Thinking in Waves Episode 16 ／ Part VIII · Differentiating the effective constants
The way around a defined value was in the dispersion relation
The effective speed of light ──
even a defined value can be differentiated
In Episode 4 we wrote that \(c\) is a defined value and cannot be differentiated. But the speed light actually travels does change.
In wave language, the speed of light is the slope of the dispersion relation, \(d\omega/dk\) ──
★ so "differentiating the speed of light" means taking the second derivative of the dispersion relation.
Claims that "\(c\) has changed" are usually suspect. But there are several changes in \(c\) that are not suspect at all.
And every one of them turned out to be a change in a dimensionless quantity ── Episode 7's conclusion, at work again.
01The speed of light is the slope of a dispersion relation
★ The group velocity is a slope ── so differentiating it means taking the second derivative of the dispersion relation
| System | Dispersion relation | \(v_p\) | \(v_g\) | Note |
|---|---|---|---|---|
| Vacuum | \(\omega=ck\) | \(c\) | \(c\) | constant slope; no curvature |
| Waveguide / plasma | \(\omega^2=\omega_c^2+c^2k^2\) | \(>c\) | \(| ★ \(v_pv_g=c^2\) (Episode 6) | |
| Klein–Gordon | \(\omega^2=(mc^2/\hbar)^2+c^2k^2\) | \(>c\) | \(| same formula; mass is the cutoff | |
| Dispersive medium | \(\omega=ck/n(\omega)\) | \(c/n\) | \(c/n_g\) | \(n_g=n+\omega\,dn/d\omega\) |
| Diffusion (Episode 11) | \(\omega=-iDk^2\) | imaginary | ── | does not propagate |
The heart of §01
"The speed of light changes" means the dispersion relation bends away from a straight line.
And as Episode 6 showed, the intercept of that curve at \(k=0\) is the mass ── ★ light speed and mass are two places on the same curve.
02In a medium ── real, and the second derivative is in industrial use
We computed the refractive index of fused silica from the Sellmeier equation.
| Wavelength [nm] | \(n\) | \(n_g\) | \(v_g\) [m/s] | Use |
|---|---|---|---|---|
| 486 | 1.46313 | 1.49238 | \(2.009\times10^8\) | hydrogen F line |
| 589 | 1.45841 | 1.47902 | \(2.027\times10^8\) | sodium D line |
| 633 | 1.45701 | 1.47538 | \(2.032\times10^8\) | He-Ne laser |
| 1310 | 1.44680 | 1.46164 | \(2.051\times10^8\) | ★ near zero dispersion |
| 1550 | 1.44402 | 1.46260 | \(2.050\times10^8\) | ★ minimum loss (long-haul) |
The heart of §02 ── telecoms infrastructure uses a second derivative
Material dispersion vanishes where \(d^2n/d\lambda^2=0\). Bisecting for it:
zero-material-dispersion wavelength \(=1272.8\) nm (published value about 1270 nm)
★ That is why optical fibre uses 1310 nm and 1550 nm ── 1310 is where dispersion vanishes, 1550 where loss is least.
And "dispersion vanishes" means the second derivative of the refractive index is zero. The rate of change of the rate of change of light speed, in daily industrial use.
03Gravity ── the effective speed of light varies with place
★ We differentiated it after all ── but what is changing is the dimensionless \(2\Phi/c^2\)
| Location | \(2GM/(rc^2)\) | \(c_{\rm eff}/c\) |
|---|---|---|
| Earth's surface | \(1.39\times10^{-9}\) | 0.999999999 |
| Solar surface | \(4.25\times10^{-6}\) | 0.999995755 |
| White dwarf surface | \(2.53\times10^{-4}\) | 0.999746864 |
| Neutron star surface | 0.345 | 0.655 |
Below \(10^{-6}\) in weak fields. But integrate along a path and it becomes measurable ── the Shapiro delay.
| Path | Impact parameter \(b\) | One-way delay | Round trip |
|---|---|---|---|
| Earth → Venus (superior conjunction) | \(1.0\,R_\odot\) | 116.3 µs | 232.6 µs |
| Earth → Mars (Viking 1976) | \(1.2\,R_\odot\) | 120.0 µs | 240.1 µs |
| Earth → Cassini (2002 conjunction) | \(1.6\,R_\odot\) | 131.2 µs | 262.4 µs |
Conclusion of §03 ── already measured to 5 digits
Cassini's round trip of 262 µs ── measuring it to \(10^{-5}\) gives the Cassini bound \(\gamma-1=(2.1\pm2.3)\times10^{-5}\) from Episode 7.
★ "Variation of the effective speed of light" is a quantity already measured to five digits.
One caveat ── this is a coordinate speed. Measure locally and you always get exactly \(c\).
It is observable as a "delay" only when two separated points are joined ── nothing has changed locally.
04Changing the vacuum itself ── two real effects
One: vacuum birefringence in a strong magnetic field (Euler–Heisenberg). A magnetised vacuum has different refractive indices for different polarisations.
| \(B\) [T] | \(\Delta n\) | Where |
|---|---|---|
| 1 | \(3.97\times10^{-24}\) | an ordinary electromagnet |
| 5 | \(9.94\times10^{-23}\) | the PVLAS experiment (still trying) |
| 45 | \(8.05\times10^{-21}\) | record-class steady fields |
| \(10^8\) | \(3.97\times10^{-8}\) | neutron star |
| \(10^{11}\) | \(3.97\times10^{-2}\) | ★ magnetar ── the vacuum is as birefringent as calcite |
Two: the Scharnhorst effect. Between Casimir plates there are fewer vacuum modes, so light goes very slightly faster.
\(1.6\times10^{-32}\) at \(d=1\) µm; even at \(d=1\) nm only \(1.6\times10^{-20}\)
Slowing light is easy; speeding it up is hard
The effective speed of light can be slowed by more than seven orders of magnitude (17 m/s in a Bose–Einstein condensate ── slower than a bicycle).
But speeding it up is limited to about \(10^{-20}\) even in theory.
★ The asymmetry is not accidental ── to slow light you place one resonance; to speed it you must thin the vacuum.
In Episode 8's language: adding absorption is easy, removing it is hard.
05The group velocity exceeds \(c\). Information does not
Reusing Episode 8's Lorentz oscillator. Near resonance the group velocity exceeds \(c\) and even goes negative.
| \(\omega/\omega_0\) | \(n\) (real part) | \(n_g\) | \(v_g/c\) | Behaviour |
|---|---|---|---|---|
| 0.900 | 1.025753 | 1.238341 | 0.8075 | ordinary |
| 0.9915 | ── | 0.5193 | 1.926 | ★ \(v_g>c\) |
| 1.000 | 1.029086 | \(-22.04\) | \(-0.0454\) | ★ negative group velocity |
| 1.0112 | ── | 0.5248 | 1.906 | ★ \(v_g>c\) |
| 5.000 | 0.999792 | 1.000226 | 0.99977 | ordinary |
Two narrow bands with \(v_g>c\) flank the resonance, and inside them the group velocity goes negative. Both are real, experimentally confirmed phenomena (anomalous dispersion).
| \(\omega/\omega_0\) | \(n(\omega)-1\) | Ratio to \(-\omega_p^2/(2\omega^2)\) |
|---|---|---|
| \(10\) | \(-5.051\times10^{-5}\) | 1.01012240 |
| \(10^2\) | \(-5.001\times10^{-7}\) | 1.00010022 |
| \(10^3\) | \(-5.000\times10^{-9}\) | 1.00000102 |
| \(10^4\) | \(-5.000\times10^{-11}\) | 1.00000008 |
Conclusion of §05
At high frequency \(n\to1\) exactly as \(-\omega_p^2/(2\omega^2)\) (8-digit agreement).
Any medium is the vacuum at high enough frequency ── so the very first wavefront always arrives at \(c\).
★ Whatever the group velocity does, information never exceeds \(c\). This is a direct consequence of Episode 8's causality (＝ analyticity) ── that a medium returns to vacuum at high frequency and that its response is causal are one and the same fact.
06So, in the end, can the speed of light be differentiated?
| What you differentiate | Verdict | Content |
|---|---|---|
| \(dc/dt\) | × cannot | \(c\) is defined (Episode 4); zero is a convention |
| \(dc_{\rm eff}/d\Phi=2/c\) | ○ can | ★ but the content is the dimensionless \(2\Phi/c^2\) |
| \(dn/d\omega\) | ◎ can | dimensionless; dispersion itself; measured daily |
| \(d^2n/d\lambda^2\) | ◎ can | ★ zero-dispersion wavelength 1272.8 nm; used by telecoms |
| \(dv_g/d\omega\) | ◎ can | group-velocity dispersion; how fast a pulse spreads |
§06 ── the answer
The speed of light cannot be differentiated. The refractive index can.
And whenever "the effective speed of light changes", what is changing is always a dimensionless quantity ── \(n\), \(2\Phi/c^2\), \(B/B_c\), \(\hbar/(m_ecd)\).
★ Episode 7's conclusion — hits are dimensionless, misses are dimensionful — is at work here too.
"Effective" is the signal that a dimensionful quantity is being rewritten as a dimensionless one.
Exercises
- Assess "the speed of light was different in the early universe" (VSL) in this episode's language.
Show answer
The first question to ask back is "different relative to what?" \(c\) is a defined value, so changing \(c\) alone is a change of units (Episode 4).
To give it content you must say a dimensionless quantity changed ── \(\alpha=e^2/(4\pi\varepsilon_0\hbar c)\), say.
★ But then you cannot tell whether \(c\) changed or \(e\) did. A difference you cannot distinguish is not a physical difference ── which is the conclusion of Episodes 4 and 7. - Is the Shapiro delay "light slowed down" or "the path got longer"?
Show answer
It is a matter of coordinates; either reading is available. In isotropic coordinates it reads as "light speed became \(1+2\Phi/c^2\) times"; in others as "space curved and the path lengthened".
★ The only observable is the 262 µs round trip; how you apportion it is a matter of phrasing.
This has the same structure as Episode 7's "\(GM/(Rc^2)=1/2\) was an identity" ── when several pretty phrasings exist, go back and count observables. - With \(v_g<0\), the pulse exits before it enters. Is that a causality violation?
Show answer
No. What emerges is something the medium reconstructed from the leading half of the incident pulse.
The incident pulse is a smooth function, so its leading half already contains (by analytic continuation) the information in its trailing half ── the medium is merely anticipating it.
★ The decisive test is the front velocity. Make the incident wave exactly zero before some time and nothing whatsoever emerges before \(x/c\). Genuinely new information can travel only at \(c\).
Summary The way around a defined value was the dispersion relation
In wave language the speed of light is the slope of a dispersion relation, \(v_g=d\omega/dk\). So "differentiating the speed of light" means taking the second derivative of that relation ── not differentiating a constant, but measuring how a curve bends.
In a medium it is real. From the Sellmeier equation for fused silica we got the zero-material-dispersion wavelength 1272.8 nm (published about 1270 nm) ── optical fibre uses 1310 nm because the second derivative of the refractive index vanishes there.
In gravity, \(c_{\rm eff}=c(1+2\Phi/c^2)\), and \(dc_{\rm eff}/d\Phi=2/c\) ── differentiable after all, though the content is the dimensionless \(2\Phi/c^2\). Integrated, it is the Shapiro delay (262 µs round trip for Cassini), already measured to \(10^{-5}\).
The vacuum itself can be changed ── birefringence \(\Delta n=3.97\times10^{-24}B^2\) (\(\Delta n\sim0.04\) in a magnetar), and the Scharnhorst effect at \(10^{-32}\). Slowing light by seven orders is easy; speeding it up is limited to \(10^{-20}\).
And the group velocity exceeds \(c\) and goes negative. Information still does not ── we confirmed \(n(\infty)=1\) to 8 digits. A medium returning to vacuum at high frequency and having a causal response are the same fact.
★ The speed of light cannot be differentiated; the refractive index can.
"Effective" is the signal that a dimensionful quantity is being rewritten as a dimensionless one.
Expand the dispersion relation about \(k=0\) and ── order 0 is mass, order 1 is speed, and order 2 is mass again (the effective mass).
★ The reciprocal of mass is the curvature of the dispersion relation. And in graphene there is a world with zero mass and an effective light speed of \(c/300\).
This document is Episode 16 of the series "Thinking in Waves". The computation is in kenshou/hikari.js. On the numbers: the Sellmeier coefficients are standard values for fused silica, and the zero-dispersion wavelength is our own computation by numerical differentiation plus bisection (material dispersion only; in a real fibre waveguide dispersion adds and the zero is engineered to sit near 1310 nm). The Shapiro delay uses the standard approximation \(\Delta t\simeq(2GM/c^3)\ln(4r_1r_2/b^2)\) with representative distances ── the actual Cassini measurement used far more precise modelling. The birefringence coefficient computed from the formula agrees with the published \(4.0\times10^{-24}\,\mathrm{T^{-2}}\). The Scharnhorst value is a bare substitution into a formula with no experimental confirmation whatsoever (nor any prospect of one). The negative and superluminal group velocities are computed in our Lorentz model and differ in setup from real experiments (Wang–Kuzmich–Dogariu 2000 and others). "Coordinate light speed" is coordinate-dependent; light measured locally is always \(c\) ── blurring that distinction is the same hole that VSL debates fall into. ── To print, use your browser's Print → Save as PDF.