Some Simple Thermodynamics

Michael C. LoPresto, Henry Ford Community College, Dearborn, MI

n his recent popular book The Universe in a 4kG 2 Nutshell, Steven Hawking gives expressions for Eq. (1) as S = M . As is lost, the 1 c Ithe entropy and temperature (often referred to 8 kG 2 3 change in entropy will be dS = mdm. as the “Hawking temperature” ) of a black hole: c kc3 Multiplying by both sides of Eq. (2) gives Tds = 2 S = A (1) c dm. Integrating from the greater initial mass M to 4 G a lesser final mass M yields Q = (M – M)c2 = 2 c3 Mc , which is the heat energy lost when the black T = , (2) hole radiates away a mass M = M – M. Note that 8 kGM M is intrinsically a negative quantity. where A is the area of the , M is the The heat lost during a temperature change can h also be written as dQ = mCdT, which allows the spe- mass,k is Boltzmann’s constant, = (h being 2 cific heat (at constant pressure) of a black hole C to 2 Planck’s constant), c is the , and G is be determined. Substituting dQ = c dm and from c3 dm 8kG the universal . These expres- Eq. (2) dT = – yields C = – M. sions can be used as starting points for some interest- 8kG m2 c ing approximations on the thermodynamics of a The negative sign is not a surprise, since it can be Schwarzschild black hole, of mass M, which by defi- clearly seen from Eq. (2) and the above temperature nition is nonrotating and spherical with an event change that as a black hole loses mass and therefore horizon of radius energy, its temperature increases. 2GM 4,5 Note that integrating Tds over the entire mass M R = . 2 c2 of a black hole would give Q = –Mc , which suggests Hawking has theorized that during pair produc- that its total energy would eventually be radiated tion occurring just outside the event horizon, a black away as heat. Therefore, by the first law of thermo- hole slowly loses mass or evaporates as particles are dynamics, E = Q – W, there is no work being done radiated away. This, now known as “Hawking radia- on the event horizon as the volume decreases. This tion,” was initially described to many in his first suggests that the pressure on the event horizon is popular book A Brief History of Time. In this negligible and verifies that the specific heat is indeed process, anti-particles with negative energy fall into at constant pressure. the black hole actually causing the mass to decrease.6 The temperature increase with loss of mass shown Using the expression for the Schwarzschild radius, in Eq. (2) suggests that over time, the rate at which the entropy of a black hole of event-horizon area the energy is radiated from the black hole should A = R2 can be written in terms of its mass using also increase. The proportionality between specific

THE PHYSICS TEACHER Vol. 41, May 2003 DOI: 10.1119/1.1571268 299 heat and mass seen above supports the assertion as namics states that the entropy of a closed system well. must increase.17 If a black hole is in a reservoir of The rate of energy loss can be approximated with volume V and temperature T, and total energy dU/dt 4 the Stephan-Boltzmann radiation law, J = E = aVT 4, where a = in an alternative form of 2k4 A c = T 4, where = , the Stephan-Boltzmann the Stephan-Boltzmann Law,18 it can be shown with 603c4 7 the first law of thermodynamics that the entropy of constant. Rearranging the radiation equation and 4 substituting the area of the event horizon gives the reservoir is S = aVT 3. 19 The total energy of 3 dU 2 4 = 4R2 T 4. Substituting for the constant , the black hole and reservoir system E = mc + aVT dt remains constant and total entropy of the system the Schwarzschild radius, and the Hawking tempera- 4kG 4 S = m2 + aVT 3 has to increase as the black ture gives, after further rearranging, a rate of mass c 3 dm c4 1 hole radiates its energy into the reservoir. loss = , which will indeed 2 4 2 2 By conservation of energy, dE = 0 = c dm + aT dt 15360 G M 3 8,9 dV + 4aVT dT. The entropy change of the system increase as mass and energy are lost. 8kG 4 Solving for dt in the last expression and integrat- 3 2 would be dS = mdm + aT dV + 4aVT dT. 5120G 2 c 3 ing yields t = M3 = 10-16 M3, the lifetime Multiplying the entropy change by the temperature c4 10,11 T and subtracting the energy equation, then dividing of a black hole once it begins to evaporate. The 8kG Hawking temperature of a black hole can be approx- the temperature back out gives dS = mdm – imated from the values of the constants as c2 1 c 3 1023 dm + aT dV. Substituting the Hawking T Х ; this is only about 10-7 K12 above T 3 M temperature in the second term (physically appropri- absolute zero even for the smallest stellar black holes ate as it was from the energy of the black hole) (approximately 3 solar ).13 Since the average shows that it is equivalent to the first term. The first 1 temperature of the universe is about 2.7 K, most two terms therefore cancel and leave dS = aT 3 dV. black holes are absorbing more energy than they 3 emit and will not begin to evaporate for some time, Since the volume of the reservoir increases by the until the universe has expanded and cooled below same amount that the event horizon’s volume their temperature.14 Even once evaporation begins, decreases, this surprisingly simple expression for dS is by the above equation, a 3-solar-mass black hole indeed a positive quantity and therefore shows an 75 68 overall increase in the entropy of the system. would last 10 s or 10 yr! However, primordial, 4 mini-black holes, theorized by Hawking to have Substituting the value a = and the increase in 15 c been created during the , with masses of the volume of the reservoir, which is the opposite of about 1012 kg would have been much hotter and 20 13 the decrease in the event horizon’s volume, would evaporate in about 10 s or 10 yr. These 32G 3 may not have evaporated yet either, but they should dV = 4R2 dR= m2dm, shows, after quite a c6 much sooner than their stellar cousins. few cancellations, that once the system reaches equi- What actually happens when a black hole finally librium (when the reservoir reaches the Hawking radiates away the last of its mass is not clear, but at temperature), the expression for the entropy change such a high temperature a huge burst of x-rays or k dm gamma rays is likely. Nothing of this nature of the simplifies to dS = – . Integrating from the expected magnitude has even been observed.16 720 m The above expression for the change in the original to the final mass of the black hole gives k M k M entropy of a black hole shows that as a black hole S = – ln or finally, S = ln. loses mass through evaporation its entropy will 720 M 720 M decrease. However, the second law of thermody- Again, since the initial mass M is greater than the

300 THE PHYSICS TEACHER Vol. 41, May 2003 final mass M, the result is a positive quantity show- 9. B.W. Carroll and D.A. Ostlie, An Introduction to ing an increase in entropy. Modern Astrophysics (Addison Wesley Longman, New Note that the entropy depends only on the mass York, 1996), p. 673, does not give an expression but -2 of the black hole. Entropy is defined as the loga- mentions the M dependence. rithm of the number of states accessible to a sys- 10. Ref. 5, pp. 112 and 114, again gives an expression of tem20 and in the case of a Schwarzschild black hole, similar form, without as much detail with the con- mass is the only “state variable.”21,22 stants, but does not give a derivation. Despite the fact that these approximations apply 11. Ref. 9, p. 674, gives a similar expression without only to the simplest of black holes (the Schwarzs- derivation. child black hole), the results are still informative and 12. Agrees with value given in Ref. 6, p. 137. intriguing, perhaps especially for those who wish to 13. Ref. 9, p. 662. begin investigating some of the fascinating properties 14. Ref. 6, p. 137. of black holes23 in a bit more detail than Hawking 15. Ref. 6, pp. 127, 138–139. and others can go into in popular books, but with- out necessarily having to delve into the details of 16. Ref. 6, p. 139. and quantum mechanics.24,25 17. Ref. 16, Hawking states that the increase in entropy from the radiation more than makes up for the entropy decrease of the black hole. Appendix 18. Ref. 7. 4 If the energy of the reservoir is E = aVT , a change 19. See the Appendix for a proof of this. in the energy dE = dQ – dW can also be written as 20. Ref. 7, p. 42. E E dE = ΂΃ dS + ΂΃ dV. This is equivalent to 21. Ref. 9, p. 668, states that a black hole can be defined S v V s in terms of its mass, angular momentum, and charge. aT 4 dV + 4aT 3 dT = TdS + aT 4 dV, which reduces 22. Ref. 5 identifies various types of black holes (Kerr, etc.) that do rotate (therefore having angular momen- 2 4 3 to dS = 4aVT dT, or integrating, S = aVT . tum) and have charge, but defines a Schwarzschild 3 black hole as nonrotating and noncharged. References 23. D.V. Schroeder, An Introduction to Thermal Physics 1. S. Hawking, The Universe in a Nutshell (Bantam (Addison Wesley Longman, New York, 2000), pp. 83, Books, New York, 2001), p. 63. 84, 92, and 304, has derivations of the entropy, tem- perature, and rate of mass loss of a black hole as end- 2. R.M. Wald, Quantum Field Theory in Curved of-chapter problems. Spacetime and Black Hole Thermodynamics (University of Chicago Press, 1994), p. 124. 24. Ref. 2 is an excellent resource if one wishes to explore these details at a higher level. 3. Ref. 1, p. 118. 25. E.F. Taylor and J.A. Wheeler, Exploring Black Holes: 4. Ref. 1, p. 111. Introduction to General Relativity (Addison Wesley 5. C.A. Pickover, Black Holes: A Traveler’s Guide (Wiley, Longman, New York, 2000) is also excellent and at a New York, 1996), p. 52. more intermediate level. 6. S. Hawking, The Illustrated Brief History of Time, PACS codes: 44.90, 95.90 updated and expanded ed. (Bantam Books, New York, 1996), pp. 136–137. 7. C. Kittell and H. Kroemer, Thermal Physics, 2nd ed. Michael C. LoPresto is currently serving as the chair of (Freeman, New York, 1980), p. 96. the physics department at Henry Ford Community College. 8. Ref. 5, p. 116, gives an expression of similar form, but Henry Ford Community College, Dearborn, MI 48128; does not go into as much detail with the constants. [email protected]

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