《力学》课程教学资源（参考资料）其它方法推导洛伦兹变换（1/2）Getting the Lorentz transformations without requiring an invariant speed.pdf_大学文库

Getting the Lorentz transformations without requiring an invariant speed Andrea Pelissetto*and Massimo Testat Dipartimento di Fisica. Universiti degli Studi di Roma "La Sapienza" and INFN-Sezione di Roma I Piazzale A.Moro 2,1-00185 Roma,Italy (Dated:April 10,2015) The structure of the Lorentz transformations follows purely from the absence of privileged inertial reference frames and the group structure(closure under composition)of the transformations-two assumptions that are simple and physically necessary.The existence of an invariant speed is not a necessary assumption,and in fact is a consequence of the principle of relativity (though the finite value of this speed must,of course,be obtained from experiment).Von Ignatowsky derived this result in 1911,but it is still not widely known and is absent from most textbooks.Here we present a completely elementary proof of the result,suitable for use in an introductory course in special 2 relativity. I.INTRODUCTION ing x and t only,setting y'=y and 2'=2.This is equivalent to choosing coordinates so that and O are In standard textbooks,the Lorentz transformation in relative motion along the r and direction.13 ⊙0 equations,which connect inertial reference frames,are deduced from the invariance of the speed of light, which implies the invariance of the Minkowski interval.! A different approach,however,was followed by von Ignatowsky.2 Only six years after the formulation of spe- cial relativity,he proved that the Lorentz transforma- tions arise under quite general conditions,without as- suming a priori the existence of an invariant speed.Von Ignatowsky showed that the only admissible transforma- C】 tions consistent with the principle of inertia,the isotropy of space,the absence of preferred inertial frames,and a group structure(i.e.,closure under composition),are the Lorentz transformations,in which c can be any veloc- ity scale,or the Galilei transformations.This surprising result shows that the Lorentz transformations are not directly related to the properties of electromagnetic ra- diation.Electromagnetism is only relevant,if present within the theory,as a way to fix the arbitrary veloc- ity scale,which is then identified with the speed of light. This deep and fascinating result,although well known in the specialized literature,3-11 is not commonly found in textbooks,12 because its usual proof is rather complicated and uninspiring.It is the purpose of the present paper to present a derivation of the von Ignatowsky result us- FIG.1.Two inertial frames in relative motion along the x- ing elementary considerations which,in our opinion,shed axis. new light on the result. We assume the validity of the principle of inertia:in an inertial frame free particles undergo rectilinear motion II.NOTATION FOR TRANSFORMATIONS with constant speed.Therefore,if the trajectory of a BETWEEN FRAMES particle is a straight line in the frame of observer and its speed is constant,the trajectory is also a straight line We wish to characterize the transformations that relate in the frame of observer O and also the speed in the new two different inertial frames.Let us consider two inertial frame is constant.This condition implies14 that the two observers O and O'.Let r =(y,z)and t be space and inertial frames are related by a linear transformation. time coordinates for O and r'=(',y',2')and t'be the We can therefore write corresponding quantities for O. In order to simplify the argument,we will restrict our 20=Ax+BT, (1) considerations to the subgroup of transformations involv- '=Cx+Dt, (2)

arXiv:1504.02423v1 [gr-qc] 8 Apr 2015 Getting the Lorentz transformations without requiring an invariant speed Andrea Pelissetto ∗ and Massimo Testa † Dipartimento di Fisica, Universit`a degli Studi di Roma “La Sapienza” and INFN – Sezione di Roma I Piazzale A. Moro 2, I-00185 Roma, Italy (Dated: April 10, 2015) The structure of the Lorentz transformations follows purely from the absence of privileged inertial reference frames and the group structure (closure under composition) of the transformations—two assumptions that are simple and physically necessary. The existence of an invariant speed is not a necessary assumption, and in fact is a consequence of the principle of relativity (though the finite value of this speed must, of course, be obtained from experiment). Von Ignatowsky derived this result in 1911, but it is still not widely known and is absent from most textbooks. Here we present a completely elementary proof of the result, suitable for use in an introductory course in special relativity. I. INTRODUCTION In standard textbooks, the Lorentz transformation equations, which connect inertial reference frames, are deduced from the invariance of the speed of light, which implies the invariance of the Minkowski interval. 1 A different approach, however, was followed by von Ignatowsky. 2 Only six years after the formulation of special relativity, he proved that the Lorentz transformations arise under quite general conditions, without assuming a priori the existence of an invariant speed. Von Ignatowsky showed that the only admissible transformations consistent with the principle of inertia, the isotropy of space, the absence of preferred inertial frames, and a group structure (i.e., closure under composition), are the Lorentz transformations, in which c can be any velocity scale, or the Galilei transformations. This surprising result shows that the Lorentz transformations are not directly related to the properties of electromagnetic radiation. Electromagnetism is only relevant, if present within the theory, as a way to fix the arbitrary velocity scale, which is then identified with the speed of light. This deep and fascinating result, although well known in the specialized literature,3–11 is not commonly found in textbooks,12 because its usual proof is rather complicated and uninspiring. It is the purpose of the present paper to present a derivation of the von Ignatowsky result using elementary considerations which, in our opinion, shed new light on the result. II. NOTATION FOR TRANSFORMATIONS BETWEEN FRAMES We wish to characterize the transformations that relate two different inertial frames. Let us consider two inertial observers O and O ′ . Let r = (x, y, z) and t be space and time coordinates for O and r ′ = ( x ′ , y ′ , z ′ ) and t ′ be the corresponding quantities for O ′ . In order to simplify the argument, we will restrict our considerations to the subgroup of transformations involving x and t only, setting y ′ = y and z ′ = z. This is equivalent to choosing coordinates so that O and O ′ are in relative motion along the x and x ′ direction.13 x’ x y’ y z’ z O’ O FIG. 1. Two inertial frames in relative motion along the x - axis. We assume the validity of the principle of inertia: in an inertial frame free particles undergo rectilinear motion with constant speed. Therefore, if the trajectory of a particle is a straight line in the frame of observer O and its speed is constant, the trajectory is also a straight line in the frame of observer O ′ and also the speed in the new frame is constant. This condition implies14 that the two inertial frames are related by a linear transformation. We can therefore write x0′ = Ax 0 + Bx, (1) x ′ = Cx 0 + Dx, (2)

2 where we have defined the time coordinates o and ro Let us consider the two events E and E2 at the same measured in units of distance,as spatial location in frame O,separated in time by r.From x0=t, Eq.(1),observer O'measures a time lapse T given by (3) T'=AT. (11) where is an arbitrary constant with units of speed.The inverse transformation follows immediately: Consider now the complementary situation in which an identical clock,at rest with respect to measures the (D0'-Br, same time interval T between two events Ea and Ea that (4) are at the same location in O.Correspondingly,from x=-C0+Ar), 1 Eq.(4),the observer O measures a time lapse T given by (5) T=D (12) with As stated above,if neither nor O'is in some way privi- △≡AD-BC≠0. (6) leged,the two time intervals T and T'should be identical, It is useful to rewrite the transformation in matrix no- i.e.,T=T,which implies tation,introducing A=D 1 (13) A B =C D) (7) A similar argument applies in the case of a rod of rest length lo,oriented along the z axis.If the rod is at rest with respect to O,Eq.(2)implies that the length l and its inverse measured by observer O satisfies =(2) (8) lo Dl. (14) Analogously,for a rod at rest in O,we have,from Eq.(5), Then,we have that the length I'measured by O satisfies ()=△() lo二 (15) (9) Again,the absence of privileged reference frames requires that l=l'and therefore and A （9=A() =D. (16) (10 Combining Eqs.(13)and (16),we have A (17) III.EQUIVALENCE OF INERTIAL 2=A. REFERENCE FRAMES The solution A =0 is not acceptable on physical grounds, because it would lead to the meaningless result o=Br. The main ingredient in the proof is the requirement so we are forced to choose that there are no privileged frames:all inertial frames are |△=1. (18) equivalent.We wish now to express this hypothesis in a more transparent way that allows us to put constraints If we restrict ourselves to proper transformations,we on the matrix A. have simply Let us consider two events,E and E2.at the same △=1， (19) spatial location in frame but separated by a time dif- ference r.In O'the two events are separated by a time so that lapse T.One natural requirement is that if we consider A=D. (20) two different events,E3 and E4,that are at the same spa- tial location in frame O',but separated by the same time These conditions imply that the transformation relating difference r in O,then these two events are separated two different inertial frames is of the form by a time lapse T=T in O.Similarly,we require that A B if observer measures the length l(along the x axis)of C A (21) a rod that is at rest with respect to O'and has length lo in the O'frame,the same result is obtained by for an with identical rod at rest with respect to always along the A2-BC=1, (22) x axis.These requirements constrain the coefficients of the transformation A. the latter condition being a consequence of Eq.(19)

2 where we have defined the time coordinates x 0 and x 0 ′ , measured in units of distance, as x 0 = ¯vt, (3) where ¯v is an arbitrary constant with units of speed. The inverse transformation follows immediately: x 0 = 1 ∆ (Dx0 ′ − Bx′ ), (4) x = 1 ∆ (−Cx0 ′ + Ax′ ), (5) with ∆ ≡ AD − BC 6= 0. (6) It is useful to rewrite the transformation in matrix notation, introducing Λ ≡ A B C D (7) and its inverse Λ −1 = 1 ∆ D −B −C A . (8) Then, we have x 0 ′ x ′ = Λ x 0 x (9) and x 0 x = Λ−1 x 0 ′ x ′ . (10) III. EQUIVALENCE OF INERTIAL REFERENCE FRAMES The main ingredient in the proof is the requirement that there are no privileged frames: all inertial frames are equivalent. We wish now to express this hypothesis in a more transparent way that allows us to put constraints on the matrix Λ. Let us consider two events, E1 and E2, at the same spatial location in frame O, but separated by a time difference τ. In O′ the two events are separated by a time lapse T ′ . One natural requirement is that if we consider two different events, E3 and E4, that are at the same spatial location in frame O′ , but separated by the same time difference τ in O′ , then these two events are separated by a time lapse T = T ′ in O. Similarly, we require that if observer O measures the length l (along the x axis) of a rod that is at rest with respect to O′ and has length l0 in the O′ frame, the same result is obtained by O′ for an identical rod at rest with respect to O, always along the x axis. These requirements constrain the coefficients of the transformation Λ. Let us consider the two events E1 and E2 at the same spatial location in frame O, separated in time by τ. From Eq. (1), observer O′ measures a time lapse T ′ given by T ′ = Aτ. (11) Consider now the complementary situation in which an identical clock, at rest with respect to O′ , measures the same time interval τ between two events E3 and E4 that are at the same location in O′ . Correspondingly, from Eq. (4), the observer O measures a time lapse T given by T = D ∆ τ. (12) As stated above, if neither O nor O′ is in some way privileged, the two time intervals T and T ′ should be identical, i.e., T ′ = T , which implies A = D ∆ . (13) A similar argument applies in the case of a rod of rest length l0, oriented along the x axis. If the rod is at rest with respect to O′ , Eq. (2) implies that the length l measured by observer O satisfies l0 = Dl. (14) Analogously, for a rod at rest in O, we have, from Eq. (5), that the length l ′ measured by O′ satisfies l0 = A ∆ l ′ . (15) Again, the absence of privileged reference frames requires that l = l ′ and therefore A ∆ = D. (16) Combining Eqs. (13) and (16), we have A ∆2 = A. (17) The solution A = 0 is not acceptable on physical grounds, because it would lead to the meaningless result x 0 ′ = Bx, so we are forced to choose |∆| = 1. (18) If we restrict ourselves to proper transformations, we have simply ∆ = 1, (19) so that A = D. (20) These conditions imply that the transformation relating two different inertial frames is of the form Λ = A B C A , (21) with A 2 − BC = 1, (22) the latter condition being a consequence of Eq. (19)

3 IV.GROUP STRUCTURE Then we obtain To further constrain the structure of the matrix A,we now add the natural requirement that the transforma- )-(（) (30) tions connecting two inertial frames constitute a group, i.e.,that the combination of two such transformations yields a third transformation of the same form. where We consider two transformations, C=CVlal. (31) =() Eq.(22)becomes now C2 A2 (23) 42-c2-1. and their product, a (32) A3=A1A2= A If a is negative,then A is an orthogonal rotation matrix. In this case,there are transformations A such that An is close to the identity for some value of n,and this is clearly = (A1A2+B1C2 A1B2+B1A2 C1A2+A1C2 A1A2+C1B2 (24) unphysical.Therefore,we can exclude this possibility and we are left with transformations of the form The matrix A3 is an element of the set of admissible ma- trices only if its diagonal elements are equal [see Eq.(21)], =( (33) that is,if B1C2=C1B2. (25) where A2-C2=1.In this case the observer O'moves In order to satisfy Eq.(25)for all transformations of with speed w with respect to observer O,where w is the form(23),we have three different possibilities: determined by (i)B=aC,where a is a nonzero constant; C'/A=w/. (34) (ii)B=0 and A=1,where the second equation fol- The condition A2-C2=1,implies lows from Eq.(22);or 1 (iii)C=0 and A 1. A=Y= (35) V1-w22 Case (iii)is easily recognized to be physically uninter- C'yw/c (36) esting. Case (ii)corresponds to the Galilean transformations: so that A is a generic Lorentz transformation,with the 0'=x° speed of light identified with c. (26 x'=Cx+Z, (27)） the parameter C being the relative velocity of the two V.CONCLUSIONS frames in units of元. In case (i)we change the definition of zo in Eq.(3) Lorentz and Galilei transformations are the only struc- introducing tures compatible with the principles of inertia and rel- 0=t= ativity,i.e.,the nonexistence of privileged reference (28) frames.This result,which is easily stated,is usually obtained by means of rather complicated proofs.In this with paper we have presented an elementary derivation,suit- able for presentation in undergraduate courses in special C= 29 relativity. a andrea.pelissettocromal.infn.it 2 W.von Ignatowsky,"Das Relativitatsprinzip,"Archiv der t massimo.testa@romal.infn.it Mathematik und Physik 17,1-24 (1911). 1J.D.Jackson,Classical Electrodynamics,third edition 3 H.Bacry and J.M.Levy-Leblond,"Possible kinematics," (Wiley,New York,1998). J.Math.Phys.9,1605-1614(1968)

3 IV. GROUP STRUCTURE To further constrain the structure of the matrix Λ, we now add the natural requirement that the transformations connecting two inertial frames constitute a group, i.e., that the combination of two such transformations yields a third transformation of the same form. We consider two transformations, Λ1 ≡ A1 B1 C1 A1 , Λ2 ≡ A2 B2 C2 A2 , (23) and their product, Λ3 = Λ1Λ2 = A1 B1 C1 A1 A2 B2 C2 A2 = A1A2 + B1C2 A1B2 + B1A2 C1A2 + A1C2 A1A2 + C1B2 . (24) The matrix Λ3 is an element of the set of admissible matrices only if its diagonal elements are equal [see Eq. (21)], that is, if B1C2 = C1B2. (25) In order to satisfy Eq. (25) for all transformations of the form (23), we have three different possibilities: (i) B = αC, where α is a nonzero constant; (ii) B = 0 and A = 1, where the second equation follows from Eq. (22); or (iii) C = 0 and A = 1. Case (iii) is easily recognized to be physically uninteresting. Case (ii) corresponds to the Galilean transformations: x 0′ = x 0 , (26) x ′ = Cx0 + x, (27) the parameter C being the relative velocity of the two frames in units of ¯v. In case (i) we change the definition of x0 in Eq. (3) introducing x˜ 0 = ¯ct = c¯ v¯ x 0 (28) with c¯ = v¯ p |α| . (29) Then we obtain x˜ 0 ′ x ′ = A α |α| C ′ C ′ A ! x˜ 0 x , (30) where C ′ = C p |α|. (31) Eq. (22) becomes now A 2 − α |α| C ′2 = 1. (32) If α is negative, then Λ is an orthogonal rotation matrix. In this case, there are transformations Λ such that Λn is close to the identity for some value of n, and this is clearly unphysical. Therefore, we can exclude this possibility and we are left with transformations of the form Λ = A C′ C ′ A , (33) where A2 − C ′2 = 1. In this case the observer O′ moves with speed w with respect to observer O, where w is determined by C ′ /A = w/c. ¯ (34) The condition A2 − C ′2 = 1, implies A = γ = 1 p 1 − w2/c¯ 2 (35) C ′ = γw/c¯ (36) so that Λ is a generic Lorentz transformation, with the speed of light identified with ¯c. V. CONCLUSIONS Lorentz and Galilei transformations are the only structures compatible with the principles of inertia and relativity, i.e., the nonexistence of privileged reference frames. This result, which is easily stated, is usually obtained by means of rather complicated proofs. In this paper we have presented an elementary derivation, suitable for presentation in undergraduate courses in special relativity. ∗ andrea.pelissetto@roma1.infn.it † massimo.testa@roma1.infn.it 1 J. D. Jackson, Classical Electrodynamics, third edition (Wiley, New York, 1998). 2 W. von Ignatowsky, “Das Relativit¨atsprinzip,” Archiv der Mathematik und Physik 17, 1–24 (1911). 3 H. Bacry and J. M. L´evy-Leblond, “Possible kinematics,” J. Math. Phys. 9, 1605–1614 (1968)