Speed of an object in a rotating frame

[JVNY]

Great. Plenty to learn here. Thanks PeterDonis, DaleSpam and SlowThinker.

 

[Laurie K]

Øyvind Grøn provides further insights and much more in his "Space geometry in rotating reference frames: A historical appraisal" paper.

http://areeweb.polito.it/ricerca/relgrav/solciclos/gron_d.pdf

Grøn's own solution (figure 9.) shows the timings required so that light pulses emitted from each of 16 points along the rolling wheel rim arrive at the same point and time as the last point touches the road. To avoid Born rigidity issues the wheel has no thickness (i.e. x,y, t dimensions with z =0) and the observer is in the same plane of rotation as the relativistically rolling wheel. The results can be easily cross checked as the differences between each emission point time and each respective axle x location should reveal the velocity of the wheel as well as the angular velocity of any point on the rim of that wheel.

 

[pervect]

Another good (IMO) and recent paper on measuring distances on a rotating platform is "The Relative Space: Space Measurements on a Rotating Platform" http://arxiv.org/abs/gr-qc/0309020. The author (Ruggerio) first gives an exact definition of what he means by "the space" of a rotating disk as a quotient manfiold. My inexact intuitive summary of this definition is as follows. If one think of painting a dot on a rotating disk, this dot traces out a worldline in 4-d space-time. The "quotient manifold" is just a mapping from the 4-d worldline to a single point in an abstract 3-d space.

The author then proceeds to demonstrate how operationally by the exchange of light signals an observer at one point in this "relative space" can measure their distance to another nearby point in this "relative space".

The algebra is somewhat messy (not terribly messy, it requires solving a quadratic equation), but the concepts are simple and easily tied to the SI definition of the meter

The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second.

The end result is a non-euclidean geometry (the authors give a specific metric) of the "relative space". References to other papers with similar results via different methods are also given.

 

[JVNY]

Øyvind Grøn provides further insights and much more in his "Space geometry in rotating reference frames: A historical appraisal" paper.

http://areeweb.polito.it/ricerca/relgrav/solciclos/gron_d.pdf

Grøn's own solution (figure 9.) shows the timings required so that light pulses emitted from each of 16 points along the rolling wheel rim arrive at the same point and time as the last point touches the road. To avoid Born rigidity issues the wheel has no thickness (i.e. x,y, t dimensions with z =0) and the observer is in the same plane of rotation as the relativistically rolling wheel. The results can be easily cross checked as the differences between each emission point time and each respective axle x location should reveal the velocity of the wheel as well as the angular velocity of any point on the rim of that wheel.

Thanks very much.

 

[JVNY]

Another good (IMO) and recent paper on measuring distances on a rotating platform is "The Relative Space: Space Measurements on a Rotating Platform" http://arxiv.org/abs/gr-qc/0309020. The author (Ruggerio) first gives an exact definition of what he means by "the space" of a rotating disk as a quotient manfiold. My inexact intuitive summary of this definition is as follows. If one think of painting a dot on a rotating disk, this dot traces out a worldline in 4-d space-time. The "quotient manifold" is just a mapping from the 4-d worldline to a single point in an abstract 3-d space.

The author then proceeds to demonstrate how operationally by the exchange of light signals an observer at one point in this "relative space" can measure their distance to another nearby point in this "relative space".

The algebra is somewhat messy (not terribly messy, it requires solving a quadratic equation), but the concepts are simple and easily tied to the SI definition of the meter
The end result is a non-euclidean geometry (the authors give a specific metric) of the "relative space". References to other papers with similar results via different methods are also given.

Thanks very much.