LOG#205. Ether wind, SR and light clocks.

From the Michelson-Morley experiment, some clever experimentalist tried to derive the light speed through the “ether wind”. It is very similar to a river being rowing by sailors in a boat. The times for journey “upstream” and “downstream” are:

    \[cT_\parallel (1)=L+vT_\parallel (1)\]

    \[cT_\parallel (2)=L-vT_\parallel (2)\]

Journey across the ether wind uses the following pythagorean theorem:

    \[c^2\left(\dfrac{T_\perp}{2}\right)^2=L^2+v^2\left(\dfrac{T_\perp}{2}\right)^2\]

From

    \[T_\parallel=T_\parallel (1)+T_\parallel (2)=\dfrac{L}{c-v}+\dfrac{L}{c+v}\]

    \[\boxed{T_\parallel=\dfrac{2Lc}{c^2-v^2}}\]

and

    \[\boxed{T_\perp=\dfrac{2L}{\sqrt{c^2-v^2}}}\]

you get the time difference

    \[\Delta T=T_\parallel-T_\perp=\dfrac{2Lc}{c^2-v^2}-\dfrac{2L}{\sqrt{c^2-v^2}}\approx \dfrac{L}{c}\left(\dfrac{V}{c}\right)^2\]

The Michelson-Morley experiment had enough resolution to detect fringes caused by the above time difference. However, there were no time difference. There is no shift in light ways through the heavens. However, the ether hypothesis was kept until Einstein times…

Einstein derived the relativity of time with the tool of light clocks. Suppose a rest frame and a moving rocket with constant speed v.. Inside the rocket a light beam perpendicularly, to measure time in its frame. Supposing it travels d upside and d upside down, the time it yields is

    \[t_{LC}=\dfrac{2d}{c}\]

That is the lab light clock time. From the outside, the rocket light clock time is different, since it follows an oblique trajectory. The distance of one side is D, so the time in this frame will be

    \[t_{RC}=\dfrac{2D}{c}\]

By trigonometry, the base the rocket travels is x=vt_{RC}, so the semibase reads x/2=vt_{RC}/2. Let \Delta t the time interval between events in lab frame, and \Delta t' the time interval the lab sees or the rocket clock measures between some events. Again, pythagorean theorem rocks:

    \[D^2=d^2+\dfrac{v^2t_{CR}^2}{4}=d^2+\dfrac{v^2D^2}{c^2}\]

so

    \[D=\dfrac{d}{\sqrt{1-v^2/c^2}}\]

and then

    \[\Delta t'=\Delta t\sqrt{1-v^2/c^2}\]

Time moves “slower” for rocket clocks seen from outside, and measured by the lab outside. Similar arguments work out for lenght and we have length contraction! If L_0 is the length of a rod measured with a light beam in the rocket frame, and L is the length of the rod as measured in the LAB frame OUTSIDE. Front of rod crosses a point P at time t'_1 in the rocket and t_1 in the lab. The back of rod crosses a the point P at time t'_2 in the rocket and t_2 in the lab. Since

    \[L=v(t_2-t_1)\]

    \[L'_0=v(t'_2-t'_1)\]

then

    \[L=v\Delta t=v\Delta t'\sqrt{1-v^2/c^2}=L'_0\sqrt{1-v^2/c^2}\]

so

    \[L=L'_0\sqrt{1-v^2/c^2}\]

Therefore, the movin rod in the LAB frame outside appears (to the lab observers) length contracted L<<L_0. The rod would be normal from the rocket inside observers. There is an invariant interval of spacetime, as it was shown in my notes on special relativity here:

    \[\Delta\tau^2=c^2\Delta t^2-\Delta x^2\]

That number is the same in all frames moving at constant speed with respect to each other. Simultaneity is also relative, as space and time measurements as well.

What happens with energy and momentum? In the lab frame, particle has at time t position x. In the particle frame (rocket frame), we have t', x'=0. Thus,

    \[\dfrac{dt'}{dt'}=1\;\;\;\dfrac{dx'}{dt'}=0\]

Then, we form the invariant

    \[m^2c^2\left(\dfrac{dt'}{dt'}\right)^2-m^2\left(\dfrac{dx'}{dt'}\right)^2=m^2c^2\]

provided the transverse momentum

    \[p_T=mc^2\left(\dfrac{dt}{dt'}\right)\]

and the canonical momentum

    \[p=m\dfrac{dx}{dt'}=mv\]

satisfy

    \[m^2c^2\left(\dfrac{dt}{dt'}\right)^2-m^2\left(\dfrac{dx}{dt'}\right)^2=\left(p_T/c^2\right)^2-p^2=m^2c^2\]

Thus,

    \[p_T=\sqrt{p^2c^2+m^2c^4}\approx mc^2+\dfrac{p^2}{2m}\]

Note that p_T=E_{total}=Mc^2. However, p=0 yields E_0=mc^2 are rest energy.

Particles of light, from the classical side, are radiation. Wave light phenomena are classical electromagnetic waves. Usually, accelerated point-like particles of matter emit electromagnetic waves. Waves are also associated to the Maxwell field described by the pair E, B. In the quantum world, things are a little different. However, we see (yet!) phenomena like interference at the classical level!

    \[\vert A\vert^2=I=\vert A_1+A_2\vert^2\neq I_1+I_2\]

The Heisenberg uncertainty principle provides \Delta x\Delta p\geq \hbar/2. Quantum physics says that probability is related to \vert A_1+A_2\vert^2. Hydrogen atoms are quantized by Bohr rules, via L=n\hbar =hh/2\pi. The interaction of light with matters surprised people again when we found out that wave physics could NOT explain the photoelectric effect! A linear relation between kinetic energy and the frequency of light is NOT expected from wave light theory! Exercise: use what you know from the harmonic oscillator or waves to prove this fact. However, quantum light theory, as Einstein taught us, solves the issue of the photoelectric effect giving us the right theory with

    \[K=hf-W\]

Photons are quanta of light, with E=hf=\hbar\omega. Classically, beyond a different dependency of kinetic energy and frequency of light, we would obtain f=f'. However, interaction with atoms or matter quanta changes this naive idea. The total momentum and energy of light and atoms are conserved. Take p=h/\lambda for light, and E=pc. You invest some of the light momentum for make electrons free of the bounding forces at the matter surface. p=hf/\lambda changes, but the total momentum and energy is conserved before and after the photon hitting the electron and metal surface in the photoelectric effect! Interactions of light and matter are quantum in nature. Quantum interactions are more complicated due to fluctuations. However, in general, energy, momentum and angular momentum are conserved. Left-handed and right-handed electrons interact in the same way. Compton scattering is another interesting phenomenon. It can be seen as a consequence of gauge U(1) invariance associated to charge conservation. Antimatter interacts in a parallel way, only changing the sign of charge and we have also the CPT theorem in any local special relativistic framework. Annihilation of matter and antimatter becomes possible at quantum level. Radiation arises from high energy physics. Particle colliders use these facts to create particles. Quantum Field Theory (QFT) is a misnomer for a quantum mechanical special relativistic theory that allows to the particle number to vary! Number of particles changes in any QFT. Particle creation/destruction phenomena is the ABC of QFT. For instante, in Q.E.D., the QFT theory for light and matter, any gauge (electromagnetic) compesating field is A_\mu(x,t), it has a potential \varphi(x,t), and matter fields are \Psi(x,t). The Heisenberg principle applies, to yield:

    \[\Delta p\Delta x\sim h\]

    \[\Delta E\geq \dfrac{\hbar c}{L}\]

    \[\Delta p\geq \dfrac{\hbar}{L}\]

    \[\Delta E>E\]

    \[L<\dfrac{\hbar}{mc}=\lambda_C\]

Beyond light, beyond photons…What happens to quanta of MATTER? The question is complex. A complete theory for quanta of matter required time, 15-20 years, in the first third of the 20th century. Using the de Broglie relation, p=h/\lambda, just as we have a wave-like equation for “light”

    \[\partial_\mu\partial^\mu\varphi(x,t)=0\]

given suitable \varphi, the wave-like theory for electrons is much more complex because it implies the square root of the wave equation to understand that. Dirac derived the next equation in 1928:

    \[\left(\partial_x-1/c\partial_t\right)\Psi_+(x,t)=\dfrac{mc}{\hbar}\Psi_-\]

    \[\left(\partial_x+1/c\partial_t\right)\Psi_-(x,t)=\dfrac{mc}{\hbar}\Psi_+\]

so

    \[\left(\partial_x-1/c\partial_t\right)\left(\partial_x+1/c\partial_t\right)\Psi_{\pm}=\left(\dfrac{mc}{\hbar}\right)^2\Psi_{\pm}\]

Matter fields follow Pauli exclusion principle (PEP), they have negative energy states and they imply the existence of antimatter. Light is its own antiparticle and photons are bosons. Electrons and other matter field are FERMIONS. Under rotations, fermions are described by spinors, they need 4\pi radians or twists to become the same object. If not, their wavefunction changes by a minus sign! Dirac equation predicts antimatter. Positrons were discovered a years after Dirac wrote its equation (a Clifford algebra structure is behind it, to be discussed here in the near future!).

See you in another blog post!

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