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Thomas rotation and Thomas precession Tam´a s Matolcsi –M´a t´e Matolcsi Department of Applied Analysis,E¨o tv¨o s Lor´a nd University Budapest February 5,2008
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Abstract
Exact and simple calculation of Thomas rotation and Thomas preces-sions along a circular world line is presented in an absolute(coordinate-
free)formulation of special relativity.Besides the simplicity of calculations
the absolute treatment of spacetime allows us to gain a deeper insight into
the phenomena of Thomas rotation and Thomas precession.
Key words:Thomas rotation,Thomas precession,gyroscope
1Introduction
The’paradoxic’phenomenon of Thomas precession has given rise to much dis-cussion ever since the publication of Thomas’seminal paper(Thomas1927) in which he made a correction by a factor1/2to the angular velocity of the spin of an electron moving in a magneticfield.Let us mention here that in the literature there seems to be no standard agreement as to the usage of the terms’Thomas precession’and’Thomas rotation’.As explained in more detail in Section10below,we prefer to use the term Thomas precession to refer to the continuous change of direction,with respect to an inertial frame,of a gyroscopic vector moving along a world line.Thomas rotation,on the other hand,will refer to the spatial rotation experienced by a gyroscopic vector having moved along a’closed’world line,and having returned to its initial frame of reference(see Section9).
One of the most studied cases(Costella et al.2001,Kennedy2002) is the fact that the application of three successive Lorentz boosts(with the relative velocities adding up to zero)results,in general,in a spatial rotation: the discrete Thomas rotation(see Section4for details).The same fact is often described as’the composition of two Lorentz boosts is equivalent to a boost and a spatial rotation’.We prefer to use three Lorentz boosts instead(with the relative velocities adding up to zero),in order to return to the initial frame of reference,in accordance with our terminology of Thomas rotation.Describing the mathematical structure of discrete Thomas rotations has motivated A.A.
Ungar to build the comprehensive theory of gyrogroups and gyrovector spaces (Ungar2001).
The other case typically under consideration comes from the original ob-servation of Thomas:the continuous change of direction,with respect to an inertial frame,of a gyroscopic vector moving along a circular orbit.This phe-nomenon has been subject to considerations from various points of view(Muller 1992(Appendix),Philpott1996,Rebilas2002(Appendix),Herrera&Di Prisco 2002,Rhodes&Shemon2003).The considerations usually involve,either ex-plicitly or implicitly,the viewpoint of the orbiting’airplane’,i.e.a rotating observer.This might lead us to believe(see Herrera&Di Prisco2002)that the calculated angle of rotation depends on the definition of the rotating observer (and this could lead to an experimental checking of what the’right’definition of a rotating observer is).From our treatment below,however,it will be clear the Thomas rotation is an absolute fact,independent of the rotating(or,any other)observer.
电子计算机专业It is also interesting to note that new connections between quantum mechan-ical phenomena and Thomas rotation have recently been pointed out(L´e vay 2004).
As it is well known,the theory of special relativity contradicts our common sense notions about space
and time in many respects.Early day’paradoxes’were usually based on our intuitive assumption of absolute simultaneity.With the resolution of paradoxes such as the’twin paradox’or the’tunnel paradox’it has become common knowledge that the concept of time must be handled very carefully.As it is also well known,the theory of special relativity implies,besides the non-existence of absolute time,the non-existence of absolute space.An expression such as’a point in space’simply does not have an absolute meaning, just as the expression’an instant in time’.However,this fact seems to be given less attention to and even overlooked sometimes.The fact that the space vectors
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of any observer are usually represented as vectors in R3leads one to forget that these spaces really are different.This conceptual error the’velocity addition paradox’(Mocanu1992).The spaces of two different inertial observers are,of course,connected via the corresponding Lorentz boost,and the non-transitivity of Lorentz boosts(which,in fact,gives rise to the notion of Thomas rotation)gave the correct explanation of this’paradox’(Ungar1989,Matolcsi &Goher2001).
To grasp the essence of the concepts related to Thomas rotation,let us mention that in some sense this intriguing phenomenon is analogous to the well known twin paradox.Consider two twins in an ine
rtial frame.One of them remains in that frame for all times,while the other goes for a trip in spacetime, and later returns to his brother.It is well-known that different times have passed for the two twins:the traveller is younger than his brother.What may be surprising is that the space of the traveller when he arrives,although he experienced no torque during his journey,will be rotated compared to the space of his brother;this is,in fact,the Thomas rotation.This analogy is illuminating in one more respect:until the traveller returns to the original frame of reference it makes no sense to ask’how much younger is the traveller compared to his brother?’and’by what angle is the traveller’s gyroscope rotated compared to that of his brother?’Different observers may give different answers.When the traveller returns to his brother,these questions suddenly make perfect sense, and there is an absolute answer(independent of who the observer is)as to how much younger and how much rotated the traveller is.
Of course,an arbitrtary inertial frame can observe the brothers continuously, and can tell,at each of the frame’s instants,what difference he sees between the ages of the brothers.More explicitly,as it is well known,given a world line, an arbitrary inertial frame can tell the relation between the frame’s time and the proper time of the world line.This relation depends on the inertial frame:
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different inertial frames establish different relations.
Similarly,an arbitrary inertial frame,observing the two brothers,can tell at each frame-instant what difference he sees between the directions of the gyro-scopes of the brothers.Different inertial frames establish different relations.
This philsophy makes a clear distinction between Thomas rotation and Thomas precession connected to a world line:
–Thomas rotation refers to an absolute fact(independent of who observes it),which makes sense only for two equal local rest frames(if such exist)of the world line,
–Thomas precession refers to a relative depending on who observes the motion),which makes sense with respect to an arbitrary inertial frame.
In this paper we use the formalism of(Matolcsi1993)to give a concise and rigorous treatment of the discrete and circular-path Thomas rotations.The Thomas rotation as well as the Thomas precession(with respect to certain inertial observers)along a circular world line are calculated.Our basic concept here is that special relativistic spacetime has a four-dimensional affine structure, and c
oordinatization(relative to some observer)is,in many cases,unnecesary in the description of physical phenomena.In fact,coordinates can sometimes lead to ambiguities in concepts and definitions,and bear the danger of leading us to overlook the fact that absolute space does not exist.
As well as providing a clear overview of the appearing concepts,the coordinate-free formulation of special relativity enables us to give simple calculations.The indispensable Fermi-Walker equation is also straightforward to derive in our formalism.
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2Fundamental notions
In this section some notions and results of the special relativistic spacetime model as a mathematical structure(Matolcsi1993,1998,2001)will be recapit-ulated.As the formalism slightly differs from the usual textbook treatments of special relativity(but only the formalism:our treatment is mathematically equivalent to the usual treatments),we will point out several relations between textbook formulae and those of our formalism.
Special relativistic spacetime is an oriented four dimensional affine space M over the vector space M;
the spacetime distances form an oriented one dimen-sional vector space I,and an arrow oriented Lorentz form M×M→I⊗I, (x,y)→x·y is given.
An absolute velocity u is a future directed element of M
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