George Ellis Challenges Penrose’s Use of Infinity in Conformal Cyclic Cosmology

George Ellis uses a focused critique of Roger Penrose’s conformal cyclic cosmology to highlight what he sees as a widespread misunderstanding of infinity in theoretical physics and cosmology. While Ellis expresses deep admiration for Penrose’s creativity and credits Penrose’s work on gravitation and black holes with helping to ground the classic text “The Large Scale Structure of Space-Time,” he argues that some of Penrose’s later proposals rest on untenable assumptions about quantum mechanics, consciousness, and the mathematical status of infinity.

Ellis agrees with Penrose on one major cosmological issue: the entropy problem at the beginning of the universe. He endorses Penrose’s view that inflation does not genuinely resolve the low-entropy, high-smoothness initial conditions, because inflation assumes an already smooth state before it begins. In his account, inflation shifts rather than solves the fine-tuning question. However, this agreement does not extend to Penrose’s conformal cyclic cosmology. Ellis describes the idea of successive eons connected by a conformal boundary as brilliant in concept and comparable in ambition to Stephen Hawking’s no-boundary proposal, but he argues that conformal cyclic cosmology lacks a concrete, physically defensible mechanism for transforming one eon into the next.

The central point of Ellis’s challenge is Penrose’s use of infinity in conformal diagrams. Drawing on his own paper “The Physics of Infinity,” Ellis insists that infinity is not a very large number, but something categorically beyond any number that can exist in a physical system. He illustrates this with the age of the universe: as cosmic time advances from 13.7 billion years to larger values, it grows without bound yet never becomes infinitely old. No matter how long the universe expands, it does not even reach the “first step” toward infinity, because infinity is not a destination achievable by finite progression.

Applied to conformal cyclic cosmology, this distinction leads Ellis to a sharp objection. If the conformal boundary between eons is truly at infinity, then signals or events originating in one eon must traverse an infinite separation. In that case, he argues, those signals are diluted by an infinite factor, reducing any finite amount of information transferred from one eon to the next to exactly zero. On his reading, a real infinity in the geometry implies that nothing finite can bridge the gap, so Penrose’s envisaged transmission of structure or information between eons does not work. For Ellis, this is not a minor technical issue but a fundamental conundrum: taking the symbol for infinity seriously blocks the very eon-to-eon transition that conformal cyclic cosmology requires.

Ellis extends this insistence on mathematical precision to other areas where infinity and asymptotic structures play a role. Asked about the physical relevance of AdS/CFT and related holographic dualities, he notes that anti-de Sitter space has a negative cosmological constant, whereas observations indicate that dark energy corresponds to a positive cosmological constant. On that basis, he concludes that AdS/CFT, however powerful as a theoretical laboratory, does not describe the actual universe. For Ellis, importing such dualities into cosmology without matching the sign of the cosmological constant misconstrues their domain of applicability.

In place of AdS/CFT, Ellis points to what he calls real holography grounded in general relativity. He recalls the null initial value problem developed in the 1970s by Ray Sachs and others, in which data specified on a three-dimensional null cone determines the four-dimensional spacetime interior. This result, he stresses, is a rigorous mathematical theorem: information on a three-dimensional boundary suffices to fix the physics within the associated four-dimensional region. Ellis regards this as a genuine holographic principle in classical general relativity, equally applicable to electromagnetism in curved spacetime, and independent of the anti-de Sitter constructions that dominate contemporary discussions.

Ellis’s broader critique is that when physicists treat infinity as a convenient stand-in for a very large number, they risk building physical theories—such as particular implementations of conformal cyclic cosmology or extrapolations of AdS/CFT—that depend on limits the universe can never actually reach. By insisting that infinity is strictly beyond any realizable physical quantity, and that our universe with positive dark energy expands forever without attaining an infinite age or size, he presses for cosmological models and holographic frameworks that respect this mathematical boundary. In doing so, Ellis separates his respect for Penrose’s foundational contributions from a clear rejection of specific proposals where, in his view, the physics of infinity has been misapplied.