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## Main Question or Discussion Point

I've looked at Taylor and Wheeler's

In Taylor and Wheeler a meter stick (spin, gyroscope), aligned along the x-axis of S, is given a velocity [itex]{\beta _x}[/itex]relative to S. Its rest frame is then S'. It is then given a velocity [itex]\beta {'_y}[/itex] relative to the S'. After invoking the "simultaneity of relativity" (i.e. as viewed from S, the right side of the meter stick gets a y-velocity later than the left side) it turns out that the angle of rotation of the meter stick, as viewed from S, is a clockwise [itex]{\beta _x}{\beta _y}[/itex]. If the meter stick is taken around a circle it turns out that the clockwise rotation is between 0 and [itex]{\beta _x}{\beta _y}[/itex] depending on the orientation of the meter stick with respect to its motion. On average the rotation is [itex]\frac{{{\beta _x}{\beta _y}}}{2}[/itex].

Eisberg and Goldstein on the other hand do not look directly at the meter stick, but instead look at the rest frames through which the meter stick passes. The original rest frame is [itex]{S_1}[/itex], the next rest frame,

[itex]{S_2}[/itex], is found by giving a boost [itex]{\beta _x}[/itex] with respect to [itex]{S_1}[/itex] and finally

[itex]{S_3}[/itex] is found by giving a boost of [itex]\beta {'_y}[/itex] relative to [itex]{S_2}[/itex]. The rest frame

[itex]{S_3}[/itex] is found to rotate through an angle of [itex]\frac{{{\beta _x}{\beta _y}}}{2}[/itex] as seen by [itex]{S_1}[/itex] and this is then equated with the rotation of the meter stick. They don't have an oscillating term depending on the orientation of the meter stick with respect to the direction of motion.

So, does the meter stick rotate by [itex]{\beta _x}{\beta _y}[/itex] or by [itex]\frac{{{\beta _x}{\beta _y}}}{2}[/itex] when it aligned parallel to the motion? Does the Eisberg/Goldstein approach use the "relativity of simultaneity"? I can't see where they've used it and yet it would be strange if they get any rotation at all without its use? What happened to the oscillating term? Wouldn't the Eisberg/Goldstein approach yield the wrong answer if we did an astronomical calculation of a partial period?

**Spacetime Physics**Example 103 on the Thomas Precession and also the discussion of Thomas precession in Eisberg and Goldstein (3rd edition). Both treat the rotation angle gotten by the addition of 2 non-collinear velocities. The answers they get are different and I'm trying to figure out why. In doing the calculations I used the usual approximations [itex]{\beta _y} < < {\beta _x} < < 1[/itex], so for instance, [itex]\left( {\gamma - 1} \right)\frac{{{\beta _y}}}{{{\beta _x}}} \approx \left( {\frac{1}{{\sqrt {1 - \beta _x^2} }} - 1} \right)\frac{{{\beta _y}}}{{{\beta _x}}} \approx \left( {\frac{{\beta _x^2}}{2}} \right)\frac{{{\beta _y}}}{{{\beta _x}}} = \frac{{{\beta _x}{\beta _y}}}{2}[/itex].In Taylor and Wheeler a meter stick (spin, gyroscope), aligned along the x-axis of S, is given a velocity [itex]{\beta _x}[/itex]relative to S. Its rest frame is then S'. It is then given a velocity [itex]\beta {'_y}[/itex] relative to the S'. After invoking the "simultaneity of relativity" (i.e. as viewed from S, the right side of the meter stick gets a y-velocity later than the left side) it turns out that the angle of rotation of the meter stick, as viewed from S, is a clockwise [itex]{\beta _x}{\beta _y}[/itex]. If the meter stick is taken around a circle it turns out that the clockwise rotation is between 0 and [itex]{\beta _x}{\beta _y}[/itex] depending on the orientation of the meter stick with respect to its motion. On average the rotation is [itex]\frac{{{\beta _x}{\beta _y}}}{2}[/itex].

Eisberg and Goldstein on the other hand do not look directly at the meter stick, but instead look at the rest frames through which the meter stick passes. The original rest frame is [itex]{S_1}[/itex], the next rest frame,

[itex]{S_2}[/itex], is found by giving a boost [itex]{\beta _x}[/itex] with respect to [itex]{S_1}[/itex] and finally

[itex]{S_3}[/itex] is found by giving a boost of [itex]\beta {'_y}[/itex] relative to [itex]{S_2}[/itex]. The rest frame

[itex]{S_3}[/itex] is found to rotate through an angle of [itex]\frac{{{\beta _x}{\beta _y}}}{2}[/itex] as seen by [itex]{S_1}[/itex] and this is then equated with the rotation of the meter stick. They don't have an oscillating term depending on the orientation of the meter stick with respect to the direction of motion.

So, does the meter stick rotate by [itex]{\beta _x}{\beta _y}[/itex] or by [itex]\frac{{{\beta _x}{\beta _y}}}{2}[/itex] when it aligned parallel to the motion? Does the Eisberg/Goldstein approach use the "relativity of simultaneity"? I can't see where they've used it and yet it would be strange if they get any rotation at all without its use? What happened to the oscillating term? Wouldn't the Eisberg/Goldstein approach yield the wrong answer if we did an astronomical calculation of a partial period?