Actually it is an extensive quantity, but not in the way extensive is used in classical thermodynamics. It is proportional to area and not mass (or equivalently for a classical object, volume). Entropy is normally (but not always) proportional to mass, or energy, which is approximately proportional to the number of elementary particles and thus the number of possible states. For a black hole (BH), the number of possible states is proportional to the number of different Planck areas there are in the BH horizon. i.e., the states information is as though it was stored in the horizon, not the bulk
We don't know how to calculate the entropy of a singularity, the physics breaks down and it requires quantum gravity to calculate it. But Hawking has calculated the temperature of a BH, through his calculations, where he discovered that a BH radiates as a black body at a given temperature given by the inverse of its mass. From there it is easy to get the entropy.
The simplistic version then is that since $dS = dQ/T$, with $Q$ the heat absorbed by the BH and $T$ its black body radiation-equivalent temperature, if you take Hawking's result as $T = k/M$ for some constant $k$, then $dS = kMdQ$. Since the heat absorbed into the BH is energy, it increases the BH mass (in natural units) as $dQ = dM$. So, substituting, $dS = kMdM$, and then integrating $S = kM^2$. You can see other derivations, but when Hawking found his BH radiation temperature dependent on mass, there's then no two ways around it that they could think of, nor anybody else so far.
This of course was proven by Hawking-Bekenstein based on finding that the Hawking radiation is black body with temperature proportional to the so called surface gravity at the horizon, which is inversely proportional to the BH mass, and with Bekenstein then led to entropy proportional to the horizon area, and in fact equal to the number of Planck areas multiplied by $k_b/4$ ($k_b$ is the Boltzmann constant), as per the Hawking-Bekenstein equation, also discussed in @Heather’s answer, and written (slightly off) in the question. This relates BH entropy and physics to thermodynamics – and indeed BH entropy can grow or stay the same, but never decrease, by the second law of thermodynamics. The two BH’s which merged on 9/14/15 had a final entropy greater than the sum of the two before, plus some entropy was radiated away in the gravitational radiation. The BH entropy laws also lead to equations for the maximum energy that may be extracted from merging BH’s. Bekenstein also showed that BH entropies are the maximum entropy possible for any volume of space with the same volume as the BH.
But the question still remains: why, and how? How is it that the possible states of the BH are encoded in the horizon, if indeed the statistical interpretation holds? For only if it holds would the statistical interpretation be on firm ground, regardless of the thermodynamic relationships. So there’s been attempts and some success (but no proof or certainty yet) in two separate results in physics.
One is the Holographic principle by 't Hooft and Susskind, and others, now a few years old, and still no proof, but some ocassional developments on it. They conjecture that the quantum gravity solution of a spacetime in d+1 dimensions is in a 1-1 correspondence to a Conformal Field Theory (CFT) without gravity, in the d dimensional boundary of the spacetime.
They based this on generalizing the proven results of the AdS/CFT correspondence found by Maldacena and others, who proved the results for a string quantum gravity in anti-de Sitter (AdS) spacetime. AdS see is a vacuum solution of the Einstein Field Equations with a negative cosmological constant (de Sitter is with a positive cosmological constant, the limit of our known universe as its age goes to infinity and the cosmological constant completely dominates). The AdS/CFT correspondence is also called gauge/gravity duality, gauge for the CFT. CFT’s are quantum field theories.
The Holographic conjecture and the AdS/CFT correspondence have led people to think that information on the quantum states of the bulk, in some cases or in general, is stored in its boundaries. Or surfaces, similar to the way holography works for a 3D object. But there are also some counter-examples, and so in general it is still an interesting approach, but not well understood or accepted. Still, if the general case is valid, the ideas is that the states of the BH would be encoded, or imprinted, on its horizon. Possibly that could provide a mechanism for the matter/energy that falls in, and the quantum information that was thought lost to the BH, to be in the horizon and not lost, and possibly later encoded in the Hawking blackbody (with something extra then) radiation.
There is another more recent finding that indicates that the information on the state of the BH may be stored at the horizon. It’s by Hawking, Perry and Strominger, from January of 2016. See the paper in arXiv, and the Phys. Rev. article in June 2016 (I’ve not read the Phys Rev version but the abstracts are the same).
What they claim is new, and based on new re-discovered asymptotic symmetries at conformal infinity. They claim that based on those symmetries BH's have conserved quantities, they call soft hair. That is, that BH’s have some hair that is over and beyond the mass, angular momentum and charge proved by Hawking years ago. Where it is not the same he 'proved' before (yes, there's reason this new symmetries break on of his assumptions back then, that the vacuum was non-degenerate, was unique) is that BHs actually have what the authors call soft hair, very low energy hair that in the limit is zero energy, but still is there. The soft hair is due to soft photons or soft gravitons, that reside in the horizon. They claim that, from part of their abstract:
This Letter gives an explicit description of soft hair in terms of soft gravitons or photons on the black hole horizon, and shows that complete information about their quantum state is stored on a holographic plate at the future boundary of the horizon. Charge conservation is used to give an infinite number of exact relations between the evaporation products of black holes which have different soft hair but are otherwise identical. It is further argued that soft hair which is spatially localized to much less than a Planck length cannot be excited in a physically realizable process, giving an effective number of soft degrees of freedom proportional to the horizon area in Planck units.
Thus they claim they have argued or shown that the entropy is due to the degrees of freedom, or possible states of, the horizon, in accord with BH thermodynamics. Still, they do state, in the body of the arXiv paper, that they have not proven that indeed there’s enough degrees of freedom to really store all the information, and that work remains. Their degrees of freedom, or the soft hair, comes from new symmetries that they re-discovered on the conformal infinity of asymptotically flat spacetime (think of a BH in asymptotically flat spacetime), and which lead to new conserved quantities, the soft hair. Since the BH horizon is one boundary of the asymptotically flat spacetime (i.e., outside the horizon), they show using Penrose diagrams and can calculate the conserved soft hair over the BH horizon.
The soft ‘charges’ (the conserved entities of the symmetries) they re-discovered had been identified by Weinberg in 1965, based on conformal symmetries at infinity called the BMS symmetries (Bondi, actually also van der Burg, Metzner and Sachs), found and published by those first 3 in a paper, and Sachs in another, in 1962. See the Living Reviews article. Those 4 demonstrated in 1962 that there are additional symmetries at conformal infinity in asymptotically flat spacetime, besides the Poincare group. The BMS group was also used by them to define the BMS mass, in asymptotically flat spacetime, at conformal infinity. They found a family of symmetries called supertranslations, and another called superrotations, generalizations of the Poincare group which also inlcudes it, basically coming out of the conformal invariant structure. They also found that those symmetries were non-trivial, that were physical, and cannot be transformed away. Those symmetries turns out were an infinite set of diffeomrphisms, and lead to the soft hair. They occur for gravitational fields and for electromagnetic fields, so for soft photons and gravitons. Hawking et al. did their calculations for electromagnetic fields in a BH, but argued for the same effect for the gravitational field. They admit there’s still a lot to calculate.
So, the most promising (with all the unknowns and unproven theories yet) explanations for the entropy of a BH to be proportional to the horizon area is that the information on the state of the BH, at a quantum level, is imprinted in its horizon. If so it has to be proportional to area and not mass.
[BTW, a personal aside, I’m proud to have had Sachs as my advisor in General Relativity, but it was about 5-6 years later, and he was not doing the gravitational wave work he did earlier anymore, not of course did I]
Kwa kweli ni wingi wa kina, lakini si kwa njia ya kina hutumiwa katika thermodynamics ya kawaida. Ni uwiano na eneo na si wingi (au sawa kwa ajili ya kitu classic, kiasi). Entropy ni kawaida (lakini si daima) inalingana na wingi, au nishati, ambayo ni takriban inalingana na idadi ya chembe za msingi na hivyo idadi ya hali iwezekanavyo. Kwa shimo jeusi, idadi ya hali zinazowezekana ni sawa na idadi ya maeneo tofauti ya Planck katika upeo wa BH. Kwa mfano, katika hali ya habari, kama kwamba ni kuhifadhiwa katika upeo wa macho, si kiasi kikubwa, hatujui jinsi ya kuhesabu entropy ya singularity, fizikia kuvunjika na inahitaji quantum mvuto kuhesabu. Hawking alijua kwamba BH inaweza kuangaza kama mwili mweusi kwa joto lililopewa na inverse ya wingi wake. Kutoka hapo ni rahisi kupata entropy. "Kama ilivyoelezwa hapo juu, ""DQT"" ni joto linalopatikana katika mwili mweusi na ""T"" ni joto linalopatikana katika mwili mweusi, kwa hivyo matokeo ya Hawking ni ""T"" kwa kitu fulani cha mara kwa mara, kwa hivyo ""DQT"" ni ""K"" kwa kitu fulani cha mara kwa mara." Kwa kuwa joto linalopatikana katika BH ni nishati, huongeza uzito wa BH (katika vitengo vya asili) kama $ dQ = dM$. Kwa hiyo, kwa kutumia kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm = kdm Hawking alijua kwamba joto la mnururisho wa BH linategemea wingi wa nishati, lakini hakujua jinsi ya kuepuka hali hiyo. Hawking-Bekenstein alionyesha kwamba radiation ya Blackbody ni joto sawa na uzito wa uso katika upeo wa macho, ambayo ni sawa na uzito wa BH, na kwa hivyo ilisababisha entropy sawa na eneo la upeo wa macho, na kwa kweli sawa na idadi ya maeneo ya Planck mara $ kb (Boltzmann mara kwa mara) kulingana na usawa wa Hawking-Bekenstein. Hii inahusiana BH entropy na fizikia kwa thermodynamics <unk> na kwa kweli BH entropy inaweza kukua au kukaa sawa, lakini kamwe kupungua, na sheria ya pili ya thermodynamics. BH2 na BH2 ambazo ziliungana tarehe 9 14 15 zilikuwa na entropy ya mwisho kubwa kuliko jumla ya mbili zilizopita, pamoja na entropy fulani ilionekana mbali katika mionzi ya mvuto. Sheria za entropy za BH pia husababisha usawa kwa nishati ya juu ambayo inaweza kuchukuliwa kutoka kwa kuunganisha BH<unk>s. Bekenstein pia alionyesha kwamba BH entropies ni entropy upeo inawezekana kwa kiasi chochote cha nafasi na kiasi sawa na BH. Lakini swali bado linabaki: kwa nini, na jinsi gani? Ni jinsi gani kwamba hali iwezekanavyo ya BH ni encoded katika upeo wa macho, kama kweli tafsiri ya takwimu inashikilia? Kwa sababu ikiwa tu inashikilia tafsiri ya takwimu itakuwa juu ya ardhi thabiti, bila kujali mahusiano ya thermodynamic. Kwa hiyo kumekuwa na majaribio na mafanikio fulani (lakini hakuna uthibitisho au uhakika bado) katika matokeo mawili tofauti katika fizikia. Holographic Principle ni nadharia ya kisayansi iliyoundwa na Hooft na Susskind, ambayo bado haijathibitishwa na wengi. Kwa mfano, kwa mfano, kuunganisha kwa quantum gravity kwa d + 1 (d + 1) ni sawa na kuunganisha kwa d + 1 (d + 1) katika d + 1 (d + 1) katika d + 1 (d + 1) katika d + 1 (d + 1) katika d + 1 (d + 1) katika d + 1 (d + 1) katika d + 1 (d + 1) katika d + 1 (d + 1) katika d + 1 (d + 1) katika d + 1 (d + 1) katika d + 1 (d + 1) katika d + 1 (d). Kwa hivyo, kwa sababu ya kuunganisha kwa kawaida, matokeo ya AdS na CFT yaligunduliwa na Maldecana na wengine, ambao walithibitisha matokeo ya mvuto wa quantum wa kamba katika nafasi ya anti-de Sitter. AdS ni ufumbuzi wa utupu wa Einstein Field Equations na mara kwa mara hasi cosmological (de Sitter ni na mara kwa mara chanya cosmological, kikomo cha ulimwengu wetu unaojulikana kama umri wake huenda infinity na mara kwa mara cosmological kabisa kutawala). "Kutokana na ""AdS"" na ""CFT,"" kuna uwezekano wa kuwa na ""gauge-gravity duality"" (GG) au ""gauge-to-gravity duality"" (GGT)." CFT ni nadharia za quantum uwanja. Holographic conjecture na AdS-CFT imewafanya watu kufikiri kwamba habari juu ya hali ya quantum ya wingi, katika baadhi ya kesi au kwa ujumla, ni kuhifadhiwa katika mipaka yake. Holography ni njia ya kuunda picha ya holographic ya 3D. Lakini pia kuna baadhi ya counter-mfano, na hivyo kwa ujumla bado ni mbinu ya kuvutia, lakini si vizuri kueleweka au kukubaliwa. Hata hivyo, kama kesi ya jumla ni halali, mawazo ni kwamba hali ya BH ingekuwa encoded, au imprinted, juu ya upeo wake. "Hiyo inaweza kuwa na uwezo wa kuunda utaratibu wa habari ya quantum ambayo ilidhaniwa kuwa imepotea kwa BH, kuwa katika upeo wa macho na sio kupotea, na labda baadaye kuandikwa katika mwangaza wa Hawking ""mwili mweusi"" (na kitu cha ziada wakati huo)." Utafiti wa hivi karibuni unaonyesha kwamba habari kuhusu hali ya BH inaweza kuhifadhiwa kwenye upeo wa macho. "Hawking, Perry na Strominger, ""Mwaka wa 2016.""" Angalia karatasi katika arXiv, na Phys. Mchungaji. "Mwaka 2016 ""Mwaka 2016"" ni kitabu cha kwanza cha Phys.ru, lakini kinaendelea kuwa na matamshi tofauti." Kile wanachosema ni kipya, na kulingana na symmetries mpya re-discovered asymptotic katika conformal infinity. Kwa sababu ya ulinganifu huo, BH ina kiasi cha kuhifadhiwa, wanaita nywele laini. Hiyo ni, kwamba BH <unk>s kuwa na baadhi ya nywele ambayo ni zaidi na zaidi ya wingi, mwendo angular na malipo kuthibitishwa na Hawking miaka iliyopita. "Hii ni kwa sababu ya ""mfano"" wa BH, ambayo ni sawa na ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano"" wa ""mfano.""" Nywele laini ni kwa sababu ya photons laini au gravitons laini, ambayo kukaa katika upeo wa macho. "Kutokana na taarifa ya ""The New York Times,"" ""Hii barua inatoa maelezo ya wazi ya nywele laini katika suala la gravitons laini au photons juu ya black-shimo upeo wa macho, na inaonyesha kwamba taarifa kamili kuhusu hali yao quantum ni kuhifadhiwa juu ya sahani holographic katika mpaka wa baadaye wa upeo wa macho." Uhifadhi wa malipo hutumiwa kutoa idadi isiyo na kikomo ya mahusiano sahihi kati ya bidhaa za mvuke wa mashimo meusi ambayo yana nywele laini tofauti lakini kwa njia nyingine ni sawa. Kwa upande mwingine, nywele laini ambazo ziko chini ya urefu wa Planck haziwezi kusisimua katika mchakato unaowezekana kimwili, ikitoa idadi ya idadi ya digrii laini ya uhuru inayolingana na eneo la upeo wa macho katika vitengo vya Planck. Kwa hivyo wanadai kwamba wamedai au wameonyesha kwamba entropy ni kwa sababu ya digrii za uhuru, au hali zinazowezekana za, upeo wa macho, kulingana na BH thermodynamics. Bado, wao kufanya taarifa, katika mwili wa karatasi arXiv, kwamba hawajaonyesha kwamba kwa kweli kuna digrii ya kutosha ya uhuru wa kweli kuhifadhi habari zote, na kwamba kazi inabaki. "Hiyo ""nywele laini"" inatokana na symmetry mpya ambazo ziligunduliwa kwenye infinity ya conformal ya asymptotically flat spacetime (fikiria BH katika asymptotically flat spacetime), na ambayo husababisha kiasi kipya cha kuhifadhiwa." Kwa kuwa upeo wa BH ni mpaka mmoja wa asymptotically gorofa nafasi-wakati (yaani, nje ya upeo wa macho), wao kuonyesha kutumia Penrose michoro na wanaweza kuhesabu kuhifadhiwa nywele laini juu ya upeo wa macho BH. "Hii ni kwa sababu ya ""mali ya nyembamba"" ambayo iligunduliwa na Weinberg mnamo 1965 na ambayo ilitokana na ""mali ya nyembamba"" inayoitwa ""Bundy, Metzner, na Sachs"" ambayo iligunduliwa na kuchapishwa na wale watatu wa kwanza katika karatasi moja na Sachs katika nyingine mnamo 1962." Tazama makala ya Living Reviews. Katika mwaka wa 1962 alionyesha kwamba kuna symmetries ziada katika asymptotically gorofa nafasi-wakati, mbali na kundi Poincaré. BMS pia hutumiwa kuelezea uzito wa BMS katika nafasi ya asymptotically gorofa wakati wa infinity. "Kama ilivyoelezwa hapo juu, ""supertranslations"" ni aina ya mzunguko wa juu wa Poincaré, ambayo ni sawa na ""conformal invariant.""" Pia waligundua kwamba ulinganifu huo haukuwa wa kawaida, ulikuwa wa kimwili, na haungeweza kubadilishwa. Symmetries hizo hutokea kuwa seti isiyo na mwisho ya diffeomorphisms, na kusababisha nywele laini. Wao kutokea kwa ajili ya mashamba ya mvuto na kwa ajili ya mashamba ya umeme, hivyo kwa ajili ya photons laini na gravitons. Hawking et al. Kwa mfano, katika uchunguzi wa uwanja wa umeme wa BH, watu wengi walidai kuwa kuna athari sawa kwa uwanja wa mvuto. Wanakubali kwamba bado kuna mengi ya kuhesabu. Kwa hiyo, ufafanuzi wa kuahidi zaidi (kwa sababu ya mambo yote yasiyojulikana na nadharia ambazo bado hazijathibitishwa) kwa entropy ya BH kuwa inalingana na eneo la upeo wa macho ni kwamba habari juu ya hali ya BH, katika kiwango cha quantum, imechapishwa katika upeo wake. Kwa hiyo ni lazima iwe inalingana na eneo na si wingi. "Sasa, kwa sababu ya kuwa na uzoefu wa kutosha, nilikubaliana na Sachs na kuamua kwamba ""mfano wa mwelekeo wa jumla"" ni muhimu kwa sababu ya kwamba, kwa kweli, kuna uwezekano mkubwa kwamba mwelekeo wa mwelekeo wa jumla ni muhimu kwa sababu ya mwelekeo wa mwelekeo wa mwelekeo."