Chapppter 11: Special Theory of Relativity Section 1: Definitions and Operation Rules of Tensors of Differen t Ran ks in the 4-Dimens iona l Space (Ref.: Marion & Heald, “Classical Electromagnetic Radiation,” The Lorentz Transformation : 3rd ed ., Ch . 14) Consider two Lorentz f rames, K and KK. Frame moves along Einstein’s special theory of relativity is based on two postulates: the common zK-axis with constant speed v0 relative to frame . 1. LfhiLaws of physics are iitifinvariant in form illLtfin all Lorentz frames (I n Assume that at tt 0, coordinate axes of frames K and K relativity, we often call the inertial frame a Lorentz frame.) overlap. Postulate 2 leads to the following Lorentz transformation 2. The speed of light in vacuum has the same value c in all Lorentz for space and time coordinates. [derived in Appendix A, Eq. (A.15), frames, independent of the motion of the source. where the relative motion is assumed to be along the x-axis.] The basics of the theory are covered in Appendix A on an xx x x elementary level with an emphasis on the Lorentz transformation and (x , yzt, , ) (,xy y , zt, ) relativistic momentum/energy. Here, we examine relativity in the yy K K z , z (1) four-dimensional space of x and t, which provides the framework for zzvt 00 v0 us to examine the laws of mechanics and electromagnetism. The v0 tt 0 ()2 z y y FKdKFrames K and K contents of the lecture notes depart considerably from Ch.11 of c coincide at tt 0. Jackson. Instead, we follow Ch. 14 of Marion. v2 1 whihffhhere () 10 2 is the Lorentz factor for the transfiformation. 0 c2 In the lecture notes, section numbers do not follow Jackson. 1 2 11.1 Definitions and Operation Rules of … (continued) 11.1 Definitions and Operation Rules of … (continued) AbA note about notat ion: IbkhlidIn many books, the relative speed between Four- Dimension Space Quantities and Operation Rules : two frames is denoted by v and the particle velocity in a given frame Define a position vector in the 4-dimensional space of x and t as idis deno tdbted by u. This even tua llllly lea dtds to two de fin itions f or the same x ()()(xyzict, , , )( x, ict) notation : 10 0 0 4-vector spatial vector 2 1 Lorentz factor for the transformation , v 2 01 0 0 ()12 c Jackson (11.17) and a 4-D matrix as a ,/00 vc 00 000i 2 1 u 2 LtftfLorentz factor of a par tiliiticle in a given frame, 14, row num ber ()12 . 00i c Jackson (11.46) and (11.51) 1 4, column number 00 0 TidfiihhiTo avoid confusion with the notation (h(e.g. when we per form a thhLhen, the Lorentz trans format ii(1)ion in (1) can bibe written Lorentz transformation of the Lorentz factor of a particle), we will xx 10 0 0 dtthltidenote the relative speed dbttf between two frames bby v andthd the par tilticle 0 yy 01 0 0 4 velocity by v throughout this chapter, and thus define or xax (2) zz 00 i 1 v2 1 000 ()10 2 [Lorentz factor for the transformation] 0 c2 ict 00i00 0 ict 2 1 4 v 2 ()12 [Lorentz factor of a particle in a given frame]. and the inverse Lorentz transformation is: xax . (3) c 3 1 4 11.1 Definitions and Operation Rules of … (continued) 11.1 Definitions and Operation Rules of … (continued) The a matrix in (2) shows that the L orentz transformation is 1A1. Any set o f4f 4 quant iiities AAAAA ((14) 1 4) or A ( 1234, , , )hih), which an orthogonal transformation because it satisfies transform in the same way as x , i.e. dfiiidefinition of ort hogonal A aA, (5) aa (4) transformation* is called a 44vector-vector (or 44tensor-tensor of the fir st rank). *See H. Goldstein, "Classical Mechanics," 2nd edition, p.134. The position vector x [ (xyzict , , , )] of a point in the 4-D xx by (2) by (2) by (4) space is obviously a 4- vector. As another example , the mome ntum 22 Thus, xaxaxaaxxx vector of a particle in the 4-D space, defined as , p ()()( p , pp, , iE)(p, iE), xyz222222ct x y 2222 z ct, xyzcc which is a statement of postulate 2 [see Eqs. (B.1) and (B.2) in is a 4-vector because it transforms as [[q(),pp]see Eq. (A.28), Appendix A.] Appendix B.] px 10 0 0 px Ju st as the 3- dimensional v ectors (a nd tensors in general) are p 01 0 0 p 4 y y defined by their transformation properties in the x-space, we may or pap pz 00 000i pz 1 define 4- vectors (and 4 -tensors in gener al) by their transformation iE iE 00i00 0 properties in the (x , t ) space and find rules for their operation. 5 c c 6 11.1 Definitions and Operation Rules of … (continued) 11.1 Definitions and Operation Rules of … (continued) 3. Define the 4 -D operator , [, , , ] , as the counterpart 2. If a quantity is unchanged under the Lorentz transformation, xyz() ict it is called a Lorentz scalar (or 4 -vector of the zeroth rank) . The of the operator in the x-space. Then, the 4-gradient of a Lorentz Lorentz scalar is also called a Lorentz invariant. scalar, [xyz, , , ()ict ], is a 4-vector. The Lorentz scalar is in general a functi on of the components x Proof : aa (6) of a 4-vector. For example, we have just shown that xx x x 22 b(3)by (3) xx a TfTransforms as a 4 4t-vector. A 2 4. The 4-divergence of a 4-vector,,, A x , is a Lorentz scalar. HiLtlHence, x is a Lorentz scalar. Proof : A: 4-vector x aA b(5)by (5) AtfA : component of A x A AA A A xxx aa xx A (7) a by (3) by (4) 7 A is unchanged under the Lorentz transformation 8 11.1 Definitions and Operation Rules of … (continued) 11.1 Definitions and Operation Rules of … (continued) 2 2221 2 Example: In frame K, a particle'sposition s position changes by dx in a 5Th5. The 4-LliLaplacian opera tor, 22222 , (8) xyzct time interval dt. Then, dx ( dx , dy, dz, icdt) is a 4-vector. Hence, 22 is a Lorentz scalar opp,erator, i.e. [[] : a Lorentz scalar]. ddxx (() dxdx ) is a Lorentz invariant, i. e. in frame Kdd, x x Proof : We have shown in item 4 that the divergence of a 4-vector ( dx dx ) is given by ddxx. is a Lorentz scalar , i .e . AA . Let be a Lorentz scalar , then AA and are both 4-vectors (see item 3). Hence, Special case: The particle is at rest in frame K (the rest frame of 22 AA . the particle) . Hence , ddx =0 = 0 and x (0 , 0 , 0 , icd ) , where we have denoted the differential time in frame Kd by instead of dt, because 6. The dot product of two 4-vector s, AB AB, is a Lorentz scalar. frame Kd is a unique frame . is called the proper time of the particle . 22 2 2 2 22 Proof : dx dx dx dx c d dx dy dz c dt AB by (4) 2 AB AB a A a B a a AB ddt 1 v dt [a Lorentz invariant] (10) c2 AB AB (9) dxdy dz where vedtxyz dt e dt e is the velocity of the particle in frame K. 22a useful property of the AA= AA AA Discussion: (i) For the special case that K i s the rest frame of the orthogonal transformation 9 particle, v is also the relative velocity of the 2 frames. Hence, 0. 10 11.1 Definitions and Operation Rules of … (continued) 11.1 Definitions and Operation Rules of … (continued) (ii) The Lorentz transformation appl ies only to inertial frames. If the 7A7. A 4t4-tensor o fthf the second rank (()T )i is a se t o f16f 16 quantiti es, particle has an acceleration, d ( dt ) in (10) is the differential time in T ( , 1 4), which transform according to the instantaneous rest frame of the particle , in w hich the particle has TaaT (1 1) zero velocity but a finite acceleration. In general, the speed of the rest , dt frame (hence ) is a function of time , i . e. d ()t [Jackson, (11. 26)]. 8Th8. The ddf4fhddot product of a 4-tensor of the second rankd4k and a 4-vector, (iii) Consider a special case of constant particle velocity. The muon ()TA TA , is a 4-vector. has a lifetime of 2.2 sec in its rest frame between birth and decay. If the lifetime is measured in a Lorentz frame in which the muon has a aaT aA constant , then b(10)hby (10), the rest- frame lifilifetime (() ) an d thdhe measured d lifetime td are related by Proof: (TA ) T A a a a T A tt decay decay dt 1 decay td ddt tt d . birth birth birth aTAa (TA ) (12) This expresses t he phenomenon of time dilation; namely, when the time interval of a clock's rest time (e.g. d above) is observed in a Transform as a 4-vector. td moving frame, it is greater by a factor of . The invariance of d () (= ) td means that will have the same value in all Lorentz frames. 11 12 11.1 Definitions and Operation Rules of … (continued) 11.1 Definitions and Operation Rules of … (continued) T 10.