The wave vector ##k_\mu## in curved spacetime

[Kostik]

TL;DR

How do we define the wave vector $k_\mu$ in curved spacetime?

A plane wave (which can be a scalar function like air pressure, a vector function like the electric field $\bf{E}$ or a tensor field like the spacetime metric $g_{\mu\nu}$) is a function of $$\xi = \omega t - {\bf{k} \cdot \bf{x}} = k_\sigma x^\sigma.$$ We call $k_0$ the wave vector. The quantity $\xi$ is the phase, which marks where $e^{i\xi}$ falls on the unit circle. This is plainly coordinate invariant, hence a scalar. Since $$k_\mu = \xi_{,\mu}$$ we know that $k_\mu$ is a vector.

How does one define the wave vector in curved spacetime? In curved space, there is not generally a plane wave solution to the covariant d’Alembert equation $$g^{\rho\sigma}g_{\mu\nu;\rho;\sigma} \quad\quad\quad (*) $$ In the plane wave case, the idea is to write a periodic ("monochromatic") wave in complex form, i.e., $$g_{\mu\nu}(x) = A_{\mu\nu}e^{ik_\sigma x^\sigma} = A_{\mu\nu}e^{i\xi}$$ (where the physical wave is the real part) and then define the phase $$\xi(x) = \arg[g_{\mu\nu}(x)] \quad\quad\quad(**)$$ Then, as before, the wave vector can be defined as $k_\mu = \xi_{,\mu}$ (now a field function of $x$).

For a periodic wave in curved space, can we extend this definition? How does one, for example, take a solution $g_{\mu\nu}(x)$ to the covariant d'Alembert equation $(*)$ and write it as the real part of a complex function, so that we can define the phase by $(**)$?

 

[martinbn]

https://uchicago.app.box.com/s/x4fi21zy1xnvja2j32vzlnjp7wo9802m

Section 17

 

[Dale]

TL;DR Summary: How do we define the wave vector $k_\mu$ in curved spacetime?

For a periodic wave in curved space, can we extend this definition?

That is a good question. The key point in the question is that in flat spacetime $x$ is a four-vector so $g_{\mu\nu}x^\mu k^\nu=\phi$ clearly requires that $k$ also be a four-vector.

In curved spacetime however $x$ is not a vector. It is an element of a manifold, which lacks the structure of a vector space.

I admit not knowing the details of this resolution, but I suspect the key lies in the fact that in curved spacetime $dx$ is a vector. So the definitions should still work, taken as local definitions in the tangent space rather than global definitions in the manifold.

 

[PeterDonis]

I admit not knowing the details of this resolution, but I suspect the key lies in the fact that in curved spacetime $dx$ is a vector. So the definitions should still work, taken as local definitions in the tangent space rather than global definitions in the manifold.

This is the correct answer, not just for the wave vector, but for every vector and tensor in curved spacetime; they are all objects in the tangent space at a point. The most complete discussion of this that I know of in a GR textbook is the one in Misner, Thorne, & Wheeler, which takes, IIRC, multiple chapters to fully develop this idea and how it works.

 

[otennert]

First of all, we are not talking about linearized GR, right?

In GR and in generally curved spacetimes, the problem actually begins earlier, i.e. in the definition of a "plane wave" per se, and in a coordinate-free manner. The resolution is given in early papers by Ehlers and Kundt 1962, and a good presentation is given in Stewart: Advanced General Relativity, section 2.9.

The idea is to look at coordinate-free characteristics a plane wave as we know it in Minkowski spactime has and take these as defining properties in curved spacetime. This leads to a definition within the context of null congruences, generated by a null vector field $l^\mu$. A plane-fronted wave is hereby characterized as a null congruence which is geodesic, and free of rotation, shear and expansion.

And if, in addition, the tangent (null) vector field $l^\mu$ is covariantly constant ($\nabla_\mu l_\nu=0$), then the wave is called a pp-wave ("plane-fronted with parallel rays"). In my eyes this is as closest to a plane wave as it can get in GR.

A newer publication is by Hogan and Puetzfeld, although I have not yet studied it:
https://link.springer.com/book/10.1007/978-3-031-16826-0