sign in sign up
<<
Home , Big Bounce, Event horizon, Hawking radiation, Hayward metric, Initial singularity, Planck star

ONDES MATIÈRE ETUNIVERS- RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS Abstract 13 interesting observational consequences. interesting observational to lead holes,mayalso which exploding and loop on Einstein’sone,based speculative more a modified holes;and black non-singular on and equations based one, conservative more a questions: these famous -loss paradox. the I will present two different these, hypothesis gravity. to address Among quantum of theory aimed fundamental questions any technical at and conceptual yet crucial phenomenon, raises observed, theoretical be to This . thermal quantum by holes into account can change the picture, leadingtoaslow evaporation of black effects quantum taking that show to first the was Hawkingtime. in bigger According toclassicalgeneralrelativity, they are eternalandcanonly grow Black holesare oneofthemostfascinating objectspopulatingtheuniverse. Centre de Physique UMR7332, de Théorique, Centre CNRS, Université d’Aix Marseille

Loop quantumgravity & Université de Toulon,& Université de 13288 France Marseille, and exploding black holes Simone Speziale Simone

385 386 two immediate confirmations of , and includes the slowing down of time, ofdowntime, theslowing andincludes relativity, of general confirmations immediate two the , of deflection the to and Mercury of perihelium the of precession the to rection a- cor to leads it particular, In gravity. of description Newton’s to corrections crucial but small introduces and a , or a around of deformation the describes This solution. solution and static symmetric found spherically simplest the Schwarzschild relativity, of general equations of the publication the after months three as early As tion. more laser interferometers around the world. of construction upcoming the with begun, just has waves gravitational via the to listening of era the and LIGO, to thanks we have, Now enough. sensitive instrument an have not we did as time, the at possible not was waves those of observation Direct waves. gravitational of form the in away carried was that if by GR predicted rate the tly time same and awas slowing losing with observed field strong energy magnetic downstars the atexac Around satellites. expe GPS for first the of seventies camethe also the firstbuilding indirect until the observation of wait gravitationalwith to waves: had an orbital pair of field neutron confirmation, rimental gravitational the of increase the with time of down slowing the is that GR, of extraordinary another weakness, same the For origin. electromagnetic of ultimately my muscles, of energy chemical the using attraction, this beat to effort a me small takes it ground, the towards I hold chalk the pulling is whole the even though forces: other to the compared of gravity weakness in extreme the more once the why reason of itthe confirmed The tookobservation, sodifficulty long,. lies Einstein’s perfectly has result extraordinary This consortium. LIGO the by led interferometers laser American two by the observed been have waves gravitational Introduction 1.

Right on time for the hundredth birthday of Einstein’s theory of general relativity, relativity, general of theory Einstein’s of birthday hundredth the for time on Right Black holes also have been predicted using Einstein’s theory long before their observa their before long theory Einstein’s using predicted been have also holes Black Résumé uniu à oce t e tos or epois hptèe u pu également peut qui conduire àd’intéressantesconséquencespotentiellementobservables. hypothèse explosifs, noirs trous des et boucle à quantique avec ;absence de singularité une seconde, plus spéculative, est basée sur la gravité est basée sur une modification des équations d’Einstein conduisant à des trous noirs différentes pour tenter d’y répondre : une première hypothèse, la plus conservatrice, fameux paradoxe de la d’information.perte Ce chapitre présente deux hypothèses gravitationfondamentalede quantique. théorie toute visant Parmiquestions, ces le à êtreobservé,encore questions cruciales,des soulève conceptuellesettechniques, a qui théorique, phénomène Ce quantique. thermique radiation par évaporation lente une à conduisant en donne, la changer peut en quantiques effets prise des la compte que montrer à premier le été a Hawking temps. le avec croître que Selon la relativité générale dans sa classique,version ils et sont ne éternels peuvent univers. notre peuplant fascinants plus les objets les parmi figurent noirs trous Les àboucleettrousGravité quantique noirs explosifs LOOP AND EXPLODINGBLACK HOLES-S. SPEZIALE - - - ONDES, MATIÈRE ET QUANTIFICATION 387

Figure 1 an effect called gravitational . Remarkably, the and associated slowing down of time can become increasingly large, the more the star is compact. If a star of M could be compressed in a region of radius smaller than the critical value 2 rS = 2GM/c , the effect would become infinite. This infinite redshift implies also an infinite redshift of any propagating wave, so that , not even light, can emerge from the region inside the surface of radius rS , thereby named the horizon: an outside observer has no access to events happening inside this surface.

Initially considered to be just a theoretical abstraction, the notion of such ‘black holes’ become experimental evidence with the observation of regions of spacetime sourcing strong (observed through the effects on the surrounding ) and being completely invisible. The first such observation dates back to 1964 and the X-ray source Cygnus X1, now understood to be caused by the matter falling into a from a orbiting around it [1]. Since then we have discovered a large number of black holes, in mass ranges that go from a few solar , to monsters the equivalent of a hundred million . We even have pictures of them! See for instance this famous image by the KECK telescope (figure 1), showing the center of our : The highly elliptical orbits of a few have been reconstructed over a 15-year period, all turning around a dark object the mass of a million Suns.

Observational confirmation of black holes, even if restricted to the region outside the , stimulates questions and theoretical thoughts on what lies beyond: what truly happens at the event horizon? What happens inside? From the mathematical viewpoint, these questions can be answered using general relativity. The theory tells us for instance that an observer falling inside the black hole will cross the event horizon in a finite time

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 388 affect the physical application of general relativity to astrophysical phenomena, since the the since phenomena, astrophysical to relativity general of application physical the affect below. question this to back come We will hole? black a inside happens what really this Is destruction. to doomed is hole black the in falls that everything and it, from escape no is there end: trajectories infalling all where region a is cuum polarisation: the quantum vacuum of the field is teeming with particle- hole. black the of mass the Mis where form of a of quantum particles at a given by given atemperature at particles quantum of radiation athermal of form quantum would effect leadindeed, to leaking energy out of the blackconclusion: hole. The leaking takes the stupendous a derive to able was Hawking setup, approximate this In content. energy little has content matter the when interme effects physical and an ideas simpler a test to as provide setting can it but but theory, Einstein’s of heart below, the thus this spacetime, on to matter of back come will We provided it This is set stationary. up it is because limited the ignores strongly back-reaction Minkowski. of on spacetime, a theory butfixed curved field quantum to it consider is step, possible diate isometries the general in not andfixedlacks is dynamical, spacetime relativity in general On the contrary, relativity. special like in isometries, associated its and spacetime freedom, Minkowski of flat and a fixed degrees requires many infinitely with systems to extension its theory, field quantum effects mechanical quantum and mechanics of quantum foundation the very issue: is a including delicate very relativity Now, in general shrinking. and mass losing hole black a to lead could then effects mechanical Quantum effects. or tunnelling of quantum of anti-particles, would one violate of of in the energy. its Think particular, positivity of hypotheses, instance relativity. general classical in eternal are holes black that proving thus hole, black the in falls matter new as increase, only but size, in decrease never can horizon event an that proved Hawking around Furthermore, horizons singularities. event form collapses gravitational that evidence mathematical supported the is by it and Conjecture, Censorship Cosmic the called was This horizons. event by hidden are singularities such all that conjectured relativity, general and holes black of scientistsIndeed, like Hawking, Penrose and many others developing the classical theory is hidden by singularity the event from horizon and our disconnected thus universe. causally This so-called where it infinite. becomes solution, the of center the we approach as unboundedly grows strength theory, gravitational the Einstein’s in himself: observer the and spaceship observer the destroy to strong so are effects tidal the until strength, in enormously increases attraction gravitational the however, in falling keeps he As avideo. on motion accelerated in like eyes, his of front in exponentially accelerate universe visible its of evolution the see would he from, coming is he direction the at back Looking conditions. extremal any experiencing without and

In any case, the strange interior region with its singularity lurking in the dark does not not does dark the in lurking singularity its with region interior strange the case, any In Intuitively, this effect can be understood as a gravitational version of Schwinger’s va Schwinger’s of version gravitational a as understood be can effect this Intuitively, Shortly after proving after Shortly the theorem, Hawking realized that quantum mechanical effects T H = 8 πG M ,

LOOP QUANTUM GRAVITY AND EXPLODINGBLACK HOLES-S. SPEZIALE curvature singularity curvature at the centre of the black hole hole black the of centre the at (1) - - ONDES, MATIÈRE ET QUANTIFICATION 389

virtual pairs, and if we apply a strong enough electric field say with a capacitor, the pairs can turn into real particles, separated from each other and attracted to the two opposite plates of the capacitor. The energy to excite real particles from the vacuum is provided by the capacitor. In the Hawking effect, it is the strength of the gravitational redshift near the event horizon that polarises the vacuum, and the energy to create real particles is subtracted to the black hole itself. What is surprising about Hawking’s result is the per- fectly thermal of the quantum radiation. A black hole then behaves exactly as a in . Hawking’s result is just a first order approximation and it ignores the back reaction. Hence his theory does not tell us how the black hole changes when a quantum of radiation is emitted. But the thermodynamical behaviour of the black hole, suggested by many other properties discovered around the same time such as the famous area law, suggests that we can use Stefan-Boltzmann law to estimate the energy loss rate. This gives

c6 M˙ = . (2) −G2M 2

If there are no important higher order or non-perturbative corrections appearing, the evaporation will make a black hole lose all its mass in a finite time: integrating (2), we get

G2 τ M 3. (3) H ∼ c6

Black holes may not be eternal after all!

Such a finite lifetime is by no means in contradiction with the observed existence of black holes, as for macroscopic black holes is much longer than the itself. For instance, the Hawking lifetime of a black hole is 1067 years! Only for microscopic black holes this finite lifetime could be tested; however, although suggested by some theorists, microscopic black holes have never been observed. Furthermore, Hawking’s radiation itself is not astrophysically observable: for a solar mass black hole, −8 1 TH 10 K is smaller than the cosmic microwave background of 2.7K. Even lacking direct testing, has been rederived in a plentiful of different ways in the passing years,∼ and very few doubt of the solidity of this theoretical result.

From a theoretical point of view, Hawking’s radiation poses a number of extraordinary challenges that go hand to hand with the issue of the central singularity. In fact, if classically we could invoke the Cosmic Censorship Conjecture and assume that all singularities are hidden from the eye, and general relativity can be applied without having to deal with them, Hawking’s result on the evaporating event horizon brings out the singularity again. To deal with the singularity, it may then be necessary to go beyond general relativity, beyond also the approximate scheme of on curved used by Hawking, and deal with a proper quantum theory of gravity.

(1) Efforts of verifying Hawking’s radiation rely on so called analogue systems [2].

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 390 question has blossomed into a whole research field that regularly produces important ad important produces regularly that field research whole a into blossomed has this question verified, be could that predictions experimental consistent to led has ideas these of none While forward. put been have proposals creative very of number large a and now, for years over 80 minds brilliant most the fascinated has quest This physics. of theoretical grail holy the bymany considered is it and physics, theoretical of foundations actual the variations. In the following, I will briefly present two different hypothesis on which I have I which on hypothesis different two present briefly Iwill following, the In variations. the equations. by Einstein’s of simply modification violated are Penrose and Hawking of theorems singularity The collapse. gravitational the halting pressure effective of sort some introducing equations, Einstein’s modify will effects gravity quantum -scale case. gravitational the in also happen will similar something that conjecture researchers many gravity, of quantum theory complete Even of in a the absence zero momentum. in with the centre localised from being the preventing pressure quantum an effective introduces principle the uncertainty effects: r = 0 at asingularity had potential classical the words, other In nucleus. the into falling inevitably and momentum angular losing thus radiation, emitted have should nuclei the charged the around spinning electromagnetism, of theory Maxwell’s to According atoms. of classical instability the namely century, 20th early the in theoreticians by faced problem analogous hole? black the in fell that matter the of information the to happens what absent, is larity vanish? it does exposed, become it Does singularity? completely? evaporate hole black the does addressed: be should that of gravity.any quantum theory candidate In particular, we three key select can questions discussed in more below. details birthday will be celebrated this year with a conference in his honour60th in Marseille)(whose Rovelli will be Carlo and Smolin Lee Ashtekar, (LQG) Abhay by gravity pioneered approach, quantum This loop by offered is relativity, general in spacetime of nature dynamical the on focuses instead but supersymmetry, nor extra-dimensions require not does which approach, orthogonal An effect. observed the see to failing experiments quantum designed on based of ideas, two these against building been has evidence experimental of foundations amount and increasing ever an and extra-dimensions, and supersymmetry of existence the mathematics as well as require theory of the foundations very the is that theory string with trouble The theory. general, in string physics is certainly approach andfollowed most thedeveloped most field In this mechanics. theoretical in vances

Applying to general relativistic systems requires thus going beyond requires systems relativistic to general mechanics quantum Applying There is a vast literature of ideas and results concerning these questions and similar similar and questions these concerning results and ideas of literature vast a is There an of reminiscent is relativity general of singularities curvature the with situation The The information paradox: The singularity problem: i.e. throughout, valid formula Hawking’s Is evaporation problem: dynamical The for ground testing perfect a provide radiation Hawking and holes black Summarizing, AND EXPLODINGBLACK HOLES-S. SPEZIALE . The solution to this problem came from the inclusion of quantum of from . quantum came to the inclusion problem this The solution If it evaporates completely, what happens to the central central the to happens what completely, evaporates it If Hawking’s radiation is perfectly thermal; if the singu the if thermal; is perfectly radiation Hawking’s [3]. [3]. - - ONDES, MATIÈRE ET QUANTIFICATION 391

worked: non-singular black holes [9, 10] and exploding black holes [11]. The two hypotheses share the same fundamental idea, that the fundamental theory of gravity resolves the central singularity, but differ in the detailed realisation and physical consequences. In the case of non-singular black holes, the quantum pressure stops the collapse to a somewhat steady state, still looking like a black hole from the outside, but non singular at its centre. Hawking radiation then comes into play, slowly evaporating away the black hole, finally exposing the inner core of the collapse, thus restoring the information lost in the collapse. In the case of exploding black holes, the quantum pressure is so strong that it leads to a ‘bounce’ of the , turning the in-falling matter into an outgoing, exploding shell. The relevant quantum effect is a tunnelling process, and Hawking radiation is treated as a sub-leading, dissipative effect.

2. Non-singular black holes

Many tentative theories of quantum gravity predict modifications to the Einstein’s equations. The modifications can be organised in a series in the quantum gravity 3 parameter L (ideally proportional to the P = hl / cG ),

1 16πG R g R = T + L...+ L2 ... + ... . (4) µν − 2 µν c4 µν 

A typical case study for instance is when the extra terms in the ellipses contain higher powers of the Riemann tensor Rµνρσ. The modification of the theory invalidates the singu- larity theorems, and solutions with non-singular black hole metrics can now be found. To be interesting, such solutions should satisfy three criteria:

(i) look like a black hole from outside the event horizon, so to be compatible with general relativity;

(ii) be regular everywhere, in particular have no central singularity;

(iii) have all curvature invariants everywhere sub-Planckian in :

2 µνρσ 4 and so on. R<P− ,RµνρσR <P− ,

The third criterium is needed to justify the use of a classical, smooth metric to describe quantum gravity effects. The point here is that nothing guarantees that a metric description of quantum gravity is possible above the Planck scale. and loop quantum gravity for instance, argue that this is not possible, and a completely different fabric of spacetime emerges at those scales. Therefore the smooth non-singular black hole metric makes sense only to describe regions of sub-Planckian curvature, albeit extremely high. These conditions by themselves turn out to impose important restrictions on the allowed metrics, so that it becomes interesting to study non-singular black hole metrics satisfying (i − iii) in their own right, regardless of the details of the underlying fundamental theory.

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 392 Figure 2 could be static in (provided he doesn’t burn!). doesn’t he (provided fall free in static be could observer one where astar, of centre the at happens what to similar is hole black new this of the situation region, physical central With a recover singularity. non-singular the central horizon will rejoin the inner horizon, and the two will merge together and disappear, leaving leaving disappeared. has horizon event the once available becomes and lost, disappear, means and in is by fallen no on matter the together information the simply: quite paradox information-loss the merge answers will it Furthermore, two Conjecture. Censorship the Cosmic the with compatible and perfectly is evolution horizon, this centre, the in asingularity more no is inner there As exposed. region the central the rejoin will horizon even the point way, some at the all through goes evaporation Hawking the that Assuming law event horizon continues to slowly accumulate. for matter evaporates Hawking radiation, Hawking’s to outer, the As horizon. inner at the accumulates hole according black the form to in falls that matter time in all the surface, as an accumulation acts horizon inner the evaporates Since spacetime. to resulting the mass the that assume limit the take we namely equations, Einstein’s from deviations the off switch we to L in proportional inversely curvature the of value finite a but singularity, of has region no the metric null The for rays. central the outgoing surface an accumulation as acting horizon, a second of presence We the see panel): (left Schwarzschild of one the also [4] (see by Hayward ago study years ten some case introduced The was literature used. the in developed most metric precise the on course of depend details the holes, black figure see for geodesics, region andan accumulation surface, it is of an horizon: event the infinite- properties site oppo the has horizon inner new The region. time-like a location central the making thus like a looks hole from away far black the event have horizon, must inside a horizon second that metric non-singular symmetric aspherically that prove to possible is it particular, In by the outcoming radiation after the complete evaporation of the event horizon. event the of evaporation complete the after radiation outcoming the by

What happens if we try to address the puzzles of evaporation in this context? We can context? this in evaporation of puzzles the address to we try if happens What (2) [5]). Its structure is shown in the figure 2 (right panel), in comparison with with incomparison panel), 2 figure (right the in shown is structure cone light [5]). Its

Technically,

what LOOP QUANTUM GRAVITY AND EXPLODINGBLACK HOLES-S. SPEZIALE

happens

is

that 2. While this is a rather generic property of non-singular non-singular of property generic rather a is this While 2.

the

thermal

Hawking

radiation

emitted

(2), and see what happens happens what see and (2),

at

early

2 times (4), so that if if that so (4),

L is ↦

purified 0 , we -

ONDES, MATIÈRE ET QUANTIFICATION 393

This qualitative picture is compelling, it provides a simple and elegant answer to both the singularity resolution and the information paradox. The difficulty is to find a consistent quantum theory of gravity confirming this picture and describing its details. In particular, properly taking into account the back-reaction of Hawking radiation on Hayward’s metric, which means also verifying the assumptions that the black hole evaporates completely, requires solving the equations (4) with the specific modification included, and withT µν on the right hand side computed as expectation value of the quantum field on the . This computation is extremely difficult and we have no explicit answer yet. Nonetheless, preliminary calculations expose the following potential problem: because of the accumulation surface, all the information is released at once, when the event horizon completes its evaporation. This release would be associated with a huge burst of energy much larger than the mass of the black hole itself, and thus incompatible with standard energy conservation. It is possible that a special form of the extra terms in (4) can be found, such that the scenario above is realized resolving this problem with energy conservation, but this we do not know yet.

In conclusions, the hypothesis of non-singular black holes is quite appealing for its simple resolution of the information loss paradox, but needs to be still improved before it can be given substantial theoretical evidence.

3. and exploding black holes

Among the alternatives, a more speculative idea has been recently put forward by and his research group [6–8, 11]. The basic idea is that when the energy of 5 2 96 3 the collapsing matter reaches the Planckian density (i.e. ρP = c /ħG 10 kg/m ), quantum gravity induces a repulsive pressure so strong not only to stop the collapse, as in non-singular black holes, but to make the collapsing matter ‘bounce back’, thus triggering∼ an explosion and the end of the black hole phase. Indications for this very exotic scenario come from loop quantum , an application of loop quantum gravity to simple cosmological models. In that context, extensive model building shows that using the dynamics of loop , the initial singularity is replaced by a ‘’: the uni- verse does not come from an , but from a previously contracting phase bouncing back once the critical Planck density is reached [12]. Rovelli’s scenario extends this possibility from cosmology to gravitational collapse. From the outside, the exploding black hole phase triggered by quantum gravity looks like a metric, namely the time reverse of a black hole, in which all matter is expelled from the white horizon; for this reason, this scenario is often called ‘black hole to white hole transition’. Notice that this process is classically impossible: the two trajectories of matter (the collapsing and the exploding phases), or equivalently the two metrics (black hole evolving into a white hole) are not connected by any classical solution of Einstein’s equations. The scenario is thus based on the well known quantum mechanical tunnelling effect: two classical configurations not connected by any trajectory could nonetheless have a non zero possibility of being joined by a tunnelling. The most famous example is the radioactive decay of an unstable isotope:

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 394 r lifetime of hole. black the lifetime a non-perturbative phenomenon, that could not be predicted in Hawking’s perturbative perturbative Hawking’s in predicted be not could that phenomenon, non-perturbative a is it words, other In evaporation. complete before occur then would explosion associated and The tunnelling to hole evaporate. make the black radiation for it time Hawking’s takes long extremely the than shorter be could half-life the particular, In sense. make to picture point scale, horizon the at happen must bounce the causing 3. figure the in complete reproduced is The matter the of metric. hole trajectory white now the in again out comes matter bouncing the and occurs, tunnelling the point some at until collapse, classical the in as down redshifted being and slowing increasingly horizon, event the towards falling matter the is see would he what Hence, density. energy in Planckian and the event falling reaching horizon the matter see not can he perspective, his From hole. black the outside observer an of viewpoint the from picture tunnelling above the consider us let here, discussed lifetime relevant the is what of isotope. the lifetime the hence, tunnelling, the of probability the estimate to is key question The barrier. potential its ofescaping haveprobability can anon-zero thenucleus to confined particle a quantum 3 Figure S is the observed observed the Tis and observer, external the of coordinates temporal and radial the are

It is this lifetime as measured from the outside that is crucial for the exploding black hole hole black exploding the for crucial is that outside the from measured as lifetime this is It effects quantum non-perturbative that concludes observer outside the that Notice understand To observer. the on depends time of notion the settings, relativistic all in Like LOOP QUANTUM GRAVITY AND EXPLODINGBLACK HOLES-S. SPEZIALE t S r S = 2m δ r S = R T R r S in the picture. Here Here picture. the δ in t S and and ONDES, MATIÈRE ET QUANTIFICATION 395

scheme. Notice that in this picture Hawking radiation plays a sub-leading role, much like a dissipative effect. It is there and affects the evolution of the black hole, but it is not the principal cause of the black hole’s finite life. This scenario shares with non-singular black holes the key point of a regular centre, with no singularity, and associated later purification of the Hawking radiation once the event horizon has disappeared. However, while in the non-singular black hole picture it is the Hawking radiation that makes the horizon evaporate completely, in this alternative picture it is the non-perturbative quantum tunnelling that intervenes before and makes the black hole explode. The other crucial difference between the two scenarios is that in the non-singular black hole, strong quantum gravity effects are only invoked near the centre of the black hole. In the exploding black hole instead, the outside observer sees strong quantum gravity effects at the horizon scale.

The idea can be tested in principle with any approach to quantum gravity that is suffi- ciently developed to tackle this issue. Loop quantum gravity has reached this level, and in the past year we have worked in a collaboration led by Rovelli [11] to set up the ground for the calculation. To understand how this ‘bounce’ could be possible, it is necessary to know some simple details about loop quantum gravity. In this description of quantum gravity, the very fabric of spacetime is made by a collection of ‘atoms of space’, much like my desk is made out of atoms. Such atoms of space would have the dizzyingly small size of 10 −35m, and would thus be impossible to see at large, macroscopic scales: their collective behaviour is indistinguishable from the continuous, smooth spacetime of general relativity and quantum field theory. Only when the matter density reaches Planckian values, it is possible to start interacting with the individual atoms of space. The calculation we have set up shows that at Planckian energy , there is a transfer of energy from the in-falling matter to the atoms of space. The latter, previously quietly sitting in a coherent state describing a smooth black hole geometry, absorbs enough energy to tunnel into a new configuration, where they settle down into a coherent state describing the white hole. Energy is exchanged back to the matter, and the classical evolution under the white hole metric is restored. The tricky point is to compute the actual probability amplitude for this phenomenon, and because of the complexity of our formula, we still do not know the answer. Nonetheless, we can already advance a few interpretations of the outcome.

Assuming for simplicity spherical symmetry and time-reversal symmetry, the lifetime can depend only on the mass M of the black hole, and can be a priori any function. However, there are strict bounds that can be derived from theory and experiments. Lifetimes pro- portional to M or M log M for instance, are incompatible with observations: astrophysical black holes of solar masses would undergo tunnelling in a matter of seconds, contrarily to their observed stability. If the loop quantum gravity calculation predicts a lifetime that is too short, the mechanism can be ruled out. One the other hand, lifetimes proportional to M3 or longer would mean that the black hole would evaporate under Hawking radiation before tunnelling, making the phenomenon not relevant, and switching interest back to non-singular black holes. Hence, the only interesting outcomes would be lifetimes of order M2. Reintroducing physical units, this means

G3 τ M 2. (5) ∼ c7

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS 396 Quantum gravity Quantum Rovelli, C. [3] com/cws/article/news/2014/oct/15/black-hole-analogue-works-like-a-laser] 15, 2014 Oct [http://physicsworld. World, Physics alaser, like works analogue Black-hole [2] [1] https://en.wikipedia.org/wiki/Cygnus_X-1 research. future in forward it developing into forward and we are looking prediction, gravity quantum be a testable could therefore holes black [14]. of public Tunnelling general the to also presented been has possibility intriguing this [13], Bursts and Radio Fast observed recently the to similar tantalisingly is signal resulting could be exploding today and yield observable signals 10 [arXiv:1407.0989 [gr-qc]]. [arXiv:1412.6015Grav Rel. [gr-qc]]. Gen. Speziale, S. and Rovelli C. Pacilio, C. T.[9] Lorenzo, De [arXiv:1404.5821 B Lett. [gr-qc]]. Phys. phenomenology,” star “Planck Rovelli, C. and Barrau A. [gr-qc]]. [8] [arXiv:1401.6562 doi:10.1142/S0218271814420267 D Phys. Mod. J. Int. stars,” “Planck F. and Vidotto, Rovelli C. [7] D Rev. Phys. tunneling,” hole white to black spark horizon the outside effects “Quantum-gravity Rovelli, C. and Haggard [6] M. H. (2014) arXiv [1402.5446]. horizon, apparent aclosed with model hole’ . a‘black and problem loss V.[5] P. Information Frolov, 6126] 50 /0 qc - gr [ Lett. Rev. 31103 Phys. holes, black ofregular evaporation and Formation Hayward, A. [4] S. of scales small 10 ridiculously the at occur law Newton’s to modifications the precision, 10 order of scales at occur law Coulomb’s to modifications the while However, law. Coulomb’s like much loops, by scales atsmall law is modified Newton’s wethat know instance, For invisible. be to virtually tiny so be to expected are effects gravity quantum that fact the appreciate should reader The in telescopes! could well be observable explosions and such be exploding, would actually holes black true, is hypothesis tunnelling hole black the If phenomenology: gravity tum hand, a lifetime proportional to M other the On quest. ma mathematical and thus theoretical domain, purely a typically gravity experimental quantum king the outside well similarly lie tests Most galaxy! the of size this cannot be a perturbative phenomenon, as expected. as phenomenon, aperturbative be cannot this

−17 −35 References Notice that the effect depends on the inverse square root of of root square inverse the on depends effect the that Notice The most interesting aspect of this possibility is that it opens a new window for quan for window anew opens it that is possibility this of aspect interesting most The m m , is like saying that we are trying to identify the position of Rome using a ruler the the a ruler using Rome of position the identify to trying we are that saying like , is , to be compared with the smallest scales accessible today, 10 accessible scales smallest the with compared be , to LOOP QUANTUM GRAVITY AND EXPLODINGBLACK HOLES-S. SPEZIALE , Cambridge University Press 2004 Press University , Cambridge

92 2 , n

implies that primordial black holes of lunar-size mass o . 10, 104020 (2015) doi:10.1103/PhysRevD.92.104020 doi:10.1103/PhysRevD.92.104020 (2015) 104020 10, . −15 m , and have been abundantly tested to high high to tested abundantly been have , and [8]. A component of the expected expected the of component A [8]. 23 ħ (2014) n , which indicates that that indicates , which −18 . m 47 . The difference, difference, The . 739 (2015) n (2015) o .12, 1442026 1442026 .12, , 405 (2014), 405

96 (2006) (2006) o .4, 41 41 .4, - - ONDES, MATIÈRE ET QUANTIFICATION 397

[10] T. De Lorenzo, A. Giusti and S. Speziale, “Non-singular with a time delay in the center,” Gen. Rel. Grav. 48 (2016) no.3, 31 [arXiv:1510.08828 [gr-qc]]. [11] M. Christodoulou, C. Rovelli, S. Speziale and I. Vilensky, “Realistic Observable in Background- Free Quantum Gravity: the Planck-Star Tunnelling-Time,” Phys.Rev. D 94 (2016) no.8, 084035 [arXiv:1605.05268 [gr-qc]]. [12] Big Bang et gravitation quantique, Pour la , avril-juin 2014 [http://www.pourlasci- ence.fr/ewb/pages/a/article-big-bang-et-gravitation-quantique-32826.php] [13] Keane E. et al. Nature 530, 453-456 (25 February 2016). [14] Les sursauts radio rapides: des explosions d’etoiles de Planck?, Futura sciences 2015, janvier 2015 [http://www.futura-sciences.com/sciences/actualites/ trou-noir-sursauts-radio-rapides-explosions-etoiles-planck-56875]

ONDES MATIÈRE ET UNIVERS - RELATIVITÉ GÉNÉRALE, PHYSIQUE QUANTIQUE ET APPLICATIONS