Extremal black hole

{{Short description|Scientific theory}}

In theoretical physics, an extremal black hole is a black hole with the minimum possible mass that is compatible with its charge and angular momentum.{{cite journal |last1=Kallosh |first1=Renata |authorlink1=Renata Kallosh |last2=Linde |authorlink2=Andrei Linde |first2=Andrei |last3=Ortín |first3=Tomás |last4=Peet |first4=Amanda |last5=Van Proeyen |first5=Antoine |title=Supersymmetry as a cosmic censor |journal=Physical Review D |date=1 December 1992 |volume=46 |issue=12 |pages=5278–5302 |doi=10.1103/PhysRevD.46.5278 |pmid=10014916 |url=http://prd.aps.org/abstract/PRD/v46/i12/p5278_1 | arxiv=hep-th/9205027|bibcode = 1992PhRvD..46.5278K |s2cid=15736500 }} Extremal black holes have zero temperature. Near-extremal black holes with mass slightly above the extremal value have a simple horizon structure that make them valuable tools for black hole research.{{Cite journal |last=Iliesiu |first=Luca V. |last2=Turiaci |first2=Gustavo J. |date=2021-05-18 |title=The statistical mechanics of near-extremal black holes |url=https://doi.org/10.1007/JHEP05(2021)145 |journal=Journal of High Energy Physics |language=en |volume=2021 |issue=5 |pages=145 |doi=10.1007/JHEP05(2021)145 |issn=1029-8479}}

The concept of an extremal black hole is hypothetical, and none have thus far been observed in nature. However, they have proven useful as theoretical constructs.

It is believed on general grounds that extremal black holes would have zero Hawking temperature and emit no Hawking radiation.{{cite journal|title=Dynamics of extremal black holes |first1=Steven B. |last1=Giddings |first2=Andrew |last2=Strominger |journal=Physical Review D |volume=46 |page=627 |date=1992-07-15 |doi=10.1103/PhysRevD.46.627 }}{{cite journal|title=Hawking effect in an extremal Kerr black hole spacetime |first1=Saumya |last1=Ghosh |first2=Subhajit |last2=Barman |journal=Physical Review D |volume=105 |page=045005 |date=2022-02-11 |doi=10.1103/PhysRevD.105.045005}} In supersymmetric theories, extremal black holes are often supersymmetric: they are invariant under several supercharges. This is a consequence of the BPS bound.{{cn|date=June 2025}} Their black hole entropy{{cite journal |last=Bekenstein |first=Jacob D. |authorlink=Jacob Bekenstein |title=Black Holes and Entropy | journal=Phys. Rev. D |year=1973 |volume=7 |issue=8 |pages=2333–2346 |doi=10.1103/PhysRevD.7.2333 |bibcode = 1973PhRvD...7.2333B |s2cid=122636624 }} can be calculated in string theory.{{cn|date=June 2025}}

One proposal, known as the "third law of black hole thermodynamics", says that no physical process can form a black hole with vanishing surface gravity.{{cite journal|last1=Bardeen |first1=J. M. |last2=Carter |first2=B. |last3=Hawking |first3=S. W. |year=1973 |title=The four laws of black hole mechanics |journal=Communications in Mathematical Physics |volume=31 |number=2 |pages=161–170 |doi=10.1007/bf01645742}} This would disallow the formation of an extremal black hole; more specifically, no process involving a finite number of steps could produce a black hole without violating the weak energy condition.{{cite journal |last1=Carroll |first1=Sean M. |authorlink=Sean M. Carroll |last2=Johnson |first2=Matthew C. |last3=Randall |first3=Lisa |authorlink3=Lisa Randall |title=Extremal limits and black hole entropy |arxiv=0901.0931 |year=2009 |doi=10.1088/1126-6708/2009/11/109 |volume=2009 |issue=11 |journal=Journal of High Energy Physics |page=109|bibcode=2009JHEP...11..109C |s2cid=73604121 }} A proof of this was published in 1986 by Werner Israel.{{Cite journal |last=Israel |first=W. |date=1986-07-28 |title=Third Law of Black-Hole Dynamics: A Formulation and Proof |url=https://link.aps.org/doi/10.1103/PhysRevLett.57.397 |journal=Physical Review Letters |language=en |volume=57 |issue=4 |pages=397–399 |doi=10.1103/PhysRevLett.57.397 |pmid=10034049 |issn=0031-9007|url-access=subscription }} However, more recent work claims it contains an error and therefore extremal black holes are indeed possible.{{cite journal|last1=Kehle |first1=Christoph |first2=Ryan |last2=Unger |title=Event horizon gluing and black hole formation in vacuum: the very slowly rotating case |journal=Advances in Mathematics |volume=452 |year=2024 |page=109816 |arxiv=2304.08455 |doi=10.1016/j.aim.2024.109816}}{{cite arXiv |last1=Kehle |first1=Christoph |title=Extremal black hole formation as a critical phenomenon |date=2024-02-15 |eprint=2402.10190 |last2=Unger |first2=Ryan|class=gr-qc }}{{Cite web |date=2024-08-21 |title=Mathematicians Prove Hawking Wrong About 'Extremal' Black Holes |url=https://www.quantamagazine.org/mathematicians-prove-hawking-wrong-about-extremal-black-holes-20240821/ |first=Steve |last=Nadis |access-date=2024-08-27 |website=Quanta Magazine |language=en}} The third law of thermodynamics for black holes has always been controversial.

A black hole whose mass is not far from the minimal possible mass that can be compatible with the given charges and angular momentum is known as a near-extremal black hole. Calculations regarding these bodies are usually performed using perturbation theory around the extremal black hole; the expansion parameter is called non-extremality. In supersymmetric theories, near-extremal black holes are often small perturbations of supersymmetric black holes. Such black holes have a very small Hawking temperature and consequently emit a small amount of Hawking radiation. Their entropy can often be calculated in string theory, much like in the case of extremal black holes, at least to the first order in non-extremality.{{cn|date=June 2025}}