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Transcript
Thermodynamics of Black Holes I Thermodynamics The field of thermodynamics began in the 19th Century as a means to enhance steam engine performance. It was soon determined that this study could be applied to a wider variety of systems. Today, thermodynamics is an important branch of physics relevant for not only engines but for material studies, astrophysics, cosmology, and as we will discuss here, black holes. The basic idea behind thermodynamics is to try to understand the properties of large collections of molecules (macroscopic systems) without the knowledge of the complete state of the system. In order to completely specify a state within physics the position and velocities of each component must be known (in addition to other quantities like electric charge etc.). In practice this is not feasible. Suitable averages over these microscopic parameters are performed and relevant macroscopic quantities are used to describe the thermodynamic state of a system. As an example, consider a volume of gas (say 1 liter at room temperature) contained within a container of volume V. As stated above, a complete description of this system would include the position and velocity of each molecule at one particular time. This alone could involve on the order of 1024 parameters to keep track of (not including the boundary conditions, electric dipoles, etc.). It is not possible with current technology to perform a measurement of the complete state of this gas. In classical thermodynamics it is assumed that the molecules within the gas are governed by classical mechanics (Newton’s 3 laws). One can then perform an average over the system to come up with thermodynamic quantities describing this gas. Some quantities are easy to determine - the mass of the gas (M) and the volume of the gas (V). The mass of the gas is often re-expressed in terms of the number of molecules in the gas (N). Knowing the mass of each molecule this can be found. The result is usually expressed in terms of the unit – moles (n), one mole of a substance is equal to 1 Avogrado’s number of them (NA = 6.02 x 1023). Other macroscopic quantities found by averaging include the pressure (P), the energy (U) of the gas which is related to the temperature of the gas (T). Temperature is an expression of the statistical average of the kinetic energies of the microscopic components. Thus, thermodynamics is a statistical theory, we know the underlying theory in describing the microscopic parameters, we know that they are there, we just do not know in exactly what state they are in. Using the fact that classical mechanics governs the interactions of the microscopic objects we can obtain useful description of the system as a whole in terms of the macroscopic parameters. A quick example, in trying to predict weather, meteorologists give probabilities that some event will happen; “there is a 20% chance of rain tomorrow”. In order to arrive at this prediction the meteorologist examines the measured macroscopic quantities (high and low fronts, wind speeds, jet streams, etc). If it were possible (though way, way beyond our current technology) to determine the exact location and velocity of every atom in the atmosphere, the amount of solar radiation penetrating the atmosphere, the ground temperature at every spot on Earth, the flapping of every butterfly, etc. at one instant in time, a prediction of the future evolution of the weather would be nearly 100%. (This leads us down the road of chaos theory, basically EVERY minute detail would need to be known precisely in order to predict with near 100% certainty (the slightest error would soon magnify and swamp out the precision). But, in principle, it would be possible to do. [This differs from quantum mechanics, which has inherent uncertainties for the most microscopic objects. There are no underlying dynamics to analyze.] Thermodynamics as Information Theory An aspect of thermodynamics that was not understood until the mid-20th Century was that thermodynamics is intimately connected with information. Consider the information content of our gas sample. The complete specification of the state of the gas (state of each molecule within the gas) yields the greatest amount of information about the gas. In performing these averages over the microscopic states a great amount of information has been lost (we do not know the positions or velocities of the molecules). It turns out for most uses that we are interested in the averaged quantities is enough to describe the system and predict its future evolution. 1 The idea of our knowledge of information of a system is closely related to a quantity called entropy. Entropy is often cloaked in mysterious double talk from those not familiar with the concept. There are three basic ways to define entropy. At the most basic level, entropy is a measure of the number of states available to a system. The most common way to describe entropy is as a measure of disorder (the greater the entropy, the greater the disorder in a system). Another way that entropy is thought of is as the amount of missing information in a macroscopic system. For example, back to our gas, the totally specified state had the maximal information (the amount of information we know about it). This state is equivalent to a very low (zero) entropy state. The same state where we consider only the macroscopic quantities (P, V, T, n), has a much larger amount of unknown information. The reason being that there are a very large number of distinct microscopic states (different positions and velocities of molecules) which give the exam same macroscopic results (same P,V,T,n). We do not know in which of these myriad microscopic states the system is in, just that it has the values P,V,T, and n. The study of Thermodynamics begins with the four laws of thermodynamics (laws 0,1,2, and 3). Zeroth Law of Thermodynamics This law states that for any system in thermal equilibrium, all points in the system are at the same temperature. (No heat flow is occurring). First Law of Thermodynamics If two systems are placed in contact there will be heat flow between them until thermal equilibrium is reached (they are at the same temperature). It also expresses how much work can be done by or on a system given an amount of heat flow. -The change in internal energy of a closed system is equal to the net heat added to the system minus the net work done by the system. Second Law of Thermodynamics There are various ways to state the 2nd law, -The total entropy of any system plus that of its environment increases as a result of any natural process. -Natural processes tend to move toward a state of greater disorder. Third Law of Thermodynamics This law states that it is not possible to lower the temperature of a system to absolute zero (0 Kelvin or –273.15 degrees Celsius) by a finite number of transformations, which cool the system. Basically, you can’t reach absolute zero. We are mostly interested in the second law of thermodynamics which states that the entropy of the universe either remains constant (for a process called reversible) or increases during any physical process. Common examples of increasing disorder (and not a decrease of disorder) are; - A glass falling and breaking into many shards of glass (We never see shards of glass assemble themselves into a glass). - Stars may explode (or spew of gas) as they age. This is a more disordered system there are many more states for the gas to be in. You might argue that freezing water into an ice cube is an example of a system going from a more disordered state to a more ordered state (there are many more states available to water than in an ordered array of atoms which are not moving much). However, you must always consider the entire universe. In creating an ice cube you expel quite a bit of heat. This heat (which can be in many equivalent states) has high entropy. When you find the net change you find that the entropy increase of the universe by the release of the heat is greater than the decrease of the H20 molecules. Viewing entropy as the lack of information we have about the system we see that the second law states that our ignorance of the universe always increases. It is this fact that we want to examine in the context of black holes. 2 II Black Holes in terms of thermodynamics An immediate problem We just introduced the second law of thermodynamics and stated it in a form relating to the lack of information about a system. The lack of information never decreases in a physical process. Recall that we stated that a black hole is one of the simplest classical objects in the universe. It only takes 3 independent parameters to describe the state completely, (M,Q,L). Now imagine the amount of information (or the number of available states) in a massive star prior to collapsing into a black hole. This corresponds to a large amount of entropy (more specifically, entropy is related to the logarithm of the number of states available). When this star collapses into a black hole it goes from a state with a lot of entropy (much unknown information or equivalently large number of equivalent states yielding the same macroscopic state) to a state completely specified by M, Q, and L. We seem to have a case here where the entropy of the universe has decreased. In fact, since everything is known about the black hole, the entropy is zero. A black hole is a much more ordered object than a star. We have a gross violation of the second law of thermodynamics, the entropy is going from a very large value (for the star) to a completely defined state (entropy zero). On a further note, consider the temperature of a classical black hole. It is black, nothing escapes, meaning it does not radiate – thus it has zero temperature. This violates the third law of thermodynamics listed above. How to resolve this problem? In the 1970s Bekenstein proposed modifying the second law by attributing some entropy to the black hole. This can be seen as merely an accounting trick to keep the entropy of the universe, now expressed as, S ' ≡ S + S BH a never decreasing function. Then we replace the ordinary second law of thermodynamics with the generalized second law (GSL): The total entropy of the universe, S’, never decreases in any physical process. The actual argument to show that this generalized law is valid is rather involved and would take us quite a bit of time. To make progress we will introduce two parameters that we can associate with a black hole (these are functions of M, Q, and L, the independent parameters describing a classical black hole). The first is simply the area of the event horizon. For simplicity we consider a Schwarzschild (Q=L=0) black hole. The area is simply ABH = 4π RS and with the definition of the Schwarzschild radius we have, 2 ABH = 4π 4G 2 M 2 G2M 2 16 = π c4 c4 , Rs = 2GM . c2 The other quantity is of interest is the surface gravity at the horizon (the acceleration due to gravity). We will label this quantity as κ (by convention). Recall that on the surface of the Earth g = GMe/Re2 =9.8 m/s2, thus we have, GM GMc 4 c4 κ= 2 = 2 2= . RS 4G M 4GM Thus as a black hole acquires more mass, its area grows and its surface gravity decreases. Note that if we take two black holes of mass M1 and M2 and allow them to coalesce the area of the resulting black hole is greater than the sum of the two original areas, A1 + A2 ~ M1 + M2 < A1+2 ~ (M1 +M2)2. This idea that the area of a black hole never decreased sounds familiar if we replace ‘area’ by entropy. In fact, this leads us to a parallel between the classical laws of thermodynamics and a new set relating to black holes. Note however that if we now attribute some entropy to the black hole then this object should exist at some non-zero temperature. So we have not completely resolved the problem yet. Notice however, that if there were some temperature associated with the horizon, then the problem with the 3rd law would go away. But how could an object that only absorbs energy be at a non-zero temperature (radiate thermal energy 3 away)? We will see how this can be accounted for by examining Hawking radiation – this requires quantum mechanics to understand. (We will return to this in a later lecture). The Laws of Black Hole Mechanics For the moment we will assume that our classical black hole has some finite temperature T (of course it can’t classically, this requires quantum mechanics to exist) and some entropy S. We have stated that we have identified the entropy of the black hole with its horizon area. Thus the second law becomes that the area of a black hole’s horizon always grows in a physical process. We would like to make one more connection between thermodynamic quantities and properties of black holes. The first law states that the temperature is constant throughout a body in thermal equilibrium. Analyzing our black hole we note that the surface gravity, κ, is the same over the spherically symmetric Schwarzschild black hole horizon. Thus it is tempting to relate the temperature of the black hole to the surface gravity at the horizon. Doing so allows us to state the 4 laws of thermodynamics in terms of the mass, surface area, and surface gravity of the black hole. It is believed that these new laws hold for black holes. Laws of Black Hole Thermodynamics Law Thermodynamics Schwarzschild black hole Zeroth T is constant throughout body in thermal equilibrium. surface gravity, k, is constant on the horizon of a black hole First dE = dQ – W or dE = TdS – pdV using E = mc2, d(mc2) ~ κdA Second dS >= 0 dA >= 0 Third T = 0 can not be reached κ = 0 can not be reached. (+ΩdJ-φdq for spinning charged BH). There are still some troubles with this scenario (we have yet to explain how a black hole can be hot) but to go further requires some knowledge of quantum mechanics. 4