Frequency Dependence of the Speed of Light in Space

Abstract

To characterize the possible dispersion of the velocity of light in space (vacuum) a Cauchytype formula, n 2 = 1 +A/v' + B v2, is used. It is shown that relativity only allows a nonzero A term, independent of the nature of the waves o r a quantization thereof. Recent experimental data provide upper bounds for A and B, limiting thereby the dispersion in the microwave, infrared, visible, and ultraviolet regions of the spectrum to less than one part in loz0.Recent observations of radio-wave,' visible,' and x-ray3 emissions f r o m p u l s a r s have been int e r p r e t e d to provide experimental bounds on the dispersion of light in i n t e r s t e l l a r The dispersion o r lack thereof h a s been discussed u sing the expression2f6x7 to r e l a t e m e a s u r e m e n t s in different regions of the s p e c t r u m giving different l i m i t s of dispersion A c in the velocity. It was pointed out by Brown7 that p clearly cannot be a good constant to characterize the variation of velocity with energy, because it would b e infinite f o r h, = A,. Even though that difficulty could be avoided by the introduction of (A2 -h1)/Al in place of x,/x, i n the definition of p, t h e r e r e m a i n s the m o r e s e r i o u s objection against the concept of p, in o u r opinion, that i t suggests a l i n e a r dependence of c on A. It a p p e a r s preferable to the p r e s e n t authors to r e p r e s e n t the dispersion (if t h e r e is any) via a modified Cauchy expression of the form8 w h e r e n is defined by c,h,,, = c,/n and c, is the velocity of light in the absence of dispersion. The corresponding group velocities, to be used in the analysis of the experimental data, are readily calculated f r o m the above expressions. In anticipation of t h e i r use in regions of the s p e c t r u m remote f r o m resonances, only the leading t e r m s in v2 and ( l /~)~, o r AZ and ( l /~)~, a r e retained i n Eqs. (2) and (3). The absence of odd powers of v o r X is a s s u r e d by the presumed s y mm e t r y with respect to r e v e r s a l of the direction of time .g Expressions (2) and (3) d e s c r i b e the frequency dependence of the speed of light in any dispersive

Neutrinos with velocities greater thanc?

“…This very small value seems to be compatible with a bound of about 10 −20 [13] on non trivial contributions to the dispersion relation (Cauchy formula) for free space.…”

Measurement of the speed of light II. Wavelength measurements and conclusion

“…Although there is good astronomical evidence to confirm th a t the speed of light in free space is the same to 2 parts in 1015 for radiations of widely different frequency (Bay & White 1972), it is interesting to consider whether, if there were any difference, the value which we have obtained corresponds to th a t for the infrared radiation of the C 02 laser, or to one of the visible radiations used in the interferometric intercomparison. If we bear in mind th a t the interferometric experiment determines the wavelength ratio R = Ag^/ A679 and the upconversion gives a frequency difference, it may be shown th at the speed of light which we have obtained is related to the values c633 and c679 at the wavelengths 633 nm and 679 nm by This equation is of the same form as th a t relating group velocity u with phase velocity vi n a dispersive medium, u = v -Adv/d A.…”

Dispersion by pulsars, magnetars, fast radio bursts and massive electromagnetism at very low radio frequencies

“…Pulsar observations [66] indicated that the speed of light was constant to within 10 −20 throughout the visible, near infrared and ultraviolet regions of the spectrum, corresponding to a rough upper limit to the photon mass of 3 × 10 −49 kg [36,65,66]. It is evident that exploring lower frequencies provides more stringent upper limits to the photon mass.…”