 |
| Quantum-Geodesics Large masses – such as a galaxy – curve space-time. Objects move along a geodesic. If we take into account that space-time itself has quantum properties, deviations arise (dashed line vs. solid line). Image Credit: © TU Wien |
A team at TU Wien combines quantum physics and general relativity theory – and discovers striking deviations from previous results.
It is something like the “Holy Grail” of physics: unifying particle physics and gravitation. The world of tiny particles is described extremely well by quantum theory, while the world of gravitation is captured by Einstein’s general theory of relativity. But combining the two has not yet worked – the two leading theories of theoretical physics still do not quite fit together.
There are many ideas for such a unification – with names like string theory, loop quantum gravity, canonical quantum gravity or asymptotically safe gravity. Each of them has its strengths and weaknesses. What has been missing so far, however, are observable predictions for measurable quantities and experimental data that could reveal which of these theories describes nature best. A new study from TU Wien may now have brought us a small step closer to this ambitious goal.
Cinderella and 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 determine the correct “shoe size” – in other words, to find measurable criteria for testing different theories – the team took a closer look at the concept of geodesics. “Practically everything we know about general relativity relies on the interpretation of geodesics,” explains Benjamin Koch.
“A geodesic is the shortest connection between two points – on a flat plane that’s simply a straight line, whereas on curved surfaces things become more complicated.” For example, if you want to move from the North Pole to the South Pole on the surface of a sphere, the shortest path is a semicircle.
In relativity theory, space and time are inseparably linked. Together, they form a four-dimensional spacetime, which is curved by masses such as stars or planets. According to general relativity, the Earth orbits the Sun because the Sun’s mass bends space and time, thereby curving the geodesic along which the Earth moves into an approximately circular path.
The quantum version of geodesics
The course of these geodesics is determined by the so-called metric – a measure of 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 fuzzier and more uncertain the other becomes.”
In a similar way to how particle positions and momenta are replaced in quantum physics by a more complicated mathematical object – a quantized wave function – one can now also try to replace the metric of general relativity with a quantized version. In that case, spacetime curvature is no longer exactly defined at every point; it is replaced by a quantum-mechanically fuzzy version of this quantity.
This approach leads to major mathematical challenges.
But together with his PhD student Ali Riahinia and Angel Rincón (Czech Republic), Benjamin Koch has now succeeded in quantizing the metric in a novel way for an important special case – that of a spherically symmetric gravitational field that does not change over time.
Such a field can be used, for example, to describe the gravity of the Sun. “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 result was an equation which the team calls the q-desic equation, in analogy to the classical concept of 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.” This means that by observing the trajectories of freely moving particles in spacetime (such as an apple falling toward Earth in outer space), one can infer the quantum properties of the metric.
Shoe size 10^(-35) or rather 10^(+21)?
So how large are the differences between a q-desic and a classical geodesic? If we consider only ordinary gravitation, the weakest of the known fundamental forces, it turns out that the difference is minimal. “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, general relativity includes another important quantity – the cosmological constant, which is also known as “dark energy”. It is responsible for the accelerated expansion of the universe on the largest scales. This cosmological constant can also be included in the q-desic equation. “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.”
Interestingly, there are deviations both at very small distances and at very large distances. While the deviations at small distances will probably remain unobservable, at length scales of around 10^(21) meters there can be substantial differences: “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 new perspective on observational data
The work, which has been published in the journal Physical Review D, is not only a novel mathematical approach to linking quantum theory and gravitation – above all, it opens up new ways of comparing the theory with 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.”
Or, to return to the Cinderella story: we may finally have identified an observable that allows us to distinguish between viable and incorrect approaches to quantum gravity. A slipper has been found – now we must find out which theory it truly fits.
Published in journal: Physical Review D
Title: Geodesics in quantum gravity
Authors: Benjamin Koch, Ali Riahinia, and Angel Rincon
Source/Credit: Technische Universität Wien
Reference Number: qs120225_01
.jpg)
