Question

Α'2 = Σ Λ Α' (4.4) V=1...2 The instructions under the summation symbol tell us to assign the values t, x, y, z to the index v and sum the four terms that result. The value of u is left unspecified: if = 1, then this equation corresponds to the first row of equation 4.1; if u = x, it corresponds to the second line of 4.1, and so on. Equations 4.4 and 4.1 are equivalent. Equation 4.4 can be made even more compact if we adopt the rule known as the Einstein summation convention: If the same Greek-letter index appears exactly once as a superscript and ex- actly once as a subscript in any single term of an equation, we will assume that term is to be summed over all four possible values of that index. Using this convention, equations 4.1 and 4.2 can be written simply as A'' = A“,A and A4 = (1-1,A” (4.5) P4.3 Show that (1“)" Nav = 1 MB AP y. P4.9 Evaluate nuc NvBF4v F«ß in terms of the compo- nents of Ể and B. res P4.10 As we will see in chapter 6, the components of the electromagnetic field tensor Fin a primed inertial refer- ence frame are related to its components in an unprimed reference frame according to the following rule: fid = 141" (4.36) where [4] is the Lorentz transformation matrix from the unprimed to the primed frame. This is a generalization of the transformation rule for four-vectors given by equation 4.5: here there is one Lorentz transformation matrix fac- tor for each upper index in F. Assuming that this is correct, use the result of problem P4.3 to prove that the quantity defined in problem P4.9 has the same numerical value in every inertial reference frame, and therefore represents something that is frame- independent about an electromagnetic field.

Answer 1

sola To prove that furpaß Mua MNB is frame independent. su inestial reference frame the electromagnetic field tenser f' is related to f as - Filed = 1 1 & fap (4.35) Now, lets calculate Mex Mvp peler flop [ie, in incestial substitute fram (4.36) frame mua Map f'lev f up Pq fmn non Map from P4.3 (A-1) 4 Mar = Mush Aur Mana YuB ( Ah ah ) ( My FPV) nh AB w q AB ✓ and we know A 2 ✓ Mup hearsanging equ Mua 1 m n up (A-174 km (A) Mop I fram eq" (2) AM FPV Amn A V e- Aam amp

mm Aam A A 1 a q т» = Aam 1 Amp At q f par f mw Eng Amw Smp Z smp dag AV Anne Emn ma م peter f up Amn so, Max Map minn mte for Emn flur Mex Morp Flev Fuß eller flap qua Map fler is invasiant there, this term Mur og up fap fler of frame.