Thermodynamics, Relativity, and Gravitation:
Evidence for a Close Link Between the Three Theories

Jean-Louis Tane  TaneJL@aol.com
Formerly with the Department of Geology,
Joseph Fourier University, Grenoble, France

Abstract: After recalling that Einstein's mass-energy relation is closely linked to the laws of thermodynamics - a proposition that has been presented in a previous article - the aim of this paper is an extension of this link to Newton's laws of gravitation. Arguments are given showing that the law of the evolution of systems, usually described as an increase in entropy, can be more generally understood as an increase in energy and a correlative decrease in mass, according to the principle of relativity. As it is derived exclusively from the macroscopic approach of Physics, the suggested theory offers the advantage - already present in Einstein's relation, Newton's laws and the classical laws of thermodynamics - of a great mathematical simplicity.

Keywords:  Laws of thermodynamics, energy, entropy, Newton's laws, relativity, Einstein's mass-energy relation, reversibility, irreversibility, gravitation, space-time state parameters.

Introduction

This article is a continuation of a first one,¹ itself followed by a comment,² that I had published last year in the Journal of Theoretics.  Some considerations already given are recalled, so that a previous reading of those papers is not strictly indispensable for the understanding of the present one. The only necessary condition is to have in mind some basic understanding of scientific theory corresponding approximately to that gained in a course taken during the first or second year of college.

As in these previous articles, the numerical examples considered in this one have been kept as simple as possible to enable the reader to focus his or her attention on the central subject without being distracted by mathematical details.

It is generally admitted that relativistic equations (Einstein's laws) are only needed for studying phenomena that imply very high velocities, while classical equations (Newton's laws) are sufficient for studying phenomena that imply ordinary velocities.  The standpoint exposed here is that Einstein's laws are potentially present in Newton's laws, so that the classical and relativistic conceptions cannot be dissociated. 

Recalls and Complements Concerning Fundamental Points Previously Presented

1 - Detection of an incoherency in the usual thermodynamic theory and suggested answer for solving the problem.

When applied to simple systems (those for which the energetic exchanges are limited to heat and work of volume) the thermodynamic tool consists of two equations that are classically written :

dU = dQ + dW     (1)

dU = TdS - PdV   (2)

Admitting that the signification of these formulas is well known by the reader (more details are given in article¹), a litigious point can be easily detected when crossing from the first to the second equation. Let us imagine a system where the energetic exchanges are limited to heat. In such case, dW and PdV are both zero, so that equations (1) and (2) are reduced to: 

dU = dQ    (3)

dU = TdS   (4)

which imply necessarily: 

dQ = TdS   (5)

The problem, precisely, comes from the fact that the efficiency of the thermodynamic tool - an efficiency which is not contestable - needs that the relation between equations (3) and (4) would not be written under the form (5) but under the form:

dQ TdS   (6)

In order to solve the problem, I have previously suggested¹ that the physical signification of dU is not the same in equations (3) and (4), nor (consequently) in equations (1) and (2). A more general equation must be substituted to equation (1), which can be given the form:

dU* = dU_e + dU_i   (7)

and whose signification, referring to a determined system, is: 

Total energy (dU*) = energy of external origin (dU_e) + energy of internal origin (dU_i)

In equations (1) and (3), the exact signification of dU is dU_e, while in equations (2) and (4), the exact signification of dU is dU*. Correlatively, dU_i represents the energetic difference between TdS and dQ, whose existence is materialized by the mathematical sign "<" in equation (6). Referring to this equation, it can be remembered that the sign "<" corresponds to real processes (called irreversible processes in thermodynamics), while the sign "=" corresponds to ideal processes (called reversible processes).

As suggested in the previous article¹, the reality of dU_i can be explained by the Einstein mass-energy relation: E = mc².  More accurately, dU_i represents its differential form dE = c²dm, which would be better written dE = - c²dm, since an increase in energy (dE) is correlated to a decrease in mass (dm) and conversely. Therefore, the equation to have in mind can be given the form:

dU_i = - c²dm         (8)

Note that the art of applying thermodynamics to practical problems consists in using the concept S, called entropy, through an equation, which is classically written:

dS = dQ/T_e + dS_i    (9)

where T_e is the external temperature, only equivalent to the internal temperature T_i when the considered process is reversible. In such conditions, the complementary term dS_i becomes zero (even for processes that are not limited to thermo-mechanical exchanges, such as chemical reactions, whose case has been rapidly discussed in paper²)

As proposed in article,¹ if we transcript equation (9) under the form:

T_edS = dQ + T_edS_i   (10)

its correspondence with equation (7) is immediately perceptible. Writing the two formulas one above the other, leads to the doublet:

dU* = dU_e + dU_i     (7)

T_edS = dQ + T_edS_i  (10)

each term of the first equation having the signification and numerical value of the corresponding term of the second equation.

By integration, equations (7) and (10) become respectively:

U* = U_e + U_i     (11)

T_eS = Q + T_eS_i (12)

Note that the coherence between all the formulas already evoked implies that the precise significations - and consequently a better writing - of equations (1) and (2) are respectively:

 dU_e = dQ + dW    (13)

 dU* = T_edS - P_edV (14)

A table is presented further (table 2) which summarizes what needs to be remembered concerning the different conversions.

2 - Exchange of heat between two parts of an isolated system.

The question has been examined in the previous article,¹ through the example of an isolated system composed of the following two parts:

- part 1: a given mass of water (0.24 grams) whose initial temperature is 293 K
- part 2: a given mass of water (0.24 grams) whose temperature is 333 K

For the two masses of water, the value 0.24 grams (g) has been chosen in order to have for each of them C = 1 (C being its heat capacity in J.K^-1), which simplifies the calculus, without reducing the interest of the problem.

Applying equations (7) and (10) to both part 1 and part 2 gives the results that are summarized in the following table and whose detailed derivation has been presented in article.¹  

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 mass-energy 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 "space-time 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.

3 - Exchange of work between two parts of an isolated system

Let us consider an isolated system composed of two gaseous parts 1 and 2, whose initial pressures P₁ and P₂ 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_edV     (16)

which, when applied to each part of the system, gives:

dW₁ = - P₂dV₁ and dW₂ = - P₁dV₂. Observing that dV₂ = - dV₁, the energetic result for the whole system is:

dW = (P₁ - P₂) dV₁    (17)

It is easy to see that this value is always positive, since dV₁ is itself positive when P₁ > P₂ and negative when P₁ < P₂. The only exception is that dW = 0 when P₁ = P₂, 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 P₂ = 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 dQ₁ and dQ₂, 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 P₂ = 0.

It is therefore clear, once again, that the answer of the problem is given by the Einstein mass-energy 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 jack-in-the-box, which is sometimes chosen as a popular illustration of the Einstein mass-energy 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.

4 - Further comments concerning the process of 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 well-known 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 P₂ being zero, equation (17) is reduced to dW = P₁dV₁ 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 dW₁ = P₁dV₁ by T₁ 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 P₁, T₁ and dV₁.

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.

5. Recapitulative summary

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.

Evidence for a Close Link Between Newton's Laws and Einstein's Laws

1 - Exchange of energy between two springs inside an isolated system.

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 F₁ on the piston and spring 2 a strength F₂. 

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:

dW₁ = F₁ dL₁     (21)

dW₂ = - F₂ dL₂   (22)

For the whole system, the work is:

dW = (F₁ - F₂ ) dL₁   (23)

whose value is positive, whether we have F₁ > F₂ or F₁ < F₂. The only exception is that dW = 0 when F₁ = F₂.

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 mass-energy 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 F₁ and F₂ are different. When the difference disappears and is substituted by the condition F₁ = F₂, 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.

2 - Concerning the link between Newton's laws and Einstein's laws.

Let us consider the well-known system consisting of two masses M₁ and M₂, 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 M₁ and M₂.  This example is directly inspired by one examined in the famous book by Frederick J. Bueche and Eugene Hecht.³

Suppose, for example, that M₁ = 9 kg and M₂ = 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:

F₁ - T = M₁a    (24) 

T - F₂ = M₂a    (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 (F₁ - T) and negative for the strength which loses (T - F₂), the corresponding values are respectively:

For the mass M₁:  dW₁ = (F₁ - T) dL₁    (28)

For the mass M₂:  dW₂ = (T - F₂) dL₂    (29) 

Having noted just above that dL₂ = - dL₁, the result for the whole system is:

dW = [(F₁ -T) - (T - F₂)] dL₁       (30)

Introducing the terms F₁* and F₂* defined as:

F₁* = F₁ -T   and   F₂* = T - F₂    (31) 

equation (30) takes the form:

dW = (F₁* - F₂*) dL₁           (32) 

As was the case for equation (23), this intermediary result calls several remarks:

a) Since dL₁ is positive when F₁* > F₂* and negative when F₁* < F₂*, 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 mass-energy relation. 

b) This creation of energy is the consequence of an irreversibility whose origin is the difference between the values M₁ and M₂. If the initial values M₁ and M₂ were equal, implying F₁ = F₂ and F₁* = F₂, 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 F₁* - F₂*) that can be looked constant (at least over the vertical distance covered by the masses M₁ and M₂ in the considered experimental context). Thus, the integration of equation (30) leads to the simple general expression:

W = (F₁* - F₂*) L₁     (33) 

As an example, if L₁ = 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 physico-chemical 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 mass-energy 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 = - (M₁ x g x L) + (M₂ 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 mv², 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 M₂ = 0, the process is reduced to the falling of the mass M₁, and we have necessarily F₂ = 0 and T = 0. Consequently, equation (30) is reduced to dW = F₁dL₁ whose integration (corresponding to the particular value L₁ = 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 L₁ = 1 m towards the Earth's surface, the mass M₁ (more precisely the mass M* = M₁ + mass of the Earth) decreases of a value m = - 88.29/c². 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 M₁ 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 M₁ 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 physico-chemical 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 well-known cosmologic theory of an expanding universe.

 

Conclusions

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 = - c²dm of the Einstein mass-energy 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 mass-energy 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 article¹ 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.

References:

1. Jean-Louis Tane, "Evidence for a close link between the laws of thermodynamics and the Einstein mass-energy relation", Journal of Theoretics, June/July 2000, Vol. 2 No. 3.

2. Journal of Theoretics, Jean-Louis 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", McGraw-Hill Companies, New York. 

Acknowledgments

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.