Are black holes stable?

Ein­stein thought that black holes didn’t exist, but (indir­ect) obser­va­tions in the 2010s proved him wrong: black holes are real. Here we take a look at a cen­tury of black holes, from Schwar­z­schild’s first cal­cu­la­tions to the recent research find­ings from Szeftel and Klainerman.

His­tor­ic­ally, black holes were ini­tially math­em­at­ic­al objects, unex­pec­ted by-products of the the­ory of gen­er­al relativ­ity. This the­ory, pub­lished in 1915 by Albert Ein­stein, revolu­tion­ised our con­cep­tion of grav­it­a­tion and space. As such, space and time form a con­tinuum that has its own geo­metry. In par­tic­u­lar, space-time can bend under mat­ter, like a sheet sup­port­ing a heavy object. To describe this curvature, Ein­stein for­mu­lated an equa­tion that today bears his name:

Just a few months after this pub­lic­a­tion, the Ger­man phys­i­cist Karl Schwar­z­schild, then serving on the Rus­si­an front in the Great War, dis­covered a par­tic­u­lar solu­tion to Einstein’s equa­tions. Called the Schwar­z­schild met­ric, it describes space-time in the vicin­ity of a spher­ic­ally sym­met­ric body, such as a star or a plan­et. If the dens­ity of the body exceeds a cer­tain threshold, it is a black hole. The dens­ity of such an object is simply insane: to turn the Earth into a black hole, you would have to fit it into a pista­chio! Black holes are so dense that noth­ing can escape them: once they pass their bound­ary, or event hori­zon, any object will remain trapped forever. This also applies to light! Black holes there­fore do not emit any light, hence their name.

Prob­lem: all our astro­nom­ic­al obser­va­tions are based on the study of light com­ing from the cos­mos, col­lec­ted by tele­scopes or simply our eyes. To observe black holes, you there­fore have to be cunning!

One solu­tion is to detect grav­it­a­tion­al waves rather than light waves. Anoth­er spec­tac­u­lar pre­dic­tion of gen­er­al relativ­ity, grav­it­a­tion­al waves are the trace of cata­clys­mic phe­nom­ena such as the fusion of two black holes in space-time. They propag­ate at the speed of light and we can detect their motion thanks to the giant LIGO-VIRGO col­lab­or­a­tion inter­fer­o­met­ers. These kilo­metre-scale detect­ors allowed for the first indir­ect obser­va­tion of black holes in 2015.

The first pho­to­graph of a black hole was taken in 2019. Here, it was not the black hole itself that was observed but the disc of glow­ing mat­ter ready to be devoured orbit­ing the giant star.

Long before indir­ect obser­va­tion could even be envis­aged, math­em­at­ics was the best tool for tack­ling the mys­ter­ies sur­round­ing black holes. An import­ant ques­tion was: how do black holes form? What mech­an­ism can pos­sibly allow for such monsters?

To answer this, the physicist/mathematician Roger Pen­rose put for­ward a sin­gu­lar­ity form­a­tion the­or­em in the 1960s¹. This states that the caus­ally cata­stroph­ic sin­gu­lar­ity at the centre of black holes neces­sar­ily forms if space­time loc­ally sat­is­fies very strong curvature con­di­tions. Such con­di­tions can be met dur­ing the grav­it­a­tion­al col­lapse of a star at the end of its life, when it has used up all its fuel. This work earned Pen­rose the 2019 Nobel Prize in Phys­ics, and led to the under­stand­ing that the death of a star can give rise to the birth of a black hole.

Tech­nic­al pro­gress in mod­ern astro­nomy allows us to observe stars that are fur­ther and fur­ther away, and thus probe their past. We can there­fore hope to observe the birth, or at least the young lives, of black holes. But anoth­er import­ant ques­tion eludes our tele­scopes: the future of black holes. In par­tic­u­lar, are they stable?

In phys­ics, we prefer stable objects to unstable ones. Ima­gine a ball on top of a hill: if you push it slightly, it will roll down the hill! The top of the hill is there­fore an unstable pos­i­tion and it is sens­it­ive to small per­turb­a­tions. In con­trast, the bot­tom of a hill can be thought of as being a stable pos­i­tion. The James Webb Space Tele­scope (JWST) will per­fectly illus­trate these con­cepts: its des­tin­a­tion is the Lag­range point L2, which is stable to angu­lar per­turb­a­tions, but unstable to radi­al ones…

We for­mu­late the sta­bil­ity of black holes as fol­lows: what hap­pens to a Schwar­z­schild solu­tion rep­res­ent­ing a black hole if we slightly dis­turb it? Does it return to equi­lib­ri­um like the ball at the bot­tom of a hill? Yes, accord­ing to two new long art­icles²³ by math­em­aticians Jérémie Szeftel and Ser­giu Klain­er­man. The res­ult of ten years of research, their demon­stra­tion is based on a detailed under­stand­ing of the geo­metry of black holes, as well as on the devel­op­ment of numer­ous tech­niques for ana­lys­ing par­tial dif­fer­en­tial equations.

Szeftel and Klain­er­man’s res­ol­u­tion of the sta­bil­ity con­jec­ture for black holes is a major math­em­at­ic­al achieve­ment, but their work is by no means the end of the story – quite the con­trary. Their ulti­mate goal is known as the final state con­jec­ture. In simple terms, this states that in the extremely dis­tant future, the uni­verse will con­sist only of black holes mov­ing away from each oth­er. In this scen­ario, if two black holes are too close, they will merge, lib­er­at­ing grav­it­a­tion­al waves.

Math­em­at­ic­ally prov­ing this con­jec­ture requires solv­ing many inter­me­di­ate ques­tions, includ­ing the black hole sta­bil­ity con­jec­ture. Indeed, if all the con­tents of our uni­verse even­tu­ally con­verge to black holes, the black holes must “con­verge to them­selves”, that is, they become stable! Anoth­er ‘sub­con­jec­ture’ is that of cos­mic cen­sor­ship, intro­duced by Pen­rose in 1969, which pre­dicts that sin­gu­lar­it­ies such as those found at the centre of black holes can­not be ‘naked’, that is, exist without an event hori­zon around them that pro­tects the uni­verse from their para­dox­ic­al nature. The final state con­jec­ture and most of its ‘sub­con­jec­tures’ are cur­rently com­pletely out of reach. They will no doubt keep math­em­aticians busy for dec­ades, if not hun­dreds of years to come.

To find out more

Voy­age au Coeur de l’Espace-Temps, Stéphane d’Ascoli et Arthur Tou­ati (2021), First Edi­tions