Geometry Behind g_μν Λμ ρ Λν φ = g_ρ φ ?

[Hymne]

What is the physical interpretation of the equation:

$$g_{\mu \nu} \Lambda^{\mu}\; _\rho \Lambda^{\nu}\; _\phi = g_{\rho \phi}$$?

I see that all the sub- and superindicies add up but what's the geometry behind it?

 

[mathman]

This may sound naive. But unless you define the physical meaning of the various symbols, there is no way to give a physical interpretation to the equation.

 

[RedX]

What is the physical interpretation of the equation:

$$g_{\mu \nu} \Lambda^{\mu}\; _\rho \Lambda^{\nu}\; _\phi = g_{\rho \phi}$$?

I see that all the sub- and superindicies add up but what's the geometry behind it?

I prefer the $$\Lambda 's$$ to have an inverse sign on them, but that's all right.

Basically that equation says two reference frames have the same metric, or that the metric tensor is invariant under Lorentz transformation.

This implies that two observers take dot products in the same way, and that both observers observe the same physics.