### The Variation of Photon Speed with Photon Frequency in Quantum Gravity

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Einstein's special relativity is Lorentz invariance; the postulate that all observers measure exactly the same speed of light in vacuum, independent of photon frequency. There is a fundamental scale the Planck scale, at which quantum effects are expected to strongly affect the nature of space-time. The commonly used ideas of space-time should break down at or before the Planck length is reached. It is then natural to question the exactness of the Lorentz invariance that is pervasive in all macroscopic theories. Quantum gravity effect could be seen from the dispersion relations violating Lorentz invariance, because the motivation for the Lorentz invariance violation is quantum gravity. Then it is expected that the energy-momentum dispersion relation could be modified to include the dependence on the ratio of the particle's energy and the quantum gravity energy. In the present work, we have derived an expression of Planck mass or Planck energy by equating the Compton wavelength with Kerr gravitational radius of the Kerr rotating body. Then we derived the modified expression for the photon energy-momentum dispersion relation and hence derived the variation of the photon propagation speed with photon frequency.