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Satellites, black holes, exoplanets: when science extends beyond our planet

Are black holes stable?

Arthur Touati, PhD Student in Mathematics at École Polytechnique (IP Paris)
On June 1st, 2022 |
4 min reading time
Arthur Touati 1
Arthur Touati
PhD Student in Mathematics at École Polytechnique (IP Paris)
Key takeaways
  • Black holes began as purely mathematical ideas, unexpected by-products of Albert Einstein's 1915 theory of general relativity.
  • If the density of a body exceeds a certain threshold, it will distort the space around it and become a black hole. For example, for the Earth to be a black hole, it would have to fit inside a pistachio.
  • Recently, two mathematicians have shown that these surprising objects are stable, a first step towards understanding the final state conjecture..
  • It is hoped that new technologies will soon allow us to observe the birth or at least the youth of a black hole in order to better understand them.

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.

Cosmic gluttons

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.

How do we observe them?

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.

Uncertain birth

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.

A conjecture becomes a theorem

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…

JWST in out­er space. 

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.

The final state of our universe

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 

1Roger Pen­rose. Tech­niques of dif­fer­en­tial topol­o­gy in rel­a­tiv­i­ty. Con­fer­ence Board of the Math­e­mat­i­cal Sci­ences Region­al Con­fer­ence Series in Applied Math­e­mat­ics, No. 7. Soci­ety for Indus­tri­al and Applied Math­e­mat­ics, Philadel­phia, Pa., 1972
2Sergiu Klain­er­man and Jérémie Szef­tel. Glob­al non­lin­ear sta­bil­i­ty of Schwarz­schild space­time under polar­ized per­tur­ba­tions, vol­ume 210 of Annals of Math­e­mat­ics Stud­ies. Prince­ton Uni­ver­si­ty Press, Prince­ton, NJ, 2020
3Sergiu Klain­er­man and Jérémie Szef­tel. Kerr sta­bil­i­ty for small angu­lar momen­tum. 2021

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