Kostik
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- TL;DR Summary
- 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 ##(**)##?
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 ##(**)##?
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