Gravitational singularity#Curvature

{{short description|Condition in which spacetime itself breaks down}}

File:Black_hole_lensing_web.gif caused by a Schwarzschild black hole passing in a line-of-sight planar to a background galaxy. Around and at the time of exact alignment (syzygy) extreme lensing of the light is observed.]]

A gravitational singularity, spacetime singularity, or simply singularity, is a theoretical condition in which gravity is predicted to be so intense that spacetime itself would break down catastrophically. As such, a singularity is by definition no longer part of the regular spacetime and cannot be determined by "where" or "when”. Gravitational singularities exist at a junction between general relativity and quantum mechanics; therefore, the properties of the singularity cannot be described without an established theory of quantum gravity. Trying to find a complete and precise definition of singularities in the theory of general relativity, the current best theory of gravity, remains a difficult problem.{{harvnb|Earman|1995|loc=Section 2.2 What is a singularity?|pp=28–31}} A singularity in general relativity can be defined by the scalar invariant curvature becoming infinite{{cite web|url=http://www.physicsoftheuniverse.com/topics_blackholes_singularities.html|title=Singularities |website=Physics of the Universe }} or, better, by a geodesic being incomplete.{{Cite journal |last=Uggla |first=Claes |year=2006 |title=Spacetime Singularities |url=http://www.einstein-online.info/spotlights/singularities |url-status=dead |journal=Einstein Online |publisher=Max Planck Institute for Gravitational Physics |volume=2 |archive-url=https://web.archive.org/web/20170124030605/http://www.einstein-online.info/spotlights/singularities |archive-date=2017-01-24 |access-date=2015-10-20 |number=1002}}

General relativity predicts that any object collapsing beyond a certain point (for stars this is the Schwarzschild radius) would form a black hole, inside which a singularity (covered by an event horizon){{cite encyclopedia|url=http://plato.stanford.edu/entries/spacetime-singularities/|title=Singularities and Black Holes|last=Curiel|first=Erik|publisher=Metaphysics Research Lab, Stanford University |name-list-style=amp|encyclopedia=Stanford Encyclopedia of Philosophy|date=2021 |access-date=1 October 2021}} would appear (although observers outside the event horizon could never see it).{{cite journal

| last1 = Narlikar

| first1 = J. V.

| author-link1 = Jayant Narlikar

| last2 = Padmanabhan

| first2 = Th.

| author-link2 = Thanu Padmanabhan

| date = June 1988

| title = The Schwarzschild solution: Some conceptual difficulties

| journal = Found Phys

| volume = 18

| publisher = Springer Nature

| pages = 659–668

| doi = 10.1007/BF00734568

}} The density would become infinite at the singularity. General relativity also predicts that the initial state of the universe, at the beginning of the Big Bang, was a singularity of infinite density and temperature.{{harvnb|Wald|1984|p=99}}.{{Obsolete source|reason=this source is forty years old - a lot has changed since then.|date=October 2024}} However, classical gravitational theories are not expected to be accurate under these conditions, and a quantum description is likely needed.{{cite web |last=Hawking |first=Stephen |title=The Beginning of Time |url=http://www.hawking.org.uk/the-beginning-of-time.html |work=Stephen Hawking: The Official Website |publisher=Cambridge University |access-date=26 December 2012 |archive-date=6 October 2014 |archive-url=https://web.archive.org/web/20141006200729/http://www.hawking.org.uk/the-beginning-of-time.html |url-status=dead }} For example, quantum mechanics does not permit particles to inhabit a space smaller than their Compton wavelengths.{{cite book |last=Zebrowski |first=Ernest |url=https://books.google.com/books?id=2twRfiUwkxYC |title=A History of the Circle: Mathematical Reasoning and the Physical Universe |date=2000 |publisher=Rutgers University Press |isbn=978-0813528984 |location=Piscataway New Jersey |page=180}}

Interpretation

Many theories in physics have mathematical singularities of one kind or another. Equations for these physical theories predict that the ball of mass of some quantity becomes infinite or increases without limit. This is generally a sign for a missing piece in the theory, as in the ultraviolet catastrophe, re-normalization, and instability of a hydrogen atom predicted by the Larmor formula.

In classical field theories, including special relativity but not general relativity, one can say that a solution has a singularity at a particular point in spacetime where certain physical properties become ill-defined, with spacetime serving as a background field to locate the singularity. A singularity in general relativity, on the other hand, is more complex because spacetime itself becomes ill-defined, and the singularity is no longer part of the regular spacetime manifold. In general relativity, a singularity cannot be defined by "where" or "when".{{Cite book |last=Ashtekar |first=Abhay |title=100 years of relativity: space-time structure, Einstein and beyond |date=2005 |publisher=World Scientific |isbn=978-981-256-394-1 |editor-last=Ashtekar |editor-first=Abhay |location=Singapore |chapter=3: "The nature of spacetime singularities" by Alan D. Randall}}

Some theories, such as the theory of loop quantum gravity, suggest that singularities may not exist.{{Cite journal |last1=Gambini |first1=Rodolfo |last2=Olmedo |first2=Javier |last3=Pullin |first3=Jorge |date=2014-05-07 |title=Quantum black holes in loop quantum gravity |journal=Classical and Quantum Gravity |volume=31 |issue=9 |pages=095009 |arxiv=1310.5996 |bibcode=2014CQGra..31i5009G |doi=10.1088/0264-9381/31/9/095009 |issn=0264-9381 |s2cid=119247455}} This is also true for such classical unified field theories as the Einstein–Maxwell–Dirac equations. The idea can be stated in the form that, due to quantum gravity effects, there is a minimum distance beyond which the force of gravity no longer continues to increase as the distance between the masses becomes shorter, or alternatively that interpenetrating particle waves mask gravitational effects that would be felt at a distance.

Motivated by such philosophy of loop quantum gravity, recently it has been shown{{Cite journal |last=Majhi |first=Abhishek |year=2022 |title=Resolving the Singularity by Looking at the Dot and Demonstrating the Undecidability of the Continuum Hypothesis |journal=Foundations of Science |language=en |volume=29 |issue=2 |pages=405–440 |doi=10.1007/s10699-022-09875-9 |issn=1233-1821 |s2cid=246942045|url=https://hal.science/hal-03528767v2/file/RSLAD.pdf }} that such conceptions can be realized through some elementary constructions based on the refinement of the first axiom of geometry, namely, the concept of a point{{Cite book |last1=Euclides |url=https://farside.ph.utexas.edu/Books/Euclid/Elements.pdf |title=Euclid's elements of geometry: the Greek text of J. L. Heiberg (1883–1885): from Euclidis Elementa, edidit et Latine interpretatus est I. L. Heiberg, in aedibus B. G. Teubneri, 1883–1885 |last2=Heiberg |first2=J. L. |last3=Fitzpatrick |first3=R. |date=2008 |publisher=s.n |isbn=978-0-615-17984-1 |editor-last=Fitzpatrick |editor-first=Richard |edition=Revised and corrected |translator-last=Fitzpatrick |translator-first=Richard}} by considering Klein's prescription of accounting for the extension of a small spot that represents or demonstrates a point,{{Cite book |last=Klein |first=Felix |title=Elementary Mathematics From A Higher Standpoint |date=2016 |publisher=Springer Berlin Heidelberg |isbn=978-3662495155}} which was a programmatic call that he called as a fusion of arithmetic and geometry.{{cite book |last=Klein |first=Felix |url=https://www.gutenberg.org/files/36154/36154-pdf.pdf |title=The Evanston Colloquium Lectures on Mathematics Delivered From August 28 to September 9, 1893 Before Members of the Congress of Mathematics Held in Connection with the World's Fair in Chicago, Illinois |date=2011 |publisher=The Project Gutenberg}} Klein's program, according to Born, was actually a mathematical route to consider 'natural uncertainty in all observations' while describing 'a physical situation' by means of 'real numbers'.{{cite book |last=Born |first=Max |date=1968 |title=Physics in My Generation |publisher=Springer New York}}

Types

There are multiple types of singularities, each with different physical features that have characteristics relevant to the theories from which they originally emerged, such as the different shapes of the singularities, conical and curved. They have also been hypothesized to occur without event horizons, structures that delineate one spacetime section from another in which events cannot affect past the horizon; these are called naked.

=Conical=

A conical singularity occurs when there is a point where the limit of some diffeomorphism invariant quantity does not exist or is infinite, in which case spacetime is not smooth at the point of the limit itself. Thus, spacetime looks like a cone around this point, where the singularity is located at the tip of the cone. The metric can be finite everywhere the coordinate system is used.

An example of such a conical singularity is a cosmic string and a Schwarzschild black hole.{{cite journal |last1=Copeland |first1=Edmund J. |last2=Myers |first2=Robert C. |last3=Polchinski |first3=Joseph |year=2004 |title=Cosmic F- and D-strings |journal=Journal of High Energy Physics |volume=2004 |issue=6 |page=13 |arxiv=hep-th/0312067 |bibcode=2004JHEP...06..013C |doi=10.1088/1126-6708/2004/06/013 |s2cid=140465}}

=Curvature=

File:Black hole details.svg and its singularity ]]

Solutions to the equations of general relativity or another theory of gravity (such as supergravity) often result in encountering points where the metric blows up to infinity. However, many of these points are completely regular, and the infinities are merely a result of using an inappropriate coordinate system at this point. To test whether there is a singularity at a certain point, one must check whether at this point diffeomorphism invariant quantities (i.e. scalars) become infinite. Such quantities are the same in every coordinate system, so these infinities will not "go away" by a change of coordinates.

An example is the Schwarzschild solution that describes a non-rotating, uncharged black hole. In coordinate systems convenient for working in regions far away from the black hole, a part of the metric becomes infinite at the event horizon. However, spacetime at the event horizon is regular. The regularity becomes evident when changing to another coordinate system (such as the Kruskal coordinates), where the metric is perfectly smooth. On the other hand, in the center of the black hole, where the metric becomes infinite as well, the solutions suggest a singularity exists. The existence of the singularity can be verified by noting that the Kretschmann scalar, being the square of the Riemann tensor i.e. $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ , which is diffeomorphism invariant, is infinite.

While in a non-rotating black hole the singularity occurs at a single point in the model coordinates, called a "point singularity", in a rotating black hole, also known as a Kerr black hole, the singularity occurs on a ring (a circular line), known as a "ring singularity". Such a singularity may also theoretically become a wormhole.If a rotating singularity is given a uniform electrical charge, a repellent force results, causing a ring singularity to form. The effect may be a stable wormhole, a non-point-like puncture in spacetime that may be connected to a second ring singularity on the other end. Although such wormholes are often suggested as routes for faster-than-light travel, such suggestions ignore the problem of escaping the black hole at the other end, or even of surviving the immense tidal forces in the tightly curved interior of the wormhole.

More generally, a spacetime is considered singular if it is geodesically incomplete, meaning that there are freely-falling particles whose motion cannot be determined beyond a finite time, being after the point of reaching the singularity. For example, any observer inside the event horizon of a non-rotating black hole would fall into its center within a finite period of time. The classical version of the Big Bang cosmological model of the universe contains a causal singularity at the start of time (t=0), where all time-like geodesics have no extensions into the past. Extrapolating backward to this hypothetical time 0 results in a universe with all spatial dimensions of size zero, infinite density, infinite temperature, and infinite spacetime curvature.

=Naked singularity=

Until the early 1990s, it was widely believed that general relativity hides every singularity behind an event horizon, making naked singularities impossible. This is referred to as the cosmic censorship hypothesis. However, in 1991, physicists Stuart Shapiro and Saul Teukolsky performed computer simulations of a rotating plane of dust that indicated that general relativity might allow for "naked" singularities. What these objects would actually look like in such a model is unknown. Nor is it known whether singularities would still arise if the simplifying assumptions used to make the simulation were removed. However, it is hypothesized that light entering a singularity would similarly have its geodesics terminated, thus making the naked singularity look like a black hole.{{Cite journal |last=Bojowald |first=Martin |year=2008 |title=Loop Quantum Cosmology |journal=Living Reviews in Relativity |language=en |volume=11 |issue=1 |pages=4 |bibcode=2008LRR....11....4B |doi=10.12942/lrr-2008-4 |issn=2367-3613 |pmc=5255532 |pmid=28163606 |doi-access=free}}{{Cite journal |last1=Goswami |first1=Rituparno |last2=Joshi |first2=Pankaj S. |year=2008 |title=Spherical gravitational collapse in N dimensions |journal=Physical Review D |language=en |volume=76 |issue=8 |pages=084026 |arxiv=gr-qc/0608136 |bibcode=2007PhRvD..76h4026G |doi=10.1103/PhysRevD.76.084026 |issn=1550-7998 |s2cid=119441682}}{{Cite journal |last1=Goswami |first1=Rituparno |last2=Joshi |first2=Pankaj S. |last3=Singh |first3=Parampreet |date=2006-01-27 |title=Quantum Evaporation of a Naked Singularity |journal=Physical Review Letters |language=en |volume=96 |issue=3 |pages=031302 |arxiv=gr-qc/0506129 |bibcode=2006PhRvL..96c1302G |doi=10.1103/PhysRevLett.96.031302 |issn=0031-9007 |pmid=16486681 |s2cid=19851285}}

Disappearing event horizons exist in the Kerr metric, which is a spinning black hole in a vacuum, if the angular momentum ( $J$ ) is high enough. Transforming the Kerr metric to Boyer–Lindquist coordinates, it can be shown{{harvnb|Hobson|Efstathiou|Lasenby|2013|pp=300-305}}. that the coordinate (which is not the radius) of the event horizon is, $r_{\pm} = \mu \pm \left(\mu^{2} - a^{2}\right)^{1/2}$ , where $\mu = G M / c^{2}$ , and $a=J/M c$ . In this case, "event horizons disappear" means when the solutions are complex for $r_{\pm}$ , or $\mu^{2} < a^{2}$ . However, this corresponds to a case where $J$ exceeds $GM^{2}/c$ (or in Planck units, {{Nowrap| $J > M^{2}$ )}}; i.e. the spin exceeds what is normally viewed as the upper limit of its physically possible values.

Similarly, disappearing event horizons can also be seen with the Reissner–Nordström geometry of a charged black hole if the charge ( $Q$ ) is high enough. In this metric, it can be shown{{harvnb|Hobson|Efstathiou|Lasenby|2013|pp=320-325}}. that the singularities occur at $r_{\pm}= \mu \pm \left(\mu^{2} - q^{2}\right)^{1/2}$ , where $\mu = G M / c^{2}$ , and $q^2 = G Q^2/\left(4 \pi \epsilon_0 c^4\right)$ . Of the three possible cases for the relative values of $\mu$ and $q$ , the case where $\mu^{2} < q^{2}$ causes both $r_{\pm}$ to be complex. This means the metric is regular for all positive values of $r$ , or in other words, the singularity has no event horizon. However, this corresponds to a case where $Q/\sqrt{4 \pi \epsilon_0}$ exceeds $M\sqrt{G}$ (or in Planck units, {{Nowrap| $Q > M$ )}}; i.e. the charge exceeds what is normally viewed as the upper limit of its physically possible values. Also, actual astrophysical black holes are not expected to possess any appreciable charge.

A black hole possessing the lowest $M$ value consistent with its $J$ and $Q$ values and the limits noted above; i.e., one just at the point of losing its event horizon, is termed extremal.

Entropy

Before Stephen Hawking came up with the concept of Hawking radiation, the question of black holes having entropy had been avoided. However, this concept demonstrates that black holes radiate energy, which conserves entropy and solves the incompatibility problems with the second law of thermodynamics. Entropy, however, implies heat and therefore temperature. The loss of energy also implies that black holes do not last forever, but rather evaporate or decay slowly. Black hole temperature is inversely related to mass.{{cite journal|last1= LoPresto|first1=M. C.|s2cid=122758428|title= Some Simple Black Hole Thermodynamics|journal= The Physics Teacher|volume= 41|issue=5|year= 2003|pages= 299–301|doi= 10.1119/1.1571268|bibcode=2003PhTea..41..299L}} All known black hole candidates are so large that their temperature is far below that of the cosmic background radiation, which means they will gain energy on net by absorbing this radiation. They cannot begin to lose energy on net until the background temperature falls below their own temperature. This will occur at a cosmological redshift of more than one million, rather than the thousand or so since the background radiation formed.{{citation needed|date=October 2018}}

References

Bibliography

{{cite book |last1=Earman |first1=John |title=Bangs, crunches, whimpers, and shrieks: Singularities and acausalities in relativistic spacetimes |date=1995 |publisher=Oxford University Press |isbn=019509591X}}
{{cite book |last1=Joshi |first1=Pankaj S |title=Gravitational collapse and spacetime singularities |date=2007 |publisher=Cambridge University Press |location=New York |isbn=9781107405363}}
{{cite book

| first1 = Charles W. | last1 = Misner | author-link = Charles W. Misner

| first2 = Kip | last2 = Thorne | author-link2 = Kip Thorne

| first3 = John Archibald | last3 = Wheeler | author-link3 = John Archibald Wheeler

| title = Gravitation | publisher = W. H. Freeman | date = 1973 | isbn = 0-7167-0344-0 | title-link = Gravitation (book) }} §31.2 The nonsingularity of the gravitational radius, and following sections; §34 Global Techniques, Horizons, and Singularity Theorems

{{cite book |last1=Wald |first1=Robert M. | author-link = Robert Wald | title = General Relativity | publisher = University of Chicago Press | date = 1984 | isbn = 0-226-87033-2 | title-link = General Relativity (book) }}
{{Cite journal |last1=Hawking |first1=S. W. |author-link=Stephen Hawking |last2=Penrose |first2=R. |author-link2=Roger Penrose |date=1970 |title=The singularities of gravitational collapse and cosmology |journal=Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences |language=en |volume=314 |issue=1519 |pages=529–548 |bibcode=1970RSPSA.314..529H |doi=10.1098/rspa.1970.0021 |issn=0080-4630 |doi-access=free}}
{{Cite journal |last1=Shapiro |first1=Stuart L. |last2=Teukolsky |first2=Saul A. |author-link2=Saul Teukolsky |date=1991 |title=Formation of naked singularities: The violation of cosmic censorship |journal=Physical Review Letters |language=en |volume=66 |issue=8 |pages=994–997 |bibcode=1991PhRvL..66..994S |doi=10.1103/PhysRevLett.66.994 |issn=0031-9007 |pmid=10043968 |s2cid=7830407|url=https://resolver.caltech.edu/CaltechAUTHORS:20180629-153305570 }}
{{Cite journal |last=Penrose |first=Roger |url=http://www.ias.ac.in/jarch/jaa/17/213-231.pdf |title=Chandrasekhar, black holes, and singularities |journal=Journal of Astrophysics and Astronomy |date=December 1996 |volume=17 |issue=3–4 |pages=213–231 |language=en |doi=10.1007/BF02702305 |bibcode=1996JApA...17..213P |issn=0250-6335}}
{{Cite journal |last=Penrose |first=Roger |url=http://www.ias.ac.in/jarch/jaa/20/233-248.pdf |title=The question of cosmic censorship |journal=Journal of Astrophysics and Astronomy |date=December 1999 |volume=20 |issue=3–4 |pages=233–248 |language=en |doi=10.1007/BF02702355 |bibcode=1999JApA...20..233P |issn=0250-6335}}
{{Cite journal |last=Singh |first=T. P. |url=http://www.ias.ac.in/jarch/jaa/20/221-232.pdf |title=Gravitational collapse, black holes and naked singularities |journal=Journal of Astrophysics and Astronomy |date=December 1999 |volume=20 |issue=3–4 |pages=221–232 |language=en |doi=10.1007/BF02702354 |arxiv=gr-qc/9805066 |bibcode=1999JApA...20..221S |issn=0250-6335}}
{{Cite book |last1=Hobson |first1=Michael P. |title=General relativity: an introduction for physicists |last2=Efstathiou |first2=George |last3=Lasenby |first3=Anthony |date=2013 |publisher=Cambridge University Press |isbn=978-0-521-82951-9 |edition=1. publ., 6. print |location=Cambridge}}