is seen under an angle \theta\neq \dfrac{\pi}{2}.

Other important consequence from the stellar aberration is when we track ultra-relativistic particles (\beta\approx 1). Then, \theta'\rightarrow \pi and then, the observer moves close to the source of light. In this case, almost every star (excepting those behind with \theta=\pi) are seen “in front of” the observer. If the source moves with almost the speed of light, then the light is “observed” as it were concentrated in a little cone with an aperture \Delta\theta\sim\sqrt{1-\beta_V^2}


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<c.

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<cB, 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.


LOG#032. Invariance and relativity.

Invariance, symmetry and invariant quantities are in the essence, heart and core of Physmatics. Let me begin this post with classical physics. Newton’s fundamental law reads:

\mathbf{F}=m\mathbf{a}=\begin{cases}m\ddot{x}, \;\; m\ddot{x}=m\dfrac{d^2x}{dt^2}\\\;\\m\ddot{y}, \;\; m\ddot{y}=m\dfrac{d^2y}{dt^2}\\\;\\ m\ddot{z}, \;\; m\ddot{z}=m\dfrac{d^2z}{dt^2}\end{cases}

Suppose two different frames obtained by a pure translation in space:

\mathbf{x}'=\mathbf{x}-\mathbf{x}_0

or

\mathbf{r}'=\mathbf{r}-\mathbf{r}_0

We select to make things simpler

\mathbf{x}_0=\mathbf{r}_0=(x_0,0,0)

We can easily observe by direct differentiation that Newton’s fundamental is invariant under translations in space, since mere substitution provides:

\mathbf{F}'=\mathbf{F}

since

m\dfrac{d^2\mathbf{r}'}{dt^2}=m\dfrac{d^2\mathbf{r}}{dt^2} \leftrightarrow \mathbf{a}'=\mathbf{a}

By the other hand, rotations around a fixed axis, say the z-axis, are transformations given by:

\boxed{\mathbf{r}'=R\mathbf{r}\leftrightarrow \begin{pmatrix}x'\\y'\\z'\end{pmatrix}=R\begin{pmatrix}x\\y\\z\end{pmatrix}\rightarrow\begin{cases}x'=x\cos\theta+y\sin\theta\\y'=-x\sin\theta+y\cos\theta\\z'=z\end{cases}}

If we multiply by the mass these last equations and we differentiate with respect to time twice, keeping constant \theta and m, we easily get

\mathbf{F}'=R\mathbf{F}\leftrightarrow \begin{cases}F_{x'}=F_x\cos\theta+F_y\cos\theta\\F_{y'}=-F_x\sin\theta+F_y\cos\theta\\F_{z'}=F_z\end{cases}

or

\boxed{\begin{pmatrix}F'_x\\F'_y\\F'_z\end{pmatrix}=\begin{pmatrix}\cos\theta & \sin\theta & 0\\ -\sin\theta & \cos\theta & 0\\ 0 & 0& 1 \end{pmatrix}\begin{pmatrix}F_x\\F_y\\F_z\end{pmatrix}\rightarrow \begin{cases}F'_x=F_x\cos\theta+F_y\sin\theta\\F'_y=-F_x\sin\theta+F_y\cos\theta\\F'_z=F_z\end{cases}}

Thus, we can say that Newton’s fundamental law is invariant under spatial translations and rotations. Its form is kept constant under those kind of transformations. Generally speaking, we also say that Newton’s law is “covariant”, but nowadays it is an abuse of language since the word covariant means something different in tensor analysis. So, be aware about the word “covariant” (specially in old texts). Today, we talk about “invariant laws”, or about the symmetry of certain equations under certain set of (group) transformations.

Newton’s law use the concept of acceleration:

\mathbf{a}=\dfrac{d\mathbf{v}}{dt}=\left(\dfrac{d}{dt}\right)\left(\dfrac{d}{dt}\right)\mathbf{r}=\dfrac{d^2\mathbf{r}}{dt^2}

with

a_x=\dfrac{dv_x}{dt}=\dfrac{d^2x}{dt^2} a_y=\dfrac{dv_y}{dt}=\dfrac{d^2y}{dt^2} a_z=\dfrac{dv_z}{dt}=\dfrac{d^2z}{dt^2}

or, in compact form

a_i=\dfrac{dv_i}{dt},\;\; i=x,y,z

And then, the following equations are invariant under translations in space and rotations:

\mathbf{F}=m\mathbf{a} or \mathbf{F}=\dfrac{d\mathbf{p}}{dt} with \mathbf{p}=m\mathbf{v}

Intrinsic components of the aceleration provide a decomposition

\mathbf{a}=\mathbf{a}_\parallel+\mathbf{a}_\perp

where we define

a_\parallel=\dfrac{dv}{dt}\leftrightarrow \mathbf{a}_\parallel=\dfrac{dv}{dt}\mathbf{u}_\parallel

where \mathbf{u}_\parallel is a unit vector in the direction of the velocity, and

\mathbf{a}_\perp=\mathbf{a}-\mathbf{a}_\parallel

In the case of motion along a general curve, we can approximate the motion in every point of the curve by a circle of radius R, and thus

s=R\Delta \theta\rightarrow \Delta \theta=\dfrac{s}{R}=\dfrac{v\Delta t}{R}\rightarrow \dfrac{\Delta \theta}{\Delta t}=\omega=\dfrac{v}{R}

By the other hand,

\Delta v_\perp=v\Delta \theta\rightarrow a_\perp=v\dfrac{\Delta \theta}{\Delta t}=\dfrac{v^2}{R}=\omega^2R

and we get the known expression for the centripetal acceleration:

\mathbf{a}_\perp=a_\perp\mathbf{u}_\perp=\dfrac{v^2}{R}\mathbf{u}_\perp

More about invariant quantities in Classical Physics: the scalar (sometimes called dot) product of two vectors is invariant, since the length of every vector is constant in euclidean spaces under rotations and translations. For instance,

\boxed{r^2=x^2+y^2+z^2=\mathbf{r}\cdot\mathbf{r}=\mathbf{r'}\cdot\mathbf{r'}=\mbox{INVARIANT}=\mbox{SQUARED LENGTH}}

In matrix form,

r^2=X^TX=\delta _{ij}x^ix^j=\begin{pmatrix}x & y & z\end{pmatrix}\begin{pmatrix}1 &0& 0\\ 0 &1& 0\\ 0& 0& 1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}

where we have introduced the \delta_{ij} symbol to be the so-called Kronecker delta as “certain object” with components: its components are “1” whenever i=j and “0” otherwise. Of course, the Kronecker delta symbol “is” the identity matrix when the symbol have two indices. However, let me remark that “generalized delta Kronocker” with more indices do exist and it is not always posible to express easily that “tensor” in a matrix way, excepting using some clever tricks.

The scalar (dot) product can be computed with any vector quantity:

\mathbf{a}\cdot\mathbf{a}=a^2=a_x^2+a_y^2+a_z^2\rightarrow \mathbf{a}\cdot\mathbf{b}=a_xb_x+a_yb_y+a_zb_z

Moreover, there is a coordinate free definition as well:

\mathbf{a}\cdot\mathbf{b}=ab\cos\theta,\;\; \theta=\mbox{angle formed by}\; \mathbf{a},\mathbf{b}

Note that the invariance of the dot product implies the invariance of classical kinetic energy, since:

K.E.=T=\dfrac{1}{2}m\mathbf{v}\cdot\mathbf{v}=\dfrac{\mathbf{p}\cdot\mathbf{p}}{2m}=\dfrac{1}{2}mv^2=\dfrac{1}{2m}p^2=\mbox{INVARIANT}

We have also the important invariant quantities:

\mbox{WORK}=W=\int \mathbf{F}\cdot d\mathbf{r}

\mbox{POWER}=P=\dfrac{dW}{dt}=\mathbf{F}\cdot\mathbf{v}

where the second equality holds if the force is constant along the trajectory. Moreover, in relativistic electromagnetism, you also get the wave-number 4-vector:

\boxed{\mathbb{K}=(K^0,\mathbf{K})}\leftrightarrow \mbox{WAVE NUMBER SPACETIME VECTOR}

and the invariant \mathbb{K}\cdot\mathbb{K}=K^2, that you can get from the plane wave solution:

A=A_0\exp (i(K^\mu X_\mu))=A_0\exp (i(\mathbb{K}\cdot \mathbb{X}))

where the phase invariant reads

\phi =\mathbb{K}\cdot \mathbb{X}=\mathbf{K}\cdot\mathbf{X}-\omega t

Therefore, we deduce that

K^0=\dfrac{\omega}{c}

and the wave number  vector satisfies the following relation with the wave-length

\vert \mathbf{K}\vert =K=\dfrac{2\pi}{\lambda}

There is another important set of transformations or symmetry in classical physics. It is related to inertial frames. Galileo discovered that the laws of motion are the same for every inertial observer, i.e., the laws of Mechanics are invariant for inertial frames! A Galilean transformation is defined by:

\boxed{\mbox{GALILEAN TRANSFORMATIONS}\begin{cases}\mathbf{x}'=\mathbf{x}-\mathbf{V}t\\t'=t\end{cases}}

where \mathbf{V}=constant. Differentiating with respect to time, we get

\boxed{\mbox{GALILEAN TRANSFORMATIONS}\begin{cases}\dfrac{d\mathbf{x}'}{dt}=\dfrac{d\mathbf{x}}{dt}-\mathbf{V}\\ \;\\ \dfrac{dt'}{dt}=1\end{cases}}

and then

\boxed{\mbox{GALILEAN TRANSFORMATIONS}\begin{cases}\dfrac{d^2\mathbf{x}'}{dt^2}=\dfrac{d^2\mathbf{x}}{dt^2}\\ \;\\\dfrac{d^2t'}{dt^2}=0\end{cases}}

And thus, the accelerations (and forces) that observe different inertial ( i.e., reference frames moving with constant relative velocity) frames are the same

\mathbf{a}'=\mathbf{a}

And now, about symmetry. What are the symmetries of Physics? There are many interesting transformations and space-time symmetries. A non-completely exhaustive list is this one:

1. Translations in space.

2. Translations in time.

3. Rotations around some axis ( and with fixed angle).

4. Uniform velocity in straight line, a.k.a., galilean transformations for inertial observers. This symmetry “becomes” Lorentz boosts in the spacetime analogue of special relativity.

5. Time reversal ( inversion of the direction of time), T.

6. Reflections in space (under “a mirror”). It is also called parity P.

7. Matter-antimatter interchange, or charge conjugation symmetry, C.

8. Interchange of identical atoms/particles.

9. Scale transformations \mathbb{X}'=\lambda\mathbb{X}.

10. Conformal transformations (in the complex plane or in complex spaces).

11. Arbitrary coordinate transformations (they are also called general coordinate transformations).

12. Quantum-mechanical (gauge) phase symmetry: \Psi\rightarrow \Psi'=\Psi \exp (i\theta).

Beyon general vectors, in classical physics we also find “axial” vectors (also called “pseudovectors”). Pseudovectors or axial vectors are formed by the 3d “cross”/outer/vector  product:

\mathbf{C}=\mathbf{A}\times \mathbf{B}=\begin{vmatrix}e_1 & e_2 & e_3\\ A_x & A_y & A_z\\ B_x & B_y & B_z \end{vmatrix}=e_1\begin{vmatrix}A_y & A_z\\ B_y & B_z\end{vmatrix}-e_2\begin{vmatrix}A_x & A_z\\ B_x & B_z\end{vmatrix}+e_3\begin{vmatrix}A_x & A_y\\ B_x & B_y\end{vmatrix}

Some examples are the angular momentum

\mathbf{L}=\mathbf{r}\times\mathbf{p}=\mathbf{r}\times m\mathbf{v}

or the magnetic force

\mathbf{F}_m=q\mathbf{v}\times \mathbf{B}

Interestingly, the main difference between axial and polar vectors, i.e., between common vectors and pseudovectors is the fact that under P (parity), pseudovectors are “invariant” while common vectors change their sign, i.e., polar vectors become the opposite vector under reflection, and pseudovectors remain invariant. It can be easily found by inspection in the definition of angular momentum or the magnetic force ( or even the general definition of cross product given above).

Now, we turn our attention to invariants in special relativity. I will introduce a very easy example to give a gross idea of how the generalization of “invariant theory” works as well in special relativity. From the classical definition of power:

P=\dfrac{dE}{dt}=\mathbf{F}\cdot\mathbf{v}

Using the relativistic definition of 4-momentum:

\mathbb{P}=\left(Mc,\dfrac{d\mathbf{p}}{d\tau}\right)=\left(E/c,M\mathbf{v}\right)

where M=m\gamma, we are going to derive a known result, since E=Mc^2. Note that, in agreement with classical physics, from this

\mathbf{F}=\dfrac{d\mathbf{P}}{dt}

Therefore, inserting the relativistic expressions for energy and force into the power equation, we obtain:

\dfrac{d(Mc^2)}{dt}=\mathbf{v}\cdot\dfrac{d(M\mathbf{v})}{dt}

Multiplying by 2M, and using the Leibniz rule for the differentiation of a product of two functions:

2M\dfrac{d(Mc^2)}{dt}=2M\mathbf{v}\cdot\dfrac{d(M\mathbf{v})}{dt}=2\mathbf{P}\cdot\dfrac{d\mathbf{P}}{dt}

or equivalently

2Mc^2\dfrac{dM}{dt}=2\mathbf{P}\cdot\dfrac{d\mathbf{P}}{dt}=\dfrac{d(\mathbf{P})^2}{dt}

and so

c^2\dfrac{dM^2}{dt}=\dfrac{d(\mathbf{P})^2}{dt}

Integrating this, we deduce that

M^2c^2=M^2v^2+\mbox{constant}

If we plug \mbox{constant}=m^2c^2=M^2c^2(v=0)

then

M^2c^2=M^2v^2+m^2c^2\rightarrow M^2=m^2\gamma^2,\;\; \gamma^2=\dfrac{1}{1-\frac{v^2}{c^2}}

and thus, we have rederived the notion of “relativistic mass”

M=m\gamma=\dfrac{1}{\sqrt{1-\frac{v^2}{c^2}}}

Special relativity generalizes the notion of dot product ( scalar product) to a non-euclidean (pseudoeuclidean to be more precise) geometry. The dot product in special relativity is given by:

\boxed{\mathbb{A}\cdot\mathbb{B}=A^x B_x+A^y B_y+A^z B_z-A^t B_t}

The sign of the temporal fourth component is conventional in the sense some people uses a minus sign for the purely spatial components and a positive sign for the temporal component. Using a more advanced notation we can write the new scalar product as follows:

\boxed{A^\mu B_\mu=A_\mu B^\mu=A^x B_x+A^y B_y+A^z B_z-A^t B_t}

where the repeated dummy index implies summation over it. This convention of understanding summation over repeated indices is called Einstein’s covention and it is due to Einstein himself. Another main point about notation is that some people prefer the use of a \mu=0,1,2,3 while other people use \mu=1,2,3,4. We will use the notation with \mu=0,1,2,3 unless we find some notational issue. Unlikely to the 3d world, the 4d world of special relativity forces us to use something different to the Kronecker delta in the above scalar product. This new object is generally called pseudoeuclidean “metric”, or Minkovski metric:

\boxed{A^\mu B_\mu=\eta_{\mu\nu}A^\mu B^\nu=\mathbb{A}\cdot\mathbb{B}=A^xB_x+A^yB_y+A^zB_z-A^tB_t}

In matrix form,

\boxed{A^\mu B_\mu=A^\mu\eta_{\mu\nu}B^\mu=A^T\eta B=\begin{pmatrix}A^t & A^x & A^y & A^z\end{pmatrix}\begin{pmatrix}-1 & 0 & 0 & 0\\ 0& 1 & 0 & 0\\ 0 & 0 & 1& 0\\ 0 & 0& 0& 1\end{pmatrix}\begin{pmatrix}B^t \\ B^x \\ B^y \\ B^z\end{pmatrix}}

Important remarks:

1st. \eta=\eta_{\mu\nu}=diag(-1,1,1,1) in our convention. The opposite convention for the scalar product would give \eta=diag(1,-1,-1,-1).

2nd. The square of a “spacetime” vector is its “lenght” in spacetime. It is given by:

\boxed{A^2=\mathbb{A}\cdot\mathbb{A}=A^\mu A_\mu=A_x^2+A_y^2+A_z^2-A_t^2=-(\mbox{SQUARED SPACETIME LENGTH})}

In particular, for the position spacetime vector

S^2=x^\mu x_\mu=-c^2\tau ^2

3rd. Unlike the euclidean 3d space, 4d noneuclidean spacetime introduces “objects” non-null that whose “squared lenght” is equal to zero, and even weirder, objects that could provide a negative dot product!

4th. For spacetime events given by a spacetime vector \mathbb{X}=x^\mu e_\mu=(ct,\mathbf{r}), and generally for any arbitrary events A and B (or 4-vectors) we can distinguish:

i) Vectors with A^2=\mathbb{A}\cdot\mathbb{A}>0 are called space-like vectors.

ii) Vectors with A^2=\mathbb{A}\cdot\mathbb{A}=0 are called null-vectors, isotropic vectors or sometimes light-like vectors.

iii) Vectos with A^2=\mathbb{A}\cdot\mathbb{A}<0 are called time-like vectors.

Thus, in the case of the spacetime (position) vector, every event can be classified into space-like, light-like (null or isotropic) and time-like types, depending on the sign of s^2=\mathbb{X}\cdot\mathbb{X}=X^T\eta X. Moreover, the metric itself allows us to “raise or lower” indices, defining the following rules for components:

x^\mu=\begin{pmatrix}ct \\ \mathbf{r}\end{pmatrix}\rightarrow x_\mu =\eta _{\mu \nu}x^\nu=x^\nu \eta_{\mu\nu}=X^T\eta=(-ct,\mathbf{r})

The minkovskian metric has a very cool feature too. Its “square” is the identity matrix. That is,

\eta^2=\eta^T\eta=\eta\eta^T=\mathbb{I}

Then, the metric is its own inverse:

\eta=\eta^{-1}

In components, it reads

\eta^{\mu\nu}\eta_{\nu\sigma}=\delta^{\mu}_{\;\;\sigma}

with

\eta^{\mu\nu}=(\eta^{-1})_{\mu\nu}

and where we have introduced the Kronecker delta symbol in four dimensions in the same manner we did in the 3d space. Therefore, the Kronecker delta has only non-null components when \mu=\sigma, so that \delta^0_{\;\;0}=\delta^1_{\;\;1}=\ldots=1

Subindices are called generally “covariant” components, while superindices are called “contravariant” components. It is evident that euclidean spaces don’t distinguish between covariant and contravariant components. The metric is the gadget we use in non-euclidean metric spaces to raise and lower indices/components from tensor quantities. Tensors are multi-oriented objects. The metric itself is a second order tensor, more precisely, the metric is a second order rank 2 covariant tensor. 4-vectors are contravariant object with a single index. Upwards single-indexed tensors are contravariant vectors, downwards single-indexed tensores are covariant vectors. When a metric is introduced, there is no need to distinguish covariant and contravariant tensors, since the components can be calculated with the aid of the metric, so we speak about n-th rank tensors. Multi-indixed objects can have some features of symmetry. The metric is symmetric itself, for instance, under the interchange of subindices ( columns and rows). So, then

\eta_{\mu\nu}=\eta_{\nu\mu}\rightarrow \eta =\eta^T

What kind of general objects can we use in Minkovski spacetime or even more general spaces? Firstly, we have scalar fields or functions, i.e., functions depending only on the spacetime coordinates:

\psi (x)=\psi' (x') \leftrightarrow \psi (x^\mu) =\psi ' (x'^\mu) \leftrightarrow \psi (x,y,z, ct)= \psi ' (x',y',z',ct')

Another objetct we have found are “vectors” or “oriented segments”. In 3d space, they transform as \mathbf{x}'=R\mathbf{x}. In 4d spacetime, we found \mathbb{X}'=L\mathbb{X}.

In 3d space, we also found pseudovectors. They are defined via the cross product, that in components read: c^i=\epsilon ^{ijk}a_jb_k, where the new symbol \epsilon^{ijk} is a completely antisymmetric object under the interchange of any pair of indices (with \epsilon^{123}=+1) is generally called Levi-Civita symbol. This symbol is the second constant object that we can use in any number of dimensions, like the Kronecker delta

\delta ^i_{\;\; j}=\begin{cases}+1,\mbox{if}\; i=j\\ 0,\mbox{otherwise}\end{cases}

The completely antisymmetric Levi-Civita symbol has some interesting identities related with the Kronecker delta. Thus, for instance, in 2d and 3d respectively:

\epsilon^{ij}\epsilon_{ik}=\delta^{j}_{\;\;k}

\epsilon^{ijk}\epsilon_{ilm}=\delta^{j}_{\;\; l}\delta^{k}_{\;\; m}-\delta^{j}_{\;\; m}\delta^{k}_{\;\; l}

or

\epsilon^{ijk}a_ib_jc_k=det(\mathbf{a},\mathbf{b},\mathbf{c})

We have also the useful identity:

\epsilon^{i_1i_2\ldots i_n}\epsilon_{i_1i_2\ldots i_n}=n!

The n-dimensional Levi-Civita symbol is defined as:

\epsilon^{i_1i_2\ldots i_n}a_1a_2\ldots a_n=det(\mathbf{a_{1}},\mathbf{a_2},\ldots,\mathbf{a_{n}})

and its product in n-dimensions

\varepsilon_{i_1 i_2 \dots i_n} \varepsilon_{j_1 j_2 \dots j_n} = \begin{vmatrix}\delta_{i_1 j_1} & \delta_{i_1 j_2} & \dots & \delta_{i_1 j_n} \\\delta_{i_2 j_1} & \delta_{i_2 j_2} & \dots & \delta_{i_2 j_n} \\\vdots & \vdots & \ddots & \vdots \\ \delta_{i_n j_1} & \delta_{i_n j_2} & \dots & \delta_{i_n j_n} \\ \end{vmatrix}

or equivalently, given a nxn matrix A=(a_{ij})

\epsilon^{i_1i_2\ldots i_n}a_{1i_i}a_{2i_2}...a_{ni_n}=det( a_{ij})=\dfrac{1}{n!}\epsilon^{i_1i_2\ldots i_n}\epsilon^{j_1j_2\ldots j_n}a_{i_1 j_1}a_{i_2 j_2}\ldots a_{i_n j_n}

This last equation provides some new quantity called pseudoscalar, different from the scalar function in the sense it changes its sign under parity in 3d, while a common 3d scalar is invariant under parity! Generally speaking, determinants (pseudoscalars) in even dimensions are parity conserving, while determinants in odd dimensions don’t conserve parity.

Like the Kronecker delta, the epsilon or Levi-Civita can be generalized to 4 dimensions (or even to D-dimensions). In 4 dimensions:

\epsilon^{\mu\nu\sigma\tau}=\epsilon_{\mu\nu\sigma\tau}=\begin{cases}+1,\mbox{if} (\mu\nu\sigma\tau)\mbox{is an even permutation of 0,1,2,3}\\-1,\mbox{if} (\mu\nu\sigma\tau)\mbox{is an odd permutation of 0,1,2,3}\\ 0,\mbox{otherwise}\end{cases}

In general, unlike the Kronecker deltas, the Levi-Civita epsilon symbols are not ordinary “tensors” (quantities with subindices and superindices, with some concrete properties under coordinate transformations) but more general entities called “weighted” tensors (sometimes they are also called tensorial densities). Indeed, the generalized Kronecker delta can be defined of order 2p is a type (p,p) tensor that is a completely antisymmetric in its ”p” upper indices, and also in its ”p” lower indices.  This characterization defines it up to a scalar multiplier.

\delta^{\mu_1 \dots \mu_p }_{\;\;\;\;\;\;\;\;\;\; \nu_1 \dots \nu_p} =\begin{cases}+1 & \quad \text{if } \nu_1 \dots \nu_p \text{ are an even permutation of } \mu_1 \dots \mu_p \\-1 & \quad \text{if } \nu_1 \dots \nu_p \text{ are an odd permutation of } \mu_1 \dots \mu_p \\ \;\;0 & \quad \text{in all other cases}.\end{cases}

Using an anti-symmetrization procedure:
\delta^{\mu_1 \dots \mu_p}_{\;\;\;\;\;\;\;\;\;\;\nu_1 \dots \nu_p} = p! \delta^{\mu_1}_{\lbrack \nu_1} \dots \delta^{\mu_p}_{\nu_p \rbrack}

In terms of an pxp determinant:
\delta^{\mu_1 \dots \mu_p }_{\;\;\;\;\;\;\;\;\;\;\nu_1 \dots \nu_p} =\begin{vmatrix}\delta^{\mu_1}_{\nu_1} & \cdots & \delta^{\mu_1}_{\nu_p} \\ \vdots & \ddots & \vdots \\ \delta^{\mu_p}_{\nu_1} & \cdots & \delta^{\mu_p}_{\nu_p}\end{vmatrix}

Equivalently, it could be defined by induction in the following way:

\delta^{\mu \rho}_{\nu \sigma} = \delta^{\mu}_{\nu} \delta^{\rho}_{\sigma} - \delta^{\mu}_{\sigma} \delta^{\rho}_{\nu}

\delta^{\mu \rho_1 \rho_2}_{\nu \sigma_1 \sigma_2} = \delta^{\mu}_{\nu} \delta^{\rho_1 \rho_2}_{\sigma_1 \sigma_2} - \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2}_{\nu \sigma_2} + \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2}_{\sigma_2 \nu}
\delta^{\mu \rho_1 \rho_2 \rho_3}_{\nu \sigma_1 \sigma_2 \sigma_3} = \delta^{\mu}_{\nu} \delta^{\rho_1 \rho_2 \rho_3}_{\sigma_1 \sigma_2 \sigma_3} - \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2 \rho_3}_{\nu \sigma_2 \sigma_3} + \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2 \rho_3}_{\sigma_2 \nu \sigma_3} - \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2 \rho_3}_{\sigma_2 \sigma_3 \nu}
and so on.

In the particular case where  p=n  (the dimension of the vector space), in terms of the Levi-Civita symbol:
\delta^{\mu_1 \dots \mu_n}_{\nu_1 \dots \nu_n} = \varepsilon^{\mu_1 \dots \mu_n}\varepsilon_{\nu_1 \dots \nu_n}

Under a Lorentz transformation, we have ( using matrix notation) the next transformations:

A'=LA \leftrightarrow (A')^T=A^TL^T

\eta A'=\eta LA\rightarrow (A')^T\eta A'=A^T(L^T\eta L)A

A'^T\eta A'=A^T \eta A iff L^T \eta L=\eta

so the metric itself is “invariant” under a Lorentz transformation (boost). I would like to remark that the metric can be built from the basis vectors in the following way:

\eta_{\mu \nu}=e_\mu e_\nu=e_\mu \cdot e_\nu= g\left( e_\mu ,e_\nu\right)=\begin{cases}-1, \mu =\nu =0\\ +1, \mu =\nu =1,2,3\\ 0,\mu \neq \nu \end{cases}

For Lorentz transformations, we get that

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

with

\Lambda^\mu _{\;\;\; \nu}=\begin{pmatrix}\Lambda^0_{\;\;\; 0}& \Lambda^0_{\;\;\; 1}& \Lambda^0_{\;\;\; 2}& \Lambda^0_{\;\;\; 3}\\ \Lambda^1_{\;\;\; 0}& \Lambda^1_{\;\;\; 1}& \Lambda^1_{\;\;\; 2}& \Lambda^1_{\;\;\; 3} \\ \Lambda^2_{\;\;\; 0}& \Lambda^2_{\;\;\; 1}&\Lambda^2_{\;\;\; 2}& \Lambda^2_{\;\;\; 3}\\ \Lambda^3_{\;\;\; 0}& \Lambda^3_{\;\;\; 1}& \Lambda^3_{\;\;\; 2}& \Lambda^3_{\;\;\; 3}\end{pmatrix}

Moreover, the equation \Lambda^{-1}=\eta^{-1}\Lambda \eta, i.e., for pseudo-orthogonal Lorentz transformations, taking the determinant, we deduce that

\det (\Lambda)=\pm 1\leftrightarrow \det (\Lambda)^2=1

We can fully classify the Lorentz transformations according to the sign of the determinant and the sign of the element \Lambda^0_{\;\;\; 0} as follows:

\Lambda\begin{cases} \mbox{Proper Lorentz transf.(e.g.,boosts,3d rotations, Id)}: L^\uparrow_+ \det (\Lambda)=+1, \Lambda^0_{\;\;\; 0}\ge 1\\ \mbox{Improper Lorentz transf.:}\begin{cases}L^\downarrow_+ \det (\Lambda)=+1, \; \Lambda^0_{\;\;\; 0}\le -1\\ L^\uparrow_- \det (\Lambda)=-1,\; \Lambda^0_{\;\;\; 0}\ge 1\\ L^\downarrow_- \det (\Lambda)=-1,\; \Lambda^0_{\;\;\; 0}\le 1\end{cases} \end{cases}

For instance, let us write 5 kind of Lorentz transformations:

1) Orthogonal rotations. They are continuous (proper) Lorentz transformations with a 3×3 submatrix \Omega +\Omega^T=0:

\Lambda=\begin{pmatrix}1 & \mathbf{0}\\ \mathbf{0} & \Omega\end{pmatrix}

2) Boosts. They are continuous (proper) Lorentz transformations mixing spacelike and timelike coordinates. The matrix has in this case the form:

\Lambda =\begin{pmatrix}\gamma & -\beta \gamma & \mathbf{0}\\ -\beta \gamma & \gamma & \mathbf{0}\\ \mathbf{0}&\mathbf{0}& \mathbb{I}\end{pmatrix}

3) PT symmetry. Discrete non-proper Lorentz transformation. It “inverts” space and time coordinates in the sense \mathbf{r}\rightarrow -\mathbf{r} and t\rightarrow -t. They belongs to L_+^\downarrow. The matrix of this transformation is:

\Lambda_{PT}=diag(-1,-1,-1,-1)

4) Parity. Discrete non-proper Lorentz transformation. It inverts only the spacelike components of true vectors ( be aware of pseudovectors!) in the sense \mathbf{r}\rightarrow -\mathbf{r}. Sometimes, it is denoted by P, parity, and this transformation belongs to L_-^\uparrow. It is defined as follows:

P=\Lambda_P=diag(1,-1,-1,-1)

5) Time reversal, T. Discrete non-proper Lorentz transformation. It inverts the direction of time in the sense that t\rightarrow -t. \Lambda_T belongs to the set L_-^\downarrow

Remark: If X^2>0, then L_+^\uparrow, L_-^\uparrow don’t change the sense of time. This is why they are called orthochronous!


LOG#026. Boosts, rapidity, HEP.

In euclidean two dimensional space, rotations are easy to understand in terms of matrices and trigonometric functions. A plane rotation is given by:

\boxed{\begin{pmatrix}x'\\ y'\end{pmatrix}=\begin{pmatrix}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}}\leftrightarrow \boxed{\mathbb{X}'=\mathbb{R}(\theta)\mathbb{X}}

where the rotation angle is \theta, and it is parametrized by 0\leq \theta \leq 2\pi.

Interestingly, in minkovskian two dimensional spacetime, the analogue does exist and it is written in terms of matrices and hyperbolic trigonometric functions. A “plane” rotation in spacetime is given by:

\boxed{\begin{pmatrix}ct'\\ x'\end{pmatrix}=\begin{pmatrix}\cosh \varphi & -\sinh \varphi \\ -\sinh \varphi & \cosh \varphi \end{pmatrix}\begin{pmatrix}ct\\ x\end{pmatrix}}\leftrightarrow \boxed{\mathbb{X}'=\mathbb{L}(\varphi)\mathbb{X}}

Here, \varphi = i\psi is the so-called hiperbolic rotation angle, pseudorotation, or more commonly, the rapidity of the Lorentz boost in 2d spacetime. It shows that rapidity are a very useful parameter for calculations in Special Relativity. Indeed, it is easy to check that

\mathbb{L}(\varphi_1+\varphi_2)=\mathbb{L}(\varphi_1)\mathbb{L}\mathbb(\varphi_2)

So, at least in the 2d spacetime case, rapidities are “additive” in the written sense.

Firstly, we are going to guess the relationship between rapidity and velocity in a single lorentzian spacetime boost. From the above equation we get:

ct'=ct\cosh \varphi -x\sinh \varphi

x'=-ct\sinh \varphi +x\cosh \varphi

Multiplying the first equation by \cosh \varphi and the second one by \sinh \varphi, we add the resulting equation to obtain:

ct'\cosh\varphi+x'\sinh \varphi =ct\cosh^2 \varphi -ct\sinh^2 \varphi =ct

that is

ct'\cosh\varphi+x'\sinh \varphi =ct

From this equation (or the boxed equations), we see that \varphi=0 corresponds to x'=x and t'=t. Setting x'=0, we deduce that

x'=0=-ct\sinh \varphi +x\cosh \varphi

and thus

ct\tanh \varphi =x or x=ct\tanh\varphi.

Since t\neq 0, and the pseudorotation seems to have a “pseudovelocity” equals to V=x/t, the rapidity it is then defined through the equation:

\boxed{\tanh \varphi=\dfrac{V}{c}=\beta}\leftrightarrow\mbox{RAPIDITY}\leftrightarrow\boxed{\varphi=\tanh^{-1}\beta}

If we remember what we have learned in our previous mathematical survey, that is,

\tanh^{-1}z=\dfrac{1}{2}\ln \dfrac{1+z}{1-z}=\sqrt{\dfrac{1+z}{1-z}}

We set z=\beta in order to get the next alternative expression for the rapidity:

\varphi=\ln \sqrt{\dfrac{1+\beta}{1-\beta}}=\dfrac{1}{2}\ln \dfrac{1+\beta}{1-\beta}\leftrightarrow \exp \varphi=\sqrt{\dfrac{1+\beta}{1-\beta}}

In experimental particle physics, in general 3+1 spacetime, the rapidity definition is extended as follows. Writing, from the previous equations above,

\sinh \varphi=\dfrac{\beta}{\sqrt{1-\beta^2}}

\cosh \varphi=\dfrac{1}{\sqrt{1-\beta^2}}

and using these two last equations, we can also write momenergy components using rapidity in the same fashion. Suppose that for some particle(objetc), its  mass is m, its energy is E, and its (relativistic) momentum is \mathbf{P}. Then:

E=mc^2\cosh \varphi

\lvert \mathbf{P} \lvert =mc\sinh \varphi

From these equations, it is trivial to guess:

\varphi=\tanh^{-1}\dfrac{\lvert \mathbf{P} \lvert c}{E}=\dfrac{1}{2}\ln \dfrac{E+\lvert \mathbf{P} \lvert c}{E-\lvert \mathbf{P} \lvert c}

This is the completely general definition of rapidity used in High Energy Physics (HEP), with a further detail. In HEP, physicists used to select the direction of momentum in the same direction that the collision beam particles! Suppose we select some orientation, e.g.the z-axis. Then, \lvert \mathbf{P} \lvert =p_z and rapidity is defined in that beam direction as:

\boxed{\varphi_{hep}=\tanh^{-1}\dfrac{\lvert \mathbf{P}_{beam} \lvert c}{E}=\dfrac{1}{2}\ln \dfrac{E+p_z c}{E-p_z c}}

In 2d spacetime, rapidities add nonlinearly according to the celebrated relativistic addition rule:

\beta_{1+2}=\dfrac{\beta_1+\beta_2}{1+\frac{\beta_1\beta_2}{c^2}}

Indeed, Lorentz transformations do commute in 2d spacetime since we boost in a same direction x, we get:

L_1^xL_2^x-L_2^xL_1^x=0

with

L_1^x=\begin{pmatrix}\gamma_1 & -\gamma_1\beta_1\\ -\gamma_1\beta_1 &\gamma_1 \end{pmatrix}

L_2^x=\begin{pmatrix}\gamma_2 & -\gamma_2\beta_2\\ -\gamma_2\beta_2 &\gamma_2 \end{pmatrix}

This commutativity is lost when we go to higher dimensions. Indeed, in spacetime with more than one spatial direction that result is not true in general. If we build a Lorentz transformation with two boosts in different directions V_1=(v_1,0,0) and V_2=(0,v_2,0), the Lorentz matrices are ( remark for experts: we leave one direction in space untouched, so we get 3×3 matrices):

L_1^x=\begin{pmatrix}\gamma_1 & -\gamma_1\beta_1 &0\\ -\gamma_1\beta_1 &\gamma_1 &0\\ 0& 0& 1\end{pmatrix}

L_2^y=\begin{pmatrix}\gamma_2 & 0&-\gamma_2\beta_2\\ 0& 1& 0\\ -\gamma_2\beta_2 & 0&\gamma_2 \end{pmatrix}

and it is easily checked that

L_1^xL_2^y-L_2^yL_1^x\neq 0

Finally, there is other related quantity to rapidity that even experimentalists do prefer to rapidity. It is called: PSEUDORAPIDITY!

Pseudorapidity, often denoted by \eta describes the angle of a particle relative to the beam axis. Mathematically speaking is:

\boxed{\eta=-\ln \tan \dfrac{\theta}{2}}\leftrightarrow \mbox{PSEUDORAPIDITY}\leftrightarrow \boxed{\exp (\eta)=\dfrac{1}{\tan\dfrac{\theta}{2}}}

where \theta is the angle between the particle momentum \mathbf{P}  and the beam axis. The above relation can be inverted to provide:

\boxed{\theta=2\tan^{-1}(e^{-\eta})}

The pseudorapidity in terms of the momentum is given by:

\boxed{\eta=\dfrac{1}{2}\ln \dfrac{\vert \mathbf{P}\vert +P_L}{\vert \mathbf{P}\vert -P_L}}

Note that, unlike rapidity, pseudorapidity depends only on the polar angle of its trajectory, and not on the energy of the particle.

In hadron collider physics,  and other colliders as well, the rapidity (or pseudorapidity) is preferred over the polar angle because, loosely speaking, particle production is constant as a function of rapidity. One speaks of the “forward” direction in a collider experiment, which refers to regions of the detector that are close to the beam axis, at high pseudorapidity \eta.

The rapidity as a function of pseudorapidity is provided by the following formula:

\boxed{\varphi=\ln\dfrac{\sqrt{m^2+p_T^2\cosh^2\eta}+p_T\sinh \eta}{\sqrt{m^2+p_T^2}}}

where p_T is the momentum transverse to the direction of motion and m is the invariant mass of the particle.

Remark: The difference in the rapidity of two particles is independent of the Lorentz boosts along the beam axis.

Colliders measure physical momenta in terms of transverse momentum p_T instead of the momentum in the direction of the beam axis (longitudinal momentum) P_L=p_z, the polar angle in the transverse plane (genarally denoted by \phi) and pseudorapidity \eta. To obtain cartesian momenta (p_x,p_y,p_z)  (with the z-axis defined as the beam axis), the following transformations are used:

p_x=P_T\cos\phi

p_y=P_T\sin\phi

p_z=P_T\sinh\eta

Thus, we get the also useful relationship

\vert P \vert=P_T\cosh\eta

This quantity is an observable in the collision of particles, and it can be measured as the main image of this post shows.