Black hole thermodynamics

Mount Tai CollegeBlack holeThermodynamicsLeiShen1-[1]Rui Yuan 2-[2]X.H. Long(Xuehao Long)3-[3]Shandong UniversityMount T ai College Physics MajorCorrespond to author: sdhuxz@Date 2012.6Abstract:Black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black holeevent horizons. It involves so many things of advanced theorem that we are far away from understanding it. We just do some simple calculation about the properties and describe the behavior of the black hole by using classical or half classical theory of thermodynamics.Keywords: black hole thermodynamics Hawking radiationTable of contents:1)The black hole1.1 The brief introduction of black hole1.2The radius of black hole(Schwarzschild radius).1.3 The entropy of the black hole.1.4The temperature of the black hole.2)The black hole thermodynamics2.1 The laws of black hole mechanics.2.2 The source of the BHT second law.2.3 Interpretation of the laws.3)Hawking radiation and the lifetime of black hole4) Authors5) References1. The black hole1.1 The brief introduction of black hole.A black hole is a region of spacetime where gravity prevents anything, including light, from escaping. The theory of general relativity predicts that a sufficiently compact mass will deformspacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon [1] that marks the point of no return. It is called "black" because it absorbs all the light that hits the horizon. It is called "black" because it absorbs all the light that hits the horizon, reflecting nothing, just like a perfect black body in thermodynamics.1.2The radius of black hole (Schwarzschild radius).First, we can just use dimensional analysis we can show that a black hole of mass M should have a radius of order GM/c2, where G is Newton’s gravitational constant and c is the speed of light.In the SI system, the units of G are N m2/kg2, so the units of G can also be written as m3/(kg∙s2). Meanwhile M has units of kg and c has units of m/s. We want a combination of these three quantities that has units of meters. We can get meters from either G or c, but eachalso involves second; the only way to make the seconds cancel is to take the combination G/c2, which has units of m/kg. Now we have meters to the first power as desired. To cancel the kg in the denominator, just multiply by M:Radius=λGMc2(λ is a purely numerical factor) (1.1)So, we can calculate the radius of a one-solar-mass black hole (M=2×1030kg).γ=λGMc2=λ×1482m(1.2)Schwarzschild radius：The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is the distance from the center of an object such that, if all the mass of the object were compressed within that sphere, the escape speed from the surface would equal the speed of light. An example of an object smaller than its Schwarzschild radius is a black hole. Once a stellar remnant collapses within this radius, light cannot escape and the object is no longer visible. It is a characteristic radius associated with every quantity of mass. The Schwarzschild radius was named after the German astronomer Karl Schwarzschild who calculated this exact solution for the theory of general relativity in 1915.Formula for the Schwarzschild radius:The Schwarzschild radius is proportional to the mass with a proportionality constant involving the gravitational constant and the speed of light:γs=2Gm c2,Where:γs is the Schwarzschild radius;G is the gravitational constant;m is the mass of the object;c is the speed of light in vacuum.The proportionality constant, 2G/c2, is approximately 1.48×10−27 m/kg, or2.95 km/solar mass.An object of any density can be large enough to fall within its own Schwarzschild radius,V s∝ρ−32,where:V s is the volume of the object;ρis its density.1.3 The entropy of the black hole.In fact, it is irreversible in the thermodynamics sense as well: Adding mass to black hole increase the black hole’s entropy. It turns out that there’s no way to tell (at least from outside) what kind of matter has gone into making a black hole. Therefore, the entropy of a black hole must be greater than the entropy of any conceivable type of matter that could have been used to create it. Knowing this, it’s not hard to estimate the entropy of a black hole.Ordinarily, the entropy of a system is of the same order as the number of particles in the system. If we take a system of N particles and compress it to form a black hole, the second law requires that when we’re done, the entropy of the black hole is still at least of order N. but since the end result is the same whether we start with a lot of particles or a few (with the same total mass), the final entropy must in fact be of the order of the maximum N, the largest possible number of particles that it could have been formed from.To make a black hole out of the maximum possible number of particles we should use particles with the lowest possible energy: long-wavelength photons (or other massless particles). But the wavelength can’t be any longer than the size of mass M.Suppose we start with N photons, each of which has a wavelength equal to the size of the blackhole: γ=λGM/c2. Each photon has energyϵ=hcγ,and the total energy of all of them mustequal M c2:M c2=N∈=Nℎcγ=Nℎc3λGM(1.3)Solving for N givesN=λG M2ℎc(1.4)And so the entropy in conventional units must be of orderS~G M2k Bℎc(1.5)Aside from a factor of8π2, our result should agree with the exact formula for the entropy of a black hole, obtain through a much more difficult calculating.That is:S=8π2G M2k Bℎc(1.6)So, we can calculate the entropy of a one-solar-mass black hole. For a one-solar-mass black hole:S k B =8π26.67×10−11N∙m2kg22×1030kg26.63×10−34J∙s3×108m/s=1.06×1077(1.7)Or in conventional units:S= 1.06×1077 1.38×10−23J/K=1.5×1054J/K(1.8)This is an enormous entropy. For comparison, an ordinary star like the sun contains of order 1057particles, so its entropy is something like 1057K. To equal the entropy of a single one-solar-mass black hole, you would need 1020ordinary stars, or enough to populate a billion (109) Milky way galaxies. Furthermore, since the entropy of a black hole is proportional to the square of its mass, a million-solar-mass black hole (as may exist at the center of our galaxy) would have a trillion (1012) times the entropy of a one-solar-mass black hole.In fact, Stephen Hawking proved that the black hole entropy was proportional to the area ofits event horizon divided by the Planck area and the constant is 14. The exact formula is:S=k B A4l P2(1.9)Where A is the area of the event horizon, calculated at 4πR2, k B is Boltzmann's constant, and l P=Għ/c3is the Planck length. That is:S=14c3k BGħA(1.10)The formula for the Bekenstein–Hawking entropy S of a black hole, which depends on the area of the black hole A. The constants are the speed of light c, the Boltzmann constant k, Newton's constant G, a nd the reduced Planck constant ħ.Use both equation we can get the radius of the event horizon R:R=2GMc2(1.11)1.4 The temperature of the black hole.We can use the result to calculate the temperature of a black hole, in terms of its mass M. (The energy is M c2.)The entropy of a black hole isS=8π2GM2k Bℎc =8π2G Mc22k Bℎc5=8π2GU2k Bℎc5(1.12)Where U is the black hole’s energy. To find the temperature, just differentiate with respect to U:1 T =dSdU=16π2GUk Bℎc5=16π2GMk Bℎc3,(1.13)T=ℎc316π2GMk B(1.14)For M=2×1030kg, T=6.1×10−8K, a temperature so tiny that for most purposes it may as well be zero. A graph of S vs. U is an upward-opening parabola:1.1The concave-up shape indicates that as energy is added to this system, its tendency to absorb energy increases—it become more and more “greedy”, or less “generous”, that is, colder.Note:[1] In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer.2. The black hole thermodynamics.2.1 The laws of black hole mechanics.The four laws of black hole mechanics are physical properties that black holes are believed to satisfy. The laws, analogous to the laws of thermodynamics, were discovered by Brandon Carter, Stephen Hawking and James Bardeen.The laws of black hole mechanics are expressed in geometrized units.(a)The Zeroth LawThe horizon has constant surface gravity for a stationary black hole.(b)The First LawChange of mass is related to change of area, angular momentum, and electric charge by:dM=k BdA+ΩdJ+ФdQ(2.1)8πwhere M is the mass, k is the surface gravity, A is the horizon area, Ω is the angular velocity, J is the angular momentum, Ф is the electrostatic potential and Q is the electric charge.(c)The Second LawThe horizon area is, assuming the weak energy condition, a non-decreasing function of time,dA≥0(2.2)dtThis "law" was superseded by Hawking's discovery that black holes radiate, which causes both the black hole's mass and the area of its horizon to decrease over time.(d)The Third LawIt is not possible to form a black hole with vanishing surface gravity. k = 0 is not possible to achieve.2.2 The source of the BHT[2] second law.The only way to satisfy the second law of thermodynamics is to admit that black holes have entropy. If black holes carried no entropy, it would be possible to violate the second law by throwing mass into the black hole. The increase of the entropy of the black hole more than compensates for the decrease of the entropy carried by the object that was swallowed.In 1971, Hawking showed under general conditions that the total area of the event horizons of any collection of classical black holes can never decrease, even if they collide and merge. This result, now known as the second law of black hole mechanics, is remarkably similar to the second law of thermodynamics, which states that the total entropy of a system can never decrease.“I had already discussed with Roger Penrose the idea of defining a black hole as the set of eventsfrom which it was not possible to escape to a large distance, which is now the generally accepted definition. It meansthat the boundary of the black hole, the event horizon, is formed by the light rays that just fail to escape from the blackhole, hovering forever. It is a bit like running away from the police and just managing tokeep one step ahead but not being able to get clear away!Suddenly I realized that the paths of these light rays could never approach one another. If they did they musteventually run into one another. It would be like meeting someone else running away from the police in the oppositedirection – you would both be caught! (Or, in this case, fall into a black hole.) But if these light rays were swallowed upby the black hole, then they could not have been on the boundary of the black hole. So the paths of light rays in theevent horizon had always to be moving parallel to, or away from, each other. Another way of seeing this is that theevent horizon, the boundary of the black hole, is like the edge of a shadow – the shadow of impending doom. If youlook at the shadow cast by a source at a great distance, such as the sun, you will see that the rays of light in the edgeare not approaching each other.If the rays of light that form the event horizon, the boundary of the black hole, can never approach each other, the areaof the event horizon might stay the same or increase with time, but it could never decrease because that would meanthat at least some of the rays of light in the boundary would have to be approaching each other. In fact, the area wouldincrease whenever matter or radiation fell into the black hole.……The non decreasing behavior of a black hole’s area was very reminiscent of the behavio r of aphysical quantity calledentropy, which measures the degree of disorder of a system.”——Stephen Hawking[3]2.1[4]2.3 Interpretation of the lawsThe four laws of black hole mechanics suggest that one should identify the surface gravity of a black hole with temperature and the area of the event horizon with entropy, at least up to some multiplicative constants. If one only considers black holes classically, then they have zero temperature and, by the no hair theorem,zero entropy, and the laws of black hole mechanics remain an analogy. However, when quantum mechanical effects are taken into account, one finds that black holes emit thermal radiation (Hawking radiation) at temperatureT H=κ(2.3)2πFrom the first law of black hole mechanics, this determines the multiplicative constant of the Bekenstein-Hawking entropy which isS BH=A(2.4)4Note:[2] Black Hole Thermodynamics.[3] A Brief History of Time, Ch.7, Stephen Hawking.[4]A Brief History of Time, Ch.7, Stephen Hawking.3. Hawking radiation and the lifetime of black hole.A black hole is a blackbody if ever there was one, so it should emit blackbody radiation, called Hawking radiation.A black hole of mass M has a total energy of M c2, a surface area of A, and a temperature of T.A=16πG2M2c4(3.1)T=h c316π2k B GM(3.2)Then we can estimate the typical wavelength of Hawking radiation emitted by a one-solar-mass (M=2×1030kg) black hole.The temperature of that black hole is:T=6×10−8K(3.3)For a blackbody at this temperature, the peak of radiation spectrum would be atϵ= 2.82kT=1.5×10−11eV. [5]This corresponds to a wavelength of λ:λ=hcϵ=84km(3.4)More generally:λ=hc2.82k B T=ℎc2.8216π2GMℎc3=(28.0)2GMc2(3.5)The quantity 2GMc is just the “radius” of the black hole.Then we can calculate the total power radiated by a one-solar-mass black hole.According to Stefan’s law[5]:Power=σA T4=(2π5k415ℎ3c2)(16πG2M2c4)(ℎc316π2GM)4=ℎc6=9×10−31W=6×10−12eV/s(3.6)Imagine a black hole in empty space, where it emits radiation but absorbs nothing. As it loses energy, its mass must decrease; one could say it “evaporates.” The power of this process is:d Mc2 dt =−σA T4=−ℎc6(30720)π6G2M2(3.7)That is,dM dt =−HM2(3.8)Where H=ℎc4(30720)π6G2=4.1×1013kg3/s. We could refer to H as Hawking’s constant.We can obtain an expression of the lifetime of the black hole in terms of its initial mass byderiving this differential equation for the mass as a function of time, and solving this equation ：M 2dM =−H dt τ00M i (3.9) M i 3=3Hτ , τ=M i 3/3H (3.10)For a one-solar-mass black hole,M i =2×1030kg the lifetime should beτ=6.5050×1076s (3.11)That ’s 2×1069 years or more than 1059 times the age of the known universe(about 1.3×1010years ).For a black hole was created early in the history of the universe (the age of the known universe is about 15 billion years or 5×1017 seconds) finishes evaporating today, its initial mass M i is:M i =(3Hτ)13=[3 4.1×1013kg 3s 5×1017s ]13 =4.0×1010kg (3.12)Therefore the peak of the blackbody radiation is atλ=hc2.82kT = 28.0 2GM c 2=7.28×10−15m (3.13) ϵ=hc λ=1.7×108eV (3.14)That ’s a very hard gamma ray, a hundred times more energetic than gamma rays emitted in nuclear reactions.Note:[5]The Stefan –Boltzmann law, also known as Stefan's law, states that the total energy radiated per unit surface area of ablack body per unit time (also known as the black -body irradiance or emissive power), j, is directly proportional to the fourth power of the black body's thermodynamic temperature T (also called absolute temperature):j =ςT 4.The constant of proportionality σ, called the Stefan–Boltzmann constant or Stefan's constant, derives from other knownconstants of nature. The value of the constant is :ς=2π5k B 4=5.670400×10−8Js −1m −2K −44. AuthorsL.Shen(Shen Lei) Shandong University Taishan College Physics MajorX.H.Long(Long Xuehao) Shandong University Taishan College Physics MajorR.Yuan(Yuan Rui) Shandong University Taishan College Physics Major.5. References1. Hawking, Stephen W., A Brief History of Time.2. Wikipedia, Black hole, /wiki/Black_Holes.3. Wikipedia, Black hole thermodynamics, /wiki/Black_hole_thermodynamics.4. Daniel V. Schroeder, Thermal Physics.5. Bardeen, J. M.; Carter, B.; Hawking, S. W. (1973). "The four laws of black hole mechanics". Communications in Mathematical Physics 31 (2): 161–170. Bibcode1973CMaPh..31..161B. DOI:10.1007/BF01645742.6. Bekenstein, Jacob D. (April 1973)."Black holes and entropy". Physical Review D 7 (8): 2333–2346. Bibcode 1973PhRvD...7.2333B.DOI:10.1103/PhysRevD.7.2333.7. Hawking, Stephen W. (1974). "Black hole explosions?". Nature 248 (5443): 30–31. Bibcode 1974Natur.248...30H. DOI:10.1038/248030a0.8. 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