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128-719: Since the second law of thermodynamics states that entropy increases as time flows toward the future, in general, the macroscopic universe does not show symmetry under time reversal. In other words, time is said to be non-symmetric, or asymmetric, except for special equilibrium states when the second law of thermodynamics predicts the time symmetry to hold. However, quantum noninvasive measurements are predicted to violate time symmetry even in equilibrium, contrary to their classical counterparts, although this has not yet been experimentally confirmed. Time asymmetries (see Arrow of time ) generally are caused by one of three categories: Daily experience shows that T-symmetry does not hold for

256-466: A b c ⋯ {\displaystyle \psi _{abc\cdots }} In this case, Covariant tensor indexes will transform as T a b = ( T − 1 ) b a {\displaystyle {T_{a}}^{b}={(T^{-1})_{b}}^{a}} and so on. For quantum fields, there is also a third T , written as T , {\displaystyle {\mathcal {T}},} which

384-403: A closed system in terms of work and heat . It can be linked to the law of conservation of energy . Conceptually, the first law describes the fundamental principle that systems do not consume or 'use up' energy, that energy is neither created nor destroyed, but is simply converted from one form to another. The second law is concerned with the direction of natural processes. It asserts that

512-449: A hypersurface of constant time (a set of points that all have the same time coordinate, such that every point on the surface has a space-like separation, giving what is called a 'space-like surface') and draw an "embedding diagram" depicting the curvature of space at that time, the embedding diagram will look like a tube connecting the two exterior regions, known as an "Einstein-Rosen bridge" or Schwarzschild wormhole . Depending on where

640-401: A particle with spin J , one can use the representation where J y is the y -component of the spin, and use of TJT = − J has been made. This has an interesting consequence on the electric dipole moment (EDM) of any particle. The EDM is defined through the shift in the energy of a state when it is put in an external electric field: Δ e = d· E + E ·δ· E , where d is called

768-420: A black hole would be a hypothetical object known as a white hole . From the outside they appear similar. While a black hole has a beginning and is inescapable, a white hole has an ending and cannot be entered. The forward light-cones of a white hole are directed outward; and its backward light-cones are directed towards the center. The event horizon of a black hole may be thought of as a surface moving outward at

896-469: A black hole, producing the expanding volume of space and matter that includes the observable universe. This black hole eventually becomes a white hole as the matter density reduces with the expansion. A related theory gives an alternative to dark energy. A 2012 paper argues that the Big Bang itself is a white hole. It further suggests that the emergence of a white hole, which was named a "Small Bang",

1024-436: A black hole. Such a configuration is not static: we start from a massive and extended body which contracts to give a black hole. The black hole therefore does not exist for all eternity, and there is no corresponding white hole. To be able to exist, a white hole must either arise from a physical process leading to its formation, or be present from the creation of the universe . None of these solutions appears satisfactory: there

1152-424: A colder to a warmer body without some other change, connected therewith, occurring at the same time. The second law of thermodynamics allows the definition of the concept of thermodynamic temperature , but this has been formally delegated to the zeroth law of thermodynamics . The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system , and expresses its change for

1280-487: A collapsing star would require a different metric. When the infalling stellar matter is added to a diagram of a black hole's history, it removes the part of the diagram corresponding to the white hole interior region. But because the equations of general relativity are time-reversible – they exhibit Time reversal symmetry – general relativity must also allow the time-reverse of this type of "realistic" black hole that forms from collapsing matter. The time-reversed case would be

1408-418: A crystallized structure of reduced disorder (sticking together in a certain order due to molecular attraction). The entropy of the system decreases, but the system approaches uniformity with its surroundings (category III). On the other hand, consider the refrigeration of water in a warm environment. Due to refrigeration, as heat is extracted from the water, the temperature and entropy of the water decreases, as

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1536-412: A factory. The robotic machinery requires electrical work input and instructions, but when completed, the manufactured products have less uniformity with their surroundings, or more complexity (higher order) relative to the raw materials they were made from. Thus, system entropy or disorder decreases while the tendency towards uniformity between the system and its environment is counteracted. In this example,

1664-577: A final value as there is no considered surface in a singularity. In quantum mechanics , the black hole emits Hawking radiation and so it can come to thermal equilibrium with a gas of radiation (not compulsory). Because a thermal-equilibrium state is time-reversal-invariant, Stephen Hawking argued that the time reversal of a black hole in thermal equilibrium results in a white hole in thermal equilibrium (each absorbing and emitting energy to equivalent degrees). Consequently, this may imply that black holes and white holes are reciprocal in structure, wherein

1792-464: A full statement of the second law: Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy. The second law has been shown to be equivalent to the internal energy U defined as a convex function of the other extensive properties of the system. That is, when a system is described by stating its internal energy U , an extensive variable, as

1920-520: A function of its entropy S , volume V , and mol number N , i.e. U = U ( S , V , N ), then the temperature is equal to the partial derivative of the internal energy with respect to the entropy (essentially equivalent to the first TdS equation for V and N held constant): The Clausius inequality, as well as some other statements of the second law, must be re-stated to have general applicability for all forms of heat transfer, i.e. scenarios involving radiative fluxes. For example,

2048-429: A heat engine operating between any two given thermal or heat reservoirs at different temperatures. Carnot's principle was recognized by Carnot at a time when the caloric theory represented the dominant understanding of the nature of heat, before the recognition of the first law of thermodynamics , and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, Carnot's analysis

2176-458: A mix of particles that fell in from either universe (and thus an observer who fell in from one universe might be able to see light that fell in from the other one), and likewise particles from the interior white-hole region can escape into either universe. All four regions can be seen in a spacetime diagram that uses Kruskal–Szekeres coordinates (see figure). In this spacetime, it is possible to come up with coordinate systems such that if you pick

2304-406: A natural process runs only in one sense, and is not reversible. That is, the state of a natural system itself can be reversed, but not without increasing the entropy of the system's surroundings, that is, both the state of the system plus the state of its surroundings cannot be together, fully reversed, without implying the destruction of entropy. For example, when a path for conduction or radiation

2432-406: A non-deterministic configuration, but does not have the status of a white hole, however, because there is no region inaccessible from a given region. In its basic conception, the Big Bang can be seen as a naked singularity in outer space, but does not correspond to a white hole. In its mode of formation, a black hole comes from a residue of a massive star whose core contracts until it turns into

2560-593: A nutshell, the Clausius inequality is saying that when a cycle is completed, the change in the state property S will be zero, so the entropy that was produced during the cycle must have transferred out of the system by heat transfer. The δ \delta (or đ) indicates a path dependent integration. Due to the inherent emission of radiation from all matter, most entropy flux calculations involve incident, reflected and emitted radiative fluxes. The energy and entropy of unpolarized blackbody thermal radiation,

2688-460: A perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines. Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on

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2816-644: A purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows: In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S. With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics . It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat

2944-465: A short transition region where the Einstein equations are violated by quantum effects. From this region, space and time emerge with the structure of a white hole interior, a possibility already suggested by John Lighton Synge . The possibility of the existence of white holes was put forward by cosmologist Igor Novikov in 1964, developed by Nikolai Kardashev . White holes are predicted as part of

3072-486: A solution to the Einstein field equations known as the maximally extended version of the Schwarzschild metric describing an eternal black hole with no charge and no rotation. Here, "maximally extended" implies that spacetime should not have any "edges". For any possible trajectory of a free-falling particle (following a geodesic ) in spacetime, it should be possible to continue this path arbitrarily far into

3200-411: A white hole that has existed since the beginning of the universe, and that emits matter until it finally "explodes" and disappears. Despite the fact that such objects are permitted theoretically, they are not taken as seriously as black holes by physicists, since there would be no processes that would naturally lead to their formation; they could exist only if they were built into the initial conditions of

3328-466: Is a function of state , while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality This is because a general process for this case (no mass exchange between the system and its surroundings) may include work being done on the system by its surroundings, which can have frictional or viscous effects inside

3456-418: Is a physical law based on universal empirical observation concerning heat and energy interconversions . A simple statement of the law is that heat always flows spontaneously from hotter to colder regions of matter (or 'downhill' in terms of the temperature gradient). Another statement is: "Not all heat can be converted into work in a cyclic process ." The second law of thermodynamics establishes

3584-423: Is a factual statement relating the old and new fields to one-another. Unlike scalar fields, spinor and vector fields ψ {\displaystyle \psi } might have a non-trivial behavior under time reversal. In this case, one has to write where T {\displaystyle T} is just an ordinary matrix . For complex fields, complex conjugation may be required, for which

3712-568: Is a holonomic process function , in other words, δ Q = T d S {\displaystyle \delta Q=TdS} . Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases

3840-429: Is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy. A statement that in a sense is complementary to Planck's principle is made by Claus Borgnakke and Richard E. Sonntag. They do not offer it as

3968-423: Is a symmetry. This is correct; if a quantum system has degenerate ground states that transform into each other under parity, then time reversal need not be broken to give EDM. Experimentally observed bounds on the electric dipole moment of the nucleon currently set stringent limits on the violation of time reversal symmetry in the strong interactions , and their modern theory: quantum chromodynamics . Then, using

T-symmetry - Misplaced Pages Continue

4096-470: Is a twofold degeneracy in the state. This result in non-relativistic quantum mechanics presages the spin statistics theorem of quantum field theory . Quantum states that give unitary representations of time reversal, i.e., have T = 1 , are characterized by a multiplicative quantum number , sometimes called the T-parity . Second law of thermodynamics The second law of thermodynamics

4224-452: Is actually an infinite dimensional operator acting on a Hilbert space. It acts on quantized fields Ψ {\displaystyle \Psi } as This can be thought of as a special case of a tensor with one covariant, and one contravariant index, and thus two T {\displaystyle {\mathcal {T}}} 's are required. All three of these symbols capture the idea of time-reversal; they differ with respect to

4352-508: Is almost customary in textbooks to speak of the "Kelvin–Planck statement" of the law, as for example in the text by ter Haar and Wergeland . This version, also known as the heat engine statement , of the second law states that Max Planck stated the second law as follows. Rather like Planck's statement is that of George Uhlenbeck and G. W. Ford for irreversible phenomena . Constantin Carathéodory formulated thermodynamics on

4480-1320: Is calculated using the spectral energy and entropy radiance expressions derived by Max Planck using equilibrium statistical mechanics, K ν = 2 h c 2 ν 3 exp ⁡ ( h ν k T ) − 1 , {\displaystyle K_{\nu }={\frac {2h}{c^{2}}}{\frac {\nu ^{3}}{\exp \left({\frac {h\nu }{kT}}\right)-1}},} L ν = 2 k ν 2 c 2 ( ( 1 + c 2 K ν 2 h ν 3 ) ln ⁡ ( 1 + c 2 K ν 2 h ν 3 ) − ( c 2 K ν 2 h ν 3 ) ln ⁡ ( c 2 K ν 2 h ν 3 ) ) {\displaystyle L_{\nu }={\frac {2k\nu ^{2}}{c^{2}}}((1+{\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})\ln(1+{\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})-({\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})\ln({\frac {c^{2}K_{\nu }}{2h\nu ^{3}}}))} where c

4608-407: Is called a wormhole . In both cases, however, it is not possible to reach the region "in" the white hole, so the behavior of it - and, in particular, what may come out of it - is completely impossible to predict. In this sense, a white hole is a configuration according to which the evolution of the universe cannot be predicted, because it is not deterministic. A "bare singularity" is another example of

4736-579: Is conventionally given as where J y {\displaystyle J_{y}} is the y-component of the angular momentum operator and K {\displaystyle K} is complex conjugation, as before. This form follows whenever the spinor can be described with a linear differential equation that is first-order in the time derivative, which is generally the case in order for something to be validly called "a spinor". The formal notation now makes it clear how to extend time-reversal to an arbitrary tensor field ψ

4864-506: Is fully converted to work) in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the right figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/ η . The net and sole effect of the combined pair of engines is to transfer heat Δ Q = Q ( 1 η − 1 ) {\textstyle \Delta Q=Q\left({\frac {1}{\eta }}-1\right)} from

4992-550: Is known as the Clausius statement : Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time. The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other. Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on

5120-401: Is limited. However, it was proved that it is possible to find other time reversal operations which preserve the dynamics and so Onsager reciprocal relations; in conclusion, one cannot state that the presence of a magnetic field always breaks T-symmetry. Most systems are asymmetric under time reversal, but there may be phenomena with symmetry. In classical mechanics, a velocity v reverses under

5248-495: Is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy change . A heat pump can reverse this heat flow, but the reversal process and the original process, both cause entropy production, thereby increasing the entropy of the system's surroundings. If an isolated system containing distinct subsystems is held initially in internal thermodynamic equilibrium by internal partitioning by impermeable walls between

T-symmetry - Misplaced Pages Continue

5376-424: Is no known astrophysical process that can lead to the formation of such a configuration, and imposing it from the creation of the universe amounts to assuming a very specific set of initial conditions which has no concrete motivation. In view of the enormous quantities radiated by quasars , whose luminosity makes it possible to observe them from several billion light-years away, it had been assumed that they were

5504-478: Is physically equivalent to the second law of thermodynamics, and remains valid today. Some samples from his book are: In modern terms, Carnot's principle may be stated more precisely: The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854,

5632-430: Is really inevitable has been considered by many physicists, often in the context of Maxwell's demon . The name comes from a thought experiment described by James Clerk Maxwell in which a microscopic demon guards a gate between two halves of a room. It only lets slow molecules into one half, only fast ones into the other. By eventually making one side of the room cooler than before and the other hotter, it seems to reduce

5760-444: Is reversed. Similarly, any operation that reverses the sense of phase, which changes the sign of i , will turn positive energies into negative energies unless it also changes the direction of time. So every antiunitary symmetry in a theory with positive energy must reverse the direction of time. Every antiunitary operator can be written as the product of the time reversal operator and a unitary operator that does not reverse time. For

5888-408: Is significant in fermionic matter at extremely high densities. Such an interaction prevents the formation of a gravitational singularity. Instead, the collapsing matter on the other side of the event horizon reaches an enormous but finite density and rebounds, forming a regular Einstein–Rosen bridge. The other side of the bridge becomes a new, growing baby universe. For observers in the baby universe,

6016-422: Is spontaneous—all the matter is ejected at a single pulse. Thus, unlike black holes, white holes cannot be continuously observed; rather, their effects can be detected only around the event itself. The paper even proposed identifying a new group of gamma-ray bursts with white holes. Unlike black holes for which there is a well-studied physical process, gravitational collapse (which gives rise to black holes when

6144-417: Is still useful to consider the time-reversal non-invariance in a local sense when the external field is held fixed, as when the magneto-optic effect is analyzed. This allows one to analyze the conditions under which optical phenomena that locally break time-reversal, such as Faraday isolators and directional dichroism , can occur.) In physics one separates the laws of motion, called kinematics , from

6272-626: Is taken separately from that due to heat transfer by conduction and convection ( δ Q C C \delta Q_{CC} ), where the temperature is evaluated at the system boundary where the heat transfer occurs. The modified Clausius inequality, for all heat transfer scenarios, can then be expressed as, ∫ cycle ( δ Q C C T b + δ S NetRad ) ≤ 0 {\displaystyle \int _{\text{cycle}}({\frac {\delta Q_{CC}}{T_{b}}}+\delta S_{\text{NetRad}})\leq 0} In

6400-517: Is the speed of light, k is the Boltzmann constant, h is the Planck constant, ν is frequency, and the quantities K v and L v are the energy and entropy fluxes per unit frequency, area, and solid angle. In deriving this blackbody spectral entropy radiance, with the goal of deriving the blackbody energy formula, Planck postulated that the energy of a photon was quantized (partly to simplify

6528-446: Is to relate it to the second law of thermodynamics, since black holes are viewed as thermodynamic objects . For example, according to the gauge–gravity duality conjecture, all microscopic processes in a black hole are reversible, and only the collective behavior is irreversible, as in any other macroscopic, thermal system. In physical and chemical kinetics , T-symmetry of the mechanical microscopic equations implies two important laws:

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6656-471: The Big Bang . Additionally, it is predicted that such a white hole would be highly "unstable" in the sense that if any small amount of matter fell towards the horizon from the outside, this would prevent the white hole's explosion as seen by distant observers, with the matter emitted from the singularity never able to escape the white hole's gravitational radius. Depending on the type of black hole solution considered, there are several types of white holes. In

6784-478: The CPT invariance of a relativistic quantum field theory , this puts strong bounds on strong CP violation . Experimental bounds on the electron electric dipole moment also place limits on theories of particle physics and their parameters. For T , which is an anti-unitary Z 2 symmetry generator where Φ is a diagonal matrix of phases. As a result, U = Φ U and U = U Φ , showing that This means that

6912-461: The T that is an involution , capturing the actual reversal of the time coordinate, the T that is an ordinary finite dimensional matrix, acting on spinors and vectors, and the T that is an operator on an infinite-dimensional Hilbert space . For a real (not complex ) classical (unquantized) scalar field ϕ {\displaystyle \phi } , the time reversal involution can simply be written as as time reversal leaves

7040-602: The entropy of the room, and reverse the arrow of time. Many analyses have been made of this; all show that when the entropy of room and demon are taken together, this total entropy does increase. Modern analyses of this problem have taken into account Claude E. Shannon 's relation between entropy and information . Many interesting results in modern computing are closely related to this problem— reversible computing , quantum computing and physical limits to computing , are examples. These seemingly metaphysical questions are today, in these ways, slowly being converted into hypotheses of

7168-446: The event horizon of a black hole from the outside, and then fall rapidly to the central region where our understanding of physics breaks down. Since within a black hole the forward light-cone is directed towards the center and the backward light-cone is directed outward, it is not even possible to define time-reversal in the usual manner. The only way anything can escape from a black hole is as Hawking radiation . The time reversal of

7296-406: The parity operator. Acting on the position, it reverses the directions of space, so that PxP = − x . Similarly, it reverses the direction of momentum , so that PpP = − p , where x and p are the position and momentum operators. This preserves the canonical commutator [ x , p ] = iħ , where ħ is the reduced Planck constant , only if P is chosen to be unitary, PiP = i . On

7424-472: The torsion tensor , as a dynamical variable. Torsion naturally accounts for the quantum-mechanical, intrinsic angular momentum ( spin ) of matter. According to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular black hole. In the Einstein–Cartan theory, however, the minimal coupling between torsion and Dirac spinors generates a repulsive spin–spin interaction that

7552-643: The 8 real components of a Dirac spinor. In the general setting, there is no ab initio value to be given for T {\displaystyle T} ; its actual form depends on the specific equation or equations which are being examined. In general, one simply states that the equations must be time-reversal invariant, and then solves for the explicit value of T {\displaystyle T} that achieves this goal. In some cases, generic arguments can be made. Thus, for example, for spinors in three-dimensional Euclidean space , or four-dimensional Minkowski space , an explicit transformation can be given. It

7680-455: The Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent. Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law. It

7808-560: The EDM and δ, the induced dipole moment. One important property of an EDM is that the energy shift due to it changes sign under a parity transformation. However, since d is a vector, its expectation value in a state |ψ⟩ must be proportional to ⟨ψ| J |ψ⟩, that is the expected spin. Thus, under time reversal, an invariant state must have vanishing EDM. In other words, a non-vanishing EDM signals both P and T symmetry-breaking. Some molecules, such as water, must have EDM irrespective of whether T

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7936-483: The Hawking radiation from an ordinary black hole is identified with a white hole's emission of energy and matter. Hawking's semi-classical argument is reproduced in a quantum mechanical AdS/CFT treatment, where a black hole in anti-de Sitter space is described by a thermal gas in a gauge theory , whose time reversal is the same as itself. In the 1930s, physicists Robert Oppenheimer and Hartland Snyder introduced

8064-438: The absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. Introducing a set of internal variables ξ {\displaystyle \xi } to describe the deviation of a thermodynamic system from a chemical equilibrium state in physical equilibrium (with the required well-defined uniform pressure P and temperature T ), one can record

8192-460: The apparent T-asymmetry of our universe is a problem in cosmology : why did the universe start with a low entropy? This view, supported by cosmological observations (such as the isotropy of the cosmic microwave background ) connects this problem to the question of initial conditions of the universe. The laws of gravity seem to be time reversal invariant in classical mechanics; however, specific solutions need not be. An object can cross through

8320-399: The behavior of bulk materials. Of these macroscopic laws, most notable is the second law of thermodynamics . Many other phenomena, such as the relative motion of bodies with friction, or viscous motion of fluids, reduce to this, because the underlying mechanism is the dissipation of usable energy (for example, kinetic energy) into heat. The question of whether this time-asymmetric dissipation

8448-423: The black hole interior region can also pass the observer at any time. Just as there are two separate interior regions of the maximally extended spacetime, there are also two separate exterior regions, sometimes called two different "universes", with the second universe allowing us to extrapolate some possible particle trajectories in the two interior regions. This means that the interior black-hole region can contain

8576-566: The canonical time reversal operation that reverses the velocities and the time t {\displaystyle t} and keeps the coordinates untouched is no more a symmetry for the system. Under this consideration, it seems that only Onsager–Casimir reciprocal relations could hold; these equalities relate two different systems, one subject to B → {\displaystyle {\vec {B}}} and another to − B → {\displaystyle -{\vec {B}}} , and so their utility

8704-479: The case of the maximally extended Schwarzschild solution , discussed below, the white hole event horizon in the past becomes a black hole event horizon in the future, so any object falling towards it will eventually reach the black hole horizon). Imagine a gravitational field, without a surface. Acceleration due to gravity is the greatest on the surface of any body. But since black holes lack a surface, acceleration due to gravity increases exponentially, but never reaches

8832-487: The case of the Schwarzschild black hole mentioned above, a geodesic coming out of a white hole comes from the "gravitational singularity" it contains. In the case of a black hole possessing an electric charge ψ ** Ώ ** ώ ( Reissner-Nordström black hole ) or an angular momentum , then the white hole happens to be the "exit door" of a black hole existing in another universe. Such a black hole - white hole configuration

8960-442: The center of every galaxy and may be essential for their formation. Stephen Hawking and others have proposed that these supermassive black holes could spawn supermassive white holes. Like black holes, white holes have properties such as mass , charge , and angular momentum . They attract matter like any other mass, but objects falling towards a white hole would never actually reach the white hole's event horizon (though in

9088-537: The combined entropy of system and surroundings accounts for the irreversibility of natural processes, often referred to in the concept of the arrow of time . Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory . Statistical mechanics provides a microscopic explanation of the law in terms of probability distributions of the states of large assemblies of atoms or molecules . The second law has been expressed in many ways. Its first formulation, which preceded

9216-414: The concept of entropy as a physical property of a thermodynamic system . It predicts whether processes are forbidden despite obeying the requirement of conservation of energy as expressed in the first law of thermodynamics and provides necessary criteria for spontaneous processes . For example, the first law allows the process of a cup falling off a table and breaking on the floor, as well as allowing

9344-637: The cooler reservoir to the hotter one, which violates the Clausius statement. This is a consequence of the first law of thermodynamics , as for the total system's energy to remain the same; Input + Output = 0 ⟹ ( Q + Q c ) − Q η = 0 {\textstyle {\text{Input}}+{\text{Output}}=0\implies (Q+Q_{c})-{\frac {Q}{\eta }}=0} , so therefore Q c = Q ( 1 η − 1 ) {\textstyle Q_{c}=Q\left({\frac {1}{\eta }}-1\right)} , where (1)

9472-452: The entries in Φ are ±1, as a result of which one may have either T = ±1 . This is specific to the anti-unitarity of T . For a unitary operator, such as the parity , any phase is allowed. Next, take a Hamiltonian invariant under T . Let | a ⟩ and T | a ⟩ be two quantum states of the same energy. Now, if T = −1 , then one finds that the states are orthogonal: a result called Kramers' theorem . This implies that if T = −1 , then there

9600-444: The entropy flux of NBR emission is farther removed from the conduction and convection q / T result, than that for BR emission. This observation is consistent with Max Planck's blackbody radiation energy and entropy formulas and is consistent with the fact that blackbody radiation emission represents the maximum emission of entropy for all materials with the same temperature, as well as the maximum entropy emission for all radiation with

9728-635: The entropy of the system of interest is defined to result from an infinitesimal transfer of heat ( δ Q {\displaystyle \delta Q} ) to the system of interest, divided by the common thermodynamic temperature ( T ) {\displaystyle (T)} of the system of interest and the auxiliary thermodynamic system: Different notations are used for an infinitesimal amount of heat ( δ ) {\displaystyle (\delta )} and infinitesimal change of entropy ( d ) {\displaystyle (\mathrm {d} )} because entropy

9856-411: The entropy spontaneously decreases by means of energy and entropy transfer. When thermodynamic constraints are not present, spontaneously energy or mass, as well as accompanying entropy, may be transferred out of a system in a progress to reach external equilibrium or uniformity in intensive properties of the system with its surroundings. This occurs spontaneously because the energy or mass transferred from

9984-433: The equality The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to

10112-433: The factory from the local electric grid. In addition, humans may directly play, in whole or in part, the role that the robotic machinery plays in manufacturing. In this case, instructions may be involved, but intelligence is either directly responsible, or indirectly responsible, for the direction or application of work in such a way as to counteract the tendency towards disorder and uniformity. There are also situations where

10240-404: The flow of heat in steam engines (1824). The centerpiece of that analysis, now known as a Carnot engine , is an ideal heat engine fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium . It represents the theoretical maximum efficiency of

10368-451: The formulation of the principle in terms of entropy. The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on

10496-410: The four combinations of either entropy (S) up or down, and uniformity (Y) – between system and its environment – up or down. This 'special' category of processes, category IV, is characterized by movement in the direction of low disorder and low uniformity, counteracting the second law tendency towards uniformity and disorder. The second law can be conceptually stated as follows: Matter and energy have

10624-431: The idea of white holes as a solution to Einstein's equations of general relativity . These equations, the foundation of modern physics, describe the curvature of spacetime due to massive objects. Whereas black holes are born from the collapse of stars, white holes represent the theoretical birth of space, time, and potentially even universes. At the center, space and time do not end into a singularity, but continue across

10752-406: The instructions, as well as the source of work may be internal or external to the system, and they may or may not cross the system boundary. To illustrate, the instructions may be pre-coded and the electrical work may be stored in an energy storage system on-site. Alternatively, the control of the machinery may be by remote operation over a communications network, while the electric work is supplied to

10880-536: The integrand (đQ/T) of the Clausius expression applies to heat conduction and convection, and the case of ideal infinitesimal blackbody radiation (BR) transfer, but does not apply to most radiative transfer scenarios and in some cases has no physical meaning whatsoever. Consequently, the Clausius inequality was re-stated so that it is applicable to cycles with processes involving any form of heat transfer. The entropy transfer with radiative fluxes ( δ S NetRad \delta S_{\text{NetRad}} )

11008-402: The internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of

11136-425: The laws of force, called dynamics . Following the classical kinematics of Newton's laws of motion , the kinematics of quantum mechanics is built in such a way that it presupposes nothing about the time reversal symmetry of the dynamics. In other words, if the dynamics are invariant, then the kinematics will allow it to remain invariant; if the dynamics is not, then the kinematics will also show this. The structure of

11264-435: The local speed of light and is just on the edge between escaping and falling back. The event horizon of a white hole is a surface moving inward at the local speed of light and is just on the edge between being swept outward and succeeding in reaching the center. They are two different kinds of horizons—the horizon of a white hole is like the horizon of a black hole turned inside-out. The modern view of black hole irreversibility

11392-416: The mapping K : ( x + i y ) ↦ ( x − i y ) {\displaystyle K:(x+iy)\mapsto (x-iy)} can be thought of as a 2x2 matrix. For a Dirac spinor , T {\displaystyle T} cannot be written as a 4x4 matrix, because, in fact, complex conjugation is indeed required; however, it can be written as an 8x8 matrix, acting on

11520-460: The mathematics), thereby starting quantum theory. A non-equilibrium statistical mechanics approach has also been used to obtain the same result as Planck, indicating it has wider significance and represents a non-equilibrium entropy. A plot of K v versus frequency (v) for various values of temperature ( T) gives a family of blackbody radiation energy spectra, and likewise for the entropy spectra. For non-blackbody radiation (NBR) emission fluxes,

11648-418: The nature of classical white holes. Some researchers have proposed that when a black hole forms, a Big Bang may occur at the core/ singularity , which would create a new universe that expands outside of the parent universe . The Einstein–Cartan–Sciama–Kibble theory of gravity extends general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part,

11776-410: The operation of T , but an acceleration does not. Therefore, one models dissipative phenomena through terms that are odd in v . However, delicate experiments in which known sources of dissipation are removed reveal that the laws of mechanics are time reversal invariant. Dissipation itself is originated in the second law of thermodynamics . The motion of a charged body in a magnetic field, B involves

11904-412: The other hand, the time reversal operator T , it does nothing to the x-operator, TxT = x , but it reverses the direction of p, so that TpT = − p . The canonical commutator is invariant only if T is chosen to be anti-unitary, i.e., TiT = − i . Another argument involves energy, the time-component of the four-momentum. If time reversal were implemented as a unitary operator, it would reverse

12032-443: The other, however, because it is impossible to enter a white hole event horizon from the outside, and anyone entering a black hole horizon from either universe will inevitably hit the black hole singularity. Note that the maximally extended Schwarzschild metric describes an idealized black hole/white hole that exists eternally from the perspective of external observers; a more realistic black hole that forms at some particular time from

12160-485: The parent universe appears as the only white hole. Accordingly, the observable universe is the Einstein–Rosen interior of a black hole existing as one of possibly many inside a larger universe. The Big Bang was a nonsingular Big Bounce at which the observable universe had a finite, minimum scale factor. Shockwave cosmology , proposed by Joel Smoller and Blake Temple in 2003, has the “big bang” as an explosion inside

12288-404: The particle's future, unless the trajectory hits a gravitational singularity like the one at the center of the black hole's interior. In order to satisfy this requirement, it turns out that in addition to the black hole interior region that particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region, which allows us to extrapolate

12416-418: The past (however, the particles or other objects experience only a finite proper time between crossing the horizon and passing the outside observer). The black hole/white hole appears "eternal" from the perspective of an outside observer, in the sense that particles traveling outward from the white hole interior region can pass the observer at any time, and particles traveling inward, which will eventually reach

12544-459: The physical sciences. The current consensus hinges upon the Boltzmann–Shannon identification of the logarithm of phase space volume with the negative of Shannon information , and hence to entropy . In this notion, a fixed initial state of a macroscopic system corresponds to relatively low entropy because the coordinates of the molecules of the body are constrained. As the system evolves in

12672-478: The presence of dissipation , the molecular coordinates can move into larger volumes of phase space, becoming more uncertain, and thus leading to increase in entropy. One resolution to irreversibility is to say that the constant increase of entropy we observe happens only because of the initial state of our universe. Other possible states of the universe (for example, a universe at heat death equilibrium) would actually result in no increase of entropy. In this view,

12800-500: The principle of detailed balance and the Onsager reciprocal relations . T-symmetry of the microscopic description together with its kinetic consequences are called microscopic reversibility . Classical variables that do not change upon time reversal include: Classical variables that time reversal negates include: Let us consider the example of a system of charged particles subject to a constant external magnetic field: in this case

12928-496: The proper definition of entropy and was based on caloric theory , is Carnot's theorem , formulated by the French scientist Sadi Carnot , who in 1824 showed that the efficiency of conversion of heat to work in a heat engine has an upper limit. The first rigorous definition of the second law based on the concept of entropy came from German scientist Rudolf Clausius in the 1850s and included his statement that heat can never pass from

13056-418: The properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature , independent of the properties of any particular reference thermometric body. The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854),

13184-470: The quantum laws of motion are richer, and we examine these next. This section contains a discussion of the three most important properties of time reversal in quantum mechanics; chiefly, The strangeness of this result is clear if one compares it with parity. If parity transforms a pair of quantum states into each other, then the sum and difference of these two basis states are states of good parity. Time reversal does not behave like this. It seems to violate

13312-442: The reader to infer. Eugene Wigner showed that a symmetry operation S of a Hamiltonian is represented, in quantum mechanics either by a unitary operator , S = U , or an antiunitary one, S = UK where U is unitary, and K denotes complex conjugation . These are the only operations that act on Hilbert space so as to preserve the length of the projection of any one state-vector onto another state-vector. Consider

13440-437: The reverse process of the cup fragments coming back together and 'jumping' back onto the table, while the second law allows the former and denies the latter. The second law may be formulated by the observation that the entropy of isolated systems left to spontaneous evolution cannot decrease, as they always tend toward a state of thermodynamic equilibrium where the entropy is highest at the given internal energy. An increase in

13568-477: The same energy radiance. Second law analysis is valuable in scientific and engineering analysis in that it provides a number of benefits over energy analysis alone, including the basis for determining energy quality (exergy content ), understanding fundamental physical phenomena, and improving performance evaluation and optimization. As a result, a conceptual statement of the principle is very useful in engineering analysis. Thermodynamic systems can be categorized by

13696-416: The same time, relatively simple tools to assess their complexity . For instance, quantum-mechanical time reversal was used to develop novel boson sampling schemes and to prove the duality between two fundamental optical operations, beam splitter and squeezing transformations. In formal mathematical presentations of T-symmetry, three different kinds of notation for T need to be carefully distinguished:

13824-690: The scalar value at a fixed spacetime point unchanged, up to an overall sign s = ± 1 {\displaystyle s=\pm 1} . A slightly more formal way to write this is which has the advantage of emphasizing that T {\displaystyle {\mathsf {T}}} is a map , and thus the "mapsto" notation ↦   , {\displaystyle \mapsto ~,} whereas ϕ ′ ( − t , x → ) = s ϕ ( t , x → ) {\displaystyle \phi ^{\prime }(-t,{\vec {x}})=s\phi (t,{\vec {x}})}

13952-441: The seat of exotic physical phenomena such as a white hole, or a phenomenon of continuous creation of matter (see the article on the steady state theory ). These ideas are now abandoned, the observed properties of quasars being very well explained by those of an accretion disk in the center of which is a supermassive black hole . A view of black holes first proposed in the late 1980s might be interpreted as shedding some light on

14080-404: The second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible." Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers , the entropy

14208-399: The sign convention of heat is used in which heat entering into (leaving from) an engine is positive (negative) and (2) Q η {\displaystyle {\frac {Q}{\eta }}} is obtained by the definition of efficiency of the engine when the engine operation is not reversed. Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e.

14336-409: The sign of the energy just as space-reversal reverses the sign of the momentum. This is not possible, because, unlike momentum, energy is always positive. Since energy in quantum mechanics is defined as the phase factor exp(– iEt ) that one gets when one moves forward in time, the way to reverse time while preserving the sign of the energy is to also reverse the sense of " i ", so that the sense of phases

14464-488: The space-like hypersurface is chosen, the Einstein-Rosen bridge can either connect two black hole event horizons in each universe (with points in the interior of the bridge being part of the black hole region of the spacetime), or two white hole event horizons in each universe (with points in the interior of the bridge being part of the white hole region). It is impossible to use the bridge to cross from one universe to

14592-453: The specific space that is being acted on: functions, vectors/spinors, or infinite-dimensional operators. The remainder of this article is not cautious to distinguish these three; the T that appears below is meant to be either T {\displaystyle {\mathsf {T}}} or T {\displaystyle T} or T , {\displaystyle {\mathcal {T}},} depending on context, left for

14720-404: The spectral entropy radiance L v is found by substituting K v spectral energy radiance data into the L v expression (noting that emitted and reflected entropy fluxes are, in general, not independent). For the emission of NBR, including graybody radiation (GR), the resultant emitted entropy flux, or radiance L , has a higher ratio of entropy-to-energy ( L/K ), than that of BR. That is,

14848-467: The statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent. The historical origin of the second law of thermodynamics was in Sadi Carnot 's theoretical analysis of

14976-596: The subsystems, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium , and its total entropy, S {\displaystyle S} , increases. In a reversible or quasi-static , idealized process of transfer of energy as heat to a closed thermodynamic system of interest, (which allows the entry or exit of energy – but not transfer of matter), from an auxiliary thermodynamic system, an infinitesimal increment ( d S {\displaystyle \mathrm {d} S} ) in

15104-455: The system may become more ordered or complex, by the combination of two things, a work or exergy source and some form of instruction or intelligence. Where 'exergy' is the thermal, mechanical, electric or chemical work potential of an energy source or flow, and 'instruction or intelligence', although subjective, is in the context of the set of category IV processes. Consider a category IV example of robotic manufacturing and assembly of vehicles in

15232-427: The system moves further away from uniformity with its warm surroundings or environment (category IV). The main point, take-away, is that refrigeration not only requires a source of work, it requires designed equipment, as well as pre-coded or direct operational intelligence or instructions to achieve the desired refrigeration effect. Before the establishment of the second law, many people who were interested in inventing

15360-504: The system to its surroundings results in a higher entropy in the surroundings, that is, it results in higher overall entropy of the system plus its surroundings. Note that this transfer of entropy requires dis-equilibrium in properties, such as a temperature difference. One example of this is the cooling crystallization of water that can occur when the system's surroundings are below freezing temperatures. Unconstrained heat transfer can spontaneously occur, leading to water molecules freezing into

15488-411: The system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ( T ) and the temperature of the surroundings ( T surr ). The equality still applies for pure heat flow (only heat flow, no change in chemical composition and mass), which is the basis of the accurate determination of

15616-417: The system, which is evident from ordinary experience of refrigeration , for example. In a refrigerator, heat is transferred from cold to hot, but only when forced by an external agent, the refrigeration system. Lord Kelvin expressed the second law in several wordings. Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work (the drained heat

15744-471: The temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states: White hole In general relativity , a white hole is a hypothetical region of spacetime and singularity that cannot be entered from the outside, although energy - matter , light and information can escape from it. In this sense, it is the reverse of a black hole , from which energy-matter, light and information cannot escape. White holes appear in

15872-404: The tendency to reach a state of uniformity or internal and external equilibrium, a state of maximum disorder (entropy). Real non-equilibrium processes always produce entropy, causing increased disorder in the universe, while idealized reversible processes produce no entropy and no process is known to exist that destroys entropy. The tendency of a system to approach uniformity may be counteracted, and

16000-439: The theorem that all abelian groups be represented by one-dimensional irreducible representations. The reason it does this is that it is represented by an anti-unitary operator. It thus opens the way to spinors in quantum mechanics. On the other hand, the notion of quantum-mechanical time reversal turns out to be a useful tool for the development of physically motivated quantum computing and simulation settings, providing, at

16128-441: The theory of eternal black holes . In addition to a black hole region in the future, such a solution of the Einstein field equations has a white hole region in its past. This region does not exist for black holes that have formed through gravitational collapse , however, nor are there any observed physical processes through which a white hole could be formed. Supermassive black holes (SMBHs) are theoretically predicted to be at

16256-416: The trajectories of particles that an outside observer sees rising up away from the event horizon. For an observer outside using Schwarzschild coordinates , infalling particles take an infinite time to reach the black hole horizon infinitely far in the future, while outgoing particles that pass the observer have been traveling outward for an infinite time since crossing the white hole horizon infinitely far in

16384-473: The velocity through the Lorentz force term v × B , and might seem at first to be asymmetric under T . A closer look assures us that B also changes sign under time reversal. This happens because a magnetic field is produced by an electric current, J , which reverses sign under T . Thus, the motion of classical charged particles in electromagnetic fields is also time reversal invariant. (Despite this, it

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