- #1

Haorong Wu

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- TL;DR Summary
- I derive the wave equations in a curved spacetime. I do not understand why it is strongly affected by the gravitational field.

A massless scalar field in a curved spacetime propagates as $$(-g)^{-1/2}\partial_\mu(-g)^{1/2}g^{\mu\nu}\partial_\nu \psi=0 .$$

Suppose the gravitational field is weak, and ##g_{\mu\nu}=\eta_{\mu\nu}+\epsilon \gamma_{\mu\nu}## where ##\epsilon## is the perturbation parameter. And let the field be ##\psi=A e^{ik(x0-x3)}##.

Then the wave equation can be solved up to the first order of ##\epsilon##, giving $$2ik\partial_3 A-\nabla ^2 A+\epsilon [-\frac k 2 (2k(\gamma_{00}+\gamma_{33})+i(\partial_3 \gamma_{00}-\partial_3 \gamma_{11}-\partial_3 \gamma_{22}+\partial_3 \gamma_{33}+2\partial_2 \gamma_{23}+2\partial_1 \gamma_{13}))A+\cdots] =0$$ where terms with the derivatives of ##A## is omitted.

I note that there are ##k^2## inside the perturbation terms for ##A##. For a light with wavelength of ##1000 ~\rm{nm}##, ##k## will be about ##6 \times 10^6##. The metric for the Earth will let ##\gamma_{00}+\gamma_{33}## be about ## \frac {mz^2}{(x^2+y^2+z^2)^{3/2}} \approx 1.4\times 10^{-9}## for ##x=y=0, z=6.37\times10^6## is the radius of the earth, ##m=8.87\times 10^{-3}## is the Schwarzschild radius of earth.

Hence it appears that the perturbation for ##A## is quite strong even in a weak gravitational field. But this should be wrong. But where did I make a mistake?

Suppose the gravitational field is weak, and ##g_{\mu\nu}=\eta_{\mu\nu}+\epsilon \gamma_{\mu\nu}## where ##\epsilon## is the perturbation parameter. And let the field be ##\psi=A e^{ik(x0-x3)}##.

Then the wave equation can be solved up to the first order of ##\epsilon##, giving $$2ik\partial_3 A-\nabla ^2 A+\epsilon [-\frac k 2 (2k(\gamma_{00}+\gamma_{33})+i(\partial_3 \gamma_{00}-\partial_3 \gamma_{11}-\partial_3 \gamma_{22}+\partial_3 \gamma_{33}+2\partial_2 \gamma_{23}+2\partial_1 \gamma_{13}))A+\cdots] =0$$ where terms with the derivatives of ##A## is omitted.

I note that there are ##k^2## inside the perturbation terms for ##A##. For a light with wavelength of ##1000 ~\rm{nm}##, ##k## will be about ##6 \times 10^6##. The metric for the Earth will let ##\gamma_{00}+\gamma_{33}## be about ## \frac {mz^2}{(x^2+y^2+z^2)^{3/2}} \approx 1.4\times 10^{-9}## for ##x=y=0, z=6.37\times10^6## is the radius of the earth, ##m=8.87\times 10^{-3}## is the Schwarzschild radius of earth.

Hence it appears that the perturbation for ##A## is quite strong even in a weak gravitational field. But this should be wrong. But where did I make a mistake?