general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

, mass inflation is a phenomenon inside

spinning Spin or spinning most often refers to: * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin quantum number, a number which defines the value of a particle's spin * Spinning (textiles), the creation of yarn or thr ...

or charged

black holes A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...

in which the interactions of outgoing and ingoing

radiation In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or a material medium. This includes: * ''electromagnetic radiation'' consisting of photons, such as radio waves, microwaves, infr ...

at the

Cauchy horizon In physics, a Cauchy horizon is a light-like boundary of the domain of validity of a Cauchy problem (a particular boundary value problem of the theory of partial differential equations). One side of the horizon contains closed space-like geodesi ...

cause the internal gravitational

mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...

parameter of the black hole to become unbounded at the Cauchy horizon. It also predicts the existence of a weak null singularity at the Cauchy horizon of a spinning or charged black hole. Mass inflation was confirmed numerically for a spherical charged black hole by Lior Burko in 1997 and for a uncharged rotating black hole by Mihalis Dafermos and Jonathan Luk in 2017.

Divergence of the gravitational mass parameter

In a collapsing star,

gravitational radiation Gravitational waves are oscillations of the gravitational field that travel through space at the speed of light; they are generated by the relative motion of gravitating masses. They were proposed by Oliver Heaviside in 1893 and then later by ...

is emitted, carrying away information about all the initial characteristics of the initial

star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...

other than its

charge Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aqu ...

, and

spin Spin or spinning most often refers to: * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin quantum number, a number which defines the value of a particle's spin * Spinning (textiles), the creation of yarn or thr ...

as per the

no-hair theorem The no-hair theorem states that all stationary black hole solutions of the Einstein–Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three independent ''externally'' observabl ...

. Some of the radiation escapes to infinity, while some is backscattered by spacetime curvature and reabsorbed by the newly-formed

black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...

. Mass inflation occurs because as the gravitational radiation approaches the

, a region of infinite

blueshift In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and increase in frequency and e ...

, it too suffers from ever-increasing blueshift and its energy density increases without bound. If one does not consider perturbations by the collapsing star within the inner horizon, the buildup of this gravitational radiation would have no effect: the Cauchy horizon, a region of infinite blueshift for observers outside the horizon, and the apparent inner horizon, a region of infinite

redshift In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and increase in frequency and e ...

for observers inside the horizon, would effectively "cancel out" and no mass inflation would occur. But as outgoing radiation from the collapsing star crosses the inner horizon, the Cauchy horizon and apparent horizon separate, stopping the cancellation from occurring and allowing mass-inflation to take place. These ingoing and outgoing streams of radiation and the energy flux of blueshifting radiation increase the

gravitational force Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the sq ...

, which further accelerates the streams, which then increases the gravitational force and so on, creating a

positive feedback loop Positive feedback (exacerbating feedback, self-reinforcing feedback) is a process that occurs in a feedback loop where the outcome of a process reinforces the inciting process to build momentum. As such, these forces can exacerbate the effects ...

that leads to runaway growth of the gravitational mass parameter (also called the internal mass function). For a realistic astrophysical black hole in classical (non-quantum)

, this mass inflation is caused only by accretion within the last few hundred black hole crossing times (rather than the distant past or far future). Counterintuitively, the accretion rate of the black hole is

inversely proportional In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called ''coefficient of proportionality'' (or ''proportionality ...

to the rate of

exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...

of the mass function, unless the accretion rate is near the maximum.

Mass-inflation singularity

The divergence of the gravitational

parameter has interesting

local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States Arts, entertainment, and media * ''Local'' (comics), a limited series comic book by Bria ...

consequences for an extended infalling object: since the mass parameter determines the Coulomb component of the local

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

inside a

, an increase in the parameter would lead to increased

tidal forces The tidal force or tide-generating force is the difference in gravitational attraction between different points in a gravitational field, causing bodies to be pulled unevenly and as a result are being stretched towards the attraction. It is the d ...

felt locally by the object. This results in a curvature singularity at the

known as the mass-inflation singularity, the Cauchy horizon singularity, the infalling singularity, or the "fat cigar" singularity. In 1991, Amos Ori confirmed that the singularity was deformationally weak: although an object would experience infinite tidal forces at the singularity, tidal distortion on an object in all three spatial dimensions would still be finite. In 1997, Lior Burko verified numerically that, for a Reissner-Nordstrom black hole, the entire null singularity is weak. Analysis of a more realistic

gravitational collapse Gravitational collapse is the contraction of an astronomical object due to the influence of its own gravity, which tends to draw matter inward toward the center of gravity. Gravitational collapse is a fundamental mechanism for structure formati ...

scenario by Hod and Piran strenghtened Burko's findings, verifying numerically that mass inflation still occurs in the gravitational collapse of a spherical charged black hole with realistic initial parameters, and the resulting singularity is still weak and null. In 2016, Burko, Khanna, and Zenginoǧlu discovered that the rate at which curvature increases near the Cauchy horizon was far lower than had previously been found via perturbative analysis. On the other hand, Hamilton and Avelino contend the idea that a realistic astrophysical black hole would not have a weak null singularity at the Cauchy horizon, arguing that the proposed singularity would not be able to exist if anything fell in. Even if the black hole remained isolated forever,

pair creation Pair production is the creation of a subatomic particle and its antiparticle from a neutral boson. Examples include creating an electron and a positron, a muon and an antimuon, or a proton and an antiproton. Pair production often refers spec ...

near the inner horizon would prevent a weak null singularity from forming. Instead, the end result of mass inflation would be a central, spacelike, BKL-type singularity.

Impact of quantum effects

Scientists differ on the extent of the impact of

quantum effects Quantum mechanics is the fundamental physical theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of a ...

on mass inflation. Some scientists, including Poisson and Israel themselves, have suggested that, even incorporating quantum effects, the mass-inflation singularity would likely still exist. Hamilton and Avelino argue, however, that mass inflation is majorly impacted by quantum gravity, and that an observer's demise in a sufficiently large black hole (as so not to spaghettify them before they reach the Cauchy horizon) would be caused by super-Planckian curvature, not by a null singularity at the Cauchy horizon. If mass inflation does exist despite the interactions of quantum gravity, it would likely be different than what is predicted by

. For one, the stress-energy tensor also diverges faster quantum mechanically than classically, which Poisson and Balbinot theorized could in turn influence the mass function. The existence of the

Higgs field The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field, one of the field (physics), fields in particl ...

also affects the dynamics of mass-inflation: in a black hole without a Higgs field, mass inflation would consist of repeated cycles of exponential growth, whereas in a black hole with a Higgs field no such cycles would occur and the mass function would grow monotonously.

Further implications

Non-determinism beyond the Cauchy horizon

After

Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics i ...

pointed out in the 1960s that the

of a

is a region of infinite

and a barrier in which causality breaks down, the inherent non-predictability of

spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

beyond the horizon puzzled scientists for decades. This was especially problematic as it violated the classical view of

determinism Determinism is the Metaphysics, metaphysical view that all events within the universe (or multiverse) can occur only in one possible way. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes ov ...

. It became the basis of Roger Penrose's 1979 strong cosmic censorship conjecture, in which he argued that the Cauchy horizon could not exist because any perturbations by passing

gravitational waves Gravitational waves are oscillations of the gravitational field that travel through space at the speed of light; they are generated by the relative motion of gravitating masses. They were proposed by Oliver Heaviside in 1893 and then later by H ...

would cause the horizon to collapse into a strong, spacelike singularity. However, the conjecture was proven false in 2018 by mathematicians Mihalis Dafermos and Jonathan Luk, who mathematically confirmed the existence of the singularity in

Kerr spacetime The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric tensor, metric is an Exact solutions in general relativity, e ...

. Despite this, the mathematicians discovered that the existence of a weak null singularity at the Cauchy horizon would prevent the existence of multiple

solutions Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solutio ...

of Einstein's equations beyond the Cauchy horizon, thus saving determinism. They instead theorized that spacetime beyond the Cauchy horizon is not smooth enough to use Einstein's equations at all.

Wormholes

Research has suggested that the central singularity in Reissner-Nordstrom and

Kerr Kerr may refer to: People *Kerr (surname) *Kerr (given name) Places ;United States *Kerr Township, Champaign County, Illinois *Kerr, Montana, A US census-designated place *Kerr, Ohio, an unincorporated community *Kerr County, Texas Kerr Co ...

black holes is

timelike In mathematical physics, the causal structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold. Lorentzian manifolds can be classified according to the types of causal structures they admit (''ca ...

, potentially allowing infalling

matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic pa ...

to avoid the singularity alltogether and "tunnel" into another universe. However, the existence of a singularity at the Cauchy horizon suggests that such a "Kerr tunnel" would be closed off, preventing any infallers from using it as a

wormhole A wormhole is a hypothetical structure that connects disparate points in spacetime. It can be visualized as a tunnel with two ends at separate points in spacetime (i.e., different locations, different points in time, or both). Wormholes are base ...

. If the singularity were to be resolved by the effects of quantum gravity, though, a wormhole leading out into a

white hole In general relativity, a white hole is a hypothetical region of spacetime and Gravitational singularity, singularity that cannot be entered from the outside, although energy, matter, light and information can escape from it. In this sense, it is ...

could still exist inside a black hole's core.

References

{{reflist, refs={{cite journal , last1= Poisson, first1= E., last2= Israel, first2= W., date= 1989, title= Inner-horizon instability and mass inflation in black holes, url= , journal= Physical Review Letters, volume= 63, issue= 16, pages= 1663–1666, doi=10.1103/PhysRevLett.63.1663 , pmid= 10040638, bibcode= 1989PhRvL..63.1663P {{cite journal , last1= Poisson, first1= Eric, last2= Israel, first2= Werner, date= 1990, title= Internal structure of black holes, url= , journal= Physical Review D, volume= 41, issue= 6, pages= 1796–1809, doi=10.1103/PhysRevD.41.1796 , pmid= 10012548, bibcode= 1990PhRvD..41.1796P {{cite journal , last1= Ori, first1= Amos, date= 1991, title= Inner structure of a charged black hole: An exact mass-inflation solution, url= , journal= Physical Review Letters, volume= 67, issue= 7, pages= 789–792, doi=10.1103/PhysRevLett.67.789 , pmid= 10044989, bibcode= 1991PhRvL..67..789O {{cite journal , last1= Herman, first1= Rhett, last2= Hiscock, first2= William A., date= 1992, title= Strength of the mass inflation singularity, url= , journal= Physical Review D, volume= 46, issue= 4, pages= 1863–1865, doi=10.1103/PhysRevD.46.1863 , pmid= 10015098, bibcode= 1992PhRvD..46.1863H {{cite journal , last1= Balbinot, first1= Roberto, last2= Poisson, first2= Eric, date= 1993, title= Mass inflation: The semiclassical regime, url= , journal= Physical Review Letters, volume= 70, issue= 1, pages= 13–16, doi=10.1103/PhysRevLett.70.13 , pmid= 10053246, bibcode= 1993PhRvL..70...13B {{cite journal , last1= Breitenlohner, first1= Peter, last2= Lavrelashvili, first2= George, last3= Maison, first3= Dieter, date= 1998, title= Mass inflation and chaotic behaviour inside hairy black holes, url= , journal= Nuclear Physics B, volume= 524, issue= 1–2, pages= 427–443, doi=10.1016/S0550-3213(98)00177-1 , arxiv= gr-qc/9703047, bibcode= 1998NuPhB.524..427B {{cite journal , last1= Burko, first1= Lior M., date= 1997, title= Structure of the Black Hole's Cauchy-Horizon Singularity, url= , journal= Physical Review Letters, volume= 79, issue= 25, pages= 4958–4961, doi=10.1103/PhysRevLett.79.4958 , arxiv= gr-qc/9710112, bibcode= 1997PhRvL..79.4958B {{cite journal , last1= Hod, first1= Shahar, last2= Piran, first2= Tsvi, date= 1998, title= Mass Inflation in Dynamical Gravitational Collapse of a Charged Scalar Field, url= , journal= Physical Review Letters, volume= 81, issue= 8, pages= 1554–1557, doi=10.1103/PhysRevLett.81.1554 , arxiv= gr-qc/9803004, bibcode= 1998PhRvL..81.1554H {{cite journal , last1= Hamilton, first1= Andrew J.S., last2= Avelino, first2= Pedro P., date= 2010, title= The physics of the relativistic counter-streaming instability that drives mass inflation inside black holes, url= , journal= Physics Reports, volume= 495, issue= 1, pages= 1–32, doi=10.1016/j.physrep.2010.06.002 , arxiv= 0811.1926, bibcode= 2010PhR...495....1H {{cite journal , last1= Burko, first1= Lior M., last2= Khanna, first2= Gaurav, last3= Zenginoǧlu, first3= Anıl, date= 2016, title= Cauchy-horizon singularity inside perturbed Kerr black holes, url= , journal= Physical Review D, volume= 93, issue= 4, page= 041501, doi=10.1103/PhysRevD.93.041501 , arxiv= 1601.05120, bibcode= 2016PhRvD..93d1501B {{cite arXiv , date= 2017, title= The interior of dynamical vacuum black holes I: The ⁰-stability of the Kerr Cauchy horizon, eprint=1710.01722 , class=gr-qc, last1= Dafermos, first1= Mihalis, last2= Luk, first2= Jonathan {{cite journal , date=2019 , title=Marolf-Ori singularity inside fast spinning black holes , arxiv=1901.03413 , last1=Burko , first1=Lior M. , last2=Khanna , first2=Gaurav, journal=Physical Review D , volume=99 , issue=8 , page=081501 , doi=10.1103/PhysRevD.99.081501 , bibcode=2019PhRvD..99h1501B {{cite journal , last1= Carballo-Rubio, first1= Raúl, last2= Di Filippo, first2= Francesco, last3= Liberati, first3= Stefano, last4= Visser, first4= Matt, date= 2024, title= Mass Inflation without Cauchy Horizons, url= , journal= Physical Review Letters, volume= 133, issue= 18, page= 181402, doi=10.1103/PhysRevLett.133.181402 , pmid= 39547177, arxiv= 2402.14913, bibcode= 2024PhRvL.133r1402C {{cite web , last=Hartnett , first=Kevin , date=17 May 2018 , title=Mathematicians disprove conjecture made to save black holes , url=https://www.quantamagazine.org/mathematicians-disprove-conjecture-made-to-save-black-holes-20180517/ , website=Quanta Magazine , access-date=14 May 2025 Black holes Gravitational singularities