“How are matter and energy distributed?” asked Peter Schweitzer, a theoretical physicist at the University of Connecticut. “We don’t know.”

Schweitzer has spent most of his career thinking about the gravitational side of the proton. Specifically, he’s interested in a matrix of properties of the proton called the energy-momentum tensor. “The energy-momentum tensor knows everything there is to be known about the particle,” he said.

In Albert Einstein’s theory of general relativity, which casts gravitational attraction as objects following curves in space-time, the energy-momentum tensor tells space-time how to bend. It describes, for instance, the arrangement of energy (or, equivalently, mass) — the source of the lion’s share of space-time twisting. It also tracks information about how momentum is distributed, as well as where there will be compression or expansion, which can also lightly curve space-time.

If we could learn the shape of space-time surrounding a proton, Russian and American physicists independently worked out in the 1960s, we could infer all the properties indexed in its energy-momentum tensor. Those include the proton’s mass and spin, which are already known, along with the arrangement of the proton’s pressures and forces, a collective property physicists refer to as the “Druck term,” after the word for pressure in German. This term is “as important as mass and spin, and nobody knows what it is,” Schweitzer said — though that’s starting to change.

In the ’60s, it seemed as if measuring the energy-momentum tensor and calculating the Druck term would require a gravitational version of the usual scattering experiment: You fire a massive particle at a proton and let the two exchange a graviton — the hypothetical particle that makes up gravitational waves — rather than a photon. But due to the extreme weakness of gravity, physicists expect graviton scattering to occur 39 orders of magnitude more rarely than photon scattering. Experiments can’t possibly detect such a weak effect.

“I remember reading about this when I was a student,” said Volker Burkert, a member of the Jefferson Lab team. The takeaway was that “we probably will never be able to learn anything about mechanical properties of particles.”

Gravity Without Gravity

Gravitational experiments are still unimaginable today. But research in the late 1990s and early 2000s by the physicists Xiangdong Ji and, working separately, the late Maxim Polyakov revealed a workaround.

The general scheme is the following. When you fire an electron lightly at a proton, it usually delivers a photon to one of the quarks and glances off. But in fewer than one in a billion events, something special happens. The incoming electron sends in a photon. A quark absorbs it and then emits another photon a heartbeat later. The key difference is that this rare event involves two photons instead of one — both incoming and outgoing photons. Ji’s and Polyakov’s calculations showed that if experimentalists could collect the resulting electron, proton and photon, they could infer from the energies and momentums of these particles what happened with the two photons. And that two-photon experiment would be essentially as informative as the impossible graviton-scattering experiment.

How could two photons know anything about gravity? The answer involves gnarly mathematics. But physicists offer two ways of thinking about why the trick works.

Photons are ripples in the electromagnetic field, which can be described by a single arrow, or vector, at each location in space indicating the field’s value and direction. Gravitons would be ripples in the geometry of space-time, a more complicated field represented by a combination of two vectors at every point. Capturing a graviton would give physicists two vectors of information. Short of that, two photons can stand in for a graviton, since they also collectively carry two vectors of information.

An alternative interpretation of the math goes as follows. During the moment that elapses between when a quark absorbs the first photon and when it emits the second, the quark follows a path through space. By probing this path, we can learn about properties like the pressures and forces that surround the path.

“We are not doing a gravitational experiment,” Lorcé said. But “we should obtain indirect access to how a proton should interact with a graviton.” 

Probing Planet Proton

The Jefferson Lab physicists scraped together a few two-photon scattering events in 2000. That proof of concept motivated them to build a new experiment, and in 2007, they smashed electrons into protons enough times to amass roughly 500,000 graviton-mimicking collisions. Analyzing the experimental data took another decade.

From their index of space-time-bending properties, the team extracted the elusive Druck term, publishing their estimate of the proton’s internal pressures in Nature in 2018.

They found that in the heart of the proton, the strong force generates pressures of unimaginable intensity — 100 billion trillion trillion pascals, or about 10 times the pressure at the heart of a neutron star. Farther out from the center, the pressure falls and eventually turns inward, as it must for the proton not to blow itself apart. “This comes out of the experiment,” Burkert said. “Yes, a proton is actually stable.” (This finding has no bearing on whether protons decay, however, which involves a different type of instability predicted by some speculative theories.)