Wigner Rotation

When I was a kid, the concept of being on a spinning Earth bothered me because it seemed like something I should be able to feel. Yet, when I laid down on a merry go round with my head on the outer rim or if I went on the spinning barrel ride at the amusement park, I would feel the centripetal force and see the rotation, but if I closed my eyes, I didn’t feel the rotation, I only felt the outward, centripetal force. It was hard for me to understand why.

Image result for merry go round
merry go round

Worries about not feeling the spin of the planet were trained out of me with the incantation – “but everything around you is accelerating along with you, so you can’t see the effect relative to anything else.” Although, this explanation didn’t fully satisfy me because I thought I should be able to feel the rotation. I mean, I can feel the rotation when I twirl around like a ballerina, why should lying on a merry go round be any different?

The automated thumbnail selection doesn’t give many good options.

I think that to really understand why I didn’t feel the rotation of the merry go round when I closed my eyes, I needed to understand something known as Wigner rotation.

In the picture below, you see the Wigner rotation of rocket ships. If they didn’t rotate, then they would point in the wrong direction as they orbit the planet.

https://www.researchgate.net/figure/The-rate-of-change-of-the-Wigner-rotation-gives-the-Thomas-precession-rate-dO-dt-The_fig2_48205023

The Wigner rotation only exists within a Lorentzian, Lagrangian framework in a curved, relative Riemmanian coordinate system. This framework best describes a system like sand blowing in the wind and the intuition required for those systems is very different from the intuition you develop from playing with water in your bathtub. That sort of physics is better described without any Wigner rotation within a Galilean, Eulerian framework in a straight, absolute Cartesian coordinate system. It is important to note that people naturally think in a Cartesian coordinate system and they get very confused when they try to apply their Cartesian intuition to a Riemannian system.

But this is technical stuff that you only need to know if you are a physicist or an engineer. A regular person only needs to understand merry go rounds.

As your merry go round spins, the particles in your head must rotate relative their centers. Whereas relative to the center of the merry go round, the particles in your head are not rotating – they are tidally locked – like the moon in orbit around the Earth. We only see one face of the moon because it is tidally locked with the Earth.

If one were to be able to sense the spin of the Earth, they would need to be able to sense the slow, rotation of the particles in their own head and we have evolved so that we can’t feel that because it would be quite distracting if we could.

A physicist might say that our motion is best approximated as linear and that makes us feel like we are in an inertial reference frame, even though we really aren’t. The acceleration we experience due to the rotation of the Earth only manifests as the Coriolis force and is something that we can see through the behaviour of a pendulum. Those sorts of explanations, while correct, never satisfied my inner child.

…..

I first learned about Wigner rotation in the context of synchrotron radiation from tilted microbunches of electrons. Above, there is a depiction of the concept in terms of rockets and if you understand the rotation in both macroscopic and microscopic terms, I believe that you understand almost everything you need to know about physics. There isn’t a magic equation, there is just a mental animation that each individual has to find through his or her own imagination. The next step is connecting that animation to Maxwell’s equations in different coordinate systems, but that is done in another post. The author below seems to have gone down this road as well.

 The rate of change of the Wigner rotation gives the Thomas precession rate dΩ/dt. The Thomas precession angle Ω T is the total infinitesimal Wigner rotation turned through after one orbit by Bob. The gray spaceships represent Bob’s unrotated rest frame (that is, observed by himself ), whilst the black ones represent the Wigner rotated one (as observed by mission control).

Matt Visser

I think that this is the most important concept to understand in physics because it illuminates the difference between sand blowing in the wind in a Lorentzian, Lagrangian approach to particle mechanics within a curved, relative Riemannian space and water flowing in a bathtub in a Galilean, Eulerian approach within a straight, absolute Cartesian space.

The difference between these two frameworks is also nicely illuminated through the question of how a gyroscope or spinning wheel works. If you’ve ever held a bicyle wheel and spun it while holding onto the axel, you will remember how it was hard to turn. It felt like it had gotten heavier in the latteral direction. Why did this happen?

The traditional answer is in terms of abstract concepts like gyroscopic force and conservation of angular momentum, but I will give an answer in terms of moving particles and conservation of energy, concepts that are easier for me to mentally animate. Of course, if physics on the macro and microscale are consistent, as they must be, then these two answers should be equivalent.

seeing is believing

In the language of rotation and vibration on a molecular level, why does a spinning bicycle wheel feel heavier in the plane perpendicular to the rotation? I believe that to explain the lateral heaviness of a current of masses flowing in a circle, as in a bicycle wheel, you need a Wigner rotation of the individual particles. Every time, they move forward in their loop, they also rotate. Rotation of a particle produces a field perpendicular to the rotation. This field will increase the lateral force between the particles in the current of mass. When those particles are not able to spread out laterally because of electromagnetic forces within the material, this lateral force will be reflected and turned into a vibration which causes the bicycle wheel to feel heavier in the lateral direction when it is spinning. Of course, given the conservation of energy, this makes the wheel lighter in the opposite plane.

One can also deliver this explanation in the language of dark matter contracting or increasing the density of the space around the wheel laterally and dark energy expanding or decreasing the density of the space around the treads of the wheel when it spins, but this would be highly unconventional. It is, however, easy to imagine if you picture a bicycle wheel spinning underwater.

In the Eulerian treatment of the system, there is no Wigner rotation because there are no particles and there is no contraction or expansion of space. Everything takes place in absolute, Cartesian space which doesn’t expand or contract when things move.

If you have been able to visualize all of this, then you will begin to understand why the Eulerian treatment is necessarily Galilean and in straight, absolute, Cartesian space while the Lagrangian treatment is necessarily Lorentzian and in curved, relative, Riemannian space.

Vortices define a particle and the centers of particles obey the Lagrangian treatment. Vortices can also be described with an Eulerian treatment, however, analytic solutions are few and far between and that is why the Lagrangian treatment is more often used, even though it removes a person’s connection to their intuition through the use of a warped coordinate system. Meanwhile, the creation and destruction of vortices is described by quantum thermodynamics and entropy, and a Lagrangian approximation thereof is given by the treatment of synchrotron radiation with the Lienard-Wiechert potential. Because singularities occur in this potential, we know that it is an approximation of something that might be more accurately described with an Eulerian numerical simulation of fluid dynamics.

Returning to the bicycle wheel. When the wheel gets heavier laterally and lighter in the plane of rotation, equipartition suggests that if the vortices in the lateral plane are vibrating more, in a quantum thermodynamic system, this would require an increase in the number of vortices in that plane. Conservation of energy demands that the increase in the number of vortices in one plane should decrease it in the other. That is why a bicycle wheel doesn’t radiate heat when it spins.

It is also why electrons don’t radiate away all of their energy when they orbit an atom, but that is another length scale entirely.

I will conclude by confusing you with a question that doesn’t have a clear answer: If thermodynamic equipartition is the law of the land, does a bicycle wheel vibrate when it is spun?

This doesn’t have a clear answer because this is a recursively defined system and such systems necessarily produce contradictions when they are investigated with yes/no questions. Socrates would understand why.

………

I first posted this material on Quora.com

The photo in the header is underwater wheelchair dance from engelli ariclari.

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