LOG#005. Lorentz transformations(I).

For physicists working with objects approaching the light speed, e.g., handling with electromagnetic waves, the use of special relativity is essential.

The special theory of relativity is based on two single postulates:

1st) Covariance or invariance of all the physical laws (Mechanics, Electromagnetism,…) for all the inertial observers ( i.e. those moving with constant velocity relative to each other).

It means that there is no preferent frame or observer, only “relative” motion is meaningful when speaking of motion with respect to certain observer or frame. Indeed, unfortunately, it generated a misnomer and a misconception in “popular” Physics when talking about relativity (“Everything is relative”). What is relative then? The relative motion between inertial observers and its description using certain “coordinates” or “reference frames”. However, the true “relativity” theory introduces just about the opposite view. Physical laws and principles are “invariant” and “universal” (not relative!).  Einstein himself was aware of this, in spite he contributed to the initial spreading of the name “special relativity”, understood as a more general galilean invariance that contains itself the electromagnetic phenomena derived from Maxwell’s equations.

2nd) The speed of light is independent of the source motion or the observers. Equivalently, the speed of light is constant everywhere in the Universe.

No matter how much you can run, speed of light is universal and invariant. Massive particles can never move at speed of light. Two beams of light approaching to each other does not exceed the speed of light either. Then, the usual rule for the addition of velocities is not exact. Special relativity provides the new rule for adding velocities.

In this post, the first of a whole thread devoted to special relativity, I will review one of the easiest ways to derive the Lorentz transformations. There are many ways to “guess” them, but I think it is very important to keep the mathematics as simple as possible. And here simple means basic (undergraduate) Algebra and some basic Physics concepts from electromagnetism, galilean physics and the use of reference frames. Also, we will limite here to 1D motion in the x-direction.

Let me begin! We assume we have two different observers and frames, denoted by S and S’. The observer in S is at rest while the observer in S’ is moving at speed $v$ with respect to S. Classical Physics laws are invariant under the galilean group of transformations:

$x'=x-vt$

We know that Maxwell equations for electromagnetic waves are not invariant under Galileo transformations, so we have to search for some deformation and generalization of the above galilean invariance. This general and “special” transformation will reduce to galilean transformations whenever the velocity is tiny compared with the speed of light (electromagnetic waves). Mathematically speaking, we are searching for transformations:

$x'=\gamma (x-vt)$

and

$x=\gamma (x'+vt')$

for the inverse transformation. There $\gamma=\gamma(c,v)$ is a function of the speed of light (denoted as c, and constant in every frame!) and the relative velocity $v$ of the moving object in S’ with respect to S. The small velocity limit of special relativity to galilean relativity imposes the condition:

$\displaystyle{\lim_{v \to 0} \gamma (c,v) =1}$

By the other hand, according to special relativity second postulate, light speed is constant in every reference frame. Therefore, the distance a light beam ( or wave packet) travels in every frame is:

$x=ct$ in S, or equivalently $x^2=c^2t^2$

and

$x'=ct'$ in S’, or equivalently $x'^2=c^2t'^2$

Then, the squared spacial  separation between the moving light-like object at S’ with respect to S will be

$x^2-x'^2=c^2(t^2-t'^2)$

Squaring the modified galilean transformations, we obtain:

$x'^2=\gamma ^2(x-vt)^2 \rightarrow x'^2=\gamma ^2 (x^2+v^2t^2-2xvt) \rightarrow x'^2-\gamma ^2x^2+2\gamma ^2xvt=\gamma ^2 v^2t^2$

$x^2=\gamma ^2 (x'+vt')^2 \rightarrow x^2-\gamma ^2x'^2-2\gamma ^2x'vt'=\gamma ^2v^2t'^2$

The only “weird” term in the above last two equations are the mixed term with “xvt” (or the x’vt’ term). So, we have to make some tricky algebraic thing to change it. Fortunately for us, we do know that $x'=\gamma(x-vt)$, so

$x'=\gamma x -\gamma vt \rightarrow \gamma x'=\gamma ^2 x-\gamma ^2 vt \rightarrow \gamma xx'=\gamma ^2 x^2-\gamma ^2 xvt$

and thus

$2\gamma xx'=2 \gamma ^2 x^2-2\gamma ^2 xvt \rightarrow 2\gamma ^2 xvt =2\gamma ^2x^2-2\gamma xx'$

In the same way,  we proceed with the inverse transformations:

$x=\gamma x'+\gamma vt' \rightarrow \gamma x=\gamma ^2x'+\gamma ^2vt' \rightarrow \gamma xx'=\gamma ^2x'^2-\gamma ^2x'vt'$

and thus

$2\gamma xx'=2\gamma^2x'^2+2\gamma^2x'vt' \rightarrow 2\gamma^2x'vt'=2\gamma^2x'^2-2\gamma^2xx'$

We got it! We can know susbtitute the mixed x-v-t and x’-v-t’ triple terms in terms of the last expressions. In this way, we get the following equations:

$x'^2=\gamma ^2(x-vt)^2 \rightarrow x'^2=\gamma ^2(x^2+v^2t^2-2xvt) \rightarrow x'^2-\gamma ^2x^2+2\gamma ^2x^2-2\gamma ^2xx'=\gamma ^2v^2t^2 \rightarrow x'^2+\gamma ^2x^2-2\gamma ^2xx'=\gamma ^2v^2t^2$

$x'^2=\gamma ^2(x'+vt')^2 \rightarrow x^2=\gamma ^2(x'^2+v^2t^2+2x'vt') \rightarrow x^2-\gamma ^2x'^2+2\gamma ^2x'^2-2\gamma ^2xx'=\gamma ^2v^2t'^2 \rightarrow x^2+\gamma ^2x'^2-2\gamma ^2xx'=\gamma ^2v^2t'^2$

And now, the final stage! We substract the first equation to the second one in the above last equations:

$x^2-x'^2+\gamma ^2(x'^2-x^2)=\gamma ^2v^2(t'^2-t^2) \rightarrow (x'^2-x^2)(\gamma ^2-1)= \gamma ^2v^2(t'^2-t^2)$

But we know that $x^2-x'^2=c^2(t^2-t'^2)$, and so

$(x'^2-x^2)(\gamma ^2-1)= \gamma ^2v^2(t'^2-t^2) \rightarrow c^2(x'^2-x^2)(\gamma ^2-1)= \gamma ^2v^2(x'^2-x^2)$

then

$c^2(\gamma ^2-1)= \gamma ^2v^2 \rightarrow -c^2= -\gamma ^2c^2+\gamma ^2v^2 \rightarrow \gamma ^2=\dfrac{c^2}{c^2-v^2}$

or, more commonly we write:

$\gamma ^2=\dfrac{1}{1-\dfrac{v^2}{c^2}}$

and therefore

$\gamma =\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}$

Moreover, we usually define the beta (or boost) parameter to be

$\beta = \dfrac{v}{c}$

To obtain the time transformation we only remember that $x'=ct'$ and $x=ct$ for light signals, so then, for time we  obtain:

$x'=\gamma (x-vt) \rightarrow t' =x' /c= \gamma (x/c-vt/c)=\gamma ( t- vx/c^2)$

Finally, we put everything together to define the Lorentz transformations and their inverse for 1D motion along the x-axis:

$x'=\gamma (x-vt)$

$y'=y$

$z'=z$

$t'=\gamma \left( t-\dfrac{vx}{c^2}\right)$

and for the inverse transformations

$x=\gamma (x'+vt)$

$y=y'$

$z=z'$

$t=\gamma \left( t'+\dfrac{vx'}{c^2}\right)$

ADDENDUM: THE EASIEST, FASTEST AND SIMPLEST DEDUCTION  of $\gamma$ (that I do know).

If you don’t like those long calculations, there is a trick to simplify the “derivation” above.  The principle of Galilean relativity enlarged for electromagnetic phenomena implies the structure:

$x'=\gamma (x-vt)$ and $x=\gamma (x'+vt')$ for the inverse.

Now, the second postulate of special relativity says that light signals travel in such a way light speed in vacuum is constant, so $t=x/c$ and $t'=x'/c$. Inserting these times in the last two equations:

$x'=\gamma (1-v/c)x$ and $x=\gamma (1+v/c)x'$

Multiplying these two equations, we get:

$x'x =\gamma ^2(1+v/c)(1-v/c)xx'$.

If we consider any event beyond the initial tic-tac, i.e., if we suppose $t\neq 0$ and $t'\neq 0$, the product $xx'$ will be different from zero, and we can cancel the factors on both sides to get what we know and expect: $\gamma^2(1-v^2/c^2)=1$ i.e.

$\gamma = \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}$

LOG#004. Feynmanity.

The dream of every theoretical physicist, perhaps the most ancient dream of every scientist, is to reduce the Universe ( or the Polyverse if you believe we live in a Polyverse, also called Multiverse by some quantum theorists) to a single set of principles and/or equations. Principles should be intuitive and meaningful, while equations should be as simple as possible but no simpler to describe every possible phenomenon in the Universe/Polyverse.

What is the most fundamental equation?What is the equation of everything? Does it exist? Indeed, this question was already formulated by Feynman himself  in his wonderful Lectures on Physics! Long ago, Feynman gave us other example of his physical and intuitive mind facing the First Question in Physics (and no, the First Question is NOT “(…)Dr.Who?(…)” despite many Doctors have faced it in different moments of the Human History).

Today, we will travel through this old issue and the modest but yet intriguing and fascinating answer (perhaps too simple and general) that R.P. Feynman found.

Well, how is it?What is the equation of the Universe? Feynman idea is indeed very simple. A nullity condition! I call this action a Feynmann nullity, or feynmanity ( a portmanteau), for brief. The Feynman equation for the Universe is a feynmanity:

$\boxed{U=0}$

Impressed?Indeed, it is very simple. What is the problem then?As Feynman himself said, the problem is really a question of “order” and a “relational” one. A question of what theoretical physicists call “unification”. No matter you can put equations together, when they are related, they “mix” somehow through the suitable mathematical structures.  Gluing “different” pieces and objects is not easy.  I mean, if you pick every equation together and recast them as feynmanities, you will realize that there is no relation a priori between them. However, it can not be so in a truly unified theory. Think about electromagnetism. In 3 dimensions, we have 4 laws written in vectorial form, plus the gauge condition and electric charge conservation through a current. However, in 4D you realize that they are indeed more simple. The 4D viewpoint helps to understand electric and magnetic fields as the the two sides of a same “coin” (the coin is a tensor). And thus, you can see the origin of the electric and magnetic fields through the Faraday-Maxwell tensor $F_{\mu \nu }$. Therefore, a higher dimensional picture simplifies equations (something that it has been remarked by physicists like Michio Kaku or Edward Witten) and helps you to understand the electric and magnetic field origin from a second rank tensor on equal footing.

You can take every equation describing the Universe set it equal to zero. But of course, it does not explain the origin of the Universe (if any), the quantum gravity (yet to be discovered) or whatever. However, the remarkable fact is that every important equation can be recasted as a Feynmanity! Let me put some simple examples:

Example 1. The Euler equation in Mathematics. The most famous formula in complex analysis is a Feynmanity $e^{i\pi}+1=0$ or $e^{2\pi i}=1+0$ if you prefer the constant $\tau=2\pi$.

Example 2. The Riemann’s hypothesis. The most important unsolved problem in Mathematics(and number theory, Physics?) is the solution to the equation $\zeta (s)=0$, where $\zeta(s)$ is the celebrated riemann zeta function in complex variable $s=\kappa + i \lambda$, $\kappa, \lambda \in \mathbb{R}$. Trivial zeroes are placed in the real axis $s=-2n$ $\forall n=1,2,3,...,\infty$. Riemann hypothesis is the statement that every non-trivial zero of the Riemann zeta function is placed parallel to the imaginary axis and they have all real part equal to 1/2. That is, Riemann hypothesis says that the feynmanity $\zeta(s)=0$ has non-trivial solutions iff $s=1/2\pm i\lambda _n$, $\forall n=1,2,3,...,\infty$, so that

$\displaystyle{\lambda_{1}=14.134725, \lambda_{2}= 21.022040, \lambda_{3}=25.010858, \lambda _{4}=30.424876, \lambda_{5}=32.935062, ...}$

I generally prefer to write the Riemann hypothesis in a more symmetrical and “projective” form. Non-trivial zeroes have the form $s_n=\dfrac{1\pm i \gamma _n}{2}$ so that for me, non-trivial true zeroes are derived from projective-like operators $\hat{P}_n=\dfrac{1\pm i\hat{\gamma} _n}{2}$, $\forall n=1,2,3,...,\infty$. Thus

$\gamma_1 =28.269450, \gamma_2= 42.044080, \gamma_3=50.021216, \gamma _4=60.849752, \gamma_5=65.870124,...$

Example 3. Maxwell equations in special relativity. Maxwell equations have been formulated in many different ways along the history of Physics. Here a picture of that. Using tensor calculus, they can be written as 2 equations:

$\partial _\mu F^{\mu \nu}-j^\nu=0$

and

$\epsilon ^{\sigma \tau \mu \nu} \partial _\tau F_{\mu\nu}=\partial _\tau F_{\mu \nu}+ \partial _\nu F_{\tau \mu}+\partial_\mu F_{\nu \tau}=0$

Using differential forms:

$dF=0$

and

$d\star F-J=0$

Using Clifford algebra (Clifford calculus/geometric algebra, although some people prefer to talk about the “Kähler form” of Maxwell equations) Maxwell equations are a single equation: $\nabla F-J=0$ where the geometric product is defined as $\nabla F=\nabla \cdot F+ \nabla \wedge F$.

Indeed, in the Lorentz gauge  $\partial_\mu A^\mu=0$, the Maxwell equations reduce to the spin one field equations:

$\square ^2 A^\nu=0$

where we defined

$\square ^2=\square \cdot \square = \partial_\mu \partial ^\mu =\dfrac{\partial^2}{\partial x^i \partial x_i}-\dfrac{\partial ^2}{c^2\partial t^2}$

Example 4. Yang-Mills equations. The non-abelian generalization of electromagnetism can be also described by 2 feynmanities:

The current equation for YM fields is $(D^{\mu}F_{\mu \nu})^a-J_\nu^a=0$

The Bianchi identities are $(D _\tau F_{\mu \nu})^a+( D _\nu F_{\tau \mu})^a+(D_\mu F_{\nu \tau})^a=0$

Example 5. Noether’s theorems for rigid and local symmetries. Emmy Noether proved that when a r-paramatric Lie group leaves the lagrangian quasiinvariant and the action invariant, a global conservation law (or first integral of motion) follows. It can be summarized as:

$D_iJ^i=0$ for suitable (generally differential) operators $D^i,J^i$ depending on the particular lagrangian (or lagrangian density) and $\forall i=1,...,r$.

Moreover, she proved another theorem. The second Noether’s theorem applies to infinite-dimensional Lie groups. When the lagrangian is invariant (quasiinvariant is more precise) and the action is invariant under the infinite-dimensional Lie group parametrized by some set of arbitrary (gauge) functions ( gauge transformations), then some identities between the equations of motion follow. They are called Noether identities and take the form:

$\dfrac{\delta S}{\delta \phi ^i}N^i_\alpha=0$

where the gauge transformations are defined locally as

$\delta \phi ^i= N^i_\alpha \epsilon ^\alpha$

with $N^i_\alpha$ certain differential operators depending on the fields and their derivatives up to certain order. Noether theorem’s are so general that can be easily generalized for groups more general than those of Lie type. For instance, Noether’s theorem for superymmetric theories (involving lie “supergroups”) and many other more general transformations can be easily built. That is one of the reasons theoretical physicists love Noether’s theorems. They are fully general.

Example 6. Euler-Lagrange equations for a variational principle in Dynamics take the form $\hat{E}(L)=0$, where L is the lagrangian (for a particle or system of particles and $\hat{E}(L)$ is the so-called Euler operator for the considered physical system, i.e., if we have finite degrees of freedom, L is a lagrangian) and a lagrangian “density” in the more general “field” theory framework( where we have infinite degrees of freedom and then L is a lagrangian density $\mathcal{L}$. Even the classical (and quantum) theory of (super)string theory follows from a lagrangian (or more precisely, a lagrangian density). Classical actions for extended objects do exist, so it does their “lagrangians”. Quantum theory for p-branes $p=2,3,...$ is not yet built but it surely exists, like M-theory, whatever it is.

Example 7.  The variational approach to Dynamics or Physics implies  a minimum ( or more generally a “stationary”) condition for the action. Then the feynmanity for the variational approach to Dynamics is simply $\delta S=0$. Every known fundamental force can be described through a variational principle.

Example 8. The Schrödinger’s equation in Quantum Mechanics $H\Psi-E\Psi=0$, for certain hamiltonian operator H. Note that the feynmanity is itself $H=0$ when we studied special relativity from the hamiltonian formalism. Even more, in Loop Quantum Gravity, one important unsolved problem is the solution to the general hamiltonian constraint for the gauge “Wilson-like” loop variables, $\hat{H}=0$.

Example 9. The Dirac’s equation $(i\gamma ^\mu \partial_\mu - m) \Psi =0$ describing free spin 1/2 fields. It can be also easily generalized to interacting fields and even curved space-time backgrounds. Dirac equation admits a natural extension when the spinor is a neutral particle and it is its own antiparticle through the Majorana equation

$i\gamma^\mu\partial_\mu \Psi -m\Psi_c=0$

Example 10. Klein-Gordon’s equation for spin 0 particles: $(\square ^2 +m^2 )\phi=0$.

Example 11. Rarita-Schwinger spin 3/2 field equation: $\gamma ^{\mu \nu \sigma}\partial_{\nu}\Psi_\sigma+m\gamma^{\mu\nu}\Psi_\nu=0$. If $m=0$ and the general conventions for gamma matrices, it can be also alternatively writen as

$\gamma ^\mu (\partial _\mu \Psi_\nu -\partial_\nu\Psi_\mu)=0$

Note that antisymmetric gamma matrices verify:

$\gamma ^{\mu \nu}\partial_{\mu}\Psi_\nu=0$

More generally, every local (and non-local) field theory equation for spin s can be written as a feynmanity or even a theory which contains interacting fields of different spin( s=0,1/2,1,3/2,2,…).  Thus, field equations have a general structure of feynmanity(even with interactions and a potential energy U) and they are given by $\Lambda (\Psi)=0$, where I don’t write the indices explicitely). I will not discuss here about the quantum and classical consistency of higher spin field theories (like those existing in Vasiliev’s theory) but field equations for arbitrary spin fields can be built!

Example 12. SUSY charges. Supersymmetry charges can be considered as operators that satisfy the condicion $\hat{Q}^2=0$ and $\hat{Q}^{\dagger 2}=0$. Note that Grassman numbers, also called grassmanian variables (or anticommuting c-numbers) are “numbers” satisfying $\theta ^2=0$ and $\bar{\theta}^2=0$.

The Feynman’s conjecture that everything in a fundamental theory can be recasted as a feynmanity seems very general, perhaps even silly, but  it is quite accurate for the current state of Physics, and in spite of the fact that the list of equations can be seen unordered of unrelated, the simplicity of the general feynmanity (other of the relatively unknown neverending Feynman contributions to Physics)

$something =0$

is so great that it likely will remain forever in the future of Physics. Mathematics is so elegant and general that the Feynmanity will survive further advances unless  a  Feynman “inequality” (that we could call perhaps, unfeynmanity?) shows to be more important and fundamental than an identity. Of course, there are many important results in Physics, like the uncertainty principle or the second law of thermodynamics that are not feynmanities (since they are inequalities).

Do you know more examples of important feynmanities?

Do you know any other fundamental physical laws or principles that can not be expressed as feynmanities, and then, they are important unfeynmanities?