Problem 2.3b3

Now we lower the indices, by expressing the upper-index coordinates (contravariant components) by lower-index coordinates (covariant components), in order to find the Lorentz transformation for the covariant components:

\displaystyle{ \begin{aligned} (x')^\mu &= L^\mu_{~\nu} x^\nu \\   \eta^{\rho \mu} (x')_\rho &= L^\mu_{~\nu} \eta^{\sigma \nu} x_\sigma \\  \sum_\rho \eta^{\rho \mu} (x')_\rho &= \sum_\sigma \sum_\nu L^\mu_{~\nu} \eta^{\sigma \nu} x_\sigma \\  \end{aligned}}

After raising the indices, we lower the indices again:

\displaystyle{ \begin{aligned}   \eta_{\alpha \mu} \eta^{\rho \mu} (x')_\rho &= \eta_{\alpha \mu} L^\mu_{~\nu} \eta^{\sigma \nu} x_\sigma \\  \eta_{\alpha \mu} \eta^{\mu \rho} (x')_\rho &= \eta_{\alpha \mu} L^\mu_{~\nu} \eta^{\sigma \nu} x_\sigma \\  \end{aligned}}

\displaystyle{ \begin{aligned}   \delta_{\alpha \rho} (x')_\rho &= \eta_{\alpha \mu} L^\mu_{~\nu} \eta^{\sigma \nu} x_\sigma \\  (x')_\alpha &= \eta_{\alpha \mu} L^\mu_{~\nu} \eta^{\sigma \nu} x_\sigma \\  \end{aligned}}

Prove that \displaystyle{\eta_{\alpha \mu} L^\mu_{~\nu} \eta^{\sigma \nu} = \left[L^{-1}\right]^\sigma_{~\alpha}}.

By index renaming, \displaystyle{ \begin{aligned}   (x')_\mu &= \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\nu \sigma} x_\nu \\  \end{aligned}}, the question becomes

Prove that \displaystyle{ \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\nu \sigma} = \left[L^{-1}\right]^\nu_{~\mu}}.

Denote \displaystyle{ \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\nu \sigma}} as \displaystyle{L^{~\nu}_{\mu}}. Then the question is simplified to

Prove that \displaystyle{ L^{~\nu}_{\mu} = \left[L^{-1}\right]^\nu_{~\mu}}.

\displaystyle{ \begin{aligned}   (x')^\mu &= L^\mu_{~\nu} x^\nu \\  (x')_\mu &= \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\beta \sigma} x_\beta \\  \end{aligned}}

\displaystyle{ \begin{aligned}  (x')^\mu (x')_\mu &= \left( L^\mu_{~\nu} x^\nu \right) \left( \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\beta \sigma} x_\beta \right) \\  (x')^\mu (x')_\mu &= \left( L^\mu_{~\nu} x^\nu \right) \left( {L^{~\beta}_{\mu}} x_\beta \right) \\  \\  \end{aligned}}

\displaystyle{ \begin{aligned}  \sum_{\mu = 0}^4 (x')^\mu (x')_\mu &= \sum_{\mu = 0}^4 \sum_{\nu = 0}^4 \sum_{\beta = 0}^4 \left( L^\mu_{~\nu} x^\nu \right) \left( {L^{~\beta}_{\mu}} x_\beta \right)  \\  \end{aligned}}

\displaystyle{ \begin{aligned}  \sum_{\mu = 0}^4 (x')^\mu (x')_\mu &= \sum_{\mu = 0}^4 \left( \sum_{\nu = 0}^4 L^\mu_{~\nu} x^\nu \right) \left( \sum_{\beta = 0}^4 {L^{~\beta}_{\mu}} x_\beta \right)  \\  \end{aligned}}

\displaystyle{ \begin{aligned}  &(x')^0 (x')_0 + (x')^1 (x')_1 + (x')^2 (x')_2 + (x')^3 (x')_3 \\  &= \sum_{\mu = 0}^4 \left( L^\mu_{~0} x^0 + L^\mu_{~1} x^1 + L^\mu_{~2} x^2 + L^\mu_{~3} x^3 \right)   \left( L^{~0}_{\mu} x_0 + L^{~1}_{\mu} x_1 + L^{~2}_{\mu} x_2 + L^{~3}_{\mu} x_3 \right)  \\  \end{aligned}}

The right hand side has 64 terms.

Since the spacetime interval is Lorentz-invariant, \displaystyle{ (x')^\mu (x')_\mu = x^\mu x_\mu }. So the left hand side can be replaced by \displaystyle{ x^\mu x_\mu }.

\displaystyle{ \begin{aligned}  &x^0 x_0 + x^1 x_1 + x^2 x_2 + x^3 x_3 \\  &= \sum_{\mu = 0}^4 \left( L^\mu_{~0} x^0 + L^\mu_{~1} x^1 + L^\mu_{~2} x^2 + L^\mu_{~3} x^3 \right)   \left( L^{~0}_{\mu} x_0 + L^{~1}_{\mu} x_1 + L^{~2}_{\mu} x_2 + L^{~3}_{\mu} x_3 \right)  \\  \end{aligned}}

Note that the 4 terms on the left side also appear on the right hand side.

\displaystyle{ \begin{aligned}  (x')^\mu &= L^\mu_{~\nu} x^\nu \\  (x')^\mu (x')_\mu &= \left( L^\mu_{~\nu} x^\nu \right) \left( \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\beta \sigma} x_\beta \right) \\  (x')^\mu (x')_\mu &= \left( L^\mu_{~\nu} x^\nu \right) \left( {L^{~\beta}_{\mu}} x_\beta \right) \\    x^\mu x_\mu &= \left( L^\mu_{~\nu} x^\nu \right) \left( {L^{~\beta}_{\mu}} x_\beta \right) \\    \end{aligned}}

\displaystyle{ \begin{aligned}  x^0 x_0 + x^1 x_1 + x^2 x_2 + x^3 x_3 &= \sum_{\mu = 0}^4 \sum_{\nu = 0}^4 \sum_{\beta = 0}^4 L^\mu_{~\nu} L^{~\beta}_{\mu} x^\nu x_\beta \\    \end{aligned}}

Since this equation is true for any coordinates, it is an identity. By comparing coefficients, we have:

1. For any terms with \displaystyle{\nu \ne \beta}, such as \displaystyle{\nu = 0} and \displaystyle{\beta=1},

\displaystyle{ \begin{aligned}  \sum_{\mu = 0}^4 L^\mu_{~0} L^{~1}_{\mu} x^0 x_1 &\equiv 0 \\  \left( \sum_{\mu = 0}^4 L^\mu_{~0} L^{~1}_{\mu} \right) x^0 x_1 &\equiv 0\\    \end{aligned}}

So

\displaystyle{ \begin{aligned}  \sum_{\mu = 0}^4 L^\mu_{~0} L^{~1}_{\mu} &= 0 \\    \end{aligned}}

2. For any terms with \displaystyle{\nu = \beta}.

\displaystyle{ \begin{aligned}  x^0 x_0 + x^1 x_1 + x^2 x_2 + x^3 x_3 &\equiv \sum_{\mu = 0}^4 \sum_{\nu = 0}^4 \sum_{\beta = \nu} L^\mu_{~\nu} L^{~\beta}_{\mu} x^\nu x_\beta \\    x^0 x_0 + x^1 x_1 + x^2 x_2 + x^3 x_3 &\equiv \sum_{\mu = 0}^4 \left( L^\mu_{~0} L^{~0}_{\mu} x^0 x_0   +  L^\mu_{~1} L^{~1}_{\mu} x^1 x_1   +  L^\mu_{~2} L^{~2}_{\mu} x^2 x_2   +  L^\mu_{~3} L^{~3}_{\mu} x^3 x_3  \right) \\    \end{aligned}}

So

\displaystyle{ \begin{aligned}  \sum_{\mu = 0}^4 L^\mu_{~0} L^{~0}_{\mu} &= 1 \\  \sum_{\mu = 0}^4 L^\mu_{~1} L^{~1}_{\mu} &= 1 \\  \sum_{\mu = 0}^4 L^\mu_{~2} L^{~2}_{\mu} &= 1 \\  \sum_{\mu = 0}^4 L^\mu_{~3} L^{~3}_{\mu} &= 1 \\  \end{aligned}}

.

Denoting \displaystyle{ \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\nu \sigma}} as \displaystyle{L^{~\nu}_{\mu}} is misleading, because that presupposes that \displaystyle{ \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\nu \sigma}} is directly related to the matrix \displaystyle{L}.

To avoid this bug, instead, we denote \displaystyle{ \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\nu \sigma}} as \displaystyle{M ^\nu_{~\mu}}. So

\displaystyle{ \begin{aligned}  (x')^\mu &= L^\mu_{~\nu} x^\nu \\  (x')^\mu (x')_\mu &= \left( L^\mu_{~\nu} x^\nu \right) \left( \eta_{\mu \rho} L^\rho_{~\sigma} \eta^{\beta \sigma} x_\beta \right) \\  (x')^\mu (x')_\mu &= \left( L^\mu_{~\nu} x^\nu \right) \left( M^{\beta}_{~\mu} x_\beta \right) \\    x^\mu x_\mu &= \left( L^\mu_{~\nu} x^\nu \right) \left( M^{\beta}_{~\mu} x_\beta \right) \\    \end{aligned}}


\displaystyle{ \begin{aligned}  \nu \neq \mu:&~~~~~~\sum_{\mu = 0}^4  L^\mu_{~\nu} M^{\beta}_{~\mu} &= 0 \\    \nu = \mu:&~~~~~~\sum_{\mu = 0}^4  L^\mu_{~\nu} M^{\beta}_{~\mu} &= 1 \\    \end{aligned}}

— Me@2020-09-12 09:33:00 PM

.

.

2020.09.13 Sunday (c) All rights reserved by ACHK