Problem 2.4

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A First Course in String Theory

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2.4 Lorentz transformations as matrices

A matrix L that satisfies (2.46) is a Lorentz transformation. Show the following.

(b) If \displaystyle{L} is a Lorentz transformation so is the inverse matrix \displaystyle{L^{-1}}.

(c) If \displaystyle{L} is a Lorentz transformation so is the transpose matrix \displaystyle{L^{T}}.

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(b)

\displaystyle{ \begin{aligned} (\mathbf{A}^\mathrm{T})^{-1} &= (\mathbf{A}^{-1})^\mathrm{T} \\ L^T \eta L &= \eta \\ \eta &= [L^T]^{-1} \eta L^{-1} \\ [L^T]^{-1} \eta L^{-1} &= \eta \\ [L^{-1}]^T \eta L^{-1} &= \eta \\ \end{aligned}}

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(c)

\displaystyle{ \begin{aligned} L^T \eta L &= \eta \\ (L^T \eta L)^{-1} &= (\eta)^{-1} \\ L^{-1} \eta^{-1} (L^T)^{-1} &= \eta \\ L^{-1} \eta (L^T)^{-1} &= \eta \\ \eta &= L \eta L^T \\ L \eta L^T &= \eta \\ \end{aligned}}

— Me@2020-12-21 04:24:33 PM

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2020.12.21 Monday (c) All rights reserved by ACHK