Understanding the "$\pi$" of a rotating disk

Question

Let us say you are in an inertial reference frame with a circular planar disk. If you take your meter measuring rods (or perhaps tape measure) you can find the diameter and circumference of the disk. If you divide the circumference by the diameter, you will get exactly $\pi$. Now you start rotating the disk. A book I have claims that now the ratio will be different (lets call it $\pi_\circ$ to avoid ambiguity.) This has caused to reevaluate my understanding of special relativity (this is a set up to general relativity, but the problem itself only requires the special theory.) My problem is that, sure, the measuring rods (or tape) would contract on the circumference, but wouldn't the circumference contract a similar amount, therefore giving a measurement of $\pi$ in every reference frame according to your own measuring rods. Of course, this reasoning goes on to make me question how I understood all of special relativity up this point.

Answer 1

Let us say you are in an inertial reference frame with a circular planar disk. If you take your meter measuring rods (or perhaps tape measure) you can find the diameter and circumference of the disk.

Then the distance ratios between constituents of the disk edge can certainly be determined;
and a midpoint of the disk may be identified (also in terms of distance ratios, involving constituents of the disk edge).

If you divide the circumference by the diameter, you will get exactly $\pi$.

Of course, the number $\pi$ is correspondingly defined (by polygonal approximation) in the first place.

Now you start rotating the disk.

Consequently, ideally: the constituents of the disk are now moving along circles (with respect to the given inertial reference frame, arond a common center), with constant duration $T$ of the rotational period (as measured by members of the given inertial reference frame). The constituents of the disk edge are moving at constant speed along a circle of the same diameter $2 R$ as had been determined earlier of the disk while it had been at rest.

The disk constituents (including the center of rotation) are therefore no longer at rest to each other; they remain (merely) rigid to each other, in the chronometric sense that any two disk constituents $A$ and $B$ keep measuring constant ratios of ping durations between each other,

$$\tau_{ABA} / \tau_{BAB} = \text{constant}.$$

The values of these ratios are generally different from $1$, and different from each other. In particular, for the center of rotation, $C$, and any constituent on the edge of the disk, $E$, the ratio of mutual ping durations is

$$\tau_{ECE} / \tau_{CEC} = \sqrt{ 1 - \beta_E^2 } = \sqrt{ 1 - \left( \frac{2~\pi~R}{c~T} \right)^2 }.$$

Further, the ratio of ping durations $\tau_{EAE} / \tau_{ECE}$ attains its maximum value if disk constituent $A$ is on the disk edge moving antipodal to $E$; and the corresponding maximum value is less than 2. In fact

$$\text{Max}[ \tau_{EAE} / \tau_{ECE} ] \approx 2 - \beta_E^2 + \frac{13}{12} \beta_E^4 - \frac{541}{360} \beta_E^6 + ...$$

Now, in order to adapt the polygonal approximation of $\pi$ to these mutually rigid disk constituents, and especially to those which make up the disk edge, we may consider some large number ($N$) of them distributed (as "marks") evenly on the disk edge, i.e. such that the ping durations between adjacent marks are equal.

Let $E$ and $F$ be two such adjacent marks on the disk edge. Then we may consider (for instance) the number

$$2~\pi^C_{\beta} := N ~ \tau_{EFE} / \tau_{ECE}.$$

The number $\tau_{ECE} / \tau_{EFE}$ also represents the number of counts obtained by $E$ of successive pings (signal round trips) to $F$ and back during just one ping to the rotational center $C$ and back.

In order to evaluate this ratio, for given numbers $N$ and $\beta_E$, we can consider separately the number $\#EFE_T$ of successive pings of $E$ to $F$ and back which $E$ is counting in the course of one full turn around the center $C$. Of course, this number can also be unambiguously determined in terms of quantities which were entirely measured by members of the given inertial frame (who were passed by the marks on the disk edge). Namely, setting

$$x := 2~\pi~R / N,$$

$$x + t_{EF} ~c ~\beta = c~t_{EF}; \, \, \, \, \, t_{EF} = \frac{x}{c} \frac{1}{1 - \beta},$$

$$x = t_{FE} ~c ~\beta + c~t_{FE}; \, \, \, \, \, t_{FE} = \frac{x}{c} \frac{1}{1 + \beta},$$

$$t_{EFE} = t_{EF} + t_{FE} = \frac{2~x}{c} \frac{1}{1 - \beta^2},$$ and

$$\#EFE_T = T / t_{EFE} = \frac{2~\pi~R}{c~\beta}~c~\frac{1 - \beta^2}{2~x} = N~\frac{1 - \beta^2}{2~\beta}.$$

The other relevant value, $\#ECE_T$, which $E$ can obtain directly by counting (and which can be likewise unambiguously evaluated from the perspective of the inertial frame) is the ratio between some (large) number of successive pings to $C$ and back and the corresponding (also large) number of successive full turns. Obviously obtains:

$$\#ECE_T = \frac{2~\pi~R}{c~\beta} / \frac{2~R}{c} = \frac{\pi}{\beta}.$$

Inserting these two values in the calculation of $2~\pi^C_{\beta}$:

$$2~\pi^C_{\beta} := N ~ \tau_{EFE} / \tau_{ECE} = N ~ \frac{\#ECE_T}{\#EFE_T} = N ~ \frac{\pi}{\beta} \times \frac{1}{N}~\frac{2~\beta}{1 - \beta^2} = \frac{2~\pi}{1 - \beta^2} \gt 2~\pi.$$

Comparing instead to the "antipodal maximum":

$$\pi^A_{\beta} := N ~ \tau_{EFE} / \tau_{ECE} \times 1 / \text{Max}[ \tau_{EAE} / \tau_{ECE} ] = \frac{2~\pi^C_{\beta}}{\text{Max}[ \tau_{EAE} / \tau_{ECE} ]} \gt \pi^C_{\beta} \gt \pi.$$

For both these cases therefore (with notation as used in the question): $$\pi_o > \pi.$$