BBlack hole essay
Enrico Brehm
Max Planck Institute for Gravitational PhysicsAlbert Einstein InstitutePotsdam-Golm, D-14476, Germany
E-mail: [email protected]
Abstract:
This essay gives a very general introduction to Schwarzschild black holes. First,it focuses on some of its classical features as solutions to Einstein’s theory of gravity. In thesecond part it discusses briefly some specific quantum aspects and how a black hole pro-cesses quantum information. No previous knowledge about black holes, gravity or quantummechanics is required. a r X i v : . [ phy s i c s . pop - ph ] J a n The classical viewpoint
Black holes are some of the strangest phenomenons in our univers. Here, we first wantto discuss them as objects of a classical physical theory, where classical means that weforget about all possible quantum effects and their consequences. In the present case theunderlying classical theory is Einstein’s theory of gravity [1]. It describes space and time interms of a collection of fields whose behaviour is dictated by the Einstein equations.One natural question to ask is how does this theory describe space and time around amassive spherical object like a star? The solution to this was found by Schwarzschild [4].It is valid all around any static round object and, remarkably, only depends on its mass.However, quite strange things can happen when all that mass is packed within a specificradius named after Schwarzschild. Then a so-called event horizon forms at the Schwarzschildradius and we start talking about a black hole.Before we shed more light on the strangeness of black holes, let us try to get some intu-ition for the circumstances under which black holes can appear. The Schwarzschild radius r s and the mass M of a spherical object have a very easy relation: they are proportionalto each other, r s = a · M , with a being very small when we measure in standard units.Let us for example consider an object with the mass of earth, then its Schwarzschild radiusis given by only about 9 mm! No physical process is known in which all of earth can becompressed that much and it seams quite unlikely that there are many (if at all) black holeswith the mass of earth. The situation changes when we consider bigger and bigger masses.This is because the volume within the Schwarzschild radius, i.e. , the space where we canstore all the mass to form a black hole, increases much faster. In fact, if we double the masswe get roughly eight times more volume to store it. The formation of black holes becameseasier the heavier they are. It is known that there are mechanisms at the end of the lifetimeof very massive stars that lead to the formation of so-called stellar black holes [3]. Evenheavier black holes can for example develop when stellar ones merge.Let us return to the promised strangeness of black holes. Actually, if we are far awayfrom the black hole, it is not much different from any other stellar object of the same mass.The only big difference is that we do not see any light originating from the black hole. Aninteresting effect, that occurs when we come closer to it, is that time for us passes slowerthan for those who stay away from the black hole. However, the effect is not tractable in adirect way. Any clock that we carry with us behaves completely fine from our perspective.Only if we return back to a place far away from the black hole and compare the timespassed we could see a difference. This effect appears, in fact, for any massive object thatwe come close to and is not only a feature of black holes. Remember that the solution toEinstein’s equations, which in particular tells us how time behaves, outside the sphericalobject only depends on its mass! However, for any stellar object that is not a black hole wewould at some point reach that object and enter it. Inside of it the solutions do depend onits specifics. One can show that effects like time dilation can not increase further arbitrarily.If we, however, come closer and closer to the black hole the effect will grow without a bounduntil we reach the previously mentioned event horizon.Passing the event horizon has some severe consequences. If we take for example the– 1 –atter observation of unbounded time dilatation serious, we come to the conclusion that inthe moment that we spend on the horizon all time passes for anything outside the blackhole. The end of all things happens in the rest of the universe, and in fact after the momentin which we enter the black hole, there is no way back. This is also visible in a remarkableand strange feature of Schwarzschild’s solution. If we compare it inside and outside of theevent horizon one observes that global time and the radial direction interchanged theirmeaning. In a world outside the black hole everything and everyone has to move forward intime. This is a fundamental feature of Einstein’s theory. Inside the event horizon the radialdirection takes the place of time with the drastic consequence that, no matter what, wehave to move forward in this direction, where forward means towards the center of the blackhole. There really is no way back! Not even the most powerful rocket that we could imaginecan prevent us from finally reaching the very center of the black hole, where gravitationalforces become immeasurably strong and at the latest there the quantum features of blackholes and gravity itself must show their face.However, other quantum aspects of black holes will be visible much earlier. Some ofthem will be discussed in what follows. We have seen before that Schwarzschild’s solution only depends on the mass of the object.But what happens with all the information about the object that collapsed into a blackhole? It persisted of many different particles, it had a temperature, a matter distribution, aspecific spectrum of radiation, and so on. If we only believe in the classical world, then allthat information is hidden behind the event horizon after the formation of the black hole.Then it is by no means tractable for anyone outside the black hole. In a classical worldthis is not a big problem. It is at most sad that no one outside the black hole can get theinformation but causes no issues concerning consistency of the theory.However, we know that our world is in fact not a classical one, and our knowledge aboutthe quantum theories that describe at least ordinary matter in our universe is rather decent.Using this knowledge Hawking could show that quantum effects near the event horizon ofa black hole lead to a constant flow of particles away from it [2]. A black hole radiates and,hence, must loose mass over time. If we wait long enough a black hole evaporates eithercompletely or until some tiny remnant of it is left.Where is all the information after the evaporation? Now that we include some quan-tumness in the description of black holes this question becomes very important. Thoughtlessprocessing of quantum information can easily lead to inconsistencies. For example, informa-tion has to be spread very fast inside the black hole. Otherwise it would be possible to copyquantum states which is strictly forbidden in any consistent quantum theory. To be honestthere is no real consensus among the physics community on how the black hole deals withquantum information. One possibility might be that it is hidden in its Hawking radiation.If we wait long enough and collect a sufficient amount of it we might be able to regain allthe information we want. However, there are many more gedankenexperiments concerningsimilar issues. Finding a convincing and self-consistent description of black holes in contact– 2 –ith the quantum world will probably be an important step towards a quantum descriptionof gravity itself. This might be one of the next big steps in theoretical physics!At last, let us try to get some intuition on how important quantum effects of a blackhole are. If we first consider an ordinary stellar black hole of, say, four times the mass ofour sun, then its Hawking radiation can be associated with a temperature only roughly ahundred millionth Kelvin above the absolute zero temperature. It, hence, plays almost norole in describing the everyday physics of that black hole. This is true for any stellar (oreven heavier) black hole. Next let us consider a coin of, say, five gram. Quantum effects ofthat coin do not play any significant role in its everyday physics. It can be described almostperfectly by classical theories. However, if we consider a black hole of the same mass, thingslook rather different. As a (partially) quantum object it radiates and evaporates within atiny fraction of a second. All its mass converts into energy which results in an explosion threetimes stronger than the bomb dropped on Hiroshima. We see that for black holes quantumeffects play a significant role much earlier than for matter under ordinary circumstances. References [1] A. Einstein. Die Grundlage der allgemeinen Relativitätstheorie.
Annalen der Physik ,354:769–822, 1916.[2] S. W. Hawking. Particle creation by black holes.
Communications in Mathematical Physics ,43(3):199–220, Aug 1975.[3] J. R. Oppenheimer and G. M. Volkoff. On Massive Neutron Cores.
Physical Review ,55:374–381, February 1939.[4] K. Schwarzschild. On the Gravitational Field of a Mass Point According to Einstein’s Theory.