General relativity is one of the most precisely confirmed theories in physics. Its predictions for the precession of Mercury’s orbit, the deflection of light around the sun, the existence of gravitational waves, and the behaviour of GPS satellites have all been verified to high precision. The theory has never made a wrong prediction in its domain.
It also predicts its own failure.
At two specific locations in the universe, inside black holes and at the origin of the Big Bang, general relativity’s equations produce results that are not merely inaccurate but mathematically meaningless. The curvature of spacetime becomes infinite. The density of matter becomes infinite. The equations break down entirely. These locations are called singularities, and their existence is not a curiosity or an edge case. It is a theorem.
Understanding singularities is essential for understanding what quantum gravity must accomplish. Any theory that correctly describes gravity at the Planck scale must say something different at singularities from what general relativity says. It must either resolve them by replacing the infinite curvature with something finite, or explain why they are physically acceptable. Singularities are the places where a theory of quantum gravity is most urgently needed, and they are the places where the three approaches examined in Group 2 make their most concrete and different predictions.
What Is a Singularity?
The word singularity is used in mathematics to mean a point where a function breaks down by producing an undefined or infinite result. A singularity in general relativity is a region where the theory’s equations produce meaningless outputs.
The most intuitive picture of a singularity is a point of infinite density and infinite spacetime curvature. This picture is largely correct but technically incomplete. General relativity defines singularities not by infinite curvature but by the property called geodesic incompleteness, which means that some paths through spacetime come to an end at a finite proper time, with no spacetime beyond them to continue into. An observer following such a path would find that their future simply stops existing. Their worldline cannot be extended. There is no point beyond the singularity in the mathematical structure of the spacetime.
This technical definition matters because it separates what general relativity actually says from what popular accounts often claim. General relativity does not say there is a point of infinite density sitting inside a black hole. It says that the spacetime manifold itself is incomplete, meaning there are paths through it that cannot be continued. The infinite curvature that physicists associate with singularities is a consequence of geodesic incompleteness in physically realistic cases, but the incompleteness is the precise mathematical statement.
The distinction has a practical consequence. Asking what happens at a singularity within general relativity is a meaningless question. There is no spacetime at the singularity. The question of what replaces it is therefore not a question general relativity can answer. It requires a different theory.
How Are Singularities Formed?
To understand how singularities arise, it is necessary to understand what prevents them from forming in ordinary circumstances and what happens when that prevention fails.
A star is, for most of its life, a balance between two opposing forces. Gravity pulls every part of the star toward its centre. The outward pressure generated by nuclear fusion in the star’s core pushes back. As long as the star has fuel, this balance holds. When the fuel runs out, the balance breaks. The outward pressure drops, and the gravity wins. What happens next depends on how massive the star is.
For stars of moderate mass, up to about eight times the mass of the sun, the collapsing core reaches a state called a white dwarf. In a white dwarf, the collapse is halted not by nuclear burning but by a quantum mechanical effect called electron degeneracy pressure. Quantum mechanics forbids two electrons from occupying the same quantum state simultaneously. When the core is compressed sufficiently, the electrons resist further compression simply because there is no available quantum state for them to occupy. The star stabilises as a white dwarf, a dense slowly cooling object roughly the size of Earth.
The Helix Nebula is a cloud of gas ejected by a dying star – a white dwarf. In the composite image, the cloud of gas strongly resembles a creature’s eye. Here, a hazy blue cloud is surrounded by misty, concentric rings of pale yellow, rose pink, and blood orange. Each ring appears dusted with flecks of gold, particularly the outer edges of the eye-shape.
Subrahmanyan Chandrasekhar showed in the 1930s that this stabilisation has a limit. If the white dwarf’s mass exceeds approximately 1.4 times the mass of the sun, a threshold now called the Chandrasekhar limit, electron degeneracy pressure cannot hold and the collapse continues.
Beyond the Chandrasekhar limit, the core collapses further until it reaches densities at which electrons and protons are forced together to form neutrons. The result is a neutron star, an object roughly the mass of the sun compressed into a sphere about 10 kilometres in radius, supported against further collapse by neutron degeneracy pressure. This too has a limit. Richard Tolman, Robert Oppenheimer, and George Volkoff calculated in 1939 that if the neutron star’s mass exceeds approximately 2 to 3 times the mass of the sun, a threshold called the Tolman-Oppenheimer-Volkoff limit, neutron degeneracy pressure also fails. At this point, no known force can halt the collapse.
The core continues to contract. Nothing stops it. The matter compresses beyond any stable configuration. The curvature of spacetime around the collapsing core grows without bound. A black hole forms, and inside it, general relativity predicts a singularity, a region where the curvature becomes infinite and the equations of the theory break down entirely.
The Big Bang singularity arises by a complementary argument. Running the observed expansion of the universe backward in time, the universe contracts. All the matter and energy in the observable universe was once concentrated in a smaller and smaller volume. Tracing this contraction to its logical conclusion under general relativity leads inevitably to a moment of infinite density and infinite curvature at a finite time in the past. This is the Big Bang singularity, not an explosion into pre-existing space, but the point at which spacetime itself begins and at which general relativity predicts its own breakdown.
The Singularity Theorems
The question of whether singularities were real physical predictions or mathematical illusions remained open for decades.
The question of a past cosmological singularity had in fact been raised long before the 1960s, in a debate that played out between two of the twentieth century’s most significant physicists. In 1927, Georges Lemaître, a Belgian physicist and Catholic priest who had trained in both theology and mathematics, independently derived what is now called Hubble’s law — the observation that galaxies recede from us at speeds proportional to their distance — two years before Hubble published his own observational confirmation. From this, Lemaître concluded that the universe is expanding. He followed this in 1931 with a more radical proposal.
If the universe is expanding now, then tracing its evolution backward in time leads inevitably to a moment when all the matter and energy of the observable universe was concentrated in a single point of extraordinary density. Lemaître called this initial state the primeval atom, and in some accounts the cosmic egg. It was the first serious scientific proposal for what we now call the Big Bang, and it carried a direct implication: the universe had a beginning, and that beginning was a singularity.
Einstein’s reaction was initially resistant. He did not dispute the mathematics. Lemaître’s solution to Einstein’s own field equations was correct, and Einstein acknowledged this. What he rejected was the physical interpretation. At a conference in Brussels in 1927, Einstein told Lemaître directly that his mathematics was correct but his physics was abominable. Einstein at that time held a deep conviction that the universe was static and eternal, that it had always existed and always would, without beginning or end. To accommodate this belief, he had introduced the cosmological constant into his field equations specifically to counteract the natural tendency of his equations to predict an expanding or contracting universe. A universe that began in a singularity was, to Einstein’s intuition, philosophically unacceptable.
The observational evidence eventually made Einstein’s position untenable. By 1929 Hubble had published systematic measurements showing that galaxies are receding at velocities proportional to their distances, and by 1931 the evidence for an expanding universe had become impossible to dismiss. Einstein visited the Mount Wilson Observatory in California and examined Hubble’s data directly. He subsequently abandoned the cosmological constant, calling it the greatest blunder of his life, a phrase that has since become one of the most quoted in the history of physics. Whether he actually used those words is disputed by historians, but the substance of his reversal is not.
In 1933, Einstein attended a seminar in California at which Lemaître presented his primeval atom hypothesis in full. According to accounts of the event, Einstein rose to his feet when Lemaître finished and applauded, declaring it the most beautiful and satisfying interpretation of creation he had ever heard. The man who had once called Lemaître’s physics abominable now regarded it as among the finest achievements in cosmological thought.
The significance of this exchange for the history of singularities is substantial. What was missing was mathematical proof that singularities form not only in idealised symmetric solutions but generally, under physically realistic conditions, in collapsing stars as much as at the origin of the universe.
Recognising that a past cosmological singularity might exist was not the same as proving that singularities form inside collapsing stars, not just at the origin of the universe, and regardless of how asymmetric or irregular the collapse is. That proof came in 1965.
Roger Penrose showed that singularities form inevitably whenever a trapped surface forms. A trapped surface is a region of spacetime where even outward-directed light rays are pulled back by gravity. Such a surface forms whenever sufficient matter collapses within a small enough region, regardless of how asymmetric or irregular that collapse is. The formation of a singularity inside a black hole is not an artifact of symmetry. It is a general consequence of general relativity and the positive energy of matter.
Stephen Hawking extended Penrose’s methods to cosmology and proved the complementary result: tracing the expansion of the universe backward in time, under the assumption that matter obeys reasonable energy conditions, leads inevitably to a past singularity. The universe did not emerge from some prior state of finite density. General relativity requires that it emerged from a singularity at which the theory itself breaks down.
Together with Robert Geroch, Penrose and Hawking established a family of singularity theorems in the late 1960s that remain among the most important results in mathematical physics.
Under physically reasonable assumptions that matter has positive energy, gravity is attractive, and no exotic energy conditions are violated, general relativity predicts that singularities are unavoidable. They are not edge cases. They are generic predictions of the theory, as common as black holes themselves.
Penrose was awarded the Nobel Prize in Physics in 2020, in part for this work.
The Types of Singularity
There are several types, each arising in different physical contexts and presenting different challenges for quantum gravity.
The point singularity is the simplest case. When a non-rotating, uncharged body collapses to form a black hole, described by the Schwarzschild solution of Einstein’s equations, the resulting singularity is a single zero-dimensional point at the centre of the black hole where the curvature of spacetime becomes infinite.
An observer falling into a Schwarzschild black hole cannot avoid this singularity. Once inside the event horizon, the singularity lies not in a particular direction in space but in the future. It is a moment in time that every infalling path must eventually reach. No manoeuvre can avoid it.
Credit: Andrew Hamilton/JILA/University of Colorado
Credit: Andrew Hamilton/JILA/University of Colorado
As the observer approaches the singularity, the difference in gravitational pull between their head and their feet grows without bound. This tidal stretching in the direction of motion combined with compression from the sides is a process physicists call spaghettification. At the singularity itself, the tidal forces become infinite and the observer is destroyed in finite proper time.
The ring singularity, also called a ringularity, arises in the more physically realistic case of a rotating black hole described by the Kerr solution. All astrophysical black holes are expected to rotate, because the stars and gas clouds from which they form carry angular momentum.
Credit: NASA’s Goddard Space Flight Center/Jeremy Schnittman
In the Kerr solution, the rotation changes the geometry of the singularity fundamentally. A point cannot carry angular momentum because a geometric point has no extent and therefore no rotation. The singularity of a rotating black hole is instead a ring, a one-dimensional circle of zero thickness but non-zero radius, lying in the equatorial plane of the black hole’s rotation. This ring singularity, or ringularity, is geometrically distinct from the point singularity of the Schwarzschild solution and has a definite shape determined by the black hole’s angular momentum.
The ring singularity has a further property that distinguishes it from its non-rotating counterpart. An observer falling into a Kerr black hole is not inevitably directed toward the singularity. The ring geometry means that paths approaching from certain directions can in principle thread through the ring and avoid it. Paths approaching from outside the plane of the ring experience intense tidal forces but are not necessarily destroyed at the singularity. This theoretical avoidability is one of the most striking differences between rotating and non-rotating black holes.
The rotating black hole drags space around with it. Outside the horizon of the black hole is a region called the ergosphere, where space is dragged around so fast that nothing can remain at rest there.
Credit: Andrew Hamilton/JILA/University of Colorado
The Kerr singularity also lies beyond an additional internal boundary called the Cauchy horizon, which is a surface beyond which the future is no longer uniquely determined by initial conditions. An observer who crosses the Cauchy horizon enters a region where general relativity’s equations lose their predictive power even before the singularity is reached. The determinism that makes physics possible breaks down at the Cauchy horizon, not at the singularity itself.
The cosmological singularity, which is the Big Bang, is a spacelike singularity extending across all of space at a moment in time. Unlike the black hole singularities which lie in the future of infalling observers, the Big Bang singularity lies in the past of all observers. It is the boundary from which the universe emerged, and general relativity cannot describe what, if anything, preceded it.
Credit: Nicole Rager Fuller/National Science Foundation
Cosmic Censorship Conjecture
Singularities inside black holes are hidden from external observers by event horizons. An observer outside the black hole cannot receive any signal from the singularity.
But what if a singularity could exist without an event horizon? Such a singularity, visible to distant observers and able to send signals into the external universe, is called a naked singularity.
Roger Penrose proposed in 1969 that naked singularities cannot form from the gravitational collapse of physically reasonable matter.
This is the cosmic conjecture. It states that all singularities formed by gravitational collapse are hidden within event horizons and therefore invisible to distant observers.
After more than half a century, the conjecture remains unproven. It is widely regarded as one of the most important unsolved problems in mathematical physics. Partial results support it in restricted settings.
Numerical simulations have found apparent counterexamples in highly idealised cases, but it remains unclear whether these counter-examples satisfy the physical reasonableness conditions the conjecture requires.
Work published in 2024 has suggested that quantum effects may support cosmic censorship. Calculations incorporating quantum mechanical corrections to black hole physics find that quantum mechanics tends to prevent the formation of naked singularities in cases where classical general relativity would permit them. Whether this quantum support for cosmic censorship is universal or holds only in specific cases is not yet established.
How Do Other Theories Explain Singularity?
Quantum Gravity
Every approach to quantum gravity must address singularities. The question of how they are resolved is one of the most concrete places where different theories make different predictions.
The general expectation, shared across most approaches, is that quantum gravity effects become significant at the Planck scale and prevent the infinite concentration of curvature that classical general relativity predicts. The Planck density, approximately 10⁹⁶ kilograms per cubic metre, is the scale at which quantum gravitational effects are expected to be of order one. No meaningful physical state can be compressed to densities exceeding this scale without quantum gravity becoming relevant.
What quantum gravity does to the singularity at Planck densities is the question on which the three approaches of Group 2 diverge most sharply.
Loop Quantum Gravity
Loop quantum cosmology is the application of loop quantum gravity’s methods to the homogeneous and isotropic universe. It inherits the theory’s central result and applies it to cosmological spacetimes.
In loop quantum cosmology, the Big Bang singularity is replaced by a Big Bounce. Tracing the universe’s evolution backward in time, instead of reaching a singularity of infinite density at a finite time in the past, the universe reaches a maximum density of approximately the Planck density and then bounces. The contracting phase that preceded our expanding universe is a valid, finite spacetime. The Big Bang was not the beginning of time. It was a transition from a prior contracting phase to the current expanding phase.
Credit: NASA, animation created by S. Perquin
This result has been demonstrated numerically across a wide range of cosmological models.
String Theory
String theory’s approach to singularities is fundamentally different. Rather than replacing the singularity with a quantum bounce, string theory uses its mathematical structures, particularly T-duality and the AdS/CFT correspondence, to argue that singularities in the gravitational description correspond to well-defined, non-singular states in the dual description.
T-duality is a symmetry of string theory that relates a string theory compactified on a circle of radius R to a string theory compactified on a circle of radius proportional to 1/R. As the radius of the compact dimension shrinks toward zero, which would classically correspond to a singularity, T-duality maps the theory to one in which the radius is growing. The singularity in one description is a perfectly regular spacetime in the dual description.
The AdS/CFT correspondence offers a complementary perspective. In the correspondence, a black hole in anti-de Sitter space corresponds to a thermal state of the boundary quantum field theory. The boundary theory has no singularity. It is a well-defined quantum mechanical system with finite entropy and unitary evolution. The black hole singularity in the gravitational description corresponds to something perfectly regular in the boundary description. The singularity is a feature of the approximate, semi-classical description rather than of the fundamental physics.
What string theory does not yet provide is a complete, explicit description of what replaces the singular geometry at the Planck scale. The duality arguments suggest that singularities are resolved, but they do not say what they are resolved into.
Asymptotic Safety
Asymptotic safety’s approach to singularities follows from the fixed point discussed in Article VI. Near the Reuter fixed point, the effective Newton’s constant runs with energy scale. At the Planck scale, instead of remaining fixed at its classical value, Newton’s constant is suppressed by the approach to the fixed point. The effective gravitational coupling becomes smaller precisely where classical general relativity predicts it becomes very large.
This running of Newton’s constant modifies the black hole metric at short distances. Rather than an infinitely curved singularity at the centre of a black hole, the asymptotic safety framework predicts a regular core, a region where the metric remains well-defined because the effective gravitational coupling has been suppressed by the renormalisation group flow. The singularity is smoothed out by the approach to scale invariance at high energies.
Whether this resolution is physically correct depends on whether the Reuter fixed point is real, which as Article VI established remains unproven. But the asymptotic safety prediction is internally consistent and produces regular, non-singular black hole metrics in the effective framework.
What Observation Can Say
Singularities inside black holes are, by definition, hidden from direct observation. No signal from inside a black hole can reach an external observer. The physics at the singularity is in principle unobservable from outside.
There are, however, indirect observational windows.
The gravitational wave observations of LIGO and Virgo have detected black hole mergers with extraordinary precision. The ringdown phase of a merger, the period after two black holes have coalesced during which the resulting black hole settles into its final state by emitting gravitational waves, is sensitive to the properties of the black hole’s exterior.
In this animation, two black holes orbit around each other and generate space-time ripples called gravitational waves. As the black holes get closer, the waves increase in frequency. Eventually, the event horizons merge into a peanut-shaped object, generating one very high-frequency wave. Within a rotation, the black holes merge completely. One lower-frequency wave, called the ring down, ripples out after the merger.
Credit: I. Markin (Potsdam University), H. Pfeiffer (Max Planck Institute for Gravitational Physics), T. Dietrich (Potsdam University and Max Planck Institute for Gravitational Physics)
The Event Horizon Telescope’s images of the supermassive black holes M87* and Sagittarius A* provide the sharpest pictures ever obtained of the region near a black hole event horizon. Current observations are consistent with classical general relativity. The precision required to detect quantum gravitational corrections is several orders of magnitude beyond present capabilities.
Credit: ESO
The image of the Sgr A* black hole is an average of the different images the EHT Collaboration has extracted from its 2017 observations.
Credit: EHT Collaboration
None of these observational windows currently has the sensitivity to detect quantum gravitational corrections to singularity physics. The most promising near-term route to indirect evidence is through the black hole information paradox, which is the question of whether information falling into a black hole is destroyed at the singularity or preserved in the outgoing radiation.
Where Singularities Stand
Singularities are not a peripheral feature of general relativity. They are a central prediction, proven to be generic by the Penrose-Hawking singularity theorems and confirmed by the existence of black holes throughout the observable universe. Every black hole contains a singularity. The universe began with one.
General relativity cannot describe what happens at singularities. This is the clearest possible signal that general relativity is incomplete and that a quantum theory of gravity is required.
The three approaches of Group 2 each propose a different resolution. All three resolutions are internally consistent. None has been confirmed by observation.
The cosmic censorship conjecture, which holds that singularities are always hidden from external observers, remains unproven after more than fifty years. A theory that renders singularities physically regular makes the question of their visibility less urgent than classical general relativity suggests.
What lies beyond the singularity, if anything, is a question that the three approaches answer differently and that observation cannot currently test.
Further Readings:
- Penrose, R. (1965). Gravitational collapse and space-time singularities. Physical Review Letters, 14(3), 57–59. https://doi.org/10.1103/PhysRevLett.14.57
- Hawking, S. W., & Penrose, R. (1970). The singularities of gravitational collapse and cosmology. Proceedings of the Royal Society of London A, 314(1519), 529–548.
- Penrose, R. (1969). Gravitational collapse: The role of general relativity. Rivista del Nuovo Cimento, 1, 252–276.
- Chandrasekhar, S. (1931). The maximum mass of ideal white dwarfs. The Astrophysical Journal, 74, 81–82.
- Oppenheimer, J. R., & Volkoff, G. M. (1939). On massive neutron cores. Physical Review, 55(4), 374–381.
- Loop quantum cosmology: Physics of singularity resolution and its implications.
https://arxiv.org/abs/2304.05426 - Planck Stars
https://arxiv.org/abs/1401.6562 - Bojowald, M. (2001). Absence of a singularity in loop quantum cosmology. Physical Review Letters, 86(23), 5227–5230.
- Maldacena, J. (1998). The large N limit of superconformal field theories and supergravity. International Journal of Theoretical Physics, 38(4), 1113–1133.
- Held, A., & Saueressig, F. (2021). Black holes in asymptotic safety: a review. Journal of Physics: Conference Series, 2168, 012038.
- Ong, Y. C. (2020). Spacetime singularities and cosmic censorship conjecture: A review with some thoughts. International Journal of Modern Physics A, 35(14), 2030007. https://doi.org/10.1142/S0217751X20300070
- Chesler, P., Curiel, E., & Narayan, R. (2024). Quantum mechanical censorship of naked singularities. Physical Review D, 110(4), 044040. https://doi.org/10.1103/PhysRevD.110.044040
- Earman, J. (1995). Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes. Oxford University Press.
- https://plato.stanford.edu/entries/quantum-gravity/
This is Article VIII of The Geometry of Reality and the second article of Group 3.
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