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Ideal rulers in relativity are generally assumed to be consistent with radar measurements. The modern definition of the unit of length is defined in terms of the speed of light. However, here is an a simple example where a ruler measurement does no coincide with the radar measurement.

To paraphrase Einstein's Carousel thought experiment, Einstein states that the rulers on the circumference of the spinning carousel length contract while the rulers placed parallel to the radius do not, so that according to measurements by these ruler's, the circumference is no longer equal to the Euclidean expectation that the circumference is equal to 2*PI*R. In fact the circumference is equal to 2*PI*R*gamma according to an accelerated observer on the carousel.

Now if the accelerated observer riding on the edge of the carousel makes measurements using radar instead of rulers, he discovers that the ruler measurement of the radius is gamma times larger than the radar measurement and that the circumference is equal to 2*PI*R*gamma^2.

Does this mean we have to be careful to always specify ruler or radar measurement in relativity, such as when defining Born rigidity?

Where does this leave the modern definition of the unit of length if the ruler measurement is not always consistent with the speed of light relationship using radar?

Interestingly, in the spinning carousel example, if we recalibrate the rulers parallel to the radius, individually using radar measurements, we find we can fit more of these recalibrated rulers along the radius and might even recover the Euclidean geometry, whereby the circumference is once again equal to 2*PI*R. However, the spatial dimensions of the disc as measured by the observer riding on the spinning carousel, will still appear larger than the measurements made by a non rotating inertial observer that is at rest with the centre of the carousel.

To paraphrase Einstein's Carousel thought experiment, Einstein states that the rulers on the circumference of the spinning carousel length contract while the rulers placed parallel to the radius do not, so that according to measurements by these ruler's, the circumference is no longer equal to the Euclidean expectation that the circumference is equal to 2*PI*R. In fact the circumference is equal to 2*PI*R*gamma according to an accelerated observer on the carousel.

Now if the accelerated observer riding on the edge of the carousel makes measurements using radar instead of rulers, he discovers that the ruler measurement of the radius is gamma times larger than the radar measurement and that the circumference is equal to 2*PI*R*gamma^2.

Does this mean we have to be careful to always specify ruler or radar measurement in relativity, such as when defining Born rigidity?

Where does this leave the modern definition of the unit of length if the ruler measurement is not always consistent with the speed of light relationship using radar?

Interestingly, in the spinning carousel example, if we recalibrate the rulers parallel to the radius, individually using radar measurements, we find we can fit more of these recalibrated rulers along the radius and might even recover the Euclidean geometry, whereby the circumference is once again equal to 2*PI*R. However, the spatial dimensions of the disc as measured by the observer riding on the spinning carousel, will still appear larger than the measurements made by a non rotating inertial observer that is at rest with the centre of the carousel.

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