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Physics 139 Relativity Relativity Notes 2002 G. F. SMOOT Oce 398 Le Conte DepartmentofPhysics, University of California, Berkeley, USA 94720 Notes to b e found at http://aether.lbl.gov/www/classes/p139/homework/homework.html 5 Four Vectors A natural extension of the Minkowski geometrical interpretation of Sp ecial Relativity is the concept of four dimensional vectors. One could also arrive at the concept by lo oking at the transformation prop erties of vectors and noticing they do not transform as vectors unless another comp onent is added. We de ne a four-dimensional vector or four-vector for short as a collection of four comp onents that transforms according to the Lorentz transformation. The vector magnitude is invariant under the Lorentz transform. 5.1 Co ordinate Transformations in 3+1-D Space One can consider co ordinate transformations manyways: If x ;x ;x ;x = x; y ; z ; ict, 1 2 3 4 then ordinary rotations in x x plane around x 1 2 3 0 x = x cos + x sin cos sin 1 2 1 0 x = x sin + x cos sin cos 1 2 2 But in x x plane: 1 4 0 x = x cos + x sin cos sin 1 4 1 0 x = x sin + x cos sin cos 1 4 4 where the angle is de ned by q p 2 2 2 cos =1= 1v =c =1= 1+tan = tan v=c q p = sin = i 2 2 2 1+tan 1 v =c tan = iv =c = i : And thus one has the trignometic identity: 2 2 2 2 cos + sin = 1 =1 1 0 x = [x +icti ] = [x ct] 1 1 1 0 x = [x i x ] 4 1 4 0 ict = [ict i x ] 1 0 ct = [ct x ] 1 0 t = [t x =c] 1 So the extension to 3+1-D includes Lorentz transformations, if angles are imaginary. Really,we are considering the set of all 4 4 orthogonal transformations matrices in which one angle may b e pure imaginary. In general all angles may b e complex, combining real rotations in 2-space with imaginary rotations relativeto t. An alternate way of writing this is 0 x = xcosh ctsinh 0 ct = xsinh + ctcosh 1 where = cosh . 0 x = xcosi+ictsini 0 ict = xsini+ ictcosi and 1 = i = icosh ; tan = i = iv =c Still another notation is with x = ict 4 0 x = x + i x 1 4 1 0 x = x i x 4 1 4 The transformation matrix is then 0 1 0 0 i B C 0 1 0 0 B C B C @ A 0 0 1 0 i 0 0 Still yet another notation is with x = ct 0 0 x = x i x 0 1 0 0 x = x + i x 1 0 1 The transformation matrix is then 0 1 2 3 0 1 0 0 0 B C 1 0 0 B C B C @ A 2 0 0 1 0 3 0 0 0 1 2 5.1.1 Generalized Lorentz Transformation For spatial co ordinates the Lorentz transform ts the linear form 4 X 0 x = x 1 =1 sub ject to the condition that the prop er length X X 2 2 0 2 2 cd = ds = x = x =ct j~xj 2 is an invariant. This condition requires that the co ecients form an orthogonal matrix: X = X = X = 3 where the Kronecker delta is de ned by = = = 1 when = and 0 otherwise. The invariance group can b e enlarged to b e the Poincare0 group by the addition of translations: 4 X 0 x = x + a 4 =1 The full group includes: translations, 3-D space rotations, and the Lorentz b o osts. 5.2 The Inner Pro duct of 3+1-D Vectors The de nition of the inner pro duct dot pro duct must b e mo di ed in 3+1 dimensions. ~ ~ A B = A B + A B + A B + A B 1 1 2 2 3 3 4 4 if x = ict. But with our usual convention 4 ~ ~ A B = A B A B A B A B 0 0 1 1 2 2 3 3 or with the opp osite signature metric one has ~ ~ A B = A B + A B + A B + A B 0 0 1 1 2 2 3 3 ~ ~ A B = A B + A B + A B A B 1 1 2 2 3 3 4 4 if x = ct which is often the convention for the opp osite sign convention. It is an 4 exercise to show that the inner pro duct is unchanged under a Lorentz transformation. Can b e done simply by substitution. This can b e extended to the general class of Lorentz transformations. 3 5.3 Four Velo city So wehave the p osition 4-vectorx ~ =x ;x ;x ;x and the displacement 4-vector 0 1 2 3 ~ dx =dx ;dx ;dx ;dx . What other 4-vectors are there? That is what other 4- 0 1 2 3 vectors are natural to construct? What we mean by a four-vector is a four-dimensional quantity that transforms from one inertial frame to another by the Lorentz transform which will then leave its length norm invariant. Consider generalizing the 3-vector velo cityv ;v ;v =dx=dt; dy =dt; dz =dt x y z what can wedotomake this into a 4-vector naturally? One clear problem is that we are dividing by a comp onent dt of a vector so that the ratio is clearly going to Lorentz transform in a complicated way.We need to take the derivative with resp ect to a quantity that will b e the same in all reference frames, e.g. d the di erential of the prop er time, and add a fourth comp onent to make the 4-vector. It is clear that the derivative of the 4-vector p osition ct; x; y ; z with resp ect to the prop er time will b e a 4-vector for Lorentz transformations since ct; x; y ; z transform prop erly and d is an invariant. So we can de ne the 4-velo cityas ! dx dx dy dz dct u = ; u~ = ; ; ; 5 d d d d d Note that ! 2 2 2 dx dy dy 2 2 2 2 2 2 2 2 2 c c d = c dt dx dy dz = dt dt dt dt 2 2 2 2 2 2 2 2 c v c v v v = dt = dt x y z or the time dilation formula we got b efore q d 1 dt 2 2 q = = = 1 v =c ; and 2 2 dt d 1 v =c So we can now explicitly write out the 4-velo city using the chain derivative rule: dx dx dt u = = d dt d u~ =u ;u ;u ;u = c; v ; v ; v = c; v ;v ;v 0 1 2 3 x y z x y z Thus three comp onents of the 4-velo city are the three comp onents of the 3-vector velo city times . Note also that the norm - the magnitude or vector invariant length - of the four-velo city is not only unchanged but it is the same for all physical ob jects matter plus energy. For 3+1 dimensions the norm or magnitude is found from the inner pro duct or dot pro duct which has the same signature as the metric see just ab ove so that 2 2 1 v =c 2 2 2 2 2 2 2 2 2 2 2 u~ u~ = u u u u = c v v v = c = c 0 1 2 3 x z z 2 2 1 v =c 4 Thus every physical thing, including light, moves with a 4-velo city magnitude of c and the only thing that Lorentz transformations do is change the direction of motion. A particle at rest is moving down its time axis at sp eed c. When it is b o osted to a xed velo city, it still travels through space-time at sp eed c but more slowly down the time axis as it is also moving in the spatial directions. One should also note that as the spatial sp eed three-velo city approaches c, all comp onents of the 4-velo city u are unb ounded as !1. One cannot then de ne a Lorentz transformation that moves to the rest frame. Thus all massless particles will have no rest frame. 5.3.1 LawofTransformation of a 4-Vector We can write the transformation in our standard algebraic Lorentz notation q 0 2 A = A A =1= 1 0 1 0 V 0 A = A A 1 0 1 c 0 0 A = A ; A = A 2 3 2 3 where and refer to the relativevelo city V of the frames. 5.3.2 LawofTransformation of a 4-Velo city 0 u = u u 1 0 1 where and are for the relativevelo city of the frames and not of the particle. But in the formula for the 4-velo city u~ =u ;u ;u ;u = c; v ; v ; v = c; v ;v ;v 0 1 2 3 x y z x y z The is for the particle! So we should have lab eled it and the and for the p frame transform and . Then wehave f f 0 0 v = v c f p x f p p x 0 So we can get out a formula for v x q 0 1 p f p 0 q q v V = v V v = x x x 0 1 1 p p f q 2 2 This is our old friend on the law of transformation of 1 u =c q q 0 2 2 2 2 q 1 u =c 1 V =c 2 2 1 u =c = 0 2 1+u V=c x 5 and q q 2 2 2 2 q 1 u =c 1 V =c 0 2 2 1 u =c = 2 1+u V=c x which is simply 1 1 = 0 2 1 u V=c p f x p So v V x 0 v = x 2 1 u V=c x as derived earlier by the di erential route.
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