In general relativity, matter tells space how to curve, determining a component of the curvature tensor. But matter does not tell every aspect of space how to curve, leaving some degrees of freedom that are incorporated into the so-called Weyl tensor.

Roger Penrose’s Weyl Curvature Hypothesis states that the Weyl curvature is zero for past singularities and can be arbitrary for future singularities. At the Big Bang, this implies a smooth, small Weyl tensor. In the scenario of the Big Crunch, the Weyl tensor grows and is eventually diverging, or infinite. Because of this growth, the Weyl tensor is tentatively associated with the gravitational component of entropy, which increases with time.

However, Penrose’s hypothesis is entirely classical. Kiefer extended it to the quantum regime.

“I tried to look for a deeper formulation of the Weyl tensor hypothesis in the language of quantum gravity because I think that’s the more fundamental language,” said author Claus Kiefer.

The problem was challenging to formulate in terms of quantum wave functions. A zero Weyl tensor, while acceptable in the classical picture, is impossible when considering the uncertainty principle. Kiefer addressed this using an oscillator ground state wave function to describe the fluctuations about zero.

Using a quantum geometrodynamics approach, he started from a low entropy state, which corresponded with low quantum entanglement. With time, the universe automatically becomes quantum entangled and there is a state of increasing entropy.

Kiefer would like to see this quantum conjecture applied to various theories of quantum gravity to see if increasing entropy corresponds with an increasing universe size.

Source: “On a quantum weyl curvature hypothesis,” by Claus Kiefer, AVS Quantum Science (2022). The article can be accessed at

This paper is part of the Celebrating Sir Roger Penrose’s Nobel Prize Collection, learn more here.