I have a couple of questions about exercise **103 (yes, a two-star problem!) in Taylor & Wheeler's(adsbygoogle = window.adsbygoogle || []).push({}); Spacetime Physics.

In part (a), it says "For an atom [itex]\beta_r \leq Z/137[/itex] (Ex. 101), and for small Z, [itex]\beta_r \ll 1[/itex]. Therefore [itex]\tan(d\phi) \approx d\phi \approx -\beta_r^2 \sin(\alpha)[/itex]." But it's also said that in Newtonian mechanics there is no Thomas precession. And Newtonian mechanics is supposed to be the mechanics of negligibly small beta. Are Taylor and Wheeler talking implicitly about two degrees of negligible smallness (one negligibly small compared to the other), that is: one in which beta is so small that [itex]d\phi = 0[/itex], and another, the subject of this exercise, in which beta is small enough to ignore some aspects of the problem (if so, which exactly?) but not small enough to ignore all precession.

[tex]\beta_a \ll \ll 1 \Rightarrow d\phi = 0[/tex]

[tex]\beta_b \ll 1 \Rightarrow \tan(d\phi) \approx d\phi \approx -b_r^2 \sin(\alpha)[/itex]

where [itex]\beta_a \ll \ll 1[/itex] and [itex]\beta_b \ll 1[/itex] means that [itex]\beta_a \ll \ll \beta_b[/itex].

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Parts (a), (b) and (c) mention two inertial frames of reference: a "laboratory frame" (unprimed), and a "rocket frame" (primed). Borrowing Kevin Brown's notation, I'll call them K and K', and use K'' for the rest frame of ball B. The rocket frame, K', is equated with the rest frame of ball A which is moving right along the x axis of K, according to figures 130 and 131. In the scenario shown in Fig. 131, is the x' axis parallel to the x axis according to K? Is the x' axis parallel to the x axis according to K'? Is the x' axis parallel to the projection of the spin of B onto the xy-plane? Does this last question have a frame-independent answer; if not, how would it be answered wrt K and K'?

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Which frame is Fig. 132 drawn according to? Since the angle alpha is defined in the laboratory frame, K, I guessed it would be K. In support of this is the fact that no obvious distortion of the polygon is shown, no squashing in the direction of motion of A. On the other hand, part (c) seems to be contrasting something being parallel (in part (a) and Fig. 131) with something being perpendicular (in part (b) and Fig. 132); and it's only in K' that the spin projection of B is parallel to the direction of movement, which suggests that Fig. 132 is according to K', perhaps with alpha representing a different angle. I'm presuming the angle alpha is changed by a boost from K to K'.

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I don't understand how the beginning of part (c)--"As the electron moves around its orbit, the projection of its spin axis onto the xy plane of the orbit will occasionally beparallelto its direction of motion (a), and occasionallyperpendicularto its direction of motion (b)"--relates to the actual examples in (a) and (b).

In part (a), the spin projection of electron B is not parallel to its direction of motion, or to that of A, whose direction of motion is said to be effectively the same as that of B when alpha is "small". This is the electron whose spin projection is analysed in part (a) and illustrated in Fig. 131. This is the electron whose spin projection changes direction when we boost from K to K', as part (a) shows. If it had been parallel to the direction of motion of A and K', then surely a boost in this direction would not rotate its spin projection, so [itex]d\phi[/itex] would be zero, not [itex]-\beta_r \sin(\theta)[/itex]. Isn't this exemplified by electron A, whose spin projectin is parallel to the direction of the boost, and indeed not rotated by the boost?

In part (b), and Fig. 132, both electrons, and in partricular B, have spin projections perpendicular to the boost from K to K' (i.e. to the direction of motion of A), and so their direction is unchanged by that boost, so it might be argued that it doesn't matter. But if both electrons, and in particular B, had been parallel to the boost in K in Fig. 131, then that wouldn't have made a difference to their direction either, and they'd have been parallel in K' too. So how do parts (a) and (b) actually contrast parallel and perpendicular?

So far, the discussion has involved only a boost from K to K'. There hasn't been any analysis of the scenario wrt K'', so presumably the fact that the spin projection of B is not perpendicular or parallel to the direction of motion of B itself is not directly significant to these questions, I guess because their approximation is going to approximate towards the motion of A rather than the motion of B.

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# Taylor & Wheeler on Thomas rotation

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