[Paper Review] Quantum gravity without space-time singularities or horizons

Gerard ’t Hooft|ArXiv.org|Sep 18, 2009

Noncommutative and Quantum Gravity Theories14 references13 citations

TL;DR

This paper proposes a quantum gravity framework that preserves space-time, causality, and locality by enforcing exact scale invariance through a conformal factor $ω(x)$ in the metric, eliminating singularities and horizons. By treating the scale function as locally unobservable and using black hole complementarity, the theory avoids emergent space-time, ensuring consistent dynamics across observers while maintaining conformal invariance in field theories.

ABSTRACT

In an attempt to re-establish space-time as an essential frame for formulating quantum gravity - rather than an "emergent" one -, we find that exact invariance under scale transformations is an essential new ingredient for such a theory. Use is made of the principle of "black hole complementarity", the notion that observers entering a black hole describe its dynamics in a way that appears to be fundamentally different from the description by an outside observer. These differences can be boiled down to conformal transformations. If we add these to our set of symmetry transformations, black holes, space-time singularities, and horizons disappear, while causality and locality may survive as important principles for quantum gravity.

Motivation & Objective

• To re-establish space-time as fundamental in quantum gravity, rather than emergent.
• To resolve the conflict between black hole singularities, horizons, and the principles of causality and locality.
• To introduce exact scale invariance as a new symmetry to eliminate singularities and horizons in quantum gravity.
• To maintain causality and locality by treating the conformal factor $\omega(x)$ as unobservable locally, while preserving the light-cone structure via $\hat{g}_{\mu\nu}$.
• To explore how information flow and Planck-scale structure may emerge from conformal invariance and quantum information principles.

Proposed method

• Decompose the metric as $g_{\mu\nu}(x) = \omega^2(x) \hat{g}_{\mu\nu}(x)$, with $\det(\hat{g}_{\mu\nu}) = -1$, separating geometry from scale.
• Use black hole complementarity to relate the descriptions of external and infalling observers, showing that their differing views arise from conformal transformations.
• Apply large Lorentz boosts to the local frame of an infalling observer, showing that they induce gravitational corrections in $\omega(x)$, not in $\hat{g}_{\mu\nu}$.
• Treat $\omega(x)$ as unobservable locally, so that observers in free fall do not perceive scale anomalies, while distant observers see $\omega(x)$ as encoding black hole evaporation.
• Use conformally invariant field theories to describe matter dynamics, ensuring consistency under scale transformations.
• Argue that the Weyl tensor, which is invariant under $\omega$-rescalings, may play a central role in the gravitational equations, replacing non-scale-invariant terms like Ricci curvature.

Experimental results

Research questions

• RQ1 How can space-time singularities and horizons be eliminated in a quantum gravity theory without discarding space-time entirely?
• RQ2 Can causality and locality be preserved in quantum gravity if space-time is not emergent?
• RQ3 What role does exact scale invariance play in resolving the black hole information paradox and avoiding singularities?
• RQ4 How do conformal transformations relate the descriptions of external and infalling observers in black hole physics?
• RQ5 Can the Planck scale and physical scales emerge from a fundamental theory that treats $\omega(x)$ as unobservable?

Key findings

• The metric decomposition $g_{\mu\nu} = \omega^2 \hat{g}_{\mu\nu}$ separates causal structure ($\hat{g}_{\mu\nu}$) from scale ($\omega$), allowing scale invariance to eliminate singularities and horizons.
• Black hole complementarity implies that the descriptions of external and infalling observers differ only by conformal transformations, which preserve light cones but not scales.
• Large Lorentz boosts in the local frame of an infalling observer generate gravitational corrections only in $\omega(x)$, not in $\hat{g}_{\mu\nu}$, preserving the conformal structure.
• The conformal factor $\omega(x)$ is unobservable locally, so no causality violation occurs for infalling observers, even when $\omega$ is used by distant observers to describe evaporation.
• The theory avoids emergent space-time by keeping causality and locality intact, with $\hat{g}_{\mu\nu}$ defining light cones and $\omega(x)$ being dynamically irrelevant locally.
• Information flow along light-like geodesics may define the Planck length and thus all scales, suggesting a deep link between quantum information and the emergence of physical scales.