Periodic Reporting for period 1 - SINGinGR (Singularities in General Relativity)
Reporting period: 2023-04-01 to 2025-09-30
General Relativity (GR), Einstein’s theory of gravity, predicts that under certain conditions the evolution of spacetime inevitably breaks down in finite time, giving rise to singularities—regions where curvature becomes unbounded and the classical description of physics ceases to apply. Such singularities appear at the Big Bang, inside black holes, and in many cosmological and collapse models. The Hawking–Penrose incompleteness theorems ensure that singularities arise generically, but they provide almost no information about their structure, stability, or dynamical formation. As a result, some fundamental questions remain unresolved:
– How do singularities actually form?
– What are their precise geometric and analytic properties?
– Are they stable under perturbations, or do small changes lead to drastically different behavior?
– Does determinism hold in GR, as conjectured by the Strong Cosmic Censorship hypothesis?
Over the last decades, partial progress has been made—e.g. on symmetric or homogeneous models, or in special matter models—but the full understanding of singularity formation in generic, non-symmetric solutions remains one of the most challenging open problems in mathematical physics. Recent breakthroughs have illuminated particular regimes (e.g. subcritical Big Bang formation, Cauchy horizon stability), revealing that singular behaviors are far richer and more subtle than previously expected. Yet core scenarios predicted by physics, such as the oscillatory BKL (Mixmaster) singularity, remain completely unverified in the general vacuum case.
This project is motivated by the need to cross this frontier by developing new mathematical techniques—analytic, geometric, and PDE-based—to attack these problems in settings far closer to the fully general Einstein equations.
Overall Objectives
The project addresses two fundamental singularity scenarios:
Big Bang singularities in vacuum
Construct the first non-symmetric solutions exhibiting oscillatory or spiky behavior, thereby testing the BKL conjecture beyond homogeneous or symmetric models.
Develop a general method to construct singularities with discontinuous asymptotic profiles (spikes).
Identify and dynamically characterize the single functional degree of freedom responsible for instabilities, proving codimension stability of Kasner-like singularities and delimiting the stable manifold within vacuum dynamics.
Spacelike singularities inside black holes
Show that spacelike singularities arise generically in one-ended black holes beyond spherical symmetry for realistic matter models (massless scalar field, Oppenheimer–Snyder dust).
Establish stable blow-up near such singularities and understand how null and spacelike singular components coexist in dynamical interiors.
Expected Impact
The project is expected to reshape our understanding of singularity formation in GR and address long-standing conjectures in the field:
The first evidence of oscillatory Big Bang dynamics without symmetry would provide a crucial step toward validating or refining the BKL conjecture.
A general construction of spike singularities could force a reformulation of the notion of “generic” singularity and the expected locality of asymptotic dynamics.
Codimension stability results would clarify which singularities are dynamically relevant and provide the first rigorous identification of the mechanism driving oscillatory instabilities.
New results on black hole interiors would show that spacelike singularities are not artifacts of symmetry but persist under perturbations—advancing our understanding of Strong Cosmic Censorship and the global structure of black holes.
Overall, the project aims to deliver foundational advances in the mathematical analysis of the Einstein equations, significantly deepening our comprehension of the most extreme and fundamental phenomena predicted by General Relativity.
– How do singularities actually form?
– What are their precise geometric and analytic properties?
– Are they stable under perturbations, or do small changes lead to drastically different behavior?
– Does determinism hold in GR, as conjectured by the Strong Cosmic Censorship hypothesis?
Over the last decades, partial progress has been made—e.g. on symmetric or homogeneous models, or in special matter models—but the full understanding of singularity formation in generic, non-symmetric solutions remains one of the most challenging open problems in mathematical physics. Recent breakthroughs have illuminated particular regimes (e.g. subcritical Big Bang formation, Cauchy horizon stability), revealing that singular behaviors are far richer and more subtle than previously expected. Yet core scenarios predicted by physics, such as the oscillatory BKL (Mixmaster) singularity, remain completely unverified in the general vacuum case.
This project is motivated by the need to cross this frontier by developing new mathematical techniques—analytic, geometric, and PDE-based—to attack these problems in settings far closer to the fully general Einstein equations.
Overall Objectives
The project addresses two fundamental singularity scenarios:
Big Bang singularities in vacuum
Construct the first non-symmetric solutions exhibiting oscillatory or spiky behavior, thereby testing the BKL conjecture beyond homogeneous or symmetric models.
Develop a general method to construct singularities with discontinuous asymptotic profiles (spikes).
Identify and dynamically characterize the single functional degree of freedom responsible for instabilities, proving codimension stability of Kasner-like singularities and delimiting the stable manifold within vacuum dynamics.
Spacelike singularities inside black holes
Show that spacelike singularities arise generically in one-ended black holes beyond spherical symmetry for realistic matter models (massless scalar field, Oppenheimer–Snyder dust).
Establish stable blow-up near such singularities and understand how null and spacelike singular components coexist in dynamical interiors.
Expected Impact
The project is expected to reshape our understanding of singularity formation in GR and address long-standing conjectures in the field:
The first evidence of oscillatory Big Bang dynamics without symmetry would provide a crucial step toward validating or refining the BKL conjecture.
A general construction of spike singularities could force a reformulation of the notion of “generic” singularity and the expected locality of asymptotic dynamics.
Codimension stability results would clarify which singularities are dynamically relevant and provide the first rigorous identification of the mechanism driving oscillatory instabilities.
New results on black hole interiors would show that spacelike singularities are not artifacts of symmetry but persist under perturbations—advancing our understanding of Strong Cosmic Censorship and the global structure of black holes.
Overall, the project aims to deliver foundational advances in the mathematical analysis of the Einstein equations, significantly deepening our comprehension of the most extreme and fundamental phenomena predicted by General Relativity.
Work toward "codimension stability of Kasner singularities" focused on resolving a key obstacle previously identified in the PI’s collaborative work. By uncovering a hidden cancellation structure in the Einstein–scalar field system, the team established approximate monotonicity at the top derivative level, enabling optimal energy estimates. This advancement removes a central technical barrier to proving codimension stability in the unstable vacuum regime.
For "spacelike singularities in the Einstein–scalar field system", recent results by other researchers completed key steps initially planned in the project. After this development, focus shifted toward related dynamical problems for the Einstein-Euler system.
Work toward "Oppenheimer–Snyder–type black hole singularities" led to a deeper investigation of fluid behavior near singularities. A previously unrecognized instability mechanism for tilted perfect fluids was identified—arising when the fluid velocity becomes asymptotically null (extreme tilt). This analysis produced two stability results, establishing the future stability of perfect fluids with extreme tilt in cosmological spacetimes with an accelerated expansion.
The techniques developed there were subsequently adapted to the Einstein vacuum equations with positive cosmological constant. This yielded a major achievement: the stability of the expanding region of Kerr–de Sitter spacetimes, settling a longstanding open problem.
Finally, a new framework for constructing localized Kasner-like singularities was developed, enabling the creation of singular solutions concentrated in spatially small regions. This provides a versatile tool for future investigations of black hole interiors and singularity localization.
For "spacelike singularities in the Einstein–scalar field system", recent results by other researchers completed key steps initially planned in the project. After this development, focus shifted toward related dynamical problems for the Einstein-Euler system.
Work toward "Oppenheimer–Snyder–type black hole singularities" led to a deeper investigation of fluid behavior near singularities. A previously unrecognized instability mechanism for tilted perfect fluids was identified—arising when the fluid velocity becomes asymptotically null (extreme tilt). This analysis produced two stability results, establishing the future stability of perfect fluids with extreme tilt in cosmological spacetimes with an accelerated expansion.
The techniques developed there were subsequently adapted to the Einstein vacuum equations with positive cosmological constant. This yielded a major achievement: the stability of the expanding region of Kerr–de Sitter spacetimes, settling a longstanding open problem.
Finally, a new framework for constructing localized Kasner-like singularities was developed, enabling the creation of singular solutions concentrated in spatially small regions. This provides a versatile tool for future investigations of black hole interiors and singularity localization.
The project has generated several advances with potential to significantly influence future research in mathematical general relativity. The results on stability of tilted fluids and on Kerr–de Sitter dynamics introduce new analytical frameworks—such as refined energy methods for systems with extreme tilt and novel geometric structures in expanding cosmologies—that can be adapted to a broad class of Einstein–matter models. These methodologies are likely to serve as foundations for future progress on long-standing conjectures, including the BKL picture and the structure of black hole interiors.
The work on approximate monotonicity and optimal top-order control has the potential to reshape the study of unstable Big Bang regimes by enabling the first rigorous attempts at codimension-stability results. Likewise, the construction of localized Kasner-type singularities provides a template for exploring localized gravitational collapse.
Potential impacts include:
• providing rigorous mathematical tools relevant to theoretical cosmology;
• advancing understanding of cosmic censorship and singularity formation;
• enabling interdisciplinary interactions between mathematics and physics through new analytic techniques.
Key needs for further uptake and success involve:
• sustained research effort to extend the new methods to fully nonlinear and less symmetric regimes;
• development of complementary numerical studies to test the new analytical predictions;
• continued support for international collaborations, workshops, and mobility, which are essential for progress in this highly specialized field;
• long-term funding for students and postdoctoral researchers, who are critical for pushing forward a technically demanding project.
The work on approximate monotonicity and optimal top-order control has the potential to reshape the study of unstable Big Bang regimes by enabling the first rigorous attempts at codimension-stability results. Likewise, the construction of localized Kasner-type singularities provides a template for exploring localized gravitational collapse.
Potential impacts include:
• providing rigorous mathematical tools relevant to theoretical cosmology;
• advancing understanding of cosmic censorship and singularity formation;
• enabling interdisciplinary interactions between mathematics and physics through new analytic techniques.
Key needs for further uptake and success involve:
• sustained research effort to extend the new methods to fully nonlinear and less symmetric regimes;
• development of complementary numerical studies to test the new analytical predictions;
• continued support for international collaborations, workshops, and mobility, which are essential for progress in this highly specialized field;
• long-term funding for students and postdoctoral researchers, who are critical for pushing forward a technically demanding project.