# Does the speed of light vary depending on its wavelength passing through empty space?

I remember that I saw the formula below somewhere that shows the dependence of light speed upon its wavelength:

$$c_\lambda=\frac{c_0}{1+27\dfrac{Gh}{8\lambda^2c_0^3}} \,.$$

In this formula, $G$ is the Gravitional constant and $h$ is Planck's constant, so that the term $\frac{Gh}{c_0^3}$ is the square of the Planck length, $l_P$, and where $c_0$ is the speed of a radio wave with a very long wavelength of infinity that appears to be approximately equal to $299792458\, \mathrm{m/s}$. The author of the Article had claimed that each photon had been trapped in its density which the denser a photon is, the lower speed it has! He/She had also claimed that the denser a photon is, the higher its frequency is. Moreover, I have recently seen a scientific show in which a scientist claimed that it was possible that photon's speed depends on its wave length. Do you think that such an equation is rational?!

• It's impossible to evaluate this without knowing exactly what the context is. It might be someone's crackpot assertion, or it might be the result of some nonlinear interaction. I suppose it doesn't even have to be nonlinear. Not enough information. – garyp Aug 19 '16 at 18:02
• I agree with you ... – Mohammad Javanshiry Aug 19 '16 at 18:04
• General answer, covered in physics.stackexchange.com/q/261328 ... Observations of pulsar emissions limit the possible dispersion from microwave to UV to 1 part in $10^{20}$. So, no, that formula is junk. – Jon Custer Aug 19 '16 at 18:18
• So, what would the dispersion be according to this formula from microwave to UV? – Mohammad Javanshiry Aug 19 '16 at 18:39
• What is G, h? Is $\lambda$ related to $c_0, c$? – jim Aug 19 '16 at 20:13

A paper you may be interested in is "A limit on the variation of the speed of light arising from quantum gravity effects" by Adbo et al, Nature 462, 331-334 (19 November 2009). In the abstract they state "A cornerstone of Einstein's special relativity is Lorentz invariance—the postulate that all observers measure exactly the same speed of light in vacuum, independent of photon-energy". They note that while special relativity assumes that there is no fundamental length-scale associated with such invariance, there is a fundamental scale (the Planck scale, $l_{Planck} \approx 10^{-33} \,\mathrm{cm}$ or $E_{Planck} = M_{Planck} \approx 10^{19} \,\mathrm{GeV}$), at which quantum effects are expected to strongly affect the nature of space–time. At such scales there is some speculation that Lorentz invariance might break down. They report the detection of emission up to $31 \,\mathrm{GeV}$ from the distant and short GRB 090510, finding no evidence for the violation of Lorentz invariance, and place a lower limit of $1.2E_{Planck}$ on the scale of any energy dependence.

Your formula may be one of the following references? Mattingly, D. Modern tests of Lorentz invariance. Living Rev. Relativity 8, 5–84 (2005)

Jacobson, T., Liberati, S. & Mattingly, D. Lorentz violation at high energy: concepts, phenomena and astrophysical constraints. Ann. Phys. 321, 150–196 (2006)

Amelino-Camelia, G. Quantum gravity phenomenology. Preprint at http://arxiv.org/abs/0806.0339 (2008)

• Thanks for your answer, but none of the references you mentioned is the reference in which I saw the formula! However, similar formulae are possibly introduced in your references. – Mohammad Javanshiry Aug 19 '16 at 21:20

I would start with the simplest. Usually if something violates this then more explanations are needed or it is just wrong. Ground oneself with reading Skeptics magazine as well.

Maybe consider Maxwell's equation (lots of references online) on constants for magnetism and electricity. $c = 1/\sqrt{\mu \epsilon}$. This is the first to show light was electromagnetic wave and speed depended on what is travelled through.

Simpler is speed energy and mass. The smaller the mass and more energy the faster something goes. What if the mass is zero? speed = energy/mass The speed is not infinity, but maybe the fastest possible, the speed of light. Simple way is photons have zero rest mass, the mass comes from their energy ($E=mc^2$).