This is from Carrol's book, page 13 -
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).
Does (ρ,σ)=(0,0)⇒ρ=σ=0?
And how can we show that |Λ0′0|≥1?
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+3∑i=1Λi0Λi0ηii=−Λ00Λ00+3∑i=1Λi0Λi0.
Combining the very left and the very right one obtains (Λ00)2=3∑i=1(Λi0)2+1.
Hence |Λ00|≥1.
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