Wave vector in curved space-time.

[MManuel Abad]

Hi there, physics lovers!

Sorry I'm bothering you again. The following may sound as stupid questions, but this is a very hard topic for me, so don't judge me that bad! :)

You see, when we're talking about light in curved spacetime, the geodesic equations (or equations of motion) use an affine parameter $$\lambda$$ to parametrize its world line (actually I know this thanks to this forum).

Indeed, searching in Section 87 "Motion of a particle in a gravitational field" of the 2nd Volume "The Classical Theory of Fields", from the Course of Theoretical Physics by Landau and Lifgarbagez, I find that these geodesic equations (which are for light, and are for null geodesics, defined as those in which ds=0) look like:

$$\frac{d^{2}x^{\mu}}{d\lambda^{2}}+\Gamma^{\mu}_{\nu\sigma}\frac{dx^{\nu}}{d\lambda}\frac{dx^{\sigma}}{d\lambda}=0$$

with $$\Gamma^{\mu}_{\nu\sigma}$$ being the Christoffel symbols. I also saw that, because the wave vector of light is always tangent to its path, we can write:

$$k^{\mu}=\frac{dx^{\mu}}{d\lambda}$$

and then, the geodesic equations look like:

$$\frac{dk^{\mu}}{d\lambda}+\Gamma^{\mu}_{\nu\sigma}k^{\nu}k^{\sigma}=0$$

In section 48, Landau defines, talking IN MINKOWSKI (FLAT) SPACE-TIME:

$$k^{\mu}=\left\{\frac{\omega}{c},\frac{\omega}{c}\textbf{n}\right\}$$

where n is a unit vector along the direction of propagation of the wave. Then it is obvious that:

$$k^{\mu}k_{\mu}=0$$

Then, returning to section 87 and the geodesic equations, Landau says that this very same equation holds, even in a gravitational field (CURVED space-time).

QUESTION 1:

Is this 'cuz $$k^{\mu}k_{\mu}$$ is a four-scalar and then it's invariant in changes of coordinates, even if they're for non-flat space-time? Or why?

QUESTION 2:

$$k^{\mu}=\left\{\frac{\omega}{c},\frac{\omega}{c}\textbf{n}\right\}$$ is the form of the wave vector ONLY IN FLAT SPACE-TIME, ISN'T IT? I mean, its form changes in curved space time, actually, the wave vector is what we want to find from the equations:

$$\frac{dk^{\mu}}{d\lambda}+\Gamma^{\mu}_{\nu\sigma}k^{\nu}k^{\sigma}=0$$

, isn't it?

Well, there it is. I hope you help me. I thank you very much your attention.

 

[Matterwave]

Which equation does Landau say holds? The contraction of 2 vectors is a scalar, and is invariant. If it is 0 in some coordinate system, then it's 0 in every other coordinate system.