Physics question about the speed of light

[Daniel]

I am imagining a wheel of variable diameter, which is rotating at a constant speed of one rotation per day.

Suppose we make the diameter really big, such that a single point on the wheel's rim must now move faster than the speed of light in order to make its way back to the starting point.

Can somebody explain how this would work, or why it wouldn't work?

[Quaremerepulisti]

It works in general relativity, where you can transform to the non-inertial rotating frame where everything is at rest.  "Not traveling faster than the speed of light" means, in GR, not the direct comparison of velocities, but that no object can outpace its own light cone in any local inertial frame.

[Michael Wilson]

I was going to post the same response, but Quare beat me to it. (NOT!) 

[Kreuzritter]

I am imagining a wheel of variable diameter, which is rotating at a constant speed of one rotation per day.

Suppose we make the diameter really big, such that a single point on the wheel's rim must now move faster than the speed of light in order to make its way back to the starting point.

Can somebody explain how this would work, or why it wouldn't work?

No, it doesn't work. In no formulation of mechanics can you just magically make the diameter of a wheel bigger, or even change the distance of a point particle from a central force, and keep the initial angular velocity without also changing the net force applied to it or increasing its kinetic energy from somewhere. No force in any frame of reference is going to bring it to the asymptote at c.

[Daniel]

I am imagining a wheel of variable diameter, which is rotating at a constant speed of one rotation per day.

Suppose we make the diameter really big, such that a single point on the wheel's rim must now move faster than the speed of light in order to make its way back to the starting point.

Can somebody explain how this would work, or why it wouldn't work?

No, it doesn't work. In no formulation of mechanics can you just magically make the diameter of a wheel bigger, or even change the distance of a point particle from a central force, and keep the initial angular velocity without also changing the net force applied to it or increasing its kinetic energy from somewhere. No force in any frame of reference is going to bring it to the asymptote at c.

So what you're saying is that if the speed is constant then it requires more force to rotate a bigger wheel than it does to rotate a smaller wheel? And that it's somehow impossible (physical impossibility? metaphysical impossibility?) to have enough force to rotate a wheel that has a circumference of 299,792,458 meters at a speed higher than 1 rotation per second?

[Kreuzritter]

I am imagining a wheel of variable diameter, which is rotating at a constant speed of one rotation per day.

Suppose we make the diameter really big, such that a single point on the wheel's rim must now move faster than the speed of light in order to make its way back to the starting point.

Can somebody explain how this would work, or why it wouldn't work?

No, it doesn't work. In no formulation of mechanics can you just magically make the diameter of a wheel bigger, or even change the distance of a point particle from a central force, and keep the initial angular velocity without also changing the net force applied to it or increasing its kinetic energy from somewhere. No force in any frame of reference is going to bring it to the asymptote at c.

So what you're saying is that if the speed is constant then it requires more force to rotate a bigger wheel than it does to rotate a smaller wheel? And that it's somehow impossible (physical impossibility? metaphysical impossibility?) to have enough force to rotate a wheel that has a circumference of 299,792,458 meters at a speed higher than 1 rotation per second?

No, that's not the problem. In your original example you are implicitly changing the radial distance of a point mass from the center of its circular motion and just assuming that its angular velocity is going to remain constant. It isn't. There's this little thing called conservation of angular momentum. The magnitude of the angular velocity is going to decrease, ceteris paribus, and the magnitude of the tangential component of the velocity isn't going to magically increase without applying an appropriate force to it. That's the "flaw" in your thought experiment.

The rest, namely that this speed is going to approach c asymptotically without reaching it as I increase the applied force needed to keep the angular speed constant while increasing the radial distance, follows directly from Special Relativity, assuming SR is accurate.

[james03]

It works in general relativity, where you can transform to the non-inertial rotating frame where everything is at rest.

I don't know if that is valid.  Rotational velocity is not relative, but is absolute.  This is because it is accelerating at all times.

An interesting thought experiment:  Two very large pulsars with tip speed of 0.6C.  Counter rotating. Observed relative tangential velocity of the system is 1.2C from Earth.   Seems somewhat related to the Sagnac compass, which is easily solved with classical physics if you allow >C velocity.  In fact no one uses GR when designing one.  And we are dealing with photons, i.e. relativistic speeds.

[Heinrich]

This pretty much sums it up:


EMS SURVEYS IN MATHEMATICAL SCIENCES
Full-Text PDF (559 KB) | Metadata | Table of Contents | EMSS summary
Volume 5, Issue 1/2, 2018, pp. 1–64DOI: 10.4171/EMSS/26
Published online: 2018-06-18

The convexification effect of Minkowski summation

Matthieu Fradelizi[1], Mokshay Madiman[2], Arnaud Marsiglietti[3] and Artem Zvavitch[4]
(1) Université Paris-Est, Marne-la-Vallée, France
(2) University of Delaware, Newark, USA
(3) California Institute of Technology, Pasadena, USA
(4) Kent State University, Kent, USA
Let us define for a compact set A?Rn the sequence
A(k)={a1+?+akk:a1,...,ak?A}=1k(A+?+A??????????????k times).
It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that A(k) approaches the convex hull of A in the Hausdorff distance induced by the Euclidean norm as k goes to ?. We explore in this survey how exactly A(k) approaches the convex hull of A, and more generally, how a Minkowski sum of possibly different compact sets approaches convexity, as measured by various indices of non-convexity. The non-convexity indices considered include the Hausdorff distance induced by any norm on Rn, the volume deficit (the difference of volumes), a non-convexity index introduced by Schneider (1975), and the effective standard deviation or inner radius. After first clarifying the interrelationships between these various indices of non-convexity, which were previously either unknown or scattered in the literature, we show that the volume deficit of A(k) does not monotonically decrease to 0 in dimension 12 or above, thus falsifying a conjecture of Bobkov et al. (2011), even though their conjecture is proved to be true in dimension 1 and for certain sets A with special structure. On the other hand, Schneider's index possesses a strong monotonicity property along the sequence A(k), and both the Hausdorff distance and effective standard deviation are eventually monotone (once k exceeds n). Along the way, we obtain new inequalities for the volume of the Minkowski sum of compact sets (showing that this is fractionally superadditive but not supermodular in general, but is indeed supermodular when the sets are convex), demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.

Keywords: Sumsets, Brunn–Minkowski, convex hull, inner radius, Hausdorff distance, discrepancy

Fradelizi Matthieu, Madiman Mokshay, Marsiglietti Arnaud, Zvavitch Artem: The convexification effect of Minkowski summation. EMS Surv. Math. Sci. 5 (2018), 1-64. doi: 10.4171/EMSS/26

[Quaremerepulisti]

The argument about asymptotically approaching c would be correct if the center of the wheel remained an inertial frame while the angular momentum of the wheel increased as the radius increased.  But it doesn't (Lense-Thirring effect) - the inertial frame is going to be one which rotates in the same direction as the wheel - and in that frame, all velocities will be lower than c, allowing for increase in angular momentum.

[Heinrich]

The argument about asymptotically approaching c would be correct if the center of the wheel remained an inertial frame while the angular momentum of the wheel increased as the radius increased.  But it doesn't (Lense-Thirring effect) - the inertial frame is going to be one which rotates in the same direction as the wheel - and in that frame, all velocities will be lower than c, allowing for increase in angular momentum.

Is that why women talk so much?