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The emergence of free energy
PHYSICA ELSEVIER
Physica A 221 (1995) 361-371
The emergence of free energy George Marx Department of Atomic Physics, E6tv6s University, Budapest H-1088, Hungary
Abstract The early Universe was hot and dense, with frequent collisions. Therefore it was in thermal equilibrium, as indicated by the properties of the observed relic black body radiation. Consequently the free energy of the early Universe was zero. Our present world, however, is out of equilibrium; it is locally alive. One has to find places and epochs in the Universe where free energy is being created or stored.
1. Heat death in the past The creation of free energy is impossible in a closed system o f fixed volume. In our world the galaxies accelerate due to gravity. By assuming a homogeneous distribution for simplicity, the equation o f motion of a galaxy at distance R is d2R dt 2
- G 47rGpR3 / 3 R2 ,
( 1)
where p is the mass density. The mass within the sphere of radius R is conserved: M = 41rpR3/3 = const.
(2)
Consequently equation (1) can be solved as R ( t ) = (4.5GM)1/3t2/3,
or taking Eq. (2) also into account,
1 p ( t ) = 67rGt2 .
This is for a world with a dust o f galaxies. A radiation-dominated Universe behaves in a very similar way: p(t) =
3 3 2 ~ G t 2"
0378-4371/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0378-4371 (95)00227-8
362
G. Marx/Physica A 221 (1995) 361-371
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Fig. 1. Spectrum of the cosmic microwavebackground from the FIRAS instrument on COBE at the north Galactic pole, comparedto a black body. It can be said that a non-empty Universe cannot be static; its density must have a singularity on the time axis. (This is true even for non-homogeneous distributions.) The singularity has been called Big Bang. The present overall temperature of the Universe is T = 2.73 K. Light can be considered to be a sine-wave drawn on a rubber sheet. As the size R of the sheet increases, the wavelength a stretches, and T drops correspondingly. This means that the expanding Universe was hot and dense in the past. For large relativistic masses even gravity is intensive. Thus frequent strong collisions produced thermal equilibrium within 10 -40 seconds after the Big Bang. The Gibbs free energy of this extreme relativistic gas was zero. This has been proved by the observed isotropy (up to 10 -4) and by the Planck shape (up to 1%) of the relic microwave radiation [1] (Fig. 1 ). The fact that, according to our present everyday experience, the Universe has fallen out of equilibrium, calls for an explanation.
2. Resurrection from the heat death In a cloud chamber the gaseous mixture of alcohol and air is under pressure and in equilibrium at room temperature at the beginning of the experiment. Then the chamber is quickly expanded, which results in adiabatic cooling. At the lower temperature the new equilibrium state should be air plus liquid alcohol. But the formation of droplets needs time, thus for a transient period there is air and overcooled alcohol vapor in the chamber. This is the short sensitivity period of the cloud chamber. When an ionizing particle passes through during this time, its trajectory will be made visible by the chain
G. Marx/Physica A 221 (1995) 361-371
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Fig. 2. See text for explanation. of alcohol droplets formed around the created ions. Let us consider two containers of the same size, one filled with helium and the other with hydrogen gas. Both contain the same number of molecules; therefore they have the same pressure of 2 atm. Due to the thermal contact between the two pistons, the two gases are at the same temperature, say, 200°C. In this equilibrium state there is no arrow of time. Let us now remove half of the weights! Both gases expand quickly. By raising the joint pistons, they perform the same work f PdV against gravity, at the cost of their internal energy. The monoatomic He cools to 76°C, the diatomic H2 cools to 121°C (Fig. 2). H2 stores energy also in molecular rotations. To perform the same work, the H2 has to cool less, because of its larger heat capacity. Expansion against gravity resulted in a transient temperature difference in the weakly coupled two-component system! But heat conduction begins immediately to level the temperatures. The arrow of time is only a transient phenomenon, but during the transient period one may even drive a steam engine by the temperature difference - at least in a thought experiment. Due to the weak thermal coupling between the two pistons, the two gases will soon reach a new equilibrium at a common temperature of 104°C. Temperature differences may be created in a closed system as well. As an illustration let us consider the following model: Einstein's Universe with non-relativistic monoatomic gas and thermal radiation in it. For a co-expanding volume V = 4zrR3/3 the total entropy is
364
G. Marx/Physica A 221 (1995) 361-371
167"r3 3 S = 1--~-(TrR) + Blog(mTgR 2) + So. The pressure of the system is p =
77-2
3
+
Tg
and the total energy amounts to 47r3 3--
3
E = --~-R T~r + B(m + ~Tg). (By using appropriate units c = h = k = 1.) If both components are in thermal equilibrium and decoupled from each other (zero gas-photon cross section), stretching the phase space in the space direction x implies shrinking in the momentum direction p to keep the phase space volume constant. This means that increasing R(t) results in an inversely proportional decreasing p (t). The average photon energy is proportional to p, and to Tr; therefore at adiabatic cooling
RTr = const, keeping the radiation entropy constant. For a non-relativistic gas the average energy is proportional to p2 and to Tg, and therefore, at adiabatic cooling,
mTgR 2 = const, keeping the gas entropy constant. (One can reach the same conclusion also in a more pictorial way. As space expands like a stretched rubber sheet, the waves behave like sine curves drawn on the sheet: the wavelengths ,A stretch proportional to R(t). The kinetic energy of a gas atom is ~ ,~-2, that of the photon is --~ A-I. Therefore, if the two disconnected components of different scaling behaviour expand adiabatically, 1 Tr '~' ~ ,
1 Tg "~ 8--5.
(3)
While R increases ten-fold, say, radiation temperature drops to 10%, gas temperature drops to 1% of its former value.) In the early hot Universe the gas was ionized, thus the charged particles were in direct contact with the electromagnetic radiation. As the temperature dropped below the ionization temperature, the neutralized non-relativistic gas decoupled from radiation, thus adiabatic expansion created a temperature difference, and consequently free energy [2] (Fig. 3). But due to a remaining weak thermal contact (small but non-vanishing photon cross section) there is a heat transfer Q from the warmer radiation to the cooler gas,
G. Marx/Physica A 221 (1995) 361-371
365
No Absorption
? ( gas
)
~,~.._,
,, ,. . . .
r'"
Fig. 3. Gas + radiation in an open Universe.
The Einstein equation /~+ P~7 = O,
(4)
is evidently valid, but entropy begins to grow in an irreversible way:
s = ° - - 9 - >o.
(5)
The temperature difference is increased by the expansion and decreased by collisions [31, Tr = -Tr R
R
Otprpg(Tr - 7"8)
(6)
Cr
7~g = - 2 T g R -t- OlprPg t T -- Tg). Cg ~ - r
(7)
The collision rate is proportional to the radiation density Pr and to the gas density pg. In order to satisfy the energy condition (3), the corresponding heat capacities C appear in the equations. The heat transfer from the warmer radiation to the cooler gas is irreversible. According to (4), the entropy increases (Fig. 4). One may imagine different scenarios: (a) ce = 0, no coupling: gas cools faster than radiation, the temperature difference increases, the entropy does not increase (Fig. 3). (b) a is small: in the original high density era the heat transfer is intensive, there is a common temperature. The entropy increases. In a later period of the expansion the densities are small, collisions become less frequent. Therefore the heat transfer decreases, and late temperature differences survive. Entropy is produced mainly in the early period (Fig. 4). This is the realistic scenario.
G. Marx/Physica A 221 (1995) 361-371
366
Weak Absorption Fast Expansion
R
~
~,~- ........... ~ - _ _ . ~ _ _ ~ e _ _ . =
)
Fig. 4. Gas + radiationin an open Universe. Strong Absorption
T(gas)
Fig. 5. Gas + radiation in an open Universe.
(c) a is large: due to the intensive thermal contact, the temperatures level up with a small delay, the entropy grows all the time (Fig. 5). (d) a is very large: immediate thermalization, common temperature, no irreversible heat transfer, no entropy production (Fig. 6). This is the case in an ionized plasma. In our actual Universe, in the later periods, at lower temperatures and lower densities the coupling gradually fades away, thus the temperature difference (and the corresponding free energy) created by the expansion and by the different cooling rates survives. The neutralized gas cools faster than radiation. Statistical density fluctuations become stabilized by their own gravity. Galaxies and stars may be born.
3. The case of the oscillating Universe
Big Bang means a mathematical singularity with infinitely large temperature and density. Mathematically and physically it does not make sense to continue functions beyond a singularity. According to scientific consensus, asking about times before the Big Bang
G. Marx/Physica A 221 (1995) 361-371
Absorption
367
j
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~
i
M
e
-
-
Fig. 6. Gas + radiation,closed Universe. is not a sensible question. At the singularity all structures, all possible "archaeological documents" have been destroyed, therefore one cannot prove or disprove any statement about "negative times". Common wisdom is that physical time values constitute an open set: 0 < t < tmax for a closed Universe, 0 < t < oo for an open Universe. There is no first event (without cause) and there is no last event (without consequence). Yakob Borisovich Zel'dovich argued, however, that even a singularity cannot destroy entropy created in the earlier period. Near the singularity, k T >> m c 2 for all the constituents, they behave like photons, thus the entropy is 4a 3
S = f~-T
16¢r3
V = f-i-~(T
R 3
) .
The constancy of this entropy enables us to transfer the entropy before the singularity to the world after the singularity. The Einstein equation (1) can be integrated as
1~2 ca4_E. 2
(8)
R
Let us investigate the case of the closed Universe (E < 0) in more detail. Originally it was argued that the dynamical Eq. (5) is symmetrical with respect to time reflection t ~ - t . Therefore the Big Crunch (R ---+0) is the time-reflected event of the Big Bang [5]. But in our two-component model the irreversible growth of entropy (4) has to be taken into account. By solving the differential equations with E < 0, asymmetric solution can be obtained, with an entropy S considerably larger at the Big Crunch than at the Big Bang (Fig. 7). According to Zel'dovich this means that more entropy is transferred to the next cosmological oscillation than received from the previous one. If one connects the oscillations with the transfer of entropy through the singularities, increasing entropy, wider oscillations, and longer regular periods can be obtained. This offers interesting speculations concerning the Antropic Principle. It might be that the
G. Marx/Physica A 221 (1995) 361-371
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Fig. 7. Oscillations of the Universe. world started with complete order, zero entropy. But as time passed, temperature differences were created. Irreversible processes increased the entropy. Increasing entropy means increasing particle number and increasing pressure, therefore each oscillation lasted longer than the previous one. During the first oscillations, time was too short for nuclear buildup, star formation, biological evolution. But after a sufficient number of oscillations, the emerging Universe inherited entropy enough to have a lifetime of billions of years. In this friendly Universe stars were formed, nuclear fusion supplied additional free energy, life evolved - and here we are.
4. Let the Sun shine! Let us consider a gas sphere (formed by gravity) in the sea of radiation. The kinetic energy of gas particles is proportional to the average temperature (E~n = 1 . 5 N k ( T g ) ) . According to the virial theorem, Epot = - - 2 E k i n . The total energy is negative: E = Ekin -+"Epot = -1.5Nk(Tg). The heat capacity of the star is negative: C = d E / d T = - 1 . 5 N k < O. As space expands, radiation performs work, and therefore it cools. The bound gas sphere is unaffected by the overall expansion, and its temperature does not change. In this way the environment gets cooler than the star. The star radiates away heat in an irreversible way. Its heat capacity is negative, therefore heat loss warms the star up. A temperature difference is created again [4]. The star becomes brighter, and in its central region nuclear fusion begins. In the early hot Universe composite nuclei could not survive: The fast cooling explains that the pre-stdlar clouds were made mostly of hydrogen. In the star hydrogen nuclei collide, H = H --+ (2He) - , H + H .
G. Marx/Physica A 221 (1995) 361-371
369
The 2He nucleus is not stable, and fast fusion is not possible. Weak /3-decays enable transitions only occasionally: H+H
--~ (2He) ___~2H + e + +~,.
From this point on, strong nuclear fusions run fast: 2H +2 H __,4 He, but the time scale of the first weak interaction is billions of years (instead of microseconds), offering free energy (warm sunshine in a cold space) and time as well for biological evolution.
5. The actual Universe
Our thesis can now be formulated: The Universe is dynamically unstable. It expands and cools: different materials with differing scaling laws. For radiation, the entropy contains the dimensionless quantity (RT) 3, corresponding to the adiabatic cooling law T ~ R -1. The entropy of a monoatomic gas contains the term log(mTR2), thus it is cooling like T ~ R -2. The entropy of a diatomic gas contains ½10g(mT5R6), cooling like T ,~ R-~2. A black hole of mass m has the entropy S = 1/167rGT 2, thus in the adiabatic case its temperature is constant: T ~ G -1/2. In the case of radiating entropy off, the star (the black hole) warms up. It can be seen, that in a Universe made of different components, one may expect different cooling rates, therefore gravitational instability leads to the emergence of free energy. The relevant periods are those in which the coupling between two components becomes so weak that the time needed for complete thermalization is longer than the actual age of the Universe. According to our present understanding, the main epochs of the creation of free energy might be the following ones: - First Day (the first 10 -35 seconds). A phase transition in the vacuum might have happened with a delay. The latent heat density p0 of the overcooled vacuum was present in the dynamical Eq. (5), giving 9
= -~- P0 = ~.~ = const, inflating the Universe according to R(t) = R(to)e t/~°, creating enough space. -Second Day (the first 10 -30 seconds). The intermediate boson X of the Grand Unification Theory decayed through weak interaction with considerable delay in an era where mxc 2 < kT. The charge asymmetry of the weak decay might have created the overweight of protons with respect to antiprotons. The conservation theorem of baryonic number explains the survival of the created nucleons, this low-entropy material, until now.
-Third Day (the first milliseconds). Neutrinos are coupled to other particles only by weak interactions. As the Universe became cool and thin enough, the collision time for neutrinos became larger than the age of the Universe. Neutrinos decoupled from other particles, and their number is constant since then. If any of the neutrino types possess a
370
G. Marx/Physica A 221 (1995)361-371
rest mass, they are also a kind of surviving low-entropy material, manifesting themselves only by their gravitation as dark matter. -Fourth Day (the first minutes). As the Universe cooled below the value kT < (mneutron-mproton)C2, the free neutrons started decaying to protons. But the neutron life time is about 15 minutes, and during this time some of the neutrons were captured to make composite light nuclei like 2H, 3He, Li, Be, B. However, soon the number of free neutrons became zero. Only positively charged nuclear matter remained. Their further fusion was prevented by the electric repulsion. In this way the nuclear non-equilibrium was frozen in. The light nuclei mentioned above may serve as nuclear fuel for future fusion reactors (in stars or perhaps on Earth as well). -Fifth Day (the first million years). The first years of the Universe were dominated by a hot plasma: light nuclei, electrons and photons in steady interaction. When the temperature dropped below the ionization temperature of hydrogen (1000 K), neutral atoms were formed, and the Universe became transparent for electromagnetic radiation. The primordial thermal radiation is untouched since then: it can be observed by microwave receivers (Fig. 1). The neutral gases cool at a faster rate than radiation, and clump into galaxies and stars. -Sixth Day (the first billion years). As stars radiate energy away, they warm up due to their negative heat capacity. All the light nuclei burn off in the stellar reactor but IH and 4He survive because the end products of the fusion reactions
1H +1 H --~ 2He, 1H +4 He ~ 5Li, 4He _+_4He ~ 8Be do not exist. This is why 1H and 4He isotopes make up about 99% of the present Universe. They may take part in nuclear fusion by the intervention of weak/3 decay or by triple collision. This is why the life time of the Sun and most stars - visible now on the sky - is prolonged to billions of years. Seventh Day (today). On planet Earth life, vegetation, the biosphere, i.e. order and organization have emerged. Earth receives Q energy from the Sun. The planet does not become warmer and warmer because the same amount Q of energy is radiated back into outer space. Visible sunlight of temperature T = 6000 K brings entropy to Earth. The infrared earth glow of temperature Te = 300 K removes disorder from Earth. Irreversible terrestrial processes produce an extra entropy o- > 0. Thus the entropy balance of the Earth says
AS=o-+
O
O
- - - - - = o" -- ~EE Zs TE
1 --
which may be even negative because Ts = 20. TE. The solar energy flowing through the biosphere washes disorder away, offering a chance for biological and social evolution.
G. Marx/Physica A 221 (1995) 361-371
References [ 1I 121 [31 141 [ 51 161 171 181
E.S. Cheng et al., Bull. Amer. Astron. Soc. 23 (1989) 896. George Marx, Acta Physica Hungarica 62 (1987) 139. George Marx, in: H.P. Diirr Festschrift (Max Planck Inst. Miinchen, 1989) p. 197. George Marx and H. Sato, J. Mod. Phys. 2 (1986) 133. S.W. Hawking, A Brief History of Time (Bantam, 1988). EJ. Dyson, Rev. Mod. Phys. 51 (1979) 447. S.W. Hawking, Phys. Rev. D 13 (1976) 191. R. Alpher and G. Marx, Vistas in Astronomy 35 (1992) 179.
371
Physica A 221 (1995) 361-371
The emergence of free energy George Marx Department of Atomic Physics, E6tv6s University, Budapest H-1088, Hungary
Abstract The early Universe was hot and dense, with frequent collisions. Therefore it was in thermal equilibrium, as indicated by the properties of the observed relic black body radiation. Consequently the free energy of the early Universe was zero. Our present world, however, is out of equilibrium; it is locally alive. One has to find places and epochs in the Universe where free energy is being created or stored.
1. Heat death in the past The creation of free energy is impossible in a closed system o f fixed volume. In our world the galaxies accelerate due to gravity. By assuming a homogeneous distribution for simplicity, the equation o f motion of a galaxy at distance R is d2R dt 2
- G 47rGpR3 / 3 R2 ,
( 1)
where p is the mass density. The mass within the sphere of radius R is conserved: M = 41rpR3/3 = const.
(2)
Consequently equation (1) can be solved as R ( t ) = (4.5GM)1/3t2/3,
or taking Eq. (2) also into account,
1 p ( t ) = 67rGt2 .
This is for a world with a dust o f galaxies. A radiation-dominated Universe behaves in a very similar way: p(t) =
3 3 2 ~ G t 2"
0378-4371/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0378-4371 (95)00227-8
362
G. Marx/Physica A 221 (1995) 361-371
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Fig. 1. Spectrum of the cosmic microwavebackground from the FIRAS instrument on COBE at the north Galactic pole, comparedto a black body. It can be said that a non-empty Universe cannot be static; its density must have a singularity on the time axis. (This is true even for non-homogeneous distributions.) The singularity has been called Big Bang. The present overall temperature of the Universe is T = 2.73 K. Light can be considered to be a sine-wave drawn on a rubber sheet. As the size R of the sheet increases, the wavelength a stretches, and T drops correspondingly. This means that the expanding Universe was hot and dense in the past. For large relativistic masses even gravity is intensive. Thus frequent strong collisions produced thermal equilibrium within 10 -40 seconds after the Big Bang. The Gibbs free energy of this extreme relativistic gas was zero. This has been proved by the observed isotropy (up to 10 -4) and by the Planck shape (up to 1%) of the relic microwave radiation [1] (Fig. 1 ). The fact that, according to our present everyday experience, the Universe has fallen out of equilibrium, calls for an explanation.
2. Resurrection from the heat death In a cloud chamber the gaseous mixture of alcohol and air is under pressure and in equilibrium at room temperature at the beginning of the experiment. Then the chamber is quickly expanded, which results in adiabatic cooling. At the lower temperature the new equilibrium state should be air plus liquid alcohol. But the formation of droplets needs time, thus for a transient period there is air and overcooled alcohol vapor in the chamber. This is the short sensitivity period of the cloud chamber. When an ionizing particle passes through during this time, its trajectory will be made visible by the chain
G. Marx/Physica A 221 (1995) 361-371
Q
•
~
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200°C
200°C
0
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121°C
N
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L
Fig. 2. See text for explanation. of alcohol droplets formed around the created ions. Let us consider two containers of the same size, one filled with helium and the other with hydrogen gas. Both contain the same number of molecules; therefore they have the same pressure of 2 atm. Due to the thermal contact between the two pistons, the two gases are at the same temperature, say, 200°C. In this equilibrium state there is no arrow of time. Let us now remove half of the weights! Both gases expand quickly. By raising the joint pistons, they perform the same work f PdV against gravity, at the cost of their internal energy. The monoatomic He cools to 76°C, the diatomic H2 cools to 121°C (Fig. 2). H2 stores energy also in molecular rotations. To perform the same work, the H2 has to cool less, because of its larger heat capacity. Expansion against gravity resulted in a transient temperature difference in the weakly coupled two-component system! But heat conduction begins immediately to level the temperatures. The arrow of time is only a transient phenomenon, but during the transient period one may even drive a steam engine by the temperature difference - at least in a thought experiment. Due to the weak thermal coupling between the two pistons, the two gases will soon reach a new equilibrium at a common temperature of 104°C. Temperature differences may be created in a closed system as well. As an illustration let us consider the following model: Einstein's Universe with non-relativistic monoatomic gas and thermal radiation in it. For a co-expanding volume V = 4zrR3/3 the total entropy is
364
G. Marx/Physica A 221 (1995) 361-371
167"r3 3 S = 1--~-(TrR) + Blog(mTgR 2) + So. The pressure of the system is p =
77-2
3
+
Tg
and the total energy amounts to 47r3 3--
3
E = --~-R T~r + B(m + ~Tg). (By using appropriate units c = h = k = 1.) If both components are in thermal equilibrium and decoupled from each other (zero gas-photon cross section), stretching the phase space in the space direction x implies shrinking in the momentum direction p to keep the phase space volume constant. This means that increasing R(t) results in an inversely proportional decreasing p (t). The average photon energy is proportional to p, and to Tr; therefore at adiabatic cooling
RTr = const, keeping the radiation entropy constant. For a non-relativistic gas the average energy is proportional to p2 and to Tg, and therefore, at adiabatic cooling,
mTgR 2 = const, keeping the gas entropy constant. (One can reach the same conclusion also in a more pictorial way. As space expands like a stretched rubber sheet, the waves behave like sine curves drawn on the sheet: the wavelengths ,A stretch proportional to R(t). The kinetic energy of a gas atom is ~ ,~-2, that of the photon is --~ A-I. Therefore, if the two disconnected components of different scaling behaviour expand adiabatically, 1 Tr '~' ~ ,
1 Tg "~ 8--5.
(3)
While R increases ten-fold, say, radiation temperature drops to 10%, gas temperature drops to 1% of its former value.) In the early hot Universe the gas was ionized, thus the charged particles were in direct contact with the electromagnetic radiation. As the temperature dropped below the ionization temperature, the neutralized non-relativistic gas decoupled from radiation, thus adiabatic expansion created a temperature difference, and consequently free energy [2] (Fig. 3). But due to a remaining weak thermal contact (small but non-vanishing photon cross section) there is a heat transfer Q from the warmer radiation to the cooler gas,
G. Marx/Physica A 221 (1995) 361-371
365
No Absorption
? ( gas
)
~,~.._,
,, ,. . . .
r'"
Fig. 3. Gas + radiation in an open Universe.
The Einstein equation /~+ P~7 = O,
(4)
is evidently valid, but entropy begins to grow in an irreversible way:
s = ° - - 9 - >o.
(5)
The temperature difference is increased by the expansion and decreased by collisions [31, Tr = -Tr R
R
Otprpg(Tr - 7"8)
(6)
Cr
7~g = - 2 T g R -t- OlprPg t T -- Tg). Cg ~ - r
(7)
The collision rate is proportional to the radiation density Pr and to the gas density pg. In order to satisfy the energy condition (3), the corresponding heat capacities C appear in the equations. The heat transfer from the warmer radiation to the cooler gas is irreversible. According to (4), the entropy increases (Fig. 4). One may imagine different scenarios: (a) ce = 0, no coupling: gas cools faster than radiation, the temperature difference increases, the entropy does not increase (Fig. 3). (b) a is small: in the original high density era the heat transfer is intensive, there is a common temperature. The entropy increases. In a later period of the expansion the densities are small, collisions become less frequent. Therefore the heat transfer decreases, and late temperature differences survive. Entropy is produced mainly in the early period (Fig. 4). This is the realistic scenario.
G. Marx/Physica A 221 (1995) 361-371
366
Weak Absorption Fast Expansion
R
~
~,~- ........... ~ - _ _ . ~ _ _ ~ e _ _ . =
)
Fig. 4. Gas + radiationin an open Universe. Strong Absorption
T(gas)
Fig. 5. Gas + radiation in an open Universe.
(c) a is large: due to the intensive thermal contact, the temperatures level up with a small delay, the entropy grows all the time (Fig. 5). (d) a is very large: immediate thermalization, common temperature, no irreversible heat transfer, no entropy production (Fig. 6). This is the case in an ionized plasma. In our actual Universe, in the later periods, at lower temperatures and lower densities the coupling gradually fades away, thus the temperature difference (and the corresponding free energy) created by the expansion and by the different cooling rates survives. The neutralized gas cools faster than radiation. Statistical density fluctuations become stabilized by their own gravity. Galaxies and stars may be born.
3. The case of the oscillating Universe
Big Bang means a mathematical singularity with infinitely large temperature and density. Mathematically and physically it does not make sense to continue functions beyond a singularity. According to scientific consensus, asking about times before the Big Bang
G. Marx/Physica A 221 (1995) 361-371
Absorption
367
j
I%N~
~
i
M
e
-
-
Fig. 6. Gas + radiation,closed Universe. is not a sensible question. At the singularity all structures, all possible "archaeological documents" have been destroyed, therefore one cannot prove or disprove any statement about "negative times". Common wisdom is that physical time values constitute an open set: 0 < t < tmax for a closed Universe, 0 < t < oo for an open Universe. There is no first event (without cause) and there is no last event (without consequence). Yakob Borisovich Zel'dovich argued, however, that even a singularity cannot destroy entropy created in the earlier period. Near the singularity, k T >> m c 2 for all the constituents, they behave like photons, thus the entropy is 4a 3
S = f~-T
16¢r3
V = f-i-~(T
R 3
) .
The constancy of this entropy enables us to transfer the entropy before the singularity to the world after the singularity. The Einstein equation (1) can be integrated as
1~2 ca4_E. 2
(8)
R
Let us investigate the case of the closed Universe (E < 0) in more detail. Originally it was argued that the dynamical Eq. (5) is symmetrical with respect to time reflection t ~ - t . Therefore the Big Crunch (R ---+0) is the time-reflected event of the Big Bang [5]. But in our two-component model the irreversible growth of entropy (4) has to be taken into account. By solving the differential equations with E < 0, asymmetric solution can be obtained, with an entropy S considerably larger at the Big Crunch than at the Big Bang (Fig. 7). According to Zel'dovich this means that more entropy is transferred to the next cosmological oscillation than received from the previous one. If one connects the oscillations with the transfer of entropy through the singularities, increasing entropy, wider oscillations, and longer regular periods can be obtained. This offers interesting speculations concerning the Antropic Principle. It might be that the
G. Marx/Physica A 221 (1995) 361-371
368
III
47
148
~'9 66 ",~ ~
1~? 184 Z~SO Z39 282
I
~_.......~__~
",
--e'---~ t I~'--)
Fig. 7. Oscillations of the Universe. world started with complete order, zero entropy. But as time passed, temperature differences were created. Irreversible processes increased the entropy. Increasing entropy means increasing particle number and increasing pressure, therefore each oscillation lasted longer than the previous one. During the first oscillations, time was too short for nuclear buildup, star formation, biological evolution. But after a sufficient number of oscillations, the emerging Universe inherited entropy enough to have a lifetime of billions of years. In this friendly Universe stars were formed, nuclear fusion supplied additional free energy, life evolved - and here we are.
4. Let the Sun shine! Let us consider a gas sphere (formed by gravity) in the sea of radiation. The kinetic energy of gas particles is proportional to the average temperature (E~n = 1 . 5 N k ( T g ) ) . According to the virial theorem, Epot = - - 2 E k i n . The total energy is negative: E = Ekin -+"Epot = -1.5Nk(Tg). The heat capacity of the star is negative: C = d E / d T = - 1 . 5 N k < O. As space expands, radiation performs work, and therefore it cools. The bound gas sphere is unaffected by the overall expansion, and its temperature does not change. In this way the environment gets cooler than the star. The star radiates away heat in an irreversible way. Its heat capacity is negative, therefore heat loss warms the star up. A temperature difference is created again [4]. The star becomes brighter, and in its central region nuclear fusion begins. In the early hot Universe composite nuclei could not survive: The fast cooling explains that the pre-stdlar clouds were made mostly of hydrogen. In the star hydrogen nuclei collide, H = H --+ (2He) - , H + H .
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The 2He nucleus is not stable, and fast fusion is not possible. Weak /3-decays enable transitions only occasionally: H+H
--~ (2He) ___~2H + e + +~,.
From this point on, strong nuclear fusions run fast: 2H +2 H __,4 He, but the time scale of the first weak interaction is billions of years (instead of microseconds), offering free energy (warm sunshine in a cold space) and time as well for biological evolution.
5. The actual Universe
Our thesis can now be formulated: The Universe is dynamically unstable. It expands and cools: different materials with differing scaling laws. For radiation, the entropy contains the dimensionless quantity (RT) 3, corresponding to the adiabatic cooling law T ~ R -1. The entropy of a monoatomic gas contains the term log(mTR2), thus it is cooling like T ~ R -2. The entropy of a diatomic gas contains ½10g(mT5R6), cooling like T ,~ R-~2. A black hole of mass m has the entropy S = 1/167rGT 2, thus in the adiabatic case its temperature is constant: T ~ G -1/2. In the case of radiating entropy off, the star (the black hole) warms up. It can be seen, that in a Universe made of different components, one may expect different cooling rates, therefore gravitational instability leads to the emergence of free energy. The relevant periods are those in which the coupling between two components becomes so weak that the time needed for complete thermalization is longer than the actual age of the Universe. According to our present understanding, the main epochs of the creation of free energy might be the following ones: - First Day (the first 10 -35 seconds). A phase transition in the vacuum might have happened with a delay. The latent heat density p0 of the overcooled vacuum was present in the dynamical Eq. (5), giving 9
= -~- P0 = ~.~ = const, inflating the Universe according to R(t) = R(to)e t/~°, creating enough space. -Second Day (the first 10 -30 seconds). The intermediate boson X of the Grand Unification Theory decayed through weak interaction with considerable delay in an era where mxc 2 < kT. The charge asymmetry of the weak decay might have created the overweight of protons with respect to antiprotons. The conservation theorem of baryonic number explains the survival of the created nucleons, this low-entropy material, until now.
-Third Day (the first milliseconds). Neutrinos are coupled to other particles only by weak interactions. As the Universe became cool and thin enough, the collision time for neutrinos became larger than the age of the Universe. Neutrinos decoupled from other particles, and their number is constant since then. If any of the neutrino types possess a
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rest mass, they are also a kind of surviving low-entropy material, manifesting themselves only by their gravitation as dark matter. -Fourth Day (the first minutes). As the Universe cooled below the value kT < (mneutron-mproton)C2, the free neutrons started decaying to protons. But the neutron life time is about 15 minutes, and during this time some of the neutrons were captured to make composite light nuclei like 2H, 3He, Li, Be, B. However, soon the number of free neutrons became zero. Only positively charged nuclear matter remained. Their further fusion was prevented by the electric repulsion. In this way the nuclear non-equilibrium was frozen in. The light nuclei mentioned above may serve as nuclear fuel for future fusion reactors (in stars or perhaps on Earth as well). -Fifth Day (the first million years). The first years of the Universe were dominated by a hot plasma: light nuclei, electrons and photons in steady interaction. When the temperature dropped below the ionization temperature of hydrogen (1000 K), neutral atoms were formed, and the Universe became transparent for electromagnetic radiation. The primordial thermal radiation is untouched since then: it can be observed by microwave receivers (Fig. 1). The neutral gases cool at a faster rate than radiation, and clump into galaxies and stars. -Sixth Day (the first billion years). As stars radiate energy away, they warm up due to their negative heat capacity. All the light nuclei burn off in the stellar reactor but IH and 4He survive because the end products of the fusion reactions
1H +1 H --~ 2He, 1H +4 He ~ 5Li, 4He _+_4He ~ 8Be do not exist. This is why 1H and 4He isotopes make up about 99% of the present Universe. They may take part in nuclear fusion by the intervention of weak/3 decay or by triple collision. This is why the life time of the Sun and most stars - visible now on the sky - is prolonged to billions of years. Seventh Day (today). On planet Earth life, vegetation, the biosphere, i.e. order and organization have emerged. Earth receives Q energy from the Sun. The planet does not become warmer and warmer because the same amount Q of energy is radiated back into outer space. Visible sunlight of temperature T = 6000 K brings entropy to Earth. The infrared earth glow of temperature Te = 300 K removes disorder from Earth. Irreversible terrestrial processes produce an extra entropy o- > 0. Thus the entropy balance of the Earth says
AS=o-+
O
O
- - - - - = o" -- ~EE Zs TE
1 --
which may be even negative because Ts = 20. TE. The solar energy flowing through the biosphere washes disorder away, offering a chance for biological and social evolution.
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References [ 1I 121 [31 141 [ 51 161 171 181
E.S. Cheng et al., Bull. Amer. Astron. Soc. 23 (1989) 896. George Marx, Acta Physica Hungarica 62 (1987) 139. George Marx, in: H.P. Diirr Festschrift (Max Planck Inst. Miinchen, 1989) p. 197. George Marx and H. Sato, J. Mod. Phys. 2 (1986) 133. S.W. Hawking, A Brief History of Time (Bantam, 1988). EJ. Dyson, Rev. Mod. Phys. 51 (1979) 447. S.W. Hawking, Phys. Rev. D 13 (1976) 191. R. Alpher and G. Marx, Vistas in Astronomy 35 (1992) 179.
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