Physicists have proposed several possible ways to merge them into a single theory. Ideas such as string theory, loop quantum gravity, canonical quantum gravity, and asymptotically safe gravity all attempt to bridge the gap. Each approach has advantages and limitations. What researchers have lacked so far is a clear observable effect that experiments could measure to determine which theory best reflects how nature actually works. A new study from TU Wien may represent a step toward solving that problem.

Searching for the “Slipper” of Quantum Gravity

“It’s a bit like the Cinderella fairy tale,” says Benjamin Koch from the Institute for Theoretical Physics at TU Wien. “There are several candidates, but only one of them can be the princess we are looking for. Only when the prince finds the slipper can he identify the real Cinderella. In quantum gravity, we have unfortunately not yet found such a slipper — an observable that clearly tells us which theory is the right one.”

To identify the right “shoe size,” meaning a measurable way to test different theories, the researchers focused on a central concept in relativity called geodesics. “Practically everything we know about general relativity relies on the interpretation of geodesics,” explains Benjamin Koch.

A geodesic describes the shortest path between two points. On a flat surface, that path is simply a straight line. On curved surfaces, the situation becomes more complicated. For instance, traveling from the North Pole to the South Pole along Earth’s surface follows a semicircle, which represents the shortest possible route on a sphere.

Einstein’s theory connects space and time into a single four dimensional structure called spacetime. Massive objects such as stars and planets curve this spacetime. According to general relativity, the Earth circles the Sun because the Sun’s mass bends spacetime and shapes the path the Earth follows into an orbit.

Creating a Quantum Version of Spacetime Paths

The exact shape of these paths depends on something called the metric, which measures how strongly spacetime is curved. “We can now try to apply the rules of quantum physics to this metric,” says Benjamin Koch. “In quantum physics, particles have neither a precisely defined position nor a precisely defined momentum. Instead, both are described by probability distributions. The more precisely you know one of them, the more fuzzy and uncertain the other becomes.”

Quantum theory replaces precise particle properties with mathematical objects known as wave functions. In a similar way, physicists can attempt to replace the classical metric of relativity with a quantum version. If this happens, spacetime curvature is no longer perfectly defined at every point. Instead, it becomes subject to quantum uncertainty.

This idea creates extremely difficult mathematical problems.

Benjamin Koch, working with his PhD student Ali Riahinia and Angel Rincón (Czech Republic), managed to quantize the metric using a new method for a specific but important case: a spherically symmetric gravitational field that remains constant over time.

Such a model can describe systems like the gravitational field of the Sun. The researchers then calculated how a small object would move in this field when the metric itself is treated as a quantum quantity.

“Next, we wanted to calculate how a small object behaves in this gravitational field — but using the quantum version of this metric,” says Koch. “In doing so, we realized that one has to be very careful — for instance, whether one is allowed to replace the metric operator by its expectation value, a kind of quantum average of the spacetime curvature. We were able to answer this question mathematically.”

The team derived a new equation called the q-desic equation, named in reference to classical geodesics. “This equation shows that in a quantum spacetime, particles do not always move exactly along the shortest path between two points, as the classical geodesic equation would predict.” By examining how freely moving objects travel through spacetime (such as an apple falling toward Earth in outer space), scientists could potentially detect quantum features of spacetime itself.

Tiny Differences and Cosmic Scale Effects

How different are these quantum paths from the ones predicted by classical relativity? If researchers consider only ordinary gravity, the difference is extremely small. “In this case, we end up with deviations of only about 10^-35 meters — far too small to ever be observed in any experiment,” says Benjamin Koch.

However, Einstein’s equations also include another factor known as the cosmological constant, often associated with “dark energy.” This component is responsible for the accelerating expansion of the universe on the largest scales. When the researchers incorporated the cosmological constant into their q-desic equation, the results changed dramatically.

“And when we did that, we were in for a surprise,” reports Benjamin Koch. “The q-desics now differ significantly from the geodesics one would obtain in the usual way without quantum physics.”

The predicted deviations appear both at extremely small distances and at very large cosmic scales. The small scale differences are likely impossible to measure. But at distances around 10²¹ meters, the effects could become substantial.

“In between, for example when it comes to the Earth’s orbit around the Sun, there is practically no difference. But on very large cosmological scales — precisely where major puzzles of general relativity remain unsolved — there is a clear difference between the particle trajectories predicted by the q-desic equation and those obtained from unquantized general relativity,” says Benjamin Koch.

A Potential Way to Test Quantum Gravity

The research, published in the journal Physical Review D, introduces a new mathematical framework for connecting quantum theory and gravity. More importantly, it may offer a path toward comparing theoretical predictions with real observations.

“At first I would not have expected quantum corrections on large scales to produce such dramatic changes,” says Benjamin Koch. “We now need to analyze this in more detail, of course, but it gives us hope that by further developing this approach we can gain a new, and observationally well testable, insight into important cosmic phenomena — such as the still unsolved puzzle of the rotation speeds of spiral galaxies.”

Returning to the Cinderella analogy, physicists may finally have identified a measurable clue that can help distinguish between competing theories of quantum gravity. The slipper may have been found. The next step is determining which theory it truly fits.