The photons that raced for seven billion years

Where every model runs out at once found the Planck length where a mass’s quantum size and its gravitational size meet. The length no experiment can resolve showed that no measurement can probe below it, because a probe energetic enough would make a horizon. The estimate that misses by a hundred and twenty found the one number computed at the Planck scale that can be compared with data, and found it wrong. The scale that may not be where it looks asked whether the scale might be lower than it seems, and followed the searches that have found no sign of it.

Each of those treated the Planck scale as unreachable by direct experiment, and for a collider it is: the Planck energy is $1.2\times10^{19}$ GeV, a million million times more than the Large Hadron Collider supplies. This essay concerns a way round that limit. It does not reach the Planck energy. It lets an effect suppressed by the Planck energy accumulate over a vast distance until it becomes large enough to measure, and a gamma-ray burst halfway across the universe provides the distance. The result is one of the few observational statements about physics at the Planck scale, and it is a negative one.

A speed that depends on energy #

The idea starts from the suspicion that spacetime, at the Planck length, is not smooth. Many approaches to quantum gravity picture it as granular, foamy or discrete there. A wave travelling through such a medium might behave like light in glass: waves much longer than the grain would travel at the usual speed, and waves approaching the grain size would be slowed, or sped up, by an amount depending on their wavelength. The simplest form of the effect is a photon speed that differs from $c$ by a fraction proportional to the photon’s energy over some scale $E_{QG}$ near the Planck energy: $v = c(1 - E/E_{QG})$.

For any photon ever detected this fraction is minuscule. A 31 GeV photon, among the most energetic seen from a gamma-ray burst, would lag by $E/E_P$, about three parts in $10^{18}$. Light is not dispersive in vacuum at any level laboratories can test, because nothing in a laboratory gives a photon $10^{18}$ times its own period to fall behind.

A distant gamma-ray burst supplies the time. The figure computes the delay such a speed difference would build up for photons of each energy from a burst at redshift 0.903, the distance of GRB 090510, seen by NASA’s Fermi satellite on 10 May 2009. At the Planck energy, the delay of a 31 GeV photon relative to a low-energy one emitted at the same moment would be 1.16 seconds. For a scale ten times higher the delay would be ten times smaller; for one ten times lower, ten times larger.

GRB 090510 was a short burst, over in about a second, and among the photons Fermi’s Large Area Telescope recorded from it was one with an energy of 31 GeV, arriving 0.83 seconds after the burst began. If it had been emitted at the burst’s start, a delay larger than 0.83 seconds is excluded, and with it any scale below 1.4 times the Planck energy, on the simplest reading of the figure. The Fermi team’s own analysis, which considered when during the burst the photon could have been emitted and used the most conservative assumption, placed the limit at 1.2 times the Planck energy. A first-order energy-dependent speed of light, at the scale where quantum gravity was expected to produce it, is excluded.

What a grainy spacetime would do to a wave #

The analogy with a medium can be made concrete, and it predicts which order to expect. The frequency a lattice cannot carry found that a row of masses joined by springs carries long waves at a fixed speed and slows short ones, because a wave whose wavelength approaches the spacing feels the discreteness. Its dispersion relation, $\omega \propto \sin(ka/2)$, gives a group velocity that falls below the long-wave value by a fraction $(ka)^2/8$ for waves still much longer than the spacing. That correction is quadratic in the wavenumber, not linear. A symmetric lattice cannot produce a linear term, because the lattice looks the same whichever way a wave travels along it, and a linear term would distinguish the two directions.

If spacetime were granular at the Planck length in a way that preserved that kind of symmetry, the natural correction to light’s speed would be quadratic, and the time-of-flight test would be blind to it. A linear correction needs a structure with a handedness, or a preferred direction of time, or a breaking of the symmetry between a photon and its reverse. Such structures were proposed in the 1990s — in particular in loop quantum gravity and in some models of string theory’s effects on the vacuum — and Amelino-Camelia and colleagues pointed out in 1998 that gamma-ray bursts could test them. The prospect was exciting because the linear effect was the one within reach.

The analogy also warns against taking it too literally. A lattice has a rest frame, the frame of the lattice, and its dispersion is a statement about waves measured there. Spacetime has no rest frame in special relativity, and a grain structure without one is hard to imagine. Continuous symmetries sit uneasily with discreteness, and whether any discrete model of spacetime can preserve Lorentz symmetry exactly on large scales is itself one of the questions quantum gravity is trying to answer. Some discrete approaches, such as causal sets, are built to preserve it statistically and predict no dispersion at all. The null result from GRB 090510 is therefore welcome to some approaches and awkward for others, and it decides between them only where they predicted a linear effect.

The distance that amplifies it #

The delay is large only because the distance is.

The figure plots the delay against the source’s redshift, together with the light-travel time. The light from GRB 090510 travelled for 7.5 thousand million years, over half the age of the universe, and the delay would have accumulated steadily along the way. A burst at redshift 3 would give 3.6 seconds.

The delay grows faster than the travel time, and the reason is cosmological. The photon’s energy is measured on arrival. Earlier in its journey, before the expansion of the universe stretched its wavelength, it had more energy, higher by a factor $1 + z$ at each redshift, and the speed deficit was proportionally larger. Jacob and Piran worked out the correct integral in 2008, weighting each stretch of the journey by $1 + z$ and by the expansion rate at that time. The figures use it, with standard values for the Hubble constant and the matter density. A fractional speed difference of a few parts in $10^{18}$ becomes a second because the photon has been travelling for half the age of the universe, and the effect was stronger when it started.

This is the same principle as the wavelength a fibre does not smear: a small dispersion accumulated over a long path spreads a pulse in time by an amount proportional to the path. There the fibre was a hundred kilometres long and the dispersion a property of glass. Here the path is seven billion light years and the dispersion would be a property of spacetime itself, if it existed.

What light travel can and cannot reach #

The result excludes the first-order effect. It says almost nothing about the next order.

The figure compares the linear effect with the quadratic one, a speed difference proportional to $(E/E_{QG})^2$. At 31 GeV the quadratic delay from the same burst is $6.5\times10^{-18}$ seconds, eighteen orders of magnitude too small to measure. Even a hundred-TeV photon, the most energetic kind now detected from astrophysical sources, would be delayed by less than a tenth of a nanosecond. The 0.83-second window excludes a quadratic effect only if its scale is below $2.8\times10^{-9}$ of the Planck energy, far below where quantum gravity was expected.

That matters because theorists have reasons to prefer the quadratic form. A linear energy dependence is odd under reversing the photon’s energy, in a sense that conflicts with some symmetries many approaches to quantum gravity retain, and in effective field theories linear terms often also cause the two polarisations of light to travel at different speeds, an effect that polarisation measurements of distant sources constrain far more tightly than timing does. So the timing result closes off the most direct form of the idea, the one that would have been visible, and leaves open the forms that are easier to believe and harder to see.

Whose energy is it? #

There is a deeper problem with an energy-dependent speed of light, and the figure on frames shows it.

A photon’s energy is not a property of the photon alone. It depends on the observer, and velocities that refuse to add make that dependence simple in terms of rapidity: an observer moving towards the photon at rapidity $\eta$ measures its energy multiplied by $e^{\eta}$. The figure plots the 31 GeV photon’s energy for observers at different rapidities. One approaching it at a rapidity of 40.5 — a Lorentz factor of about $2\times10^{17}$ — measures it at the Planck energy. One receding at the same rapidity measures it at less than $10^{-16}$ GeV, a radio wave.

So a law saying that photons near the Planck energy travel slower cannot hold in every frame, because whether a photon is near the Planck energy depends on the frame. Either the law holds in one preferred frame — and the natural candidate is the frame in which the bath of microwave background radiation is at rest — or relativity itself must be modified so that the Planck energy is the same for every observer. Theories of the second kind, called doubly special relativity, deform the rules for combining velocities and energies at high energy so that two scales, the speed of light and the Planck energy, are invariant. A minimum length raises the same issue: the length that depends on when showed that lengths contract under boosts, so a length that is the minimum in one frame is shorter in another.

The time-of-flight test addresses both options without telling them apart. A preferred-frame theory with a linear effect would have produced a delay, and so would the simplest deformed-relativity models. Neither delay appeared.

What makes a burst a good clock #

The test’s power depends on something ordinary: how sharp the burst is.

The figure simulates a burst to show what the observation looks for. A sharp spike of low-energy photons marks the moment of emission. A handful of high-energy photons emitted in the same spike arrive, with no energy-dependent speed, inside it. With a first-order Planck-scale delay they would arrive spread out after it, the most energetic ones latest, by up to a second. What makes a burst a good clock is the sharpness of its spike compared with the delay being sought. GRB 090510 was a short burst whose bright phase lasted about a second, and its highest-energy photon arrived within it.

The weak point is the assumption that the photons were emitted together. Nobody knows in detail how a gamma-ray burst produces its highest-energy photons, and they might be emitted later than the low-energy ones for reasons internal to the source, or earlier. A delay intrinsic to the source could mimic an effect of propagation or cancel one. The standard response is statistical: an effect of propagation grows with distance and energy in a definite way, while an effect of the source should not care how far away the burst is. Studies of many bursts at different redshifts look for the characteristic dependence on distance, and have found none.

Other ways the same idea has been tested #

Photons are not the only particles whose dispersion can be constrained, and some of the other tests are far more sensitive, at the price of more assumptions.

If electrons had a modified dispersion of their own, very energetic ones could in some models move faster than light. They would then emit the vacuum version of the radiation the cone the source leaves behind describes — Cherenkov radiation, from a particle outrunning its own waves — and lose energy rapidly. The Crab Nebula shines by synchrotron radiation from electrons of at least hundreds of TeV, so such electrons exist and are not losing energy that way. Jacobson, Liberati and Mattingly used this in 2003 to constrain electron dispersion at first order far beyond the Planck scale, within a specific framework in which all the modifications are described by a consistent effective theory.

Polarisation is more sensitive still. Many models in which light’s speed depends on energy at first order also make it depend on polarisation, so that the plane of polarisation of light from a distant source rotates by an amount growing with energy and distance. Polarised light from gamma-ray bursts and from the afterglows of distant galaxies arrives with its polarisation intact, which constrains that birefringence at first order to many orders of magnitude beyond the Planck scale. The loophole is that birefringence is not a universal consequence of dispersion: models can be built that have one without the other.

Each of these tests closes off a class of models rather than an idea, and the time-of-flight test remains the most direct because it assumes least. It needs only that photons of different energies left together and arrived apart, or not.

The strength of the argument is that no model of the burst is needed if the high-energy photon arrives during the low-energy spike. Whatever the source did, it cannot have emitted the photon before the burst began, so the observed arrival time bounds the delay from above. A late-emitted photon only strengthens the bound. That is why a single photon, well timed, carries so much weight, and why the most useful bursts are the shortest and most distant ones with the most energetic photons.

What the time-of-flight test assumes #

Three assumptions stand behind the conclusion, and each is a place where it could be evaded.

The form of the effect. The analysis assumes a speed that changes smoothly and systematically with energy. Proposals in which spacetime’s graininess makes the speed fluctuate randomly, photon to photon, would broaden the arrival times instead of shifting them, and bursts constrain that too, but differently.

The sign. The figures assume high-energy photons are slower. Some models make them faster. The bound from GRB 090510 applies to both, because a photon arriving early would also have appeared outside the bright phase.

The cosmology. The distance integral uses the standard expansion history. Changing it within the range the data allow alters the delay by a few per cent, which does not affect the conclusion.

Still open: where the next order is #

Since 2009 larger bursts and higher-energy detections have tightened the linear bound. The brightest gamma-ray burst ever recorded, GRB 221009A in 2022, was seen by ground-based detectors at energies above ten TeV, and analyses of it push the linear scale well above the Planck energy. Observations of flaring active galaxies and pulsars at TeV energies provide independent limits with different systematic uncertainties. None reaches the quadratic regime at the Planck scale, and none can with photons: the energies needed are beyond any known source. Proposals to reach it use high-energy neutrinos from distant sources, whose energies reach the PeV range, or the absence of reactions that a modified dispersion would allow or forbid in cosmic rays. Whether Lorentz symmetry holds exactly at the Planck scale, fails at second order, or is deformed rather than broken, is not known, and the absence of any signal so far is consistent with all three.

The habit worth carrying away is to look for effects that accumulate. A correction suppressed by an enormous scale can still be measured if something multiplies it by an enormous length, and a photon crossing half the universe is such a multiplier. It turned a speed difference of three parts in $10^{18}$ into a second. The Planck scale remains out of reach of any accelerator, but one of the most natural guesses about what happens there has already been tested and found wanting, by a single photon that arrived on time.