# Sources on relativistic rotational mechanics

Rotational mechanics is a very interesting subject. However, in the common relativity textbooks not much discussions on rotational mechanics can be found. Is there any source/books where I can find a very detailed discussion on relativistic rotational mechanics/dynamics?

• As far as I know there is no good theory of special-relativistic rotational mechanics. Hardly anything from the Newtonian theory carries over into special relativity because there's no useful analog of a rigid body. Uniformly rotating coordinates exist, but are practically useless because they aren't equivalent to inertial coordinates plus fictitious forces. It's easier to analyze the motion in an inertial frame. Nov 3, 2020 at 19:41

I can give a bit of a partial answer, but I am not really an expert in this part of physics. I debated whether this should be a comment but I figure if someone had a more comprehensive resource they will take the upvotes, heh.

The most important term to google if you are starting to investigate this is “Born rigidity”. This was the subject of a major theorem, that there are Born-rigid configurations, but with only a small family of dynamic parameters ($$\mathbb Z_2\times\mathbb R^3$$), so that the angular velocity cannot be allowed to vary with time if the axis of rotation is also specified—that requires four degrees of freedom but you only have three. So there are Born-rigid rotations, but no Born-rigid torques to get you there.

In terms of a physical intuition why things must go wrong, the usual explanation is that if you put rulers around a rotating body, they should be length-contracted in their direction of motion, whereas if you put rulers along the radius, those should not be length contracted because their motion is perpendicular to the ruler itself. For me, this never seemed like a huge obstacle because one could maybe argue that during acceleration the straight radial lines perhaps cannot stay straight and radial but must start to curve and spiral out from the center: that is what “the outside is smaller than the inside” means to me. But I suppose the details must be more complicated than this.

So I prefer a different physical intuition, which has to do with taking the Lorentz transformations to first order. In the first order theory you see only relativity of simultaneity, but not length contraction or time dilation. In the first order theory, two clocks on opposite sides of a rotating disc should be seeing each other tick anomalously faster which would seem to generate a paradox, presumably resolved by an inability to synchronize (by synchronizing with your nearest neighbor) across the rigid disk. So, if you were to do this synchronization all the way around this circle clockwise, your nearest neighbor counterclockwise would be discontinuous with your clock. Something funny like that is going on.

I recommend the following article, published 2004, by Olaf Wucknitz. Sagnac effect, twin paradox and space-time topology - Time and length in rotating systems and closed Minkowski space-times

Here I discuss one aspect of the article: the Sagnac effect. The Sagnac effect is mostly known for its essential role in ring interferometry.

To explain wat the Sagnac effect is about I will use the following thought demonstration: a series of relay stations, positioned around the equator of the Earth. The number of relay stations is arbitrary, let's put it at twelve. The relay stations are in radio contact. These relay stations proceed as follows: they start two counterpropagating relay transmissions, consisting of pulses. Let's say twelve pulses. Each time a pulse is received it is retransmitted to the next relay station.

Each pulse train, the clockwise and the counter-clockwise, has the same amount of pulses. The clockwise propagating pulses on one hand and the counterclockwise propagating pulses on the other hand hare treated independently

The timing of the pulses is adjusted as follows:

For each relay station the time interval between consecutive pulses of the clockwise propagating pulses is the same.

For each relay station the time interval between consecutive pulses of the counter-clockwise propagating pulses is the same.

Under those circumstances the clockwise and counter-clockwise pulse trains will not have the same time interval between consecutive pulses, as measured by the relay stations.

We have that the two counterpropagating pulses are moving at the same speed: the speed of radio transmission. The relay stations are co-rotating with the Earth, clockwise as seen from the south pole. Each pulse takes some time to travel from one relay station to the next, so in the time between a relay pulse being emitted and a relay pulse being received the relay stations have moved in the clockwise direction. This lengthens the measured time interval between the pulsed of the clockwise pulse train, and it shortens the measured time interval between the pulses of the counter-clockwise pulse train.

From the amount of difference between clockwise and counter-clockwise pulse-to-pulse interval the rotation rate of the Earth can be inferred.

The above process is the operating principle of ring interferometry. Ring interferometry achieves measurement of rotation rate, using the constancy of the speed of light as reference.

This procedure allows the relay stations to maintain synchronized time keeping.

By constrast: using the Einstein synchronization procedure would fail to produce a self-consistent result.

Olaf Wucknitz argues that the perimeter of an Ehrenfest disk should be thought of as a spacetime that loops back on itself.

While it is the case that the perimeter of an Ehrenfest disk is accelerating (towards the center of rotation), the acceleration is perpendicular to the direction of motion that is probed by the Sagnac effect, and Olaf Wucknitz argues that because of that the acceleration does not affect the Sagnac effect.

Olaf Wucknitz expresses the opinion that understanding of rotation in Minkowski spacetime is an underdeveloped area of physics. Different authors come to different and incompatible conclusions.

Suitalbe keywords:

-relativistic rotation,