Thermodynamics, Relativity, and Gravitation:

Table
1 
T* 
DU*
(= T_{e}DS) 
DU_{e}
(= DQ) 
DU_{i}
(=T_{e}DS_{i}) 
DS_{i} 
part
1 
302. 89 K 
21. 32 J 
20
J 
1. 32 J 
0. 004 
part
2 
322. 89 K 
 18. 76 J 
 20 J 
1. 24 J 
0. 004 
total 

2. 56 J 
0

2. 56 J 

These data call several remarks:
a) The total value U_{e} = 0 is a simple transcription of the first law of thermodynamics, which states that an isolated system does not exchange energy with its surroundings. Referring to equation (11) we see that such a system is concerned by its reduced form:
U* = U_{i }(15)
Note that in the usual conception of thermodynamics, where nor U* neither U_{i } are taken into account, the equation corresponding to an isolated system is solely written U = 0, whose translation in the new suggested language is U_{e} = 0.
b) The value U* = Ui = 2.56 J represents the total energy created inside the system, according to the Einstein massenergy relation. This result cannot be obtained by the usual practice of thermodynamics, since the concepts U_{i} and U* are not recognized.
c) The obtainment of the intermediate values 1.32 and 1.24 (whose addition gives the total value 2.56) has been explained in my previous article.^{1} It needs the use of a term which can be given the general name "spacetime state parameter" and whose definition, in the present context, is T* = Q/S. Since we are in conditions where both S and Q are variations of a state function, T* appears really as a state parameter. It can be recalled that, in thermodynamic language, the designation "function" is generally used for extensive quantities (such as Q and S) and the designation "parameter" for intensive quantities (such as T*).
Note that S_{i} (right hand column of the table) have the same value for part 1 and part 2. For avoiding an excessive length of the present paper, this interesting point will be discussed in another article.
Let us consider an isolated system composed of two gaseous parts 1 and 2, whose initial pressures P_{1} and P_{2} are different. If the piston separating the two parts is liberated, it will move inside the system until the pressures become equal.
The process can be analyzed using the well known equation:
dW =  P_{e}dV (16)
which, when applied to each part of the system, gives:
dW_{1} =  P_{2}dV_{1} and dW_{2} =  P_{1}dV_{2}. Observing that dV_{2} =  dV_{1}, the energetic result for the whole system is:
dW = (P_{1}  P_{2}) dV_{1} (17)
It is easy to see that this value is always positive, since dV_{1} is itself positive when P_{1} > P_{2} and negative when P_{1} < P_{2}. The only exception is that dW = 0 when P_{1} = P_{2}, but this case corresponds to the ideal conditions of a reversible process, not to the real conditions of an irreversible one.
The system we are dealing with being isolated, if we confront the result dW > 0, given by equation (17), with the usual conception of thermodynamics, we can be tempted to have in mind the following reasoning: Having dU = 0 and dW > 0, it seems evident, referring to equation (1), that the only possible solution is in admitting the condition dQ < 0.
In fact, such hypothesis can rapidly be seen as unrealistic. Imagine that we are in the context (often called "Joule's experiment") where, part 2 is a vacuum, so that the gas is only present in part 1 and that P_{2} = 0. Applying equation (17), we have evidently dW > 0, but a problem arises when we search how to explain the possibility of a negative value for dQ. Trying to estimate separately dQ_{1} and dQ_{2}, we discover the difficulty to conceive an exchange of heat between a gas and a vacuum, so that the first conclusion we retain takes the form dQ = 0.
Then turning to equation (1) and being comforted in the necessity of writing dU = 0 (since the considered system is isolated), we would probably think that the key to the dilemma is in admitting dW = 0 and concluding that equation (17) is not convenient for the process in question.
Such solution is far from being satisfying, because equation (17) shows precisely that dW is maximum when P_{2} = 0.
It is therefore clear, once again, that the answer of the problem is given by the Einstein massenergy relation and needs equation (1) to be substituted by equation (7) as already explained.
Note that the process of the gas expansion, just discussed, is identical to that of the jackinthebox, which is sometimes chosen as a popular illustration of the Einstein massenergy relation. The commentary accompanying the figure generally calls the attention of the reader on the fact that the mass of the spring increases when it is compressed and decreases when it is released. Substituting the jack in place of the gas, and applying to it equation (17)  a little conversion is needed between terms in PdV and terms in FdL  leads to the conclusion, already obtained, that an energy has been created inside an isolated system. Before going deeply into this kind of matter, and crossing from thermodynamics to mechanics, let us examine some complementary details concerning the gas expansion.
It is interesting to recall the classical interpretation that is given to this process. It consists in defining the system as the gas itself and in applying to it equations (1) and (2), taken in their usual conception. Admitting dU = 0 (since the whole system is isolated), the result given by equation (2) is:
dS = P/T dV (18)
Observing that P, T, and dV are all positive, dS is positive too, and we are led to the wellknown conclusion that the entropy of the system increases.
In comparison, and as already seen, the interpretation given by the new suggested theory consists in using equation (17) at the scale of the whole system, defined as including both the gas (part 1) and the vacuum (part 2). The value P_{2} being zero, equation (17) is reduced to dW = P_{1}dV_{1} which gives dW a positive value. From the conceptual point of view, and noting that dW has the dimensions of an energy, this result is evidently different from that given by equation (18), but the numerical value obtained remains the same. Dividing dW_{1} = P_{1}dV_{1} by T_{1} leads effectively to the exact result previously got, since the quantities P, T, and dV present in equation (18) refer to the gas and consequently can be written P_{1}, T_{1} and dV_{1}.
Reversing the procedure, that is multiplying equation (18) times T, equals crossing from the classical conception to the new one. The result is an equation that has the form:
TdS = PdV (19)
and whose physical signification is:
dU_{i} = TdS = PdV (20)
Equation (20) shows clearly that the energy dU_{i} created inside the system has the same numerical value whether it is presented under the designation TdS or PdV. This very observation calls the following remark:
When, according to the classical conception, we write TdS  PdV = 0, the expression is mathematically right but is physically unsuitable and a possible reason is that TdS and PdV don't exist simultaneously. To the contrary, dU_{i} represents any form of energy and, consequently, has a permanent reality. While being outside the topic of the present article, we can suspect an implicit relation between the question in matter and the problem of waves propagation and quantum theory.
All the considerations discussed above are summarized in the following table, which shows the correspondence between the basic equations of the classical thermodynamic theory, and those of the new suggested one.
Table 2 
Irreversible process
dU* =
dU_{e}
+ dU_{i}
= T_{e}dS 
P_{e}dV
dU* =
dQ + dW +
 c^{2}dm
= T_{e}dS
 P_{e}dV 
Reversible process
dU* =
dU_{e}_{ }
=
dQ + dW
dU* = dU_{e}
= TdS  PdV
(with dQ =
TdS, dW =  PdV, T = T_{e}
= T_{i}
and P = P_{e} = P_{i}) 
Let us consider an isolated system composed of two compartments, each of them being held by a spring. We suppose that the springs are separated by a piston and have been initially compressed, so that spring 1 exerts a strength F_{1} on the piston and spring 2 a strength F_{2}.
If the piston (that we suppose initially locked) is liberated, it will move on a distance dL until the two strengths become equal. Considering the displacement dL positive for the strength which wins and negative for the strength which loses, the mechanical analysis of the process is very similar to that examined for the gas in the previous section. We can write that the works released are respectively:
dW_{1} = F_{1} dL_{1} (21)
dW_{2} =  F_{2} dL_{2} (22)
For the whole system, the work is:
dW = (F_{1}  F_{2} ) dL_{1} (23)
whose value is positive, whether we have F_{1} > F_{2} or F_{1} < F_{2}. The only exception is that dW = 0 when F_{1} = F_{2}.
The system being defined as isolated, the situation is identical to that previously examined in the sense that the result dW > 0 means that an energy has been created inside the system. Once again, there is no other possibility for explaining the situation than referring to the Einstein massenergy relation, which states that an energy can be released by a disintegration of mass and conversely. At this stage of the discussion, three important points must be emphasized:
a) the first one, just recalled above, is that dW is always positive in the present context, so that the natural evolution of the system is always an increase in energy by disintegration of mass and never the contrary.
b) the second important point is that this production of energy is the result of an irreversibility, corresponding itself to the fact that the initial values F_{1} and F_{2} are different. When the difference disappears and is substituted by the condition F_{1} = F_{2}, the system is in conditions of reversibility, so that its internal evolution stops.
c) the third important point concerns the fact that contrary to what is possible for equation (17), which divided by T gives equation (18), we cannot divide equation (23) by T with the hope to substitute it by an expression giving the evolution of the system in terms of entropy. Even supposing that T, in the present context, can be correctly estimated, we don't see what would be the scientific interest of such a procedure. We are really confronted to the fact that the condition dS > 0 needs to be substituted by the condition dU_{i} > 0, which gives the same information, but in a more exact and more general way.
The classical use of the thermodynamic tool being mainly centered on the concept of entropy, a separation has been maintained for a long time, in physics, between the thermodynamic processes and the mechanical ones. Roughly speaking, a process falls into the first group when it can be easily described in terms of entropy and in the second group when it cannot. Referring to the different analysis presented above, we see that the two kinds of processes obey the same general law of evolution, which implies a decrease in mass and a correlative increase in energy. As will be shown now, the laws of Newton and the laws of Einstein are intimately associated.
Let us consider the wellknown system consisting of two masses M_{1} and M_{2}, hanging by a string over a frictionless pulley. Such a mechanism is the basis of the so called Atwood's machine and an exercise often proposed in college courses of physics is the calculus of the string tension (noted T) and of the string acceleration (noted a) as functions of M_{1} and M_{2}. This example is directly inspired by one examined in the famous book by Frederick J. Bueche and Eugene Hecht.^{3}
Suppose, for example, that M_{1} = 9 kg and M_{2} = 7 kg and that the pulley is initially locked before being liberated. Knowing that the acceleration of gravity near the Earth's surface is g = 9.81 m.s^{2}, the application of Newton's laws leads to a system of two equations, whose general expression is:
F_{1}  T = M_{1}a (24)
T  F_{2} = M_{2}a (25)
and whose corresponding numerical expression is:
(9 x 9.81)  T = 9 x a (26)
T  (7 x 9.81) = 7 x a (27)
Solving the system gives T = 77.253 N and a = 1.225 ms^{2}
This result having been obtained by the only use of Newton's laws, is it possible to imagine that the considered mechanical process is also related to Einstein's laws? The answer is yes and the argument for this statement is the following:
By analogy with the reasoning adopted in the previous section, we can evaluate the energy which is released inside the system when each mass moves over a vertical distance dL. Counting dL positive for the strength which wins (F_{1}  T) and negative for the strength which loses (T  F_{2}), the corresponding values are respectively:
For the mass M_{1}: dW_{1} = (F_{1}  T) dL_{1} (28)
For the mass M_{2}: dW_{2} = (T  F_{2}) dL_{2} (29)
Having noted just above that dL_{2} =  dL_{1}, the result for the whole system is:
dW = [(F_{1} T)  (T  F_{2})] dL_{1} (30)
Introducing the terms F_{1}* and F_{2}* defined as:
F_{1}* = F_{1} T and F_{2}* = T  F_{2} (31)
equation (30) takes the form:
dW = (F_{1}*  F_{2}*) dL_{1} (32)
As was the case for equation (23), this intermediary result calls several remarks:
a) Since dL_{1} is positive when F_{1}* > F_{2}* and negative when F_{1}* < F_{2}*, the value of dW is always positive. It corresponds to an energy created inside the system by disintegration of a mass dm, according to the Einstein massenergy relation.
b) This creation of energy is the consequence of an irreversibility whose origin is the difference between the values M_{1} and M_{2}. If the initial values M_{1} and M_{2} were equal, implying F_{1} = F_{2} and F_{1}* = F_{2}, the system would have been in conditions of reversibility, and according to Newton's laws, the acceleration of the two masses would have been zero. Note that, contrary to the processes previously examined, the present one shows a direct and evident relevance to gravitation, and is characterized by a level of irreversibility (difference F_{1}*  F_{2}*) that can be looked constant (at least over the vertical distance covered by the masses M_{1} and M_{2} in the considered experimental context). Thus, the integration of equation (30) leads to the simple general expression:
W = (F_{1}*  F_{2}*) L_{1} (33)
As an example, if L_{1} = 1m, the value obtained for W is:
W = 2.45 J (34)
whose corresponding value  m is evidently very small, but is not zero.
c) As already seen, we don't imagine that expressing equation (32), (33) and (34) in terms of entropy, that is dividing each of them by T, would represent a scientific advantage. The reason of this situation is that the more general law of evolution of the physicochemical systems, at least in our near universe, is not a law of entropy but of energy. Whether they are thermodynamic or mechanical, the processes evolve according to the Einstein massenergy relation, taken in the sense of an increase in energy and a correlative decrease in mass.
Now, applying to the considered process the classical concepts of potential energy variation (E_{p}) and kinetic energy variation (E_{c}), we can write successively:
E_{p} =  (M_{1} x g x L) + (M_{2} x g x L)
that is
E_{p} =  (9 x 9.81 x 1) + (7 x 9.81 x 1) =  19.62 J
According to the law of conservation of energy, we have E_{p} + E_{c} = 0 so that E_{c} = 19.62 J. Note that this last value can also be calculated using the formula 1/2 mv^{2}, where v (the speed reached by the two masses when they have covered the distance L = 1 m) can itself be determined by the usual laws of motion.
For the present purpose, the interesting point lies in the fact that the various results just obtained can be interpreted under the light of equation (7) or of its integrated form (11). As was done for the thermodynamic examples previously considered, we can write here:
U* = U_{e} + U_{i }(11)
that is
U* = (E_{p} + E_{c}) + U_{i} (35)
or
2.45 = 0 + 2.45 (36)
Each term of equations (35) and (36) having the same signification and the same value as the corresponding term of equation (11).
Through this observation, equations (7) and (11) appear as representing a general law that covers both mechanical and thermodynamic processes.
Turning to the mechanical example examined above, some complementary remarks can be made:
a) When M_{2} = 0, the process is reduced to the falling of the mass M_{1}, and we have necessarily F_{2} = 0 and T = 0. Consequently, equation (30) is reduced to dW = F_{1}dL_{1} whose integration (corresponding to the particular value L_{1} = 1 m) gives:
W = 9 x 9.81 x 1 = 88.29 J
If we refer exclusively to the classical concept of mechanics (law of conservation of the energy), the interpretation of this result is:
E_{p} =  88.29 J, E_{c} = 88.29 J and E = E_{p} + E_{c} = 0
If we refer to the enlarged context of equation (11), the interpretation becomes:
U* = U_{e} + U_{i} (11)
that is
U* = (E_{p} + E_{c}) + U_{i} (37)
or
88.29 = ( 88.29 + 88.29) + 88.29 (38)
It means that when falling over a vertical distance L_{1} = 1 m towards the Earth's surface, the mass M_{1} (more precisely the mass M* = M_{1} + mass of the Earth) decreases of a value m =  88.29/c^{2}. This value is evidently too small for being measurable, but the correlative increase in energy (88.89 J) is far from being negligible.
b) We can be tempted to think that before falling, the mass M_{1} has been necessarily lifted of the same quantity. Thus, the value  m, lost during the falling, balances exactly the value + m, gained during the lifting. In an experimental context (and considering negligible the gravitational effect of the heat exchanges between the mass M_{1} and the air), this is probably true, but the problem can be different in some other contexts. The case of a meteorite which reaches the Earth is a significant example, since its falling is not preceded by a lifting. Referring to equation (11) and remembering what has been seen concerning U_{i}, we are led to the idea that the sum M_{E} + M_{M} (mass of the Earth + mass of the meteorite) is lower after the falling than it was before. As a first consequence, it can be expected that the contribution of the two bodies to the gravitational field of the larger system to which they belong (the Solar system) is lowered. This gravitational field being attractive, a possible second consequence is that the distance separating the Solar system from the neighboring stellar systems would increase.
If we agree with the idea that every physicochemical process occurring in our near environment obeys equation (7), taken in the sense of an increase in dU_{i} , we are led to conclude that it contributes to lower the mass of the very system to which it belongs, and consequently the mass of the Earth and that of the Solar system. Note that the correlative increase in their mutual distance is equivalent to an increase in potential energy for the Earth, in the same manner as the lifting of a given mass M above the ground increases its potential energy. Extending the concept to other stellar systems, we easily conceive the possibility of a close convergence between the suggested theory and the wellknown cosmologic theory of an expanding universe.
The main ideas that can be held from the present article are the following :
a) A close link exists between the laws of thermodynamics, relativity, and gravitation. It can be summarized under the general equation:
dU* = dU_{e} + dU_{i} (7)
where dU_{e} corresponds to the term usually noted dU = dQ + dW in thermodynamics (reduced to dW in mechanics) and dU_{i} to the derivative dE =  c^{2}dm of the Einstein massenergy relation. A more detailed correspondence is given in Table 2 and in equations (35) to (38).
b) The condition of evolution of a system (at least in our near Universe) can be written dU_{i} > 0, which means that the Einstein massenergy relation goes in the sense of an increase in energy (dE > 0) and a correlative decrease in mass (dm < 0). This precision concerning the sense is argued by:
the fact that the condition dS > 0, which is itself the law of evolution in the thermodynamic theory, implies dE > 0, that is dU_{i} > 0 (this assertion has been justified in my previous article^{1} and recalled in paragraphs 1 and 2 of this article),
the fact that Newton's laws (which constitute the central laws of mechanics and gravitation) when analyzed under the light of equation (7), implies dU_{i} > 0, that is dE > 0 (this assertion has been justified in paragraph 3 of this article).
c) While exclusively derived from the macroscopic approach of physics, these considerations are indirectly relevant to fundamental problems that, at present, are actively debated by scientists (gravitational waves, black holes, antimatter, spring theory, great unified theory [GUT], theory of everything [TOE], etc...). Among the questions that the above discussion has opened is that of knowing precisely what are the gravitational consequences of an evolution implying an increase in energy and a correlative decrease in mass. The hypothesis of an expanding Universe seems to be in good accordance with this condition of evolution. Conversely, the eventuality of an evolution going in the opposite direction is theoretically possible, and an interesting question, related to this problem, is that of the of the exact direction of evolution in anomalous contexts such as black holes and antimatter.
1. JeanLouis Tane, "Evidence for a close link between the laws of thermodynamics and the Einstein massenergy relation", Journal of Theoretics, June/July 2000, Vol. 2 No. 3.
2. Journal of Theoretics, JeanLouis Tane, Comments Section, Aug/Nov. 2000, Vol 2 No. 4.
3. F.J. Bueche E. Hecht, "Schaum's Outline of Theory and Problems of College Physics 1997", McGrawHill Companies, New York.
I would like to thank all the persons who contacted me after the publishing of those previous articles. Concerning this new paper, I hope that the given recalls would correctly answer some questions that I have been asked and that the suggested extension of the theory can be a useful contribution to the discussion.
?Journal of Theoretics, Inc. 2001