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My Question is: If 2 rays of light are emitted with different frequencies from the same spacetime point, does an observer see them in the same worldline?

I know that the worldline of light behaves as ligtht-like curves in spacetime (i.e. $g(v_{\gamma},v_{\gamma})=0$ and $u^{\nu}u_{\nu}$=0). But where exactly is the dependence of the emitted frequency?

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  • $\begingroup$ Are these light rays travelling in a perfect vacuum? $\endgroup$
    – PM 2Ring
    Jan 22, 2019 at 19:54
  • $\begingroup$ No dependence in vacuum. In most real media, the speed of light is dependent on the wavelength, this is called dispersion and it depends on the medium. $\endgroup$ Jan 22, 2019 at 19:54
  • $\begingroup$ I am mostly trying to understand how worldlines work, and especially the wordline of light $\endgroup$
    – relaxon
    Jan 23, 2019 at 6:21

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The short answer: Lightlike geodesics do not depend on the energy (frequency) of the lightlike particle. That is lightlike particles all follow the same path. In particular, gravitational lensing does not produce a prismatic effect where one color light is deflect more than others.

The above answer ignores any effects due to the fact that the particle itself modifies the gravitational field. In principle, since a lightlike particle being deflected in a gravitational field leads to a changing energy quadrupole moment, it will generate (minute) gravitational waves. The backreaction due to the emission of gravitational waves will, in principle, lead to a small correction to the particle's trajectory, which will depend on the particle's energy/frequency.

It is worth stressing that this backreaction (also called the "gravitational self-force) will be ridiculously small for any realistic lightlike particle. (The relative correction to the scattering angle will be proportional to the ratio of the particle's energy and the rest energy of the object being scattered off). Effects due to the finite size of the wavepacket are generally going to be much larger.

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