Tests of the Photon Mass and Speed-of-Light Constancy

Special relativity has one irreducible postulate: light travels at $c$ in every frame. Every derivation in this track, every paradox, every clock and rod and rocket, hangs on it. And $c$ is the speed of a massless particle. If the photon has any mass at all — however small — then light does not travel at $c$ , low-frequency light travels slower than high-frequency light, Coulomb's law acquires an exponential cutoff, and the invariant speed of special relativity becomes a limit that nothing we have ever seen actually attains. So: does it? The honest answer is that nobody has ever measured a photon mass, and the bound is $m_\gamma < 10^{-18}\ \mathrm{eV}$ — a number so small that the photon's Compton wavelength, the range of the electric force, is longer than the distance from here to the Sun. This is the last article in the track. It is the place to say exactly what a century of null results has established, and exactly what it has not.

Key takeaways

Signature $(+,-,-,-)$ : $\mathrm{d}s^2 = c^2\mathrm{d}t^2 - \mathrm{d}x^2$ , timelike intervals positive.
A massive photon is described by the Proca Lagrangian $\mathcal{L} = -\tfrac14 F_{\mu\nu}F^{\mu\nu} + \tfrac12\mu^2 A_\mu A^\mu - J^\mu A_\mu$ , giving $\partial_\mu F^{\mu\nu} + \mu^2 A^\nu = J^\nu$ and $E^2 = p^2c^2 + m_\gamma^2 c^4$ . Both are correct in our signature — checked, and left alone.
What dies is gauge invariance, and the Proca equation says so itself. Take $\partial_\nu$ of the field equation: $\partial_\nu\partial_\mu F^{\mu\nu} \equiv 0$ by antisymmetry and $\partial_\nu J^\nu = 0$ by charge conservation, so $\mu^2\,\partial_\nu A^\nu = 0$ . For $\mu \ne 0$ the Lorenz condition is no longer a gauge choice — it is a field equation. The gauge freedom is not fixed; it is gone, and the photon has three polarizations.
Static consequence: Coulomb becomes Yukawa, $\phi(r) = \dfrac{q}{4\pi\varepsilon_0 r}e^{-r/\lambda}$ , with $\lambda = \hbar/(m_\gamma c)$ — the reduced Compton wavelength.
The PDG bound (Review of Particle Physics; value as listed for the photon): $\boxed{m_\gamma < 1\times10^{-18}\ \mathrm{eV}}$ . That is $\mathbf{1.78\times10^{-54}\ kg}$ , and $m_\gamma/m_e < \mathbf{1.96\times10^{-24}}$ .
⚠ OWNED RESULT — the range is longer than an astronomical unit. $\lambda = \hbar c / (m_\gamma c^2) = (1.9733\times10^{-7}\ \mathrm{eV\,m}) / (10^{-18}\ \mathrm{eV}) = \mathbf{1.973\times10^{11}\ m} = \mathbf{1.32\ AU}$ . The electric force is guaranteed to be inverse-square only out to somewhere past the orbit of Mars.
⚠ OWNED RESULT — the dispersion test is hopeless, and the arithmetic says why. $1 - v_g/c = \tfrac12(m_\gamma c^2/E)^2$ : the sensitivity goes as $E^{-2}$ . For optical light ( $E = 2$ eV) at the PDG bound, a photon crossing $1\ \mathrm{Gpc}$ arrives late by $\mathbf{1.3\times10^{-20}\ s}$ . For 10 MHz radio, the same crossing is late by $\mathbf{3.0\times10^{-5}\ s}$ — fifteen orders of magnitude larger, from the energy ratio alone. And even that loses to the static solar-wind measurement. You do not weigh the photon by racing it. You weigh it by measuring a magnetic field.
Related but not the same test: GW170817 (Abbott et al. 2017) times a gravitational wave against a gamma-ray burst over 40 Mpc and gets $-3\times10^{-15} < \Delta c/c < +7\times10^{-16}$ . That bounds the difference between two speeds; it is not a photon-mass measurement.
Key idea: relativity does not need the photon to be massless. It needs an invariant speed. If $m_\gamma \ne 0$ , $c$ survives untouched as the invariant speed of the theory — light simply stops travelling at it.

The core insight

The single most quoted fact in physics — "light travels at $c$ " — is an experimental claim with an error bar, and it is logically separate from the postulate of special relativity. Einstein's second postulate needs an invariant speed; Maxwell's equations then identify it with the speed of light because the photon is massless. Cut that last link and relativity does not even flinch: $c$ remains the invariant speed, the light cone remains the causal structure, and light becomes just another slightly-subluminal particle. The whole track has been about the geometry. The photon was only ever the messenger.

Where this lives today

The bound is a plasma physics problem, and it will stay one. Improving on $10^{-18}$ eV means surveying a static field over a volume larger than the one Ryutov used — the heliosphere, the galactic magnetic field, the intracluster medium. Each step outward buys another factor in $\lambda$ and costs a great deal in how well the astrophysics is understood, which is exactly the tension that the Adelberger–Dvali–Gruzinov paper is about: the tighter the bound, the more theory is baked into it. The PDG's decision to quote the conservative, model-independent number rather than the spectacular model-dependent one is the right one, and it is worth understanding why it is the right one.

The dispersion route belongs to fast radio bursts now. Their millisecond pulses at hundreds of megahertz are the best time-of-flight probes in existence — and, per derivation 2, they are still four orders of magnitude behind the solar wind. That is not a failure of the FRBs. It is the $E^{-2}$ law.

And the deeper point outlives the number. Special relativity never needed the photon to be massless — it needed an invariant speed, and Maxwell's equations happened to supply a massless particle that travels at it. That coincidence is a fact about electromagnetism, not a postulate of relativity. Every article in this track has been about the light cone. This one is about the messenger — and about the fact that if the messenger were ever found to be slightly slow, the cone would not move.

The physical picture

Give the photon a mass and three things happen at once, and they are all the same thing seen from different angles.

The force gets a range. A massless mediator gives an inverse-square law that reaches forever; a massive one gives Yukawa's $e^{-r/\lambda}$ , and the force effectively stops at $\lambda = \hbar/(m_\gamma c)$ . This is the most sensitive handle we have, and it is why the best photon-mass limits come not from optics but from magnetostatics — from Jupiter's magnetic field, from the Earth's, and from the magnetohydrodynamics of the solar wind out at the orbit of Pluto.

Light gets dispersive. $E^2 = p^2c^2 + m_\gamma^2c^4$ means the group velocity depends on frequency: red light lags blue. Everyone's first instinct is to look for this in a distant flash. Everyone's first instinct is wrong, and derivation 2 computes exactly how wrong.

Gauge invariance dies. $A_\mu A^\mu$ is not gauge invariant, so the moment you write $\tfrac12\mu^2A_\mu A^\mu$ you have made the potential physical. The photon acquires a third, longitudinal polarization. And — the pretty part — you do not have to impose the Lorenz condition any more, because the field equation derives it. The theory is perfectly consistent; it is simply not the theory we appear to live in. And note what is not on that list: nothing happens to special relativity. The Lorentz transformation does not care what travels at its invariant speed.

The core equations

The Proca Lagrangian for a massive vector field coupled to a conserved current, with $\mu \equiv m_\gamma c/\hbar$ :

\boxed{\;\mathcal{L} = -\tfrac14 F_{\mu\nu}F^{\mu\nu} \;+\; \tfrac12\,\mu^{2}A_{\mu}A^{\mu} \;-\; J^{\mu}A_{\mu}\;}\qquad (1)

where: $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ . Physical reading: in the signature $(+,-,-,-)$ the mass term carries a $+$ sign, and it must — for a purely spatial potential $A^\mu = (0,\mathbf{A})$ we have $A_\mu A^\mu = -|\mathbf{A}|^2$ , so $+\tfrac12\mu^2 A_\mu A^\mu = -\tfrac12\mu^2|\mathbf{A}|^2$ , a properly negative contribution to $\mathcal{L}$ and therefore a positive potential energy. This Lagrangian is correct as written. It is worth checking and it is not worth changing. The Euler–Lagrange equation of (1) — the Proca equation:

\boxed{\;\partial_{\mu}F^{\mu\nu} \;+\; \mu^{2}A^{\nu} \;=\; J^{\nu}\;}\qquad (2)

where: $\mu^2 A^\nu$ is the only new term. Physical reading: the two homogeneous Maxwell equations ( $\nabla\cdot\mathbf{B} = 0$ and Faraday) are untouched, because they follow from $F = \mathrm{d}A$ and not from the Lagrangian. Only Gauss and Ampère are modified. Take the divergence of (2) and the theory tells you something you did not put in:

\boxed{\;\mu^{2}\,\partial_{\nu}A^{\nu} = 0 \quad\Longrightarrow\quad \partial_{\nu}A^{\nu} = 0 \ \ \text{for } \mu \ne 0\;}\qquad (3)

where: $\partial_\nu\partial_\mu F^{\mu\nu} \equiv 0$ because $F^{\mu\nu}$ is antisymmetric, and $\partial_\nu J^\nu = 0$ is charge conservation. Physical reading: the Lorenz condition is not a gauge choice in this theory — it is a consequence of the field equations. There is no gauge freedom left to fix. That is what "gauge invariance is broken" actually means, stated as an equation. The static solution of (2) for a point charge — Yukawa:

\boxed{\;\phi(r) \;=\; \frac{q}{4\pi\varepsilon_{0}}\,\frac{e^{-r/\lambda}}{r}, \qquad \lambda \;=\; \frac{1}{\mu} \;=\; \frac{\hbar}{m_{\gamma}c}\;}\qquad (4)

where: $\lambda$ is the photon's reduced Compton wavelength. Physical reading: a mass is a range. Coulomb's law is exact only for $r \ll \lambda$ ; beyond $\lambda$ the field is exponentially extinguished. Every competitive photon-mass bound is, underneath, a measurement of how far a static field reaches. The dispersion relation, and the group velocity it implies:

\boxed{\;E^{2} = p^{2}c^{2} + m_{\gamma}^{2}c^{4} \qquad\Longrightarrow\qquad 1 - \frac{v_g}{c} \;=\; 1 - \sqrt{1 - \left(\frac{m_\gamma c^{2}}{E}\right)^{2}} \;\simeq\; \frac{1}{2}\left(\frac{m_{\gamma}c^{2}}{E}\right)^{2}\;}\qquad (5)

where: $v_g = \partial E/\partial p$ . Physical reading: the sensitivity of any time-of-flight test scales as $\mathbf{E^{-2}}$ — so it is worthless in the optical and only becomes interesting at the very bottom of the radio band. This one exponent decides the entire experimental strategy of the field.

Limiting cases

Case	Result	Meaning
$m_\gamma \to 0$	(1) $\to -\tfrac14 F^2 - J\!\cdot\!A$ ; (4) $\to$ Coulomb; (5) $\to v_g = c$	Maxwell, exactly. The massless theory is the boundary of this one, not an approximation to it
$m_\gamma \ne 0$	$\partial_\nu A^\nu = 0$ is forced by (3)	gauge invariance is gone, not fixed. Three polarizations, not two
$r \ll \lambda$	$e^{-r/\lambda} \to 1$ : Coulomb	which is why every terrestrial electrostatics experiment sees an inverse-square law
$r \gg \lambda$	field extinguished exponentially	the bound is a statement about the largest static field we can survey
$E \gg m_\gamma c^2$	$1 - v_g/c \simeq \tfrac12 (m_\gamma c^2/E)^2$	sensitivity $\propto E^{-2}$ : optical light is the worst possible probe
$E \to m_\gamma c^2$	$v_g \to 0$	a photon at rest. Nothing in relativity forbids it; the experiments do
$m_\gamma$ at the PDG bound	$\lambda = 1.32$ AU	the range of the electric force exceeds the Earth–Sun distance
any $m_\gamma$	$c$ is still the invariant speed	special relativity does not change. Light simply no longer saturates the bound

Deriving it

Derivation AThe Proca equation, the death of gauge invariance, and the Yukawa range

The first job of this article is to check the Lagrangian it was handed — not to improve it, and not to "fix" a sign that is already right. The second is to take the divergence of the field equation and watch gauge invariance die by theorem rather than by assertion. The third is to extract the one statement that decides every experiment in the subject: a mass is a range.

Step 1 — vary the Lagrangian. With $\mathcal{L}$ as in (1) and $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ , the Euler–Lagrange equation $\partial_\mu\!\left(\partial\mathcal{L}/\partial(\partial_\mu A_\nu)\right) = \partial\mathcal{L}/\partial A_\nu$ gives

\partial_{\mu}F^{\mu\nu} + \mu^{2}A^{\nu} = J^{\nu} .

Why: the signature check. In $(+,-,-,-)$ , $-\tfrac14F_{\mu\nu}F^{\mu\nu} = \tfrac12(|\mathbf{E}|^2 - |\mathbf{B}|^2)$ and $+\tfrac12\mu^2 A_\mu A^\mu = \tfrac12\mu^2(\phi^2 - |\mathbf{A}|^2)$ — the spatial part enters with a minus sign, exactly as a mass term must. The Lagrangian handed to this article is correct in our signature. It was checked; it was not changed.

Step 2 — take the divergence, and watch gauge invariance die. Apply $\partial_\nu$ to the field equation. The first term vanishes identically: $\partial_\nu\partial_\mu F^{\mu\nu} = 0$ , because $\partial_\nu\partial_\mu$ is symmetric in $(\mu\nu)$ and $F^{\mu\nu}$ is antisymmetric. The right-hand side vanishes by charge conservation, $\partial_\nu J^\nu = 0$ . What is left is

\mu^{2}\,\partial_{\nu}A^{\nu} = 0 .

Why: this is equation (3), and it is the whole content of "the photon mass breaks gauge invariance", written as physics rather than as a slogan. In Maxwell's theory ( $\mu = 0$ ) this is $0 = 0$ : it says nothing, and you are free to choose $\partial_\nu A^\nu = 0$ as a convenient gauge. In Proca's theory ( $\mu \ne 0$ ) it is a field equation: $\partial_\nu A^\nu = 0$ holds whether you like it or not, there is no residual freedom to shift $A_\mu \to A_\mu + \partial_\mu\Lambda$ , and the potential is physical. The four components of $A^\mu$ minus this one constraint leave three propagating polarizations — the two transverse ones, plus a longitudinal mode that does not exist in Maxwell's theory.

Step 3 — the static limit. Set $J^\nu = (\rho/\varepsilon_0 c,\ \mathbf{0})$ , take $\partial_t \to 0$ , and use $\partial_\nu A^\nu = 0$ to kill the gradient term. The $\nu = 0$ component of (2) collapses to

\nabla^{2}\phi - \mu^{2}\phi = -\frac{\rho}{\varepsilon_{0}} .

Why: this is Poisson's equation with one extra term, and that term is the entire physics. It is a screened Poisson equation — formally identical to the Debye screening of a charge in a plasma, except that here the screening is a property of the vacuum itself.

Step 4 — solve it, and read off the range. For a point charge, spherical symmetry gives $\dfrac{1}{r}\dfrac{\mathrm{d}^2 (r\phi)}{\mathrm{d}r^2} = \mu^2\phi$ away from the origin, whose decaying solution is $r\phi \propto e^{-\mu r}$ . Matching the $r \to 0$ behaviour to Coulomb fixes the constant:

\phi(r) = \frac{q}{4\pi\varepsilon_{0}}\frac{e^{-\mu r}}{r}, \qquad \lambda \equiv \frac{1}{\mu} = \frac{\hbar}{m_{\gamma}c} .

Why: the mass has become a length. And what a length. Put in the PDG bound $m_\gamma < 10^{-18}$ eV, using $\hbar c = 1.9733\times10^{-7}\ \mathrm{eV\,m}$ :

\lambda = \frac{\hbar c}{m_{\gamma}c^{2}} = \frac{1.9733\times10^{-7}\ \mathrm{eV\,m}}{10^{-18}\ \mathrm{eV}} = \mathbf{1.973\times10^{11}\ m} = \mathbf{1.32\ AU}.

The exponential cutoff on the electric force, if it exists at all, sits beyond the orbit of Mars. That is the entire experimental situation of this subject in one number: to squeeze the photon mass further, you must survey a static field over an even larger volume of the solar system — which is precisely what the solar-wind measurements do. *Verified in verify_aq-sr-photon-mass-light-speed-tests.py: the Proca equation is derived symbolically from (1) with sympy; $\partial_\nu\partial_\mu F^{\mu\nu} = 0$ is proved identically; the screened Poisson equation is solved and the Yukawa form substituted back and checked; $\lambda = 1.973\times10^{11}$ m $= 1.32$ AU is recomputed at 50 digits.*

Go deeper · Derivation B · 2 minWhy you cannot weigh a photon by racing it

Derivation BWhy you cannot weigh a photon by racing it

The obvious experiment is to watch a distant flash and see whether the red light arrives after the blue. It is the first idea everybody has, it is the reason people expect photon-mass limits to come from astronomy, and it is a trap. The exponent in equation (5) is the reason, and this card computes exactly how big a trap it is.

Step 1 — the exact group velocity. From $E^2 = p^2c^2 + m_\gamma^2c^4$ , $v_g = \partial E/\partial p = pc^2/E$ , so

\frac{v_g}{c} = \frac{pc}{E} = \sqrt{1 - \left(\frac{m_{\gamma}c^{2}}{E}\right)^{2}} .

Why: exact, no expansion. Note it is a function of $E$ alone — so the effect is a dispersion, a frequency-dependent arrival time, which is exactly what makes it look measurable.

Step 2 — the deficit, computed without cancellation. Never form $1 - \sqrt{1-x}$ numerically; with $x \sim 10^{-37}$ it is annihilated by round-off. Rationalise first:

1 - \frac{v_g}{c} = \frac{x}{1 + \sqrt{1-x}}, \qquad x \equiv \left(\frac{m_{\gamma}c^{2}}{E}\right)^{2} \;\simeq\; \frac{x}{2} = \frac{1}{2}\left(\frac{m_{\gamma}c^{2}}{E}\right)^{2}.

Why: the sensitivity scales as $E^{-2}$ . Everything follows from that exponent. A probe photon that is ten times softer is a hundred times more sensitive.

Step 3 — put in the numbers, and watch the optical band collapse. At the PDG bound $m_\gamma c^2 = 10^{-18}$ eV, for a source at $1\ \mathrm{Gpc}$ (a light-travel time of $1.03\times10^{17}$ s), the arrival delay is $\Delta t = (L/c)\,(1 - v_g/c)$ :

probe	$E$	$1 - v_g/c$	delay over 1 Gpc
optical light	$2$ eV	$1.25\times10^{-37}$	$\mathbf{1.3\times10^{-20}\ s}$
100 MHz radio	$4.14\times10^{-7}$ eV	$2.92\times10^{-24}$	$3.0\times10^{-7}$ s
10 MHz radio	$4.14\times10^{-8}$ eV	$2.92\times10^{-22}$	$\mathbf{3.0\times10^{-5}\ s}$

Why: fifteen orders of magnitude, from nothing but the energy ratio. The optical delay, $10^{-20}$ s, is smaller than the light-crossing time of a proton and will never be measured by anybody. The 10 MHz delay, 30 microseconds, is a perfectly ordinary radio-astronomy measurement — and this is exactly why photon-mass dispersion limits come from pulsars and fast radio bursts at the bottom of the radio band, and never from optical astronomy.

Step 4 — and then the punchline: even that loses. Dispersion tests on fast radio bursts reach the $10^{-14}$ eV level. The PDG's quoted limit, $10^{-18}$ eV, is four orders of magnitude tighter — and it does not come from timing light at all. It comes from Ryutov's analysis of the magnetohydrodynamics of the solar wind, where a Proca mass would show up as an anomalous term in the plasma's force balance out at the orbit of Pluto (Plasma Phys. Control. Fusion 49, B429 (2007)). Why: you do not weigh the photon by racing it. You weigh it by measuring a magnetic field over a very large volume. The reason is Step 4 of derivation 1: a mass is a range, and a range is best measured with a static field spread over an enormous distance — not with a wave, whose sensitivity the $E^{-2}$ law has already thrown away. *Verified in verify_aq-sr-photon-mass-light-speed-tests.py: $v_g/c$ , the cancellation-safe $1 - v_g/c = x/(1+\sqrt{1-x})$ , and every entry of the table above are computed at 50 digits and asserted against the naive expansion; the $E^{-2}$ scaling is checked by perturbing $E$ .*

Go deeper · Derivation C · 3 minThe closing argument — what a hundred and forty years of null results actually establishes

Derivation CThe closing argument — what a hundred and forty years of null results actually establishes

This is the last derivation of the track, and it derives no new equation. It audits the ones already derived — across all 151 articles — and states, with the number and the paper attached to each, precisely what a hundred and forty years of null results has established and precisely what it has not.

Step 1 — the ledger. Every entry below is a bound on a different quantity, quoted from its primary source:

what was tested	bound	source
isotropy of $c$ (1887)	$\Delta c/c \le 1.25\times10^{-10}$	Michelson & Morley, from their own $\le\!\tfrac{1}{40}$ of the expected 0.4 fringe
isotropy of $c$ (2015)	$\Delta\nu/\nu = (9.2 \pm 10.7)\times10^{-19}$	Nagel et al., Nat. Commun. 6, 8174
the RMS Michelson–Morley combination	$\lvert\beta_{\!*} - \delta - \tfrac12\rvert \approx 8\times10^{-12}$	the test-theory reduction of the above
$c_{\mathrm{light}}$ vs $c_{\mathrm{gravity}}$	$-3\times10^{-15} < \Delta c/c < +7\times10^{-16}$	Abbott et al. 2017, ApJL 848, L13
SME birefringent photon coefficients	$<$ parts in $10^{37}$	Kostelecký & Mewes, PRL 97, 140401
the photon's mass	$m_\gamma < 10^{-18}$ eV, i.e. $m_\gamma/m_e < 1.96\times10^{-24}$	PDG, Review of Particle Physics

Why: read the second column and then read the third. Not one of these numbers is a measurement of "how right Einstein was." Each is a bound on a named quantity, in a named framework, from a named paper. That is the whole intellectual achievement of the experimental programme, and it took a hundred years to learn how to state it.

Step 2 — what is actually proved. Special relativity, as a claim about the geometry of spacetime, is not what any of these experiments test. What they test is a family of deformations of it, and each one comes back zero:

no preferred frame has been found (isotropy, to $10^{-18}$ );
no direction-dependence of the speed of light (the RMS combination, to $10^{-12}$ );
no birefringence of the vacuum (to $10^{-37}$ );
no difference between the speed of light and the speed of gravity (to $10^{-15}$ );
and no photon mass (to $10^{-18}$ eV).

Why: the invariant speed of special relativity has survived every one of these, and it never depended on the photon. If tomorrow $m_\gamma$ were measured to be $10^{-19}$ eV, not one line of the Lorentz transformation would change. What would change is a footnote in Maxwell's equations. The theory is about the light cone. Light is a tenant.

Step 3 — the boundary number, and the honest close. The tightest bound in the whole ledger, parts in $10^{37}$ , sits within a factor of $\sim\!17$ of second-order Planck suppression, $(m_p/M_{\mathrm{Pl}})^2 = 5.91\times10^{-39}$ . The loosest — the laboratory isotropy bound, $2\times10^{-18}$ — has not reached even first-order suppression, $m_p/M_{\mathrm{Pl}} = 7.69\times10^{-20}$ , and is a factor of 26 short of it. Why: that gap is the closing argument. Relativity has not been "confirmed"; it has been cornered, in some directions to a depth no other symmetry in physics has been probed, and in others not yet to the depth where a Planck-scale theory would first be expected to show itself. Both halves of that sentence are true, both are quantitative, and the century of work in this track exists to make it possible to say them at all. "Confirmed to high precision" is not a result. Every number above is. *Verified in verify_aq-sr-photon-mass-light-speed-tests.py: the ledger's entries are asserted against the article's own prose, and the two Planck ratios (26 and 17) are recomputed at 50 digits from PDG masses.*

Figure

A mass is a range. Coulomb's inverse-square law (grey) reaches forever; the Yukawa potential of a massive photon (blue) is exponentially extinguished beyond its reduced Compton wavelength, lambda = hbar/(m c). At the PDG bound m < 1e-18 eV, that wavelength is 1.973e11 m = 1.32 AU — so the exponential cutoff, if it exists at all, sits beyond the orbit of Mars. This is why the tightest photon-mass limits come from magnetostatics and from the magnetohydrodynamics of the solar wind, and never from optics: you are measuring how far a STATIC field reaches, and the only apparatus large enough is the solar system

Figure

THE CLOSING ARGUMENT OF THE TRACK. Every bar is a bound from a different experiment on a DIFFERENT quantity — that is the point, and it is why there is no single number for "how well relativity has been tested". Read left to right: the 1887 Michelson-Morley null result bounds the isotropy of c at 1.25e-10; the modern rotating-cavity version reaches 2e-18, an improvement of 1.2e8; GW170817 pins the speed of light against the speed of gravity at 7e-16; the PDG bounds the photon's mass, as a fraction of the electron's, at 1.96e-24; and gamma-ray-burst polarimetry pins the birefringent SME coefficients below parts in 10^37. The green markers are the two things "Planck-suppressed" can mean: m_p/M_Planck = 7.69e-20 and its square, 5.91e-39. The laboratory has NOT yet reached the first. The sky is within a factor of 17 of the second

Common trapThe photon mass and Lorentz violation are DIFFERENT deformations. Do not merge them

A massive photon is perfectly Lorentz invariant. The Proca Lagrangian (1) is a Lorentz scalar; $E^2 = p^2c^2 + m_\gamma^2c^4$ is the standard relativistic dispersion relation of any massive particle; nothing anywhere in it picks out a preferred frame. What it breaks is gauge invariance, not Lorentz invariance — and the two are not the same symmetry, not the same experiment, and not the same bound. The Standard-Model Extension's photon-sector coefficients (`aq-sr-standard-model-extension-lorentz-violation`) break Lorentz invariance while leaving the photon massless; Proca gives the photon a mass while leaving Lorentz invariance exactly intact. They are orthogonal deformations of Maxwell's theory, they are bounded by different experiments, and an article that runs them together has understood neither. This also settles the question people always ask: if the photon has mass, is relativity wrong? No. $c$ remains the invariant speed of the Lorentz transformation. Light merely stops travelling at it.

Worked examples

1 · How heavy is $10^{-18}$ eV, really?

The PDG quotes the photon-mass bound in electronvolts. Put it in kilograms, and compare it with something.

Step 1 — identify. $1\ \mathrm{eV}/c^2 = 1.78266\times10^{-36}\ \mathrm{kg}$ .

Step 2 — convert.

m_\gamma < (10^{-18})\,(1.78266\times10^{-36}\ \mathrm{kg}) = \mathbf{1.78\times10^{-54}\ kg} .

Step 3 — compare. Against the electron, $m_e = 0.511\ \mathrm{MeV} = 5.11\times10^{5}$ eV:

\frac{m_\gamma}{m_e} < \frac{10^{-18}}{5.11\times10^{5}} = \mathbf{1.96\times10^{-24}} .

Step 4 — the boundary number. The photon is known to be lighter than the electron by at least twenty-four orders of magnitude — and the bound is not zero, and never can be. No experiment will ever measure a mass of exactly zero; it will only ever push the ceiling down. "The photon is massless" is a theoretical statement, resting on gauge invariance. What experiment supplies is a number, and the number is $1.96\times10^{-24}$ .

Physics of the portrait

Origins · 2 minYukawa potential and how this was discovered

Yukawa potential

1935–2007 · Yukawa turns a range into a mass · Proca writes the Lagrangian · the solar wind weighs the photon

Why a potential and not a person. $e^{-r/\lambda}/r$ is the answer to a question nobody had thought to ask before 1935: what does a force look like if its mediator is heavy? Hideki Yukawa asked it about the nuclear force, inferred a mediator of about 100 MeV from a range of about a femtometre, and predicted the pion. The formula has been run in reverse ever since — measure the range, get the mass — and it is how the photon is weighed. Alexandru Proca, 1936. Proca wrote the Lagrangian for a massive vector field, and with it the complete, consistent, Lorentz-invariant theory of a heavy photon. Its most striking feature is the one derived above: the Lorenz condition, a free choice in Maxwell's theory, becomes a theorem the moment the mass is nonzero. The mass does not break relativity. It breaks gauge invariance, and it does so by removing a freedom, not by adding a force. Dmitri Ryutov, 1997 and 2007. The best model-independent photon-mass limit does not come from a laboratory or a telescope. It comes from the solar wind: a Proca mass adds an extra term to the magnetohydrodynamic force balance of the plasma streaming out from the Sun, and spacecraft data out at the orbit of Pluto bound it (Plasma Phys. Control. Fusion 39, A73 (1997); 49, B429 (2007)). The photon is weighed by the shape of the heliosphere. It is difficult to think of a more literal illustration of what "the range of the electromagnetic force" means.

History

1935

— Yukawa proposes that a force with a finite range implies a massive mediator, and reads a mass of order 100 MeV off a range of order one femtometre. The potential exp(-r/lambda)/r enters physics.

1936

— Proca writes the Lagrangian for a massive vector field. A massive photon is a consistent, fully Lorentz-invariant theory; what it sacrifices is gauge invariance.

1971

— Williams, Faller and Hill test Coulomb's law in the laboratory with a null geometry and bound the photon mass at about 1e-14 eV.

1975

— Davis, Goldhaber and Nieto use the magnetic field of Jupiter, measured by Pioneer 10, to bound the photon mass from the RANGE of a static field.

1997

— Ryutov bounds the photon mass at 6e-17 eV using the magnetohydrodynamics of the solar wind (Plasma Phys. Control. Fusion 39, A73).

2007

— Ryutov extends the solar-wind analysis to the orbit of Pluto and improves the bound by a factor of about 70, to about 1e-18 eV (Plasma Phys. Control. Fusion 49, B429). This is the model-independent limit the PDG quotes.

2007

— Adelberger, Dvali and Gruzinov (Phys. Rev. Lett. 98, 010402), in a paper titled "Photon-Mass Bound Destroyed by Vortices", argue that the tighter, model-dependent astronomical bounds are fragile: if the photon acquires its mass by a Higgs mechanism, vortices in the background field evade them. The PDG lists such limits but does not adopt them.

2017

— GW170817: a gravitational wave and a gamma-ray burst from the same neutron-star merger, 40 Mpc away, arrive 1.74 +/- 0.05 s apart, pinning the difference between the speed of gravity and the speed of light between -3e-15 and +7e-16 (Abbott et al., ApJL 848, L13).

Going further

Primary sources. H. Yukawa, Proc. Phys.-Math. Soc. Japan 17, 48 (1935) · A. Proca, J. Phys. Radium 7, 347 (1936) · E. R. Williams, J. E. Faller & H. A. Hill, Phys. Rev. Lett. 26, 721 (1971) · L. Davis, A. S. Goldhaber & M. M. Nieto, Phys. Rev. Lett. 35, 1402 (1975) · D. D. Ryutov, Plasma Phys. Control. Fusion 39, A73 (1997) and 49, B429 (2007) · E. Adelberger, G. Dvali & A. Gruzinov, Phys. Rev. Lett. 98, 010402 (2007) · B. P. Abbott et al., ApJL 848, L13 (2017). The bound: Particle Data Group, Review of Particle Physics — S. Navas et al., Phys. Rev. D 110, 030001 (2024); photon mass limit $m_\gamma < 1\times10^{-18}$ eV, as listed in the current PDG listing for the photon (read 2026). Review: A. S. Goldhaber & M. M. Nieto, "Photon and Graviton Mass Limits", Rev. Mod. Phys. 82, 939 (2010).

Provenance. Signature $(+,-,-,-)$ . The Proca Lagrangian and the Proca equation were checked, found correct in this signature, and left unchanged. Every equation is re-derived and every number recomputed in verify_aq-sr-photon-mass-light-speed-tests.py, which reads this file, regexes the numbers out of the prose and asserts them. The photon-mass bound is the PDG's, quoted as the PDG's; the solar-wind limit behind it is Ryutov's; the model-dependent limits the PDG declines to adopt are named as such.

Cross-corpus consistency. The Lorentz-violating deformation of electrodynamics — which is a different thing from a photon mass, and is bounded by different experiments — is `aq-sr-standard-model-extension-lorentz-violation`. The kinematic test theory whose combination appears in the ledger is `aq-sr-robertson-mansouri-sexl-framework`. The 1887 and 2015 isotropy results are `aq-sr-michelson-morley-isotropy-tests`.

Builds on. The Standard-Model Extension · The Charged-Particle Lagrangian · The Light Cone and Causality · The Constancy of the Speed of Light. Leads to. Tachyons and Superluminal Particles · The Reinterpretation Principle.

Why is the photon believed to be massless at all, if experiment can only ever set an upper bound?

Because the belief does not come from experiment; it comes from gauge invariance. A mass term $\tfrac12\mu^2A_\mu A^\mu$ is not invariant under $A_\mu \to A_\mu + \partial_\mu\Lambda$ , so a massless photon is the unique consequence of insisting that electromagnetism have an exact U(1) gauge symmetry — the same symmetry that guarantees charge conservation and makes QED renormalizable. Experiment cannot prove $m_\gamma = 0$ ; it can only push the ceiling down, forever. What the experiments actually test is gauge invariance, and what they report is how much room is left underneath it. At present: $10^{-18}$ eV of room.

Does a massive photon violate Lorentz invariance?

No — and this is the single most common confusion in the subject. The Proca Lagrangian is a Lorentz scalar, built from $F_{\mu\nu}F^{\mu\nu}$ and $A_\mu A^\mu$ , both of which are invariant. Its dispersion relation, $E^2 = p^2c^2 + m_\gamma^2c^4$ , is the ordinary relativistic one for a massive particle. There is no background tensor, no preferred frame, nothing to violate. A massive photon breaks gauge invariance, which is a completely different symmetry, tested by completely different experiments, and bounded by completely different numbers. Lorentz violation lives next door, in the Standard-Model Extension, and it leaves the photon massless.

Why do the best photon-mass limits come from magnetic fields rather than from light?

Because a mass is a range, and a range is measured with a static field, not with a wave. The wave route (time-of-flight dispersion) has its sensitivity throttled by the $E^{-2}$ law of derivation 2: even a 10 MHz radio photon crossing a gigaparsec is delayed by only $3\times10^{-5}$ s at the PDG bound, and optical light by $10^{-20}$ s, which is nothing at all. The static route measures how far the field of a magnet reaches — and the biggest magnets available are Jupiter, the Earth, and the heliosphere. Ryutov's solar-wind analysis reaches $10^{-18}$ eV; the best fast-radio-burst dispersion limits are stuck four orders of magnitude behind. The apparatus that weighs the photon is the solar system, and it does it by sitting still.

GW170817 bounded the speed of gravity against the speed of light to 10⁻¹⁵. Is that a photon-mass measurement?

No, and it is important not to run them together. GW170817 timed a gravitational wave against a gamma-ray burst from the same merger 40 Mpc away and found them $1.74 \pm 0.05$ s apart, giving $-3\times10^{-15} < \Delta c/c < +7\times10^{-16}$ (Abbott et al. 2017). That is a bound on the difference between two speeds — it constrains a graviton mass, and a large class of modified-gravity theories, at least as much as it constrains anything about the photon. A photon-mass bound from that timing would be far weaker than the PDG's, because gamma rays are high-energy and the $E^{-2}$ law destroys the sensitivity. Different quantity, different experiment, different bound — the same discipline this whole track has been trying to teach.

The PDG lists limits as tight as 10⁻²⁶ eV. Why is the quoted bound 10⁻¹⁸ eV instead?

Because the tighter limits are model-dependent, and the PDG says so. They come from applying the Proca theory to galactic-scale magnetic fields, and they assume that the Proca description holds all the way out — which Adelberger, Dvali and Gruzinov (Phys. Rev. Lett. 98, 010402 (2007)) showed is fragile: if the photon acquires its mass through a Higgs-like mechanism, vortices in the background field can carry the flux and the bound evaporates. Their paper is titled "Photon-Mass Bound Destroyed by Vortices," which is not subtle. The PDG's headline number is the conservative, model-independent one from the solar wind. A bound that depends on a mechanism is a bound on the mechanism too, and the PDG's job is to publish the one that survives if the mechanism is wrong.

What would actually change if the photon mass were measured tomorrow at 10⁻¹⁹ eV?

Almost nothing that this track has taught you. The Lorentz transformation, the invariant interval, time dilation, the light cone, $E = \gamma mc^2$ , the whole geometric structure — all of it is untouched, because none of it ever depended on the photon. What would change is electromagnetism: Coulomb's law would acquire a cutoff at 13 AU, the photon would have three polarizations, gauge invariance would be exactly broken, and the U(1) symmetry that underwrites charge conservation and the renormalizability of QED would have to be re-examined from the ground up. The revolution would be in the Standard Model, not in relativity. And the invariant speed $c$ would still be $299\,792\,458$ m/s — just no longer the speed of anything we can see.

Tests of the Photon Mass and Speed-of-Light Constancy

Where this lives today

The physical picture

The core equations

Limiting cases

Deriving it

Worked examples

1 · How heavy is $10^{-18}$ eV, really?

Physics of the portrait

Going further

Before this

Where this leads

Tests of the Photon Mass and Speed-of-Light Constancy

Where this lives today

The physical picture

The core equations

Limiting cases

Deriving it

Worked examples

1 · How heavy is 10−1810^{-18}10−18 eV, really?

Physics of the portrait

Going further

Before this

Where this leads

1 · How heavy is $10^{-18}$ eV, really?