Thursday, 10 November 2016

homework and exercises - Definition of the Lorentz group


This is from Carrol's book, page 13 - enter image description here


This Sort of notation is new to me, and i'm having trouble understanding the claim on the bootom part (second sentence from the end).




  1. Does (ρ,σ)=(0,0)ρ=σ=0?





  2. And how can we show that |Λ00|1?




  3. Does it refer to the RHS or the LHS of 1.29?




Also, can anyone give an example to a time reversal? Why don't we allow those?



Answer



For η=diag(1,1,1,1) take the zeroth component of ηρσ=ΛμρΛνσημν setting ρ=σ=0:



1=η00=Λμ0Λν0ημν=Λ00Λ00η00+3i=1Λi0Λi0ηii=Λ00Λ00+3i=1Λi0Λi0.

Combining the very left and the very right one obtains (Λ00)2=3i=1(Λi0)2+1.


Hence |Λ00|1.


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