How to show ${\Lambda^\mu}_\rho\Lambda^{\nu\rho}=\eta^{\mu\nu}$?

Question

For the Lorentz transfomation matrix elements ${\Lambda^\mu} _\nu$ and inverse Minkowski metric elements $\eta^{\mu\nu}$, how can the relation

$${\Lambda^\mu}_\rho\Lambda^{\nu\rho}=\eta^{\mu\nu}$$ be shown to be true?

I first tried to create a $\eta$ element:$${\Lambda^\mu}_\rho\Lambda^{\nu\rho}={\Lambda^\mu}_\rho(\eta^{\nu\beta}{\Lambda_\beta}^\rho)={\Lambda^\mu}_\rho{\Lambda_\beta}^\rho\eta^{\nu\beta}.$$ The next step is to show that ${\Lambda^\mu}_\rho{\Lambda_\beta}^\rho=\delta^\beta_\mu$, so that $${\Lambda^\mu}_\rho{\Lambda_\beta}^\rho\eta^{\nu\beta}=\delta^\beta_\mu\eta^{\nu\beta}=\eta^{\nu\mu}=\eta^{\mu\nu}.$$ One relationship of the form ${\Lambda^\mu}_\rho{\Lambda_\beta}^\rho=\delta^\beta_\mu$ I can think of is $${(\Lambda^T)^\beta}_\rho{\Lambda^\rho}_\mu={\Lambda_\rho}^\beta{\Lambda^\rho}_\mu=\delta^\beta_\mu,$$ which is the relationship that says $\Lambda$ is an orthogonal matrix (i.e. $\Lambda^T\Lambda=1$).

But how can I show ${\Lambda^\mu}_\rho{\Lambda_\beta}^\rho=\delta_\beta^\mu?$

Answer 1

The relation $\Lambda^{\mu}_{\ \ \sigma}\,\Lambda^{\nu\sigma}=\eta^{\mu\nu}$, or more properly

$$ \Lambda^{\mu}_{\ \ \sigma}\,\eta^{\sigma\tau}\,\Lambda^{\nu}_{\ \ \tau}=\eta^{\mu\nu}\ , $$

actually is the definition of a Lorentz transformation $\Lambda^{\mu}_{\nu}$. As such it cannot be proven true, unless you rest on some alternative definition.

In case you were wondering about where the definition comes from, it comes from the requirement that the Minkowski norm $\eta_{\mu\nu}x^{\mu}x^{\nu}$ of a four-vector be invariant under such transformations, i.e.

$$ \eta_{\mu\nu}\,\Lambda^{\mu}_{\ \ \sigma}\,\Lambda^{\nu}_{\ \ \tau}\,x^{\sigma}x^{\tau}=\eta_{\mu\nu}x^{\mu}x^{\nu}\ . $$

In order to be verified for all $x$, $\Lambda$ must have the above-mentioned property (then and only then it is called a Lorentz transformation). This requirement justifies but does not prove the property.

Answer 2

After some more thought:

${\Lambda_\beta}^\rho$ is the inverse Lorentz transformation matrix and also satisfy the orthogonal matrix requirement:

$${(\Lambda^T)_\beta}^\rho{\Lambda_\rho}^\mu={\Lambda^\rho}_\beta{\Lambda_\rho}^\mu=1$$ Taking the transpose of both sides, $${\Lambda^\mu}_\rho{\Lambda_\beta}^\rho=1=\delta^\mu_\beta$$ which is what I needed to show that ${\Lambda^\mu}_\rho\Lambda^{\nu\rho}=\eta^{\mu\nu}$ is true.