Conformal Boundary Structure and Cyclic Extension of Spacetime: A Proposal for a Multi-Epoch Cosmology

Conformal Boundary Structure and Cyclic Extension of Spacetime: A Proposal for a Multi-Epoch Cosmology Cameron Brogan-Higgins Independent Researcher; University of London (BSc), Oxford Department for Continuing Education (CertHE) Contact: [email protected] Abstract This paper proposes a generalised cosmological conjecture in which successive universes are structured via conformal boundary matching, extending the framework of Penrose’s Conformal Cyclic Cosmology (CCC). Rather than assuming asymptotic future infinity as the launch point for a subsequent aeon, this model introduces a tiled structure of conformally compactified spacetime regions—each bounded by lightlike and spacelike conformal boundaries—stitched together in a higher-order topological fabric. We explore the theoretical implications of such a structure on information propagation, entropy scaling, and the global arrow of time. The work builds on existing conformal techniques in general relativity while incorporating heuristic arguments motivated by causal set theory and holographic encoding across null boundaries. Introduction The concept of a conformal boundary has played a central role in the understanding of asymptotic structure in general relativity. By compactifying spacetime using a conformal factor Omega, it becomes possible to represent infinite distances and durations as finite structures, revealing the global causal architecture of the universe. This procedure forms the foundation of Penrose diagrams, which represent entire spacetimes in bounded domains. In Conformal Cyclic Cosmology (CCC), Penrose proposes that the remote future of one universe—the exponentially stretched tail of de Sitter expansion—can be conformally rescaled to become the Big Bang of the next. This model intriguingly links thermodynamic asymmetry with geometrical structure, particularly the vanishing of the Weyl tensor at the bounce. This paper proposes a variation on this idea: instead of mapping one infinite future to one Big Bang, we consider an array of conformally bounded epochs—each finite but stitched together via null and spacelike conformal boundaries. Each 'tile' represents a single, compactified universe; the global structure is a quilt of causally distinct but conformally continuous regions. Theoretical Background Conformal Boundaries A conformal boundary is the mathematical boundary obtained by applying a conformal transformation to a Lorentzian manifold (M, g_ab) such that: g~(ab) = (Omega)^2 * g(ab), where Omega approaches 0 at the boundary. This boundary includes: I+ (future null infinity), I- (past null infinity), i^0 (spatial infinity), and i^+/i^- (future/past timelike infinities). These points and surfaces encode the causal behaviour of light and matter as they approach or recede from the manifold. CCC and the Matching Problem Penrose's CCC suggests that as conformal time tau approaches infinity, the rescaled metric yields a smooth conformal boundary that can be analytically continued into the Big Bang of the next aeon. However, such a mapping relies on deep assumptions about entropy dissipation, matter decay, and the asymptotic smoothness of spacetime. These assumptions, while elegant, are also restrictive. Proposal: Tiled Conformal Epochs We propose a discretised, non-linear extension to CCC in which multiple aeons are not arranged linearly in conformal time, but instead tiled adjacently—each compactified and bounded by its own conformal surfaces. The boundaries serve as shared interfaces between epochs, allowing null and possibly quantum information to transit while preserving conformal structure. Each compactified spacetime region (U_n) is bounded by a conformal boundary. Causal continuity is enforced at the shared edges. Stated plainly: for all n, the boundary of U_n intersects the boundary of U_(n+1) in a non-empty set. That is, each region connects to its neighbour via a shared interface. This produces a global configuration analogous to: causal set foam (each region is a causal cell), holographic tiles (boundaries encode the state of adjacent regions), and spacetime tessellation (the universe becomes a patchwork of conformal zones). Implications for Entropy and Time Under this framing: Each region may reset local entropy while the global entropy continues to increase. The arrow of time becomes a local property, defined by boundary conditions rather than originating from a single singularity. Time admits cyclicity in conformal space, even if local thermodynamics remains irreversible. Visualisation and Mathematical Extensions The proposed structure can be visualised as a tessellated Penrose diagram, where each triangle or diamond represents a single conformal epoch. The edges of these shapes—null and spacelike boundaries—serve as shared interfaces between epochs. Light rays and causal trajectories are continued across tiles via analytic continuation along null geodesics. This model suggests compatibility with: Geroch-Kronheimer-Penrose causal boundary theory; matching conditions for spinor fields across conformal interfaces; and analytic continuation techniques in quantum field theory and AdS/CFT correspondence. Conclusion This conjecture extends the Penrosian notion of conformal cyclicity into a more general, multidirectional framework in which spacetime is partitioned into bounded, conformally continuous epochs. Each region may evolve under its own thermodynamic constraints while remaining causally entangled with neighbouring tiles. This framework invites formal development through causal diagrammatics, differential geometry, and quantum information theory. It may also yield novel insights into black hole information transfer, relic radiation, and holographic encoding at cosmic boundaries. References Penrose, R. (2010). Cycles of Time: An Extraordinary New View of the Universe. Bodley Head. Hawking, S. W., & Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge University Press. Wald, R. M. (1984). General Relativity. University of Chicago Press. Geroch, R., Kronheimer, E. H., & Penrose, R. (1972). 'Ideal points in space-time.' Proc. R. Soc. Lond. A 327(1571): 545–567. Maldacena, J. (1999). 'The Large N Limit of Superconformal Field Theories and Supergravity.' Adv. Theor. Math. Phys. 2: 231–252.