真实之路 宇宙法则的完全指南 (彭罗斯)27宇宙大爆炸及.pdf

27 The Big Bang and its thermodynamic legacy 27 1Time symmetry in dynamical evolution What sorts of laws shape the universe with all its contents The answer provided by practically all successful physical theories from the time of Galileo onwards would be given in the form of a dynamics that is a speciWcation of how a physical system will develop with time given the physical state of the system at one particular time These theories do not tell us what the world is like they say instead if the world was like such and such at one time then it will be like so and so at some later time Such a theory will not tell us how the world is shaped unless we tell it how the world was shaped There have been important exceptions to this form of things such as Kepler s wonderful conclusion in 1609 that the orbits of the planets about the Sun have certain geometrical shapes ellipses with the Sun at one focus described with speeds satisfying speciWc rules That was an assertion about how the universe is rather than how its state might develop from moment to moment in accordance with some dynamical law But our present perspective on Kepler s geometrical motions is that they are mere consequences of 17th century gravitational dynamics as Wrst shown by Newton and published in his great Principia of 1687 and Kepler s laws are not to be thought of as directly fundamental to the ways of Nature Indeed it could be argued that Kepler and science as a whole was immensely fortunate that the nature of the speciWc law of force governing Newton s gravity the inverse square law 17 3 has the property that all the orbits of small bodies about a central force are actually simple and elegant mathematical shapes and indeed shapes that had been intensively studied by the ancient Greeks some eighteen centuries earlier For this is a very exceptional property shared by hardly any other simple central force law In general our modern perspective holds that it is the dynamical laws that we expect to have an elegant mathematical form and it is a matter of good fortune for us if we happen to Wnd simple mathematical shapes as consequences of these laws 686 Theusualwayofthinkingabouthowthesedynamicallawsactisthatitis the choice of initial conditions that determines which particular realization ofthedynamicshappenstooccur Normally onethinksintermsofsystems evolvingintothefuture fromdataspeciWedinthepast wheretheparticular evolution that takes place is determined by diVerential equations These would be partial diVerential equations Weld equations when there are dynamically evolving Welds or wavefunctions see 10 2 19 2 6 21 3 Exercise 19 2 and Note 21 1 One does not on the other hand tend to thinkofevolvingthesesameequationsintothepast despitethefactthatthe dynamical equations of classical and quantum mechanics are symmetrical under a reversal of the direction of time As far as the mathematics is concerned one can just as well specify Wnal conditions at some remote futuretime andevolvebackwardsintime Mathematically Wnalconditions are just as good as initial ones for determining the evolution of a system Some comments are called for concerning this time symmetrical dy namical determinism First the reader may be reassured that it is not substantially invalidated by the framework of either special or general relativity Data deWning the state of the system are speciWed at some initial time which is some initial spacelike 3 surface and these data evolve according to the dynamical equations to determine the physical state of the system to the future and also to the past of that 3 surface There are however some new issues that are raised by general relativity because the very structure of the spacetime into which the evolution Xows is part of the physical state to be determined This has particular implications in the context of black holes that we shall need to confront later see 28 8 30 4 9 In the case of quantum mechanics the determinism refers to the U part of that theory only the quantum state being taken to be governed by Schro dinger s equation or equivalent Under time reversal the T re ferred to in 25 4 the time derivative operator i h t of Schro dinger s equation 21 3 must be replaced by i h t since t 7 t Provided that the Hamiltonian is an ordinary one which goes to itself under the action of T we see that Schro dinger evolution also goes to itself so long as we accompany the time reversal t 7 t by a reversal of the sign of the imaginary unit i 7 i Indeed this is how we think of the action of T in quantum mechanics We may note that a positive frequency function f t is converted back to a positive frequency function under the combined replacements t 7 t and i 7 i so all is well in this respect 27 1 The behaviour of quantum state reduction R under the action of T is another matter however and it will provide an important issue for our deliber ations in Chapter 30 30 3 27 1 Why Also explain why spatial momentum is handled consistently by this replacement 687 The Big Bang and its thermodynamic legacy 27 1 27 2Submicroscopic ingredients There are however other questions that might worry the knowing reader even just with regard to classical dynamics Time reversal symmetry is certainly true of the submicroscopic dynamics of individual particles and their accompanying Welds in classical mechanics But in practice one has little knowledge of the behaviour of the individual ingredients of a system A knowledge of the detailed position and momentum of every particle is normally deemed to be both unobtainable and unnecessary the overall behaviour of the system being well enough described in terms merely of some appropriate averages of the physical parameters of indi vidual particles These would be things like the distribution of mass momentum and energy the location and velocity of the centre of mass the temperature and pressure at diVerent places the elasticity properties the moments of inertia the detailed overall shape and its orientation in space etc An important issue therefore is whether or not a good initial knowledge of such averaged overall parameters will in practice suYce for determining the dynamical behaviour of the system to an adequate degree This is certainly not always the case Systems known as chaotic have the property that the Wnal behaviour depends critically on exactly how they are started oV As a familiar example there is an executive toy in which a magnetic pendulum swings just above a collection of magnets placed in some arrangement on the base See Fig 27 1 The dynamical behaviour is well enough governed in a deterministic way by Newton s laws and the laws of magnetostatics together with the slowing down due to frictional resistance of the air Yet the Wnal resting place of the pendulum depends so critically upon the initial state that it is eVectively unpredictable although a fully detailed knowledge of this initial state with all the constituent particles and Welds would certainly Wx this evolution uniquely 1Many other examples of such chaotic systems are known A good measure of the vagary of weather prediction is commonly attributed to the chaotic nature of the dynamical systems involved Even the highly ordered and very predictable Newtonian gravitational motion of bodies in the solar Fig 27 1Chaotic motion An executive toy consisting of a magnetic pendulum swinging just above a collection of fixed magnets The actual path taken by the pendulum depends extremely sensitively on its initial position and velocity 688 27 2CHAPTER 27 system probably constitutes technically a chaotic system although the timescales that are relevant to such chaos are vastly longer than those of astronomical observation What about evolution into the past rather than the future It would be a fair comment that such chaotic unpredictability is normally much worse for the retrodiction that is involved in past directed evolution than for the prediction of the normal future directed evolution This has to do with the Second Law of thermodynamics which in its simplest form basically asserts 2 Heat Xows from a hotter to a colder body In accordance with this law if we connect a hot body to a cold one using some heat conducting material then the hot body will become cooler and the cold body warmer until they settle down to the same temperature This is the expectation of prediction and this evolution has a deterministic character If on the other hand we view this process in the reverse time direction then we Wnd the two bodies at eVectively the same temperature spontaneously evolving to bodies of unequal temperature and it would be a practical impossibility to decide which body will get hotter and which colder how much and when This procedure of dynamical retrodiction for this system is clearly a hopeless prospect in practice In fact this diYculty would apply to the retrodiction of almost any macroscopic system with large numbers of constituent particles behaving in accordance with the second law For this kind of reason physics is normally concerned with prediction rather than retrodiction 3As another aspect of this the Second Law is considered an essential ingredient to the predictive power of physics as it removes those problems that we just encountered with retrodiction Nevertheless many physicists would take the view that this law is not fundamental in the same sense that say the law of conservation of energy the principle of linear superposition in quantum mechanics and perhaps the standard model of particle physics are fundamental They would argue that the second law is an almost obvious necessary ingredient to any sensible physical theory Many would take the view that it is something vague and imprecise and that it in no way can compare with the extraordinary precision that we Wnd in the dynamical laws that control fundamental physics I wish to argue very diVerently and to demonstrate the almost mind blowing precision that lies behind that seemingly vague statistical principle that we usually simply refer to as the Second Law 689 The Big Bang and its thermodynamic legacy 27 2 27 3Entropy Let us examine somewhat more exactly what the Second Law actually states As a preliminary I should inform the reader of the First Law of thermodynamics The First Law is simply the statement that the total energy is conserved in any isolated system The reader might well complain that this is hardly something new 18 6 20 4 21 4 But when this law was put forward initially by Sadi Carnot in the early 1820s although not published by him4 it had not been clear previously that heat is just a form of energy nor was the ordinary macroscopic notion of energy itself completely clear The Wrst law makes it explicit that the total energy is not lost when say a body loses its kinetic energy 18 6 as it slows down because of air resistance For this energy is simply taken up in heating the air and the body This heat energy is understood as primarily kinetic energy in the motions of air molecules and vibrations of particles compos ing the body Moreover temperature is simply a measure of energy per degree of freedom so the thermodynamic notions of heat and temperature are basically the same as previously understood dynamical notions but applied at the level of the individual constituents of materials and treated in a statistical way The First Law has the kind of precision that we are familiar with the value of something namely the total energy remains constant despite the fact that all kinds of complicated processes may be taking place The total energy after the process is equal to the total energy before the process Whereas the Wrst law is an equality the second law is an inequality It tells us that a diVerent quantity known as the entropy has a larger or at least not smaller value after some process takes place than it had before Entropy is very roughly speaking a measure of the randomness in the system Our body moving through the air starts with its energy in an organized form its kinetic energy of motion but when it slows down from air resistance this energy gets distributed in the random motions of air particles and individual particles in the body The randomness has increased more speciWcally the entropy has increased The notion of entropy was introduced by Clausius in 1865 but it was the outstanding Austrian physicist Ludwig Boltzmann who in 1877 made the deWnition of entropy clear or at least as clear as it seems possible to make it To understand Boltzmann s idea for a classical system we need the notion of the phase space 12 1 14 1 8 20 1 2 4 which we recall for a classical system of n featureless particles is a space P of 6n dimensions each of whose points represents the entire family of positions and mo menta of all n particles In order to make the notion of entropy precise we require a concept of what is called coarse graining 5We can think of this as a division of the phase space P into a number of subregions which I shall 690 27 3CHAPTER 27 Macroscopically indistinguishable Phase space P coarse grained refer to as boxes See Fig 27 2 The idea is that collections of points of P that represent states of the system that are indistinguishable from one another with regard to macroscopic observations are considered to be grouped together in the same box but points of P belonging to diVerent boxes are deemed to be macroscopically distinguishable The Boltzmann entropy S for the state of the system represented by some point x of P is S klogV where V is the volume of the box V that contains x this being a natural logarithm see 5 3 and where k is Boltzmann s constant 6having the value k 1 38 10 23J K 1 where J means joules and K 1means per degree kelvin I have said that Boltzmann s deWnition makes the notion of entropy clear But for the above formula for S to represent something physically precise it would be necessary to have a clear cut prescription for the coarse graining that our family of boxes is supposed to represent There is undoubtedly something arbitrary in the particular division into boxes that one might happen to select The deWnition seems to depend upon how closely one chooses to examine a system Two states that are macroscopic ally indistinguishable to one experimenter might be distinguishable to another Moreover exactly where the boundary between two boxes happens to be drawn is again very arbitrary since two neighbouring points of P with one on each side of the boundary might be assigned quite diVerent entropies despite their being virtually identical There is still something very subjective about this deWnition of S despite its being a distinct advance on earlier notions of more limited applicability and an undoubted improvement on the idea of just a measure of the randomness in a system Fig 27 2Boltzmann entropy This involves the division of phase space P into sub regions boxes called a coarse graining of P where the points of a given box represent physical states that are macroscopically indistinguishable Boltzmann s deWnition of the entropy of a state x in a box V of volume V is S klogV where k is Boltz mann s constant 691 The Big Bang and its thermodynamic legacy 27 3 My own position concerning the physical status of entropy is that I do not see it as an absolute notion in present day physical theory although it is certainly a very useful one There is however the possibility that it might acquire a more fundamental status in the future For this quantum physics would certainly need to be taken into consideration and in any case it is quantum mechanics that provides an absolute measure to any particular phase space region V contained in P where units may be chosen so that h 1 as with Planck units see 27 10 27 2 Be that as it may it is remarkable how little eVect the arbitrariness in coarse graining has in the calculations of thermodynamics It seems that the reason for this is that in most considerations of interest one is concerned with absolutely enormous ratios between the sizes of the relevant phase space box volumes and it makes little diVerence where the boundaries are drawn provided that the coarse graining reasonably reXects the intuitive idea of when systems are to be considered to be macroscopically distinguishable Since the entropy is deWned as a logarithm of the box volume it would indeed need a stupendous redrawing of boundaries to get any signiWcant change in S 27 3 In my view entropy has the status of a convenience in present day theory rather than being fundamental though there are indications that in a deeper context where quantum gravitational considerations become important especially in relation to black hole entropy there may be a more fundamental status for this kind of notion We shall be coming to this issue later in this chapter 27 10 and in 30 4 8 31 15 and 32 6 27 4The robustness of the entropy concept A simple illustration may make the role of Boltzmann s entropy formula a little clearer Consider a closed container in which a region R is marked oV as special say a bulb shaped protuberence of one tenth of the container s volume having access to the rest of the container through a small opening see Fig 27 3 Suppose that there is a gas in this container consisting of m molecules We are going to ask for the entropy S to be assigned to the situation in which the entire gas Wnds itself in R in comparison with the entropy to be assigned to the situation in which the gas is randomly distributed throughout the container We shall have S klogVR by Boltzmann s formula where VRis the volume of the phase space region VRrepresenting all molecules being in R 27 2 Show how to assign an absolute measure to a phase space volume if units are chosen so that h 1 27 3 How does the logarithm in Boltzmann s formula relate to the stupendous discrepancies in box volumes 692 27 4CHAPTER 27 R For simplicity we assume what is called Boltzmann statistics as opposed to the Bose Einstein statistics of bosons and the Fermi Dirac statistics of