Mathematical Preparations


 Bartholomew Griffin Dean
 4 years ago
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1 1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the velocty of EM radaton was found to be unaffected by the moton of source of radaton wth respect to the coordnate system,.e. the space through whch the EM radaton travels. On the other hand, Newton postulated that n the absence of forces, the spatal coordnates of a movng pont are lnear n tme. Thus the poston of a partcle measured n a rest frame of reference, X, related to ts poston n a movng frame, X, s gven by X = X + V 0 t. Usng ths coordnate transformaton, the veloctes n two systems movng wth constant velocty, V 0, relatve to each other s; dx dt = dx dt + V 0 V = V + V 0 The above equatons represent a spatal transformaton between coordnate frames movng wth constant velocty, V 0, wth respect to each other. It s presumed that there s a fxed, unversal coordnate frame (nertal frame), and all frames movng wth constant velocty wth respect to ths frame have the same acceleraton; d 2 X dt 2 = a = a The nvarance of the laws of physcs n dfferent coordante frames s a symmetry called the Prncple of Relatvty. In the above case, Newton s laws of moton are the same n all nertal frames, as the force (acceleraton) F = M a s ndependent of the nertal frame. However, the ndependence of the velocty of EM waves between dfferent coordnate frames s not consstent wth Neutonan physcs. 2 Gallean transformaton To be consstent, the mathematcal form of all physcs laws cannot be changed by a coordnate transformaton. In the case of Newtonan physcs, a transformaton between nertal frames preserves Newton s laws of mechancs, and s called a Gallean transformaton. The Gallean transformaton transformaton s defned below. 1
2 Suppose 2 reference frames related to each other by a constant velocty along the Z axs, (X, Y, Z) and (X, Y, Z ). A system not subject to a force experences no force n any nertal system. Thus f the force n one frame s gven by F = M a then the force n the other frame s; F = F = M a wth a = a ; d 2 X dt 2 = d2 X dt 2 = a The Gallean transformaton between nertal systems must take the form; X Y Z t = X = Y = Z + V 0 t = t Although Newton s laws of mechancs are nvarent under a Gallean transformaton, Maxwell s equatons whch descrbe electrodynamcs are not, and ths was recognzed long before the theory of relatvty. Thus the descrpton of electromagnetc radaton was nconsstent wth a Galelan transformaton. It was orgnally thought that Maxwell s equatons were ncomplete, and theores were proposed to correct EM under the assumpton that a Galelan transformaton correctly descrbed the coordante transformaton between movng bodes. We now know of course, that EM was correct and Newtonan mechancs requred modfcaton. 2.1 Generalzed coordnates Because 4 rather that 3 dmenson (3 spatal and one tme coordnate) are necessary to descrbe the relatve moton of systems, t s mportant to frst dscuss geometry and transformatons n a generalzed set of coordnates. Most students have been mnmaly exposed to ths mathematcs. However, only the parts of tensor analyss requred for specal relatvty are developed here. General relatvty requres more ndepth development whch s not necessary for the study of classcal electrodynamcs. All coordnate systems are defned relatve to a Cartesan set of axes. For 3D wrte (x 1, x 2, x 2 ), although extenson to more spatal dmensons s trval. Thus there s a 3 D functon of the coordnates whch locates some pont n space. Ths pont can also be located n a dfferent coordnate frame, ζ ( = 1, 2, 3); ζ (x 1, x 2, x 3 ) = 1, 2, 3 2
3 There also exsts a unque nverse of the transformaton functon between the coordnates. Mathematcally, ths s descrbed by a onetoone mappng of each pont n one frane to one pont n the other. Ths mappng must have a unque nverse so that each pont has only one locaton n all frames of reference. x (ζ 1, ζ 2, ζ 3 ) = 1, 2, 3 Now at the ntersecton of the planes; ζ = constant = 1, 2, 3 defne a set of unt vectors, â, perpendcular to each surface. If these vectors are mutually orthorgonal, an orthorgonal coordnate system s defned. A reference frame wth orthogonal coordnates s not necessary n general, but orthogonal coordnates greately smplfes the mathematcs. The drecton cosnes of the coordnates wth respect to the set of Cartesan unt vectors are; â 1 ˆx = α = γ 1 â 2 ˆx = β = γ 2 â 3 ˆx = γ = γ 3 For an orthorgonal system, the 3 nontrval drecton cosnes are related, as may be shown by calculatng â â j for, j = 1, 2, 3. Then; 3 s=1 γ ms γ ns = 3 s=1 γ sm γ sn = δ mn â n = j γ nj ˆx j ˆx j = n γ nj â n Now consder the dfferental element of length, ds. In the Cartesan system, the square of ths element s; ds ds = 3 dx 2 =1 Suppose a general curvlnear set of coordnates s ntroduced as defned above. 3
4 dx = 3 j=1 dζ j The square of the length elements s then ds 2 = 3 3 j,k=1 =1 Ths s rewrtten as ; g jk = 3 =1 ζ k ζ k dζ j dζ k where g jk are the metrc elements whch defne the space. Therefore; ds 2 = jk g jk dζ j dζ k In the case of an orthorgonal system g jk = 0 f j k, so defne a scale factor, h 2 j = Note that h j dζ j s the length element for the j th coordnate. ( ) 2. ds 2 = (h dζ ) 2 Usng ths, one can obtan the dfferental volume and area elements; dτ = (h 1 dζ 1 )(h 2 dζ 2 )(h 3 dζ 3 ) dσ k = (h dζ )(h j dζ j ) To obtan the varous surface areas n the above, apply cyclc permentatons of, j, k. In general h vares at each pont n the coordnate space. The drecton cosnes along the new coordnate axes (ONLY for an orthorgonal system) are; γ n = (1/h n ) ζ n = h n ζ n (no sum) Not only do the scale factors change wth poston, but also the unt vectors change drectons, Fg. 1. For example, ˆx h j ζj â j = â j = ζ k ζ k Whch can be reduced to; ˆx h j 4
5 a^ 2 α a^ 1 d ζ 1 dh 2 dζ 1 d ζ 2 h 2 dζ 2 a^ 1 h 1 dζ 1 a^2 a^ 1 Fgure 1: A cross secton of an area element n a generalzed coordnate system â j ζ = â h h j It s then nterestng to apply these equatons to a famlar coordnate system. Use sphercal coordnates for ths example. x = rcos(φ)sn(θ) â 1 = sn(θ) cos(φ) ˆx + sn(θ) sn(φ) ŷ + cos(θ) ẑ y = rsn(φ)sn(θ) â 2 = cos(θ) cos(φ) ˆx + cos(θ) sn(φ) ŷ sn(θ) ẑ x = rcos(θ) â 3 = sn(φ) ˆx + cos(φ) ŷ Take the partal dervatves to show that an orthorgonal system s produced ( 0). The square of the metrc length s; ζ k = ds 2 = dr 2 + r 2 dθ 2 + r 2 sn 2 (θ)dφ 2 as expected. Unt vectors, volume/area elements, and the vector operatons gradent, dv, and curl can be obtaned from the physcal defnton of these operators. 2.2 Tensors Tensors are defned by consderng the transformaton propertes of functons under a coordnate rotaton and reflecton. Thus a scalar functon does not change value under rotaton or reflecton. As an example the functon f = 3 (x x 0, ) 2 remans constant and for ths example, s the magntude of a vector. On the other hand f we consder; =1 5
6 f = 3 =1 f ˆx then f transforms as a vector whch preserves magntude but changes drecton. It also changes sgn upon reflecton x x. All these propertes are preserved when a coordnate transformaton s appled so that the representaton of a vector s ndependent of the coordnate frame. A true scalar remans the same under all coordnate transformatons, ncludng reflectons. However, f a scalar functon s constant under rotaton but changes sgn under reflecton t s a pseudoscalar. Smlarly f a vector does not change sgn under reflecton t s a pseudovector. As an example, a pseudovector s the result of the cross product of 2 true vectors as wll be observed below. Now generalze ths descrpton of functons by defnng a scalar functon as a tensor of rank 0, and a vector functon as a tensor of rank 1. Ths can be generalzed by extendng the transformaton propertes to hgher rank. To help wth notaton, the summaton conventon s employed unless t leads to ambgutes. The summaton conventon suppresses the symbol and s represented by a repeated ndex on the varables. Thus the defnton; x k = x k Suppose an ndnemsonal space, wth N ndependent varables x = 1,, n. The set of x defne a pont n ths space. Now defne a set of n lnearly ndependent functons ζ (x 1,, x n ) = 1,, n. The Jacoban of a set of lneraly ndependent functons does not vansh. J = ζ 1 x 1 ζ n x 1 0 ζ 1 ζ n x n x n The functons, ζ, defne a new coordnate system. Make the substtuton x = ζ, and evaluate x k for future use. x k In addton; = δ j = x k x dx = x dx j 6
7 3 Tensor contracton and drect product In the followng, use the results of the dfferental operatons between the prmed and unprmed frame whch were obtaned n the last secton. The dfferental quanttes dx and dx j are related by a lnear transformaton, x. A tensor functon s defned by the lnear transformaton of ts dfferental form between two coordnate frames. Thus a tensor, A, of rank 1 (a vector) has the transformaton propertes; A = j A j For the record, ths s a contravarent tensor ndcated by the superscrpted ndex. A subscrpted ndex ndcates a covarent tensor, and hgher order tensors wth both super and subscrpts are called a mxed tensor. A = j x A j For Cartesan coordnates covarent and contravarent tensors are dentcal snce; x k = δ jk The contracton of any tensor by a vector for example (for 2 vectors ths s the dot product) reduces the order of the tensor by one (the rank of the tensor less the rank of the vector). In the case of contractng 2 vectors a scalar s produced. A B = jk x A k = k j A j B j On the other hand, the drect product of 2 tensors multples each element of a tensor by the elements of the other tensor. Ths ncreases the rank of the tensor by the sum of the ranks of each tensor. Thus the drect product of a tensor of rank 1 (a vector) wth another tensor of rank 1, produces a tensor of rank 2 (a matrx). A B l = j,m x x l A B m m Note the above form transforms lke a tensor of rank 2. 4 The metrc tensor As prevously, the square of the length element s; 7
8 ds 2 = dx dx = g jk dx j dx k The g j form a tensor of 2 nd rank called the metrc tensor of the space. The determnant s g = g j 0. It s possble n general to have ds 2 < 0, however, ths would not be consstent wth length, so the measure of the space s taken as the absolute value of ds 2. Note that ds 2 s a tensor of rank 0, e a scalar quantty. 5 LevCvta tensor It s useful to defne the followng tensor of rank 3 or hgher. ǫ jk = d The constant d takes on the followng values. ǫ jk = 1 when, j, k = 1, 2, 3 f j k and wth an even permutaton of 1, 2, 3. The tensor equals 1 f the ndces are an odd permutaton of 1, 2, 3 and the tensor s 0 f any of the ndces have the same value. Ths tensor s a pseudotensor, e a tensor wth nverted symmetry upon nterchange of ndces. A conjugate tensor wth the same propertes can also be defned. The contracton of a pseudoscalar tensor tensor wth another tensor produces another pseudotensor, perhaps a pseudoscalar or a pseudovector. Ths leads to the defnton of dual tensors to be defned below. The vector cross product (vector product) s a tensor of rank 2 but t has a dual representaton as a pseudovector. 6 Contracton wth the LevCvta tensor Suppose an antsymmetrc tensor of rank 2, A j = A j. We contract ths tensor wth the LevCvta tensor of rank 3, ǫ jk 0 A 12 A 13 [A j ] = A 21 0 A 23 A 31 A 32 0 Thus; A j ǫ jk = [A j A j ] k j It s obvous that ths results n a form whch transforms lke a tensor of rank 1, but does not change sgn under a coordnate nverson. It s then a pseudovector and a dual of the tensor of second rank. It s also clear that ths represents the cross product of two vectors. 8
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