Question

Derive the inverse Lorentz transformation for the partial deriva- tives, (5 и д с2 Әt! (5) (6) а дх а ду а дz а Әt д 7 Әr? ә ay' а дz! а 7 де? (7) ә - (8) Әr! Hint: you need to use the chain rule. Write down analogous expression to equations (5)-(8), assuming a Galilean transformation: x = x – ut, y' = y, z' = z and t = t.

Answer 1

Consider Lorenta transformation equaticis be': v (2-ut) u - relatiue velouty of frame s' with y' y z' t z их 2 с respect to ta's. (sc' ,9 , z', E) and (x, y, z, t) are olt space-time co-ordinates with respect to s' and s respectively. The inverse horenia transformation equations are r(x'+ut) 2 y y z ux r(Et ㅏ с 2. ax the partial derivative atx a ot' ax o nc -? 2 с = Now consider Using chain rule, о ax' + Әх ax from Lorent} transformation equations, at -Yu and r 2 20 Әс 6 becomes ju a 7 aac' doc d u o 7 ax οχ d 2 c² at d at π

the Since boosts along considering але. we X-axis, е Бе е ау 2 Oz" az де е pe је e 1e + E Ә20 Ә at Lorentą Әк" ot transformation equatiens әЕ! from V yu де әЕ (2) becomes e е на Y ax' 1 е. 7 e te 2 e e Υ [ Әс" ite је 2 ) e + ч at ах ee Note : desive To дх! е Бе е а oy' 2 да а 2 ƏZ е ta Υ | e де хе ite 아 х е te е + write e хе art' can .X ' 2х : we Әх әх 2 ay' Use inverse Lorentz traust. to compate at әх", ән, әЕ he e ах Эх , 22. 2. 22 ӘE X 2 + хе 1e 아 е te 아 до