February 21, 2023

Is "Dark energy" just contorted space-time?

A mathematically inspired extension of Einstein's general theory of relativity provides new insights into space and time. In their recent publications, David Vasak et al. of Frankfurt Institute for Advanced Studies (FIAS) describe how the torsion of spacetime can explain the dark energy of standard cosmology.

The archetypal model for the physics of space, time and matter is the General Theory of Relativity, which Einstein postulated more than a hundred years ago. His field equation describes gravity as the curvature of space-time, which in turn is caused by the presence of matter. It looks simple at first glance, but houses dozens of coupled non-linear partial differential equations - a mathematical nightmare. Nevertheless, it is still the measure of all things for most astrophysicists and cosmologists, and dominant in the perception of the media and the interested public.

But the Theory of Relativity encounters its limits when one wants to explain the observed large structures in space. These are, for example, the rotation of galaxies and galaxy clusters and, last but not least, the explosive dynamics of the universe as a whole. The "bare" Einsteinian approach is no longer sufficient. The model had to be improved by adding fictitious "dark matter" and by the mysterious "dark energy". What these invisible, impalpable "dark" components of our world are, however, has not yet been conclusively explained despite enormous efforts. In particular, the "cosmological constant problem" is considered the greatest discrepancy in theoretical physics to date: The measured value of the cosmological constant - a quantity introduced by Einstein to explain “dark energy” - differs from the value calculated from the field theory of the elementary particles by 120 orders of magnitude!

The research group Canonical Theory of Gravitation at FIAS follows a different path: deviating from Einstein's intuitive approach and also beyond arbitrary modifications of physical theories, the group led by FIAS Fellow Jürgen Struckmeier consistently applies the stringent language of mathematics to derive a new, mathematically consistent and physically well-founded theory of gravitation. 

In line with Einstein, two of his fundamental principles are first adopted, the principle of general relativity, and the equivalence principle. The former states that the dynamics of a physical system is independent of who observes it. For example, the location of a cup standing on the table is independent of whether an observer is standing to its right or to its left and describes its location in his or her "coordinate system". Coordinates are relative, even if the observer is moving along an arbitrary, possibly even curved, path. The equivalence principle, on the other hand, is the result of Einstein's thought experiment: an observer in a windowless lift cannot determine whether the lift is being accelerated upwards in weightless space with the Earth's acceleration "g" or whether the lift is at rest in the Earth's gravitational field. In both cases, the observer is pressed to the bottom of the lift with the same force. Gravity and acceleration are locally indistinguishable. In addition to these physical postulates, there are further assumptions that are required for the mathematical description of physical systems in accordance with the phenomenological characteristics we know.

With these premises, it is now possible to apply the "Canonical Transformation Theory" to enforce physically motivated symmetries of the systems to be described. The novelty of the approach developed in FIAS is that it generalises this initially simple methodology to local symmetries in relativistic field theories. Amazingly it can be used to deliver the entire standard model of elementary particles developed by Chen Yang and Robert Mills, and to formulate the three basic forces of nature, electromagnetic, weak and strong interaction, as so called gauge fields.

For the group, however, showing this was only the “proof of concept” on their way to apply it to the fourth fundamental force of nature, gravity. The "connection coefficients" were identified as gauge fields and thus as carriers of the gravitational interaction. These quantify the deviation of the path of a particle in curved geometry from a straight line. This also clearly determins the coupling of matter and gravity.

From this gravitation model of the FIAS group, Covariant Canonical Gauge Gravity (CCGG), a new dynamics of spacetime emerges. Spacetime becomes an inertial elastic medium that, contrary to Einstein's formulation, does not immediately follow the distribution of matter. Moreover, torsion is allowed as an additional degree of freedom of the four-dimensional space-time geometry. For a simple image of torsion one might imagine a balloon with its envelope representing space-time reduced to two dimensions. Pressing now a match with a sticky tip firmly onto the balloon skin and twisting it contorts the skin locally. When you let go, the balloon surface springs back to its original smooth state by itself. Obviously, energy was stored and released by the twisting and the surface tension of the balloon.

Furthermore - contrary to Einstein's conviction - the CCGG formalism shows that that the total energy of the universe must be zero. The energy densities of the medium spacetime and the matter distributed therein must cancel out, similarly to how deformations of a solid body cause a back-reaction of its crystal structure. The universe can therefore have arisen from nothing, possibly in the form of a "big bang", whereby the (positive) energy of matter must always be compensated by the (negative) energy of space-time. This explains why the previous calculation of the cosmological constants is incomplete. Spacetime itself has a vacuum energy that nullifies the vacuum energy of matter because of the compensation mechanism. The small measured cosmological constant is just a residual value that can largely be explained by the torsion of space-time. Detailed calculations at FIAS show that the effects of torsion can explain the dark energy of standard cosmology, which provides the repulsive force needed for the currently observed accelerated expansion of the universe. 

The group has begun further numerical analyses of the parameters of the FIAS theory. For a broader analysis of the available data, they are involving the network of specialists from Goethe University and FIAS to deploy innovative methods such as artificial intelligence (deep learning). In this way, the team wants to meet the surprises they expect from the James Webb Space Telescope in the future - and thus get closer to unrevelling the mystery of space-time.


  • David Vasak, Johannes Kirsch, Jürgen Struckmeier, Horst Stöcker: On the cosmological constant in the deformed Einstein-Cartan gauge gravity in De Donder-Weyl Hamiltonian formulation, Astron. Nachr., DOI:10.1002/asna.20220069 (2022)
  • Armin van de Venn, David Vasak, Johannes Kirsch, Jürgen Struckmeier: Torsional dark energy in quadratic gauge gravity, submitted to Eur. Phys. J. (2022), arXiv:2211.11868 [gr-qc].

FIAS scientists give new insights into space and time: Johannes Kirsch, Jürgen Struckmeier, Armin van de Venn, David Vasak, Vladimir Denk (from left to right).

Images for use in the context of this press release.