Further development of the Mie-de Broglie theory of Quantum Gravity and the implications for the General Theory of Relativity. E. P. J. de Haas E-mail: [email protected] Abstract. The Mie-de Broglie theory of quantum gravity, derived in a previous paper by the author, had a restricted value because it seemed rather disconnected from main stream modern physics, due to the circumstance that both Mie’s and de Broglie’s theories have become ”losing” or forgotten theories in the history of physics. But the Mie-de Broglie QG, incorporating a unification at the level of electrons, atoms and nuclei, is less isolated than it seems. With the use of von Laue’s relativistic tensor dynamics I will connect the Mie-de Broglie QG to modern physics. A relation derived by Yarman, will prove to be a key ingredient. As a result, we claim that one of the basic axioms of General Theory of Relativity, the principle of equivalence, is incompatible with the existence of de Broglie’s wave-lengths in Quantum Mechanics. So GTR and QM based on the wavelength postulate are non-unifiable. If we choose de Broglie’s phase harmony as fundamental, a new theory of gravity is needed. The Mie-Yarman theory of gravity seems to qualify as such.

PACS numbers: 01.65.+g, 03.65.Ta, 04.20.Cv, 04.60.-m

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1. Introduction In an earlier paper we connected the theory of gravity of Gustav Mie to the quantum postulates of Louis de Broglie, thus realizing a theory of quantum-gravity [1]. This theory of QG had a restricted value because both Mie’s and de Broglie’s theories have become ”losing theories” in the history of physics. Mie’s theory of gravity seems to have been forgotten almost completely. In the domain of gravity, it is the competing theory of Einstein that ”won” and that has since then dominated the scene. In the area of the quantum of action, Louis de Broglie’s approach has been set aside since the Solvay Conference of 1927, where the Copenhagen interpretation of Quantum Mechanics proved to be the strongest one. So our theory of Mie-de Broglie QG is something very strange. To my knowledge, it is the first theory in which gravity and quantum meet on such a simple and basic level of physics. As such, the Mie-de Broglie theory of QG, although composed of two ”loosing” theories, cannot be a ”loosing” theory itself, because there are no competitors on this level, it is unique. It’s uniqueness is increased through the level at which Mie-de Broglie QG integrates quantization and gravitation, at the level of electrons, atoms and nuclei, whereas modern approaches aim at integration at the Planck length, out of reach of our present day experimental capacities [2]. So our Mie-de Broglie QG seems highly disconnected from main stream modern physics. Since the writing of the first Mie-de Broglie paper, I have been searching for a way to integrate Mie-de Broglie QG in modern physics. A relation derived independently by Tolga Yarman, ug = (1/γ 2 )ui , and then rediscovered by Yarman in the papers of Gustav Mie, made this possible. In this paper I will show how to connect de Mie-de Broglie QG to Laue’s tensor dynamics. The Mie-de Broglie QG is related to, can be derived from, the mechanical stress-energy tensor and it’s trace. In this derivation, the only thing we need from Quantum Mechanics is Planck’s basic postulate U = hν. The connection I found can easily be related to the matter input side of the GTR. It’s connection to the metric side of the GTR will be outside the scope of this paper. Neither will I try to connect the Mie-de Broglie QG to modern (Copenhagen) Quantum Mechanics. 2. The combination of Mie and de Broglie as a starting point of QG In Gustav Mie’s 1912-1913 papers ([3],[4],[5]) I found the relations between the inertial mass and the rest mass mi =p γm0 and between the gravitational mass and the rest mass 2 mg = (1/γ )m0 , with γ = ( 1 − v 2 /c2 )−1 , ([5],p. 40). In terms of the energy U and using U = mc2 we can write: Ui = γU0 Ug =

1 U0 . γ

(1) (2)

The first equation (1) is forever connected to Einstein’s STR ([6]; [7], p. 313), the second equation (2) was Mie’s own result.

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In the year 1923 Louis de Broglie discovered the basic relations of modern WaveMechanics. In his foundational paper and in his thesis he compared the inertial frequency νi , which he postulated himself, with the clock frequency νc , postulated by Einstein ([8], [9]): νi = γν0 νc =

1 ν0 . γ

(3) (4)

The first νi = γν0 was based on Einstein’s Ui = γU0 and on Planck’s quantum postulate U = hν [10]. The second νc = (1/γ)ν0 was based on Einstein’s 1907 analysis of moving clocks ([7], p. 212), but remained, in de Broglie’s analysis, disconnected from an energy consideration. For a fast moving particle, the inertial frequency increased and the clock frequency decreased. De Broglie showed that by attributing an inertial wave-length λi to the moving particle, the two phenomena remained in phase, giving ϕc = ϕi , a property he called the ”Harmony of the Phases”. In my previous paper I posed the question what kind of energy to connect to the clock frequency of a particle [1]. I will present the argumentation of this paper in a somewhat different fashion, by using the concept of clock energy. The idea to use clock energy stems from Yarman’s concept of clock mass [11]. If we multiply both sides of equation (4) by Planck’s constant h, we get 1 hνc = h ν0 . (5) γ If we write the quantized clock energy as Uc = hνc and use Planck’s postulate U0 = hν0 , then equation (5) leads to 1 Uc = U0 . (6) γ Together with Mie’s equation (2) for the gravitational energy this results in the identification of the quantum clock energy with the gravitational energy, Ug = Uc .

(7)

If we presume gravitational energy to exist as quantized according to the relation Ug = hνg then we are allowed to identify the clock frequency with the gravitational frequency of a particle: νg = νc . It would seem that we have a starting point for a theory of Quantum Gravity. But we have a difficulty with the GTR, the theory of gravity that dominated the twentieth century. In our derivation we used Mie’s relation equation(2) as a one of the starting points, so the resulting QG depends on the validity of this relation. Einstein based his GTR on the principle of equivalence mi = mg , which he formulated for the first time in 1907 [12]. This principle functions as an axiom of GTR, not in a strict mathematical sense but as a guiding principle in the process of its creation and its subsequent application [13]. But the principle of equivalence mi = mg is clearly not in

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accordance with Mie’s relation. So we cannot integrate our Mie-de Broglie theory of QG with Einstein’s GTR. What to do, abandon our Mie-de Broglie QG or try to develop it further, if necessary as a competing theory? We choose for the latter option. We can derive Mie’s relations from Laue’s 1911 relativistic tensor dynamics, the same dynamics that is used by Einstein in as the matter input side of his GTR [15]. This will enable us to compare Einstein’s GTR with Mie-de Broglie QG. 3. Max von Laue’s relativistic conservation laws. From the beginning, the STR as a kinematical theory was in need of a dynamics. This dynamics was searched for by the avant garde in between 1905 and 1910 and was finally formulated by von Laue in 1911 as a relativistic tensor dynamics [14]. But Laue’s dynamics did not incorporate gravity, nor did it reflect the quantized aspect of matter. This creates the impression that von Laue did not add something of his own, a circumstance that makes von Laue rather unknown in the textbooks of standard college physics. But the originality of his work, especially his thermodynamic deviation from Einstein’s U = mc2 , is still there to be rediscovered. 3.1. Basic definitions and the mechanical stress energy tensor The basic definitions we use are quite common in STR and GTR [17], [18]. We start with an observer who has a given three as v, a rest mass as m0 and an p vector velocity −1 2 2 inertial mass mi = γm0 with γ = ( 1 − v /c ) . Then we can define the coordinate velocity four vector as " # d ic Vµ = Rµ = . (8) v dt The proper velocity four vector on the other hand will be defined using the proper time τ as " # d icγ Uµ = Rµ = = γVµ . (9) γv dτ The momentum four vector will be # " # " i icm U i i = mi Vµ = m0 Uµ . Pµ = c = mi v pi We further define the rest mass density as dm0 , ρ0 = dV0 so with 1 dV = dV0 γ

(10)

(11)

(12)

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and the inertial mass density as dmi (13) ρi = dV we get, in accordance with Arthur Haas’ 1930 exposition on relativity ([19], p. 365), dmi dγm0 ρi = = 1 = γ 2 ρ0 . (14) dV dV 0 γ The momentum density four vector will be defined as # " i d dmi u i = Gµ = c Pµ = Vµ = ρi Vµ = γ 2 ρ0 Vµ = γρ0 Uµ . gi dV dV

(15)

The mechanical stress energy tensor, introduced by Max von Laue in 1911, was defined by him as ([20], p.150) Tµν = ρ0 Uµ Uν .

(16)

Pauli gave the same definition in his standard work on relativity ([17], p. 117). With the definitions just given we get Tµν = ρ0 Uµ Uν = γ 2 ρ0 Vµ Vν = ρi Vµ Vν = Vµ ρi Vν = Vµ Gν .

(17)

So the mechanical stress energy tensor can also be written as Tµν = Vµ Gν .

(18)

In the exposition on relativity of Arthur Haas, the first definition ρ0 Uµ Uν is described as the ”Materie-tensor” of General Relativity, while ρi Vµ Vν is described as the ”Materietensor” of Special Relativity ([19], p. 395 and p. 365). Although the derivation seems to demonstrate an equivalence between the two formulations of equation (16) and equation (18), the difference between the two is fundamental. Equation (16) is symmetric by definition, while equation (18) can be asymmetric, because, as von Laue already remarked in 1911, Vµ and Gµ do not have to be parallel all the time ([20], p. 167) This crucial difference between ρ0 Uµ Uν and Vµ Gν was also discussed by de Broglie in connection with his analysis of electron spin ([21], p. 55). In the modern GTR approach, given for example by Weinberg ([22], p. 48) and Rindler ([18], p. 156), the stress energy tensor of a perfect fluid, with a rest-system pressure p0 , is given as p0 Tµν = (ρ0 + 2 )Uµ Uν − p0 gµν . (19) c According to Rindler, in such a symmetric situation we do not have ρi = γ 2 ρ0 in the frame of a moving observer ([18], p. 156). But we are not studying perfect fluids, we are interested in quantum mechanical situations. We want to apply the result to quantized matter. The most basic model we can begin with is a perfect dust, as a quantized perfect fluid without a pressure in the rest frame of the particles themselves, giving p0 = 0. This leads us, starting with the formulation of Weinberg and Rindler, to the same stress energy tensor as given by Laue, Pauli and Haas, Tµν = ρ0 Uµ Uν . In those situations we

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are allowed to use ρi = γ 2 ρ0 , implying the validity of Tµν = Vµ Gν . But we believe Vµ Gν to be more fundamental than ρ0 Uµ Uν . Because ρ0 Uµ Uν is symmetric by definition, it can never incorporate a torque as a changing angular momentum. The asymmetry of Vµ Gν cannot possibly be a property added to ρ0 Uµ Uν , whereas Vµ Gν can be made symmetric by adding the property Gν = ρi Vµ . As already observed by de Broglie, the asymmetry of Vµ Gν will be needed in the future to connect Laue’s relativistic tensor dynamics to quantum mechanics [21]. Especially a quantum jump, understood as a sudden change of angular momentum of the quantized system, incorporates a torque and thus, in its relativistic formulation, necessarily an asymmetric stress energy tensor. Using Tµν = Vµ Gν we can write the SE-tensor as # " # " #" i ic u −u icg c Tµν = = i . (20) g uv v ⊗ g v c This form of the stress energy tensor can already be found in de Broglie’s expos´e ([21], p. 55). 3.2. The conservation laws for energy and momentum For closed systems energy and momentum are conserved and the conservation laws can be expressed as ∂µ Tµν = 0 with the four vector partial derivative defined as " # − ci ∂t ∂µ = . (21) ∇ For closed systems this leads to " #" # " # " # i − ci ∂t −u icg (∂ u + ∇ · uv) 0 t ∂µ Tµν = = c = . (22) i ∇ uv v ⊗ g ∂t g + ∇(v ⊗ g) c Open systems, on the other hand, have " # " # i i (∂ u + ∇ · uv) P t ∂µ Tµν = c = c . ∂t g + ∇(v ⊗ g) f

(23)

We have used P for the power density and f for the force density. If we write S = uv for the energy density current or Umov’s vector, this results in the conservation equation for energy ∇ · S + ∂t u = 0,

(24)

which can also be written as ∂µ Sµ = 0, and the conservation equation for momentum ∂t g + ∇(v ⊗ g) = 0.

(25)

Open systems have a non-zero four force density, so the energy and momentum are not conserved for those systems. An electron in free space should be a closed system but according to relativistic electrodynamics it isn’t. This paradox is connected to the Coulomb-self-force and -energy of the free electron. The problem of the Coulomb selfforce and -energy of a free electron already troubled Lorentz, Abraham and Poincar´e and

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transformed through the work of Einstein, Minkowsky and Laue into a conflict between relativistic electrodynamics and relativistic mechanics [23]. In Laue’s tensor-dynamics, conservation-law considerations require the stress-energy tensor of a free particle in mech space to have a zero divergence: ∂µ Tµν = 0. This ensures action to equal reaction and energy to be conserved. According to von Laue, the electron in free space is a static system, so the divergence of its stress-energy tensor should be zero [20]. But in Minkowsky’s relativistic electrodynamics, the divergence of the stress-energy tensor of the electromagnetic field is zero only in charge-free space and equals the electromagnetic four-force when charges are present. In electrodynamics, the electron in free space is em a charge in its own field, so we have ∂µ Tµν = fνem [24]. The electron in free space feels its own electromagnetic four-force density, which is not zero and not balanced by a reaction force density. As a result, the electron in free space acts a net EM-force and EM-power on itself. We could call it a Baron von M¨ unchausen situation. A free electron with a non-zero power must have energy flowing in or out without compensation and an infinite amount of energy, positive or negative, will be assembled. The net force that this free particle acts on itself will create a runaway situation and its momentum will become infinite. Two mayor strategies to solve the problem have been developed in history. The first strategy postulates a Poincar´e-mechanism and adds a Poincar´e-tensor to the electromagnetic one. The second strategy tries to change the EM-stress-energy tensor. The second strategy is incorporated in the 1912 theory of Mie [3], the first in Einstein’s 1919 attempt to solve the problem within general relativity [25]. I studied Mie’s papers in the context of this problem, of the non-conservation of electrodynamic linear momentum, saw Mie’s equations (1) and (2) and realized the possibility to connect them to de Broglie’s matter wave treatment of the relativistic electron. Eventually, solving the problem of the electron will be a mayor challenge for our Mie-de Broglie QG. This will involve the introduction of the concept of phase harmony in relativistic electrodynamics. If this can be done, it will, through the phase harmony concept, fuse quantum waves, gravity and electrodynamics on a very fundamental level of physics. 3.3. The conservation law for angular momentum A relativistic system also has a mechanical torque-tensor Nµν = Tµν − Tνµ = Vµ Gν − Vν Gµ .

(26)

If we use the abbreviation a for anti-symmetric we can define na = v × g and ga = (1/c2 )uv − g. This allows us to write the torque-tensor in full as: 0 −icga,1 −icga,2 −icga,3 0 na,3 −na,2 icga,1 (27) Nµν = . 0 na,1 icga,2 −na,3 icga,3 na,2 −na,1 0 A closed system, for example a particle in empty space, should not acquire any additional angular momentum, free as it is from external influences. The conservation

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of the intrinsic angular momentum requires the stress-energy tensor to be symmetric, Vµ Gν = Vν Gµ , and the torque-density tensor to vanish, so Nµν = 0. This leads to na = v × g = 0 and ga = (1/c2 )uv − g = 0, which gives 1 g = 2 uv (28) c for a moving particle in free space. According to von Laue, electromagnetic systems do not automatically have a symmetric stress energy tensor [26]. In such cases, the angular momentum cannot be a conserved quantity. The correct formulation of the electrodynamic stress energy tensor in case of material media remains an unsolved problem of modern physics [27], [28]. So the conservation of angular momentum cannot always be assumed. Because the matter input side of the GTR involves symmetric stress energy tensors only, non symmetric electromagnetic systems describing material media cannot be integrated into the GTR. These non-symmetric systems also surpass the phase harmony condition [29]. 4. Deriving Mie’s relation from the symmetric stress energy tensor and the source of gravitational energy. After the publication of the paper in which I connected Mie to de Broglie, Yarman, a colleague from Istanbul, send me an email in which he pointed me to the fact that Mie’s relation equation (2), together with Einstein’s equation (1), lead to ug = (1/γ 2 )ui . This relation can be found, as I checked afterwards, in encrypted form in Mie’s third paper ([5], p. 53). The relation ug = (1/γ 2 )ui was independently found by Yarman, though through a straightforward derivation [30]. Yarman based himself solely on Einstein’s mass-energy equivalence and the energy conservation law. By applying these relations to the gravitational binding energy, treated as any other binding energy and thus capable of changing the rest mass of the particle in consideration, he concluded that in the case of moving particles, gravitational mass could not equal inertial mass. Instead of the principle of equivalence, Yarman found the relation mg = (1/γ 2 )mi . A few weeks after Yarman emailed me this result, I stumbled upon this relation while analyzing the electrodynamic version of Laue’s tensor dynamics [29]. Being on the alert, I realized the opportunity to connect Laue’s tensor dynamics to the Mie-de Broglie theory of QG. Then Yarman send me two new papers in which he further developed his principle guiding and unifying idea of the quantum mechanical architecture of matter as being the fundamental unified cause of relativistic clock behavior in both the kinematic Special Relativity as in the gravitational General Relativity [11], [31]. Together these developments provided me with the mathematical and conceptual substance necessary to write this paper. 4.1. From Laue to Yarman and Mie To begin with, I will show how Yarman’s result, written in the form ug = (1/γ 2 )ui , can be derived from Laue’s relativistic tensor dynamics. This leads us directly to Mie’s relation written as Ug = (1/γ)U0 .

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Let’s start with the gravitational energy density as it is generally defined, through the trace of the mechanical stress energy tensor ug = −Vµ Gµ = ui − v · gi

(29)

If we assume a context with Nµν = 0, we have gi = (1/c2 )ui v. In these situations, the gravitational energy can be written as 1 v2 1 u v = (1 − )ui = 2 ui . i 2 2 c c γ is symmetric and ug = −Vµ Gµ we have

ug = −Vµ Gµ = ui − v · gi = ui − v · So in all cases where Tµν

(30)

1 ui . (31) γ2 If we eliminate the densities we get 1 Ug = 2 Ui . (32) γ If we further assume Ui = γU0 , from Einstein’s STR, we get Mie’s relation 1 Ug = U0 . (33) γ So Mie’s relation is valid in all cases where the stress energy tensor Vµ Gν is symmetric, where the gravitational energy density is defined as ug = −Vµ Gµ , and where we are allowed to use the relation Ui = γU0 . ug =

4.2. The invariance of the gravitational energy density We can arrive at this result through a slightly different reasoning. Without proof, Gustav Mie assumed the gravitational energy density to be a Lorentz Invariant scalar or a four scalar. As we showed in our previous paper on the relation of Mie to de Broglie, Mie’s gravitational energy density can be identified as the trace of the mechanical stress energy tensor [1]. This can be used to prove that ug , written as −ρ0 Uµ Uµ , is a four-scalar: 1 ) = ρ0 c2 = u0 . (34) 2 γ Relativistic tensor dynamics gives ug = u0 , in accordance with Mie’s assumption. So we don’t have mi = mg , but we have ρg = ρ0 , an invariant gravitational mass density. Using the familiar dV = (1/γ)dV0 , we get Mie’s mg = (1/γ)m0 . When we add the relation mi = γm0 we get Yarman’s mg = (1/γ 2 )mi . ug = −ρ0 Uµ Uµ = −ρ0 γ 2 Vµ Vµ = (−ρ0 γ 2 )(v 2 − c2 ) = (−ρ0 γ 2 )(−c2

4.3. The quantization of gravitational energy If we add Planck’s basic postulate and apply it, as de Broglie did, to the rest energy as U0 = hν0 , Mie’s relation Ug = (1/γ)U0 results in Ug =

1 hν0 . γ

(35)

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According to Einstein’s STR, the right hand side of the last equation equals hνc , so Ug = hνc .

(36)

This implicates that we have derived Mie’s relations and our connection of Mie to de Broglie for all situations where the stress energy tensor Vµ Gν = −ρ0 Uµ Uν is symmetric, where the gravitational energy is defined as ug = −Vµ Gµ = −ρ0 Uµ Uµ , and where we are allowed to use the relations Ui = γU0 , U0 = hν0 and νc = (1/γ)ν0 . 5. Some implications of the Mie-de Broglie theory of Quantum Gravity 5.1. Falsification of the principle of equivalence In the perspective created by equation (36), the decrease of gravitational energy for moving particles has already been measured and thus experimentally verified. It has been proven beyond doubt that moving clocks slow down ([18], p. 66), so according to the perspective created by equation (36), the decrease of gravitational energy of these particle-like quantum-clocks has also been confirmed. And because the increase of inertial energy for moving particles has also been verified experimentally, the principle of equivalence has already been falsified in the case of moving particles. 5.2. Quantum Gravity as the causational factor replacing space curvature. There is another interesting point of view, formulated by Yarman as: both gravitation and uniform translational motion, affects the clocks, in exactly the same respect. Yarman added: it must require quite a captivation, not to consider a light phenomenon such as [gravitational] red shift in mere terms of quantum mechanics; amazingly though the history, now for many very decades, gave birth to the unexpected. [11]. In Pais’ biography of Einstein, a reason is given: First, (in the 1911 paper) Einstein derives equation 11.2 for the energy shift; then he starts ”all over again” and derives the frequency shift (equation 11.5). It is no accident, I am sure, that he did not derive only one of these equations and from there go to the other one with the help of U = hν. He had had something to do with U = hν. It cannot have slipped his mind; the quantum theory never slipped his mind. However, it was Einstein’s style forever to avoid the quantum theory if he could help it - as in the case of the energy and frequency shift ([32], p. 197). It seems that Einstein’s avoidance has become the norm amongst specialists in GTR. We on the other hand do not pretend to be able to derive Planck’s quantum postulate, we just accept it as a basic principle. Then the question becomes unavoidable: ”Why do clocks slow down?” Because of their velocity? Because of the curvature? Because of gravity? Because of the quantum architecture of matter? Why? If we inverse the logics of the previous derivation and start with the end results Ug = hνc and Ug = (1/γ)U0 , we can interpret the decrease of gravitational energy and the slowing down of quantum clocks as coinciding events. Thus the slowing down of

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moving quantum clocks and the slowing down of quantum clocks who are deeper in a gravitational field can be related to one single physical circumstance: the decrease of the gravitational energy of the quantum particle used for time measurement in a clock quantum-mechanism. A moving quantum-clock has less Ug , because Ug = (1/γ)U0 , and thus also a lower clock frequency, because Ug = hνc . A quantum-clock which is positioned lower in a g-field has less Ug and thus also a lower quantum-clock frequency, because Ug = hνc . Both phenomena, the first in STR and considered to be purely kinematic in its origin, the second in GTR and considered to be part of space-time deformation, are in this way related to one and the same circumstance in the Mie-de Broglie QG. In our interpretation, the gravitational red shift has to be interpreted as a direct effect of Quantum Gravity and not as something caused by the curvature of the metric. So Mie-de Broglie QG and Einstein’s GTR are competing theories. At the same time, Mie-de Broglie QG adds a dynamics to Einstein’s purely kinematical STR. Our conclusions based on equation (35) and equation (36) are in accordance with the results of Yarman in his 2004 paper [30]. In this paper, presenting a summary of years of research on the subject, Yarman wrote : A wave-like clock [=a quantum-clock] in a gravitational field, retards via quantum mechanics, due to the mass deficiency it develops in there, and this, as much as the binding energy it displays in the gravitational field;... and Note that, according to our approach, the classical gravitational red shift and a related mass decrease, occur to be concomitant quantum mechanical effects. 6. Implications for the Unification program 6.1. Wave Mechanics and Einstein’s GTR are non-unifiable As we already stated, in all cases where Tµν = Vµ Gν is symmetric and ug = −Vµ Gµ we have 1 ug = 2 ui . (37) γ To get Einstein’s mg = mi , we must further assume v · g = 0, which, because of gi = (1/c2 )ui v and ui 6= 0, implies v = 0. So according to Laue’s tensor dynamics, the principle of equivalence mg = mi is only valid for completely static situations. In 1992 John Norton came to the same conclusion [15]. The stress energy tensors Tµν that may be used in the GTR must satisfy two criteria, ∂µ Tµν = 0 and Nµν = 0. The principle of equivalence mg = mi is an additional demand that further restricts the set of allowed stress energy tensors in GTR. But in order to have a wave-length λi , particles must move. Stationary quantum states, described by the Schr¨odinger Equation, are about standing waves, so with moving particles. Without movement of the particles involved, we have νi = νc , so no wave-length is needed to harmonize the phases. If we have νi = νc we can multiply it by h to get hνi = hνc or Ui = Uc . Because we identified clock energy as gravitational energy this implies Ui = Ug and mi = mg . We conclude that the principle of equivalence as an axiom of

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GTR excludes wave-lengths and thus wave-mechanics or Quantum Mechanics. Because their basic principles are mutually exclusive, the validity of the principle of equivalence for the one and the existence of particles with wave-lengths for the other, we proclaim that GTR and QM are non-unifiable. The General Theory of Relativity as a theory of gravity seems much to restricted to be unifiable with the wavelength postulate of Louis de Broglie. I emphasized the word restricted because I do not claim GTR to be a wrong theory. In those regions of reality where the axioms of GTR are valid, the theory of GTR, being bases on those axioms, is valid. What I claim is that in the regions where de Broglie’s wavelength postulate can be successfully applied, Einstein’s restrictions put on his stress-energy tensors are such that they cannot be fulfilled and so GTR cannot be applied. 6.2. A new theory of gravity is needed. Einstein’s GTR cannot be integrated with de Broglie’s quantum postulates. If we choose de Broglie’s phase harmony as fundamental, a new theory of gravity is needed. Einstein’s principle of equivalence is only valid for static masses and cannot be applied to moving objects. Because a particle on a geodesic is not really moving, GTR’s space curvature has been applied to moving objects. But it can only work for so called ”test particles” or point objects with a negligible mass or for objects with to small a motion to make a difference. For fast moving objects with considerable mass, thus influencing the local g-field, we do need a new theory. This theory of gravity must be in accordance with 1 Ug = 2 Ui . (38) γ If, in contrast to Einstein, we directly apply Planck’s postulate U = hν, the condition for quantum gravitational phase harmony becomes 1 νg = 2 νi . (39) γ By identifying gravitational frequency as clock frequency we arrive at de Broglie’s 1923 result. In the words of de Broglie: ... the phase agreement is realized if one has 1 ν1 = 2 ν. γ a condition that is clearly satisfied by the definitions of ν1 and ν. The demonstration of this important result rests uniquely on the principle of special relativity and on the correctness of the quantum relationship as much for the fixed observer as for the moving observer. In the original text, de Broglie used the notations ν for the inertial frequency νi and ν1 for the internal periodic phenomenon or the clock frequency νc [8]. In the notation we use this gives 1 νc = 2 νi (40) γ as the condition for de Broglie’s 1923 Harmony of the Phases.

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Mie’s theory of gravity is compatible with de Broglie’s phase harmony and it’s basic assumptions can be proven in relativistic tensor dynamics. But Mie was not successful in predicting the cosmological experimental facts. Yarman’s theory of gravity is compatible with de Broglie’s phase harmony and it does predict the crucial cosmological experimental facts [31]. Yarman’s theory of gravity seems to a high degree unifiable with Mie’s theory of gravity. So the Mie-Yarman theory of gravity might be the candidate needed. As a scalar theory of gravity, Yarman’s approach should be translated into the framework of relativistic tensor dynamics in order to fully grasp its potentials in the domain of QG. 7. Conclusion We started this paper with the observation that our Mie-de Broglie theory of quantum gravity had a restricted value due to the fact that it was disconnected from modern physics. This isolation was due to the circumstance that both Mie’s and de Broglie’s theories have become ”losing theories” in the history of physics. But with the use of von Laue’s relativistic tensor dynamics we managed to connect the Mie-de Broglie theory of QG to the matter input side of the General Theory of Relativity. We derived Mie’s relations and the connection of Mie to de Broglie from the relations ug = −ρ0 Uµ Uµ = u0 and U0 = hν0 . So the basic assumptions of relativistic tensor dynamics, as they are common in modern GTR, together with Planck’s quantum postulate, lead to de Broglie’s postulates as they are given in his first paper and in his thesis. De Broglie’s work cleared the way for Schr¨odinger to find his basic equation and thus to formulate a wave-mechanical version of Quantum Mechanics. Since then, the unification of Einstein’s Equations with the Schr¨odinger Equation has been a major challenge for modern physics. Up until the present day all attempts to unify GTR with QM failed. By connecting de Broglie’s postulates to relativistic tensor dynamics, we were able to show that one of the basic axioms of the General Theory of Relativity, the principle of equivalence mg = mi , is incompatible with the existence of wave-lengths λi in Quantum Mechanics. We come to the conclusion that GTR and QM are non-unifiable. If we choose the combination of relativistic tensor dynamics and Planck’s postulate as fundamental, a combination which leads to de Broglie’s phase harmony, a new theory of gravity is needed. The Mie-Yarman theory of gravity seems to qualify as such, but it has to be modified into the language of tensor dynamics in order to fully judge its potentials. Acknowledgments The author would like to thank Tolga Yarman, Alexander L. Kholmetskii and Michael C. Duffy for their support and encouragement.

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

[26] [27] [28] [29]

[30] [31] [32]

Haas de E P J 2004 Les Annales de la Fondation Louis de Broglie 29 707-726 Rovelli C 2004 Quantum Gravity (Cambridge: Cambridge University Press) Mie G 1912 Ann. Phys., Lpz. 37 511-534 Mie G 1912 Ann. Phys., Lpz. 39 1-40 Mie G 1913 Ann. Phys., Lpz. 40 1-66 Einstein A 1905 Ann. Phys., Lpz. 17 891-921 Miller A I 1998 Albert Einstein’s Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905-1911) (New York: Springer-Verlag) Broglie de L 1923 Comptes Rendus 177 507-510 Broglie de L 1925 Recherches sur la th´eorie des quanta (Paris these 25 novembre 1924); Annales de Physique 10 Tombe III 22-128 Planck M 1900 Verhandl. Dtsch. Physik. Ges. Berlin 2 237 Yarman T 2005 The meaningfullness of arbitrary changing the mass of an object: the quantum mechanical mechanism behind the relativistic occurrences, Private communication; An extendend revision of the article in: 1999 Chimica Acta Turcica 27 Einstein A 1907 Jahrbuch der Radioaktivit¨ at und Elektronik 4 411-461 Norton J D 1985 Stud. Hist. Phil. Sci. 16 203-246 M. von Laue 1911 Ann. Phys., Lpz. 35 524-542 Norton J D 1992 Archive for History of Exact Sciences 45 17-94 Einstein A 1911 Ann. Phys., Lpz. 35 898-908 Pauli W 1958 Theory of Relativity (New York: Dover) Rindler W, 2001 Relativity. Special, General and Cosmological. (New York: Oxford University Press) Haas A 1930 Einf¨ urung in die Theoretische Physik II (Berlin: Walter de Gruyter and Co.) Laue von M 1955 Die Relativit¨ atstheorie, 6th ed. (Braunschweig) ´ Broglie de L 1952 La th´eorie des particule de spin 1/2. (Electrons de Dirac.), (Paris: GauthierVillars) Weinberg S 1972 Gravitation and cosmology (New York: J. Wiley and Sons) Haas de E P J 2004 A renewed theory of electrodynamics in the framework of a Dirac-ether in: Duffy M.C. (ed.) PIRT IX proceedings (Liverpool: PD Publications) p 95-123 Rohrlich F 1982 Phys. Rev. D 25 3251-3255 Einstein 1919 A Sitz. Preus. Ak. d. Wiss. 349-356 ; Do gravitational fields play an essential part in the structure of the elementary particles of matter? In: H.A. Lorentz, A. Einstein, H. Minkowsky The Priciple of Relativity (Dover Publ. 1952) Laue von M 1950 Z. Phys. 128 p 387-394 Antoci S Mihich L 1997 Nuovo Cim. B 112 991-1001 Obukhov Y N Hehl F W 2003 Phys. Lett. A 311 277-284 Haas de E P J 2005 From Laue’s stress-energy tensor to Maxwell’s Equations and the implications for Einstein’s GTR Accepted paper for the Moscow PIRT Conference at the Bauman Moscow State Technical University, to appear in the proceedings. Yarman T 2004 Les Annales de la Fondation Louis de Broglie 29 459-491 Yarman T 2005 The End Results of the General Theory of Relativity via Just Energy Conservation and Quantum Mechnanics Private communication Pais A 1982 ’Subtle is the Lord...’ The Science and the life of Albert Einstein (Oxford and New York: Oxford University Press)

PACS numbers: 01.65.+g, 03.65.Ta, 04.20.Cv, 04.60.-m

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1. Introduction In an earlier paper we connected the theory of gravity of Gustav Mie to the quantum postulates of Louis de Broglie, thus realizing a theory of quantum-gravity [1]. This theory of QG had a restricted value because both Mie’s and de Broglie’s theories have become ”losing theories” in the history of physics. Mie’s theory of gravity seems to have been forgotten almost completely. In the domain of gravity, it is the competing theory of Einstein that ”won” and that has since then dominated the scene. In the area of the quantum of action, Louis de Broglie’s approach has been set aside since the Solvay Conference of 1927, where the Copenhagen interpretation of Quantum Mechanics proved to be the strongest one. So our theory of Mie-de Broglie QG is something very strange. To my knowledge, it is the first theory in which gravity and quantum meet on such a simple and basic level of physics. As such, the Mie-de Broglie theory of QG, although composed of two ”loosing” theories, cannot be a ”loosing” theory itself, because there are no competitors on this level, it is unique. It’s uniqueness is increased through the level at which Mie-de Broglie QG integrates quantization and gravitation, at the level of electrons, atoms and nuclei, whereas modern approaches aim at integration at the Planck length, out of reach of our present day experimental capacities [2]. So our Mie-de Broglie QG seems highly disconnected from main stream modern physics. Since the writing of the first Mie-de Broglie paper, I have been searching for a way to integrate Mie-de Broglie QG in modern physics. A relation derived independently by Tolga Yarman, ug = (1/γ 2 )ui , and then rediscovered by Yarman in the papers of Gustav Mie, made this possible. In this paper I will show how to connect de Mie-de Broglie QG to Laue’s tensor dynamics. The Mie-de Broglie QG is related to, can be derived from, the mechanical stress-energy tensor and it’s trace. In this derivation, the only thing we need from Quantum Mechanics is Planck’s basic postulate U = hν. The connection I found can easily be related to the matter input side of the GTR. It’s connection to the metric side of the GTR will be outside the scope of this paper. Neither will I try to connect the Mie-de Broglie QG to modern (Copenhagen) Quantum Mechanics. 2. The combination of Mie and de Broglie as a starting point of QG In Gustav Mie’s 1912-1913 papers ([3],[4],[5]) I found the relations between the inertial mass and the rest mass mi =p γm0 and between the gravitational mass and the rest mass 2 mg = (1/γ )m0 , with γ = ( 1 − v 2 /c2 )−1 , ([5],p. 40). In terms of the energy U and using U = mc2 we can write: Ui = γU0 Ug =

1 U0 . γ

(1) (2)

The first equation (1) is forever connected to Einstein’s STR ([6]; [7], p. 313), the second equation (2) was Mie’s own result.

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In the year 1923 Louis de Broglie discovered the basic relations of modern WaveMechanics. In his foundational paper and in his thesis he compared the inertial frequency νi , which he postulated himself, with the clock frequency νc , postulated by Einstein ([8], [9]): νi = γν0 νc =

1 ν0 . γ

(3) (4)

The first νi = γν0 was based on Einstein’s Ui = γU0 and on Planck’s quantum postulate U = hν [10]. The second νc = (1/γ)ν0 was based on Einstein’s 1907 analysis of moving clocks ([7], p. 212), but remained, in de Broglie’s analysis, disconnected from an energy consideration. For a fast moving particle, the inertial frequency increased and the clock frequency decreased. De Broglie showed that by attributing an inertial wave-length λi to the moving particle, the two phenomena remained in phase, giving ϕc = ϕi , a property he called the ”Harmony of the Phases”. In my previous paper I posed the question what kind of energy to connect to the clock frequency of a particle [1]. I will present the argumentation of this paper in a somewhat different fashion, by using the concept of clock energy. The idea to use clock energy stems from Yarman’s concept of clock mass [11]. If we multiply both sides of equation (4) by Planck’s constant h, we get 1 hνc = h ν0 . (5) γ If we write the quantized clock energy as Uc = hνc and use Planck’s postulate U0 = hν0 , then equation (5) leads to 1 Uc = U0 . (6) γ Together with Mie’s equation (2) for the gravitational energy this results in the identification of the quantum clock energy with the gravitational energy, Ug = Uc .

(7)

If we presume gravitational energy to exist as quantized according to the relation Ug = hνg then we are allowed to identify the clock frequency with the gravitational frequency of a particle: νg = νc . It would seem that we have a starting point for a theory of Quantum Gravity. But we have a difficulty with the GTR, the theory of gravity that dominated the twentieth century. In our derivation we used Mie’s relation equation(2) as a one of the starting points, so the resulting QG depends on the validity of this relation. Einstein based his GTR on the principle of equivalence mi = mg , which he formulated for the first time in 1907 [12]. This principle functions as an axiom of GTR, not in a strict mathematical sense but as a guiding principle in the process of its creation and its subsequent application [13]. But the principle of equivalence mi = mg is clearly not in

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accordance with Mie’s relation. So we cannot integrate our Mie-de Broglie theory of QG with Einstein’s GTR. What to do, abandon our Mie-de Broglie QG or try to develop it further, if necessary as a competing theory? We choose for the latter option. We can derive Mie’s relations from Laue’s 1911 relativistic tensor dynamics, the same dynamics that is used by Einstein in as the matter input side of his GTR [15]. This will enable us to compare Einstein’s GTR with Mie-de Broglie QG. 3. Max von Laue’s relativistic conservation laws. From the beginning, the STR as a kinematical theory was in need of a dynamics. This dynamics was searched for by the avant garde in between 1905 and 1910 and was finally formulated by von Laue in 1911 as a relativistic tensor dynamics [14]. But Laue’s dynamics did not incorporate gravity, nor did it reflect the quantized aspect of matter. This creates the impression that von Laue did not add something of his own, a circumstance that makes von Laue rather unknown in the textbooks of standard college physics. But the originality of his work, especially his thermodynamic deviation from Einstein’s U = mc2 , is still there to be rediscovered. 3.1. Basic definitions and the mechanical stress energy tensor The basic definitions we use are quite common in STR and GTR [17], [18]. We start with an observer who has a given three as v, a rest mass as m0 and an p vector velocity −1 2 2 inertial mass mi = γm0 with γ = ( 1 − v /c ) . Then we can define the coordinate velocity four vector as " # d ic Vµ = Rµ = . (8) v dt The proper velocity four vector on the other hand will be defined using the proper time τ as " # d icγ Uµ = Rµ = = γVµ . (9) γv dτ The momentum four vector will be # " # " i icm U i i = mi Vµ = m0 Uµ . Pµ = c = mi v pi We further define the rest mass density as dm0 , ρ0 = dV0 so with 1 dV = dV0 γ

(10)

(11)

(12)

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and the inertial mass density as dmi (13) ρi = dV we get, in accordance with Arthur Haas’ 1930 exposition on relativity ([19], p. 365), dmi dγm0 ρi = = 1 = γ 2 ρ0 . (14) dV dV 0 γ The momentum density four vector will be defined as # " i d dmi u i = Gµ = c Pµ = Vµ = ρi Vµ = γ 2 ρ0 Vµ = γρ0 Uµ . gi dV dV

(15)

The mechanical stress energy tensor, introduced by Max von Laue in 1911, was defined by him as ([20], p.150) Tµν = ρ0 Uµ Uν .

(16)

Pauli gave the same definition in his standard work on relativity ([17], p. 117). With the definitions just given we get Tµν = ρ0 Uµ Uν = γ 2 ρ0 Vµ Vν = ρi Vµ Vν = Vµ ρi Vν = Vµ Gν .

(17)

So the mechanical stress energy tensor can also be written as Tµν = Vµ Gν .

(18)

In the exposition on relativity of Arthur Haas, the first definition ρ0 Uµ Uν is described as the ”Materie-tensor” of General Relativity, while ρi Vµ Vν is described as the ”Materietensor” of Special Relativity ([19], p. 395 and p. 365). Although the derivation seems to demonstrate an equivalence between the two formulations of equation (16) and equation (18), the difference between the two is fundamental. Equation (16) is symmetric by definition, while equation (18) can be asymmetric, because, as von Laue already remarked in 1911, Vµ and Gµ do not have to be parallel all the time ([20], p. 167) This crucial difference between ρ0 Uµ Uν and Vµ Gν was also discussed by de Broglie in connection with his analysis of electron spin ([21], p. 55). In the modern GTR approach, given for example by Weinberg ([22], p. 48) and Rindler ([18], p. 156), the stress energy tensor of a perfect fluid, with a rest-system pressure p0 , is given as p0 Tµν = (ρ0 + 2 )Uµ Uν − p0 gµν . (19) c According to Rindler, in such a symmetric situation we do not have ρi = γ 2 ρ0 in the frame of a moving observer ([18], p. 156). But we are not studying perfect fluids, we are interested in quantum mechanical situations. We want to apply the result to quantized matter. The most basic model we can begin with is a perfect dust, as a quantized perfect fluid without a pressure in the rest frame of the particles themselves, giving p0 = 0. This leads us, starting with the formulation of Weinberg and Rindler, to the same stress energy tensor as given by Laue, Pauli and Haas, Tµν = ρ0 Uµ Uν . In those situations we

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are allowed to use ρi = γ 2 ρ0 , implying the validity of Tµν = Vµ Gν . But we believe Vµ Gν to be more fundamental than ρ0 Uµ Uν . Because ρ0 Uµ Uν is symmetric by definition, it can never incorporate a torque as a changing angular momentum. The asymmetry of Vµ Gν cannot possibly be a property added to ρ0 Uµ Uν , whereas Vµ Gν can be made symmetric by adding the property Gν = ρi Vµ . As already observed by de Broglie, the asymmetry of Vµ Gν will be needed in the future to connect Laue’s relativistic tensor dynamics to quantum mechanics [21]. Especially a quantum jump, understood as a sudden change of angular momentum of the quantized system, incorporates a torque and thus, in its relativistic formulation, necessarily an asymmetric stress energy tensor. Using Tµν = Vµ Gν we can write the SE-tensor as # " # " #" i ic u −u icg c Tµν = = i . (20) g uv v ⊗ g v c This form of the stress energy tensor can already be found in de Broglie’s expos´e ([21], p. 55). 3.2. The conservation laws for energy and momentum For closed systems energy and momentum are conserved and the conservation laws can be expressed as ∂µ Tµν = 0 with the four vector partial derivative defined as " # − ci ∂t ∂µ = . (21) ∇ For closed systems this leads to " #" # " # " # i − ci ∂t −u icg (∂ u + ∇ · uv) 0 t ∂µ Tµν = = c = . (22) i ∇ uv v ⊗ g ∂t g + ∇(v ⊗ g) c Open systems, on the other hand, have " # " # i i (∂ u + ∇ · uv) P t ∂µ Tµν = c = c . ∂t g + ∇(v ⊗ g) f

(23)

We have used P for the power density and f for the force density. If we write S = uv for the energy density current or Umov’s vector, this results in the conservation equation for energy ∇ · S + ∂t u = 0,

(24)

which can also be written as ∂µ Sµ = 0, and the conservation equation for momentum ∂t g + ∇(v ⊗ g) = 0.

(25)

Open systems have a non-zero four force density, so the energy and momentum are not conserved for those systems. An electron in free space should be a closed system but according to relativistic electrodynamics it isn’t. This paradox is connected to the Coulomb-self-force and -energy of the free electron. The problem of the Coulomb selfforce and -energy of a free electron already troubled Lorentz, Abraham and Poincar´e and

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transformed through the work of Einstein, Minkowsky and Laue into a conflict between relativistic electrodynamics and relativistic mechanics [23]. In Laue’s tensor-dynamics, conservation-law considerations require the stress-energy tensor of a free particle in mech space to have a zero divergence: ∂µ Tµν = 0. This ensures action to equal reaction and energy to be conserved. According to von Laue, the electron in free space is a static system, so the divergence of its stress-energy tensor should be zero [20]. But in Minkowsky’s relativistic electrodynamics, the divergence of the stress-energy tensor of the electromagnetic field is zero only in charge-free space and equals the electromagnetic four-force when charges are present. In electrodynamics, the electron in free space is em a charge in its own field, so we have ∂µ Tµν = fνem [24]. The electron in free space feels its own electromagnetic four-force density, which is not zero and not balanced by a reaction force density. As a result, the electron in free space acts a net EM-force and EM-power on itself. We could call it a Baron von M¨ unchausen situation. A free electron with a non-zero power must have energy flowing in or out without compensation and an infinite amount of energy, positive or negative, will be assembled. The net force that this free particle acts on itself will create a runaway situation and its momentum will become infinite. Two mayor strategies to solve the problem have been developed in history. The first strategy postulates a Poincar´e-mechanism and adds a Poincar´e-tensor to the electromagnetic one. The second strategy tries to change the EM-stress-energy tensor. The second strategy is incorporated in the 1912 theory of Mie [3], the first in Einstein’s 1919 attempt to solve the problem within general relativity [25]. I studied Mie’s papers in the context of this problem, of the non-conservation of electrodynamic linear momentum, saw Mie’s equations (1) and (2) and realized the possibility to connect them to de Broglie’s matter wave treatment of the relativistic electron. Eventually, solving the problem of the electron will be a mayor challenge for our Mie-de Broglie QG. This will involve the introduction of the concept of phase harmony in relativistic electrodynamics. If this can be done, it will, through the phase harmony concept, fuse quantum waves, gravity and electrodynamics on a very fundamental level of physics. 3.3. The conservation law for angular momentum A relativistic system also has a mechanical torque-tensor Nµν = Tµν − Tνµ = Vµ Gν − Vν Gµ .

(26)

If we use the abbreviation a for anti-symmetric we can define na = v × g and ga = (1/c2 )uv − g. This allows us to write the torque-tensor in full as: 0 −icga,1 −icga,2 −icga,3 0 na,3 −na,2 icga,1 (27) Nµν = . 0 na,1 icga,2 −na,3 icga,3 na,2 −na,1 0 A closed system, for example a particle in empty space, should not acquire any additional angular momentum, free as it is from external influences. The conservation

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of the intrinsic angular momentum requires the stress-energy tensor to be symmetric, Vµ Gν = Vν Gµ , and the torque-density tensor to vanish, so Nµν = 0. This leads to na = v × g = 0 and ga = (1/c2 )uv − g = 0, which gives 1 g = 2 uv (28) c for a moving particle in free space. According to von Laue, electromagnetic systems do not automatically have a symmetric stress energy tensor [26]. In such cases, the angular momentum cannot be a conserved quantity. The correct formulation of the electrodynamic stress energy tensor in case of material media remains an unsolved problem of modern physics [27], [28]. So the conservation of angular momentum cannot always be assumed. Because the matter input side of the GTR involves symmetric stress energy tensors only, non symmetric electromagnetic systems describing material media cannot be integrated into the GTR. These non-symmetric systems also surpass the phase harmony condition [29]. 4. Deriving Mie’s relation from the symmetric stress energy tensor and the source of gravitational energy. After the publication of the paper in which I connected Mie to de Broglie, Yarman, a colleague from Istanbul, send me an email in which he pointed me to the fact that Mie’s relation equation (2), together with Einstein’s equation (1), lead to ug = (1/γ 2 )ui . This relation can be found, as I checked afterwards, in encrypted form in Mie’s third paper ([5], p. 53). The relation ug = (1/γ 2 )ui was independently found by Yarman, though through a straightforward derivation [30]. Yarman based himself solely on Einstein’s mass-energy equivalence and the energy conservation law. By applying these relations to the gravitational binding energy, treated as any other binding energy and thus capable of changing the rest mass of the particle in consideration, he concluded that in the case of moving particles, gravitational mass could not equal inertial mass. Instead of the principle of equivalence, Yarman found the relation mg = (1/γ 2 )mi . A few weeks after Yarman emailed me this result, I stumbled upon this relation while analyzing the electrodynamic version of Laue’s tensor dynamics [29]. Being on the alert, I realized the opportunity to connect Laue’s tensor dynamics to the Mie-de Broglie theory of QG. Then Yarman send me two new papers in which he further developed his principle guiding and unifying idea of the quantum mechanical architecture of matter as being the fundamental unified cause of relativistic clock behavior in both the kinematic Special Relativity as in the gravitational General Relativity [11], [31]. Together these developments provided me with the mathematical and conceptual substance necessary to write this paper. 4.1. From Laue to Yarman and Mie To begin with, I will show how Yarman’s result, written in the form ug = (1/γ 2 )ui , can be derived from Laue’s relativistic tensor dynamics. This leads us directly to Mie’s relation written as Ug = (1/γ)U0 .

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Let’s start with the gravitational energy density as it is generally defined, through the trace of the mechanical stress energy tensor ug = −Vµ Gµ = ui − v · gi

(29)

If we assume a context with Nµν = 0, we have gi = (1/c2 )ui v. In these situations, the gravitational energy can be written as 1 v2 1 u v = (1 − )ui = 2 ui . i 2 2 c c γ is symmetric and ug = −Vµ Gµ we have

ug = −Vµ Gµ = ui − v · gi = ui − v · So in all cases where Tµν

(30)

1 ui . (31) γ2 If we eliminate the densities we get 1 Ug = 2 Ui . (32) γ If we further assume Ui = γU0 , from Einstein’s STR, we get Mie’s relation 1 Ug = U0 . (33) γ So Mie’s relation is valid in all cases where the stress energy tensor Vµ Gν is symmetric, where the gravitational energy density is defined as ug = −Vµ Gµ , and where we are allowed to use the relation Ui = γU0 . ug =

4.2. The invariance of the gravitational energy density We can arrive at this result through a slightly different reasoning. Without proof, Gustav Mie assumed the gravitational energy density to be a Lorentz Invariant scalar or a four scalar. As we showed in our previous paper on the relation of Mie to de Broglie, Mie’s gravitational energy density can be identified as the trace of the mechanical stress energy tensor [1]. This can be used to prove that ug , written as −ρ0 Uµ Uµ , is a four-scalar: 1 ) = ρ0 c2 = u0 . (34) 2 γ Relativistic tensor dynamics gives ug = u0 , in accordance with Mie’s assumption. So we don’t have mi = mg , but we have ρg = ρ0 , an invariant gravitational mass density. Using the familiar dV = (1/γ)dV0 , we get Mie’s mg = (1/γ)m0 . When we add the relation mi = γm0 we get Yarman’s mg = (1/γ 2 )mi . ug = −ρ0 Uµ Uµ = −ρ0 γ 2 Vµ Vµ = (−ρ0 γ 2 )(v 2 − c2 ) = (−ρ0 γ 2 )(−c2

4.3. The quantization of gravitational energy If we add Planck’s basic postulate and apply it, as de Broglie did, to the rest energy as U0 = hν0 , Mie’s relation Ug = (1/γ)U0 results in Ug =

1 hν0 . γ

(35)

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According to Einstein’s STR, the right hand side of the last equation equals hνc , so Ug = hνc .

(36)

This implicates that we have derived Mie’s relations and our connection of Mie to de Broglie for all situations where the stress energy tensor Vµ Gν = −ρ0 Uµ Uν is symmetric, where the gravitational energy is defined as ug = −Vµ Gµ = −ρ0 Uµ Uµ , and where we are allowed to use the relations Ui = γU0 , U0 = hν0 and νc = (1/γ)ν0 . 5. Some implications of the Mie-de Broglie theory of Quantum Gravity 5.1. Falsification of the principle of equivalence In the perspective created by equation (36), the decrease of gravitational energy for moving particles has already been measured and thus experimentally verified. It has been proven beyond doubt that moving clocks slow down ([18], p. 66), so according to the perspective created by equation (36), the decrease of gravitational energy of these particle-like quantum-clocks has also been confirmed. And because the increase of inertial energy for moving particles has also been verified experimentally, the principle of equivalence has already been falsified in the case of moving particles. 5.2. Quantum Gravity as the causational factor replacing space curvature. There is another interesting point of view, formulated by Yarman as: both gravitation and uniform translational motion, affects the clocks, in exactly the same respect. Yarman added: it must require quite a captivation, not to consider a light phenomenon such as [gravitational] red shift in mere terms of quantum mechanics; amazingly though the history, now for many very decades, gave birth to the unexpected. [11]. In Pais’ biography of Einstein, a reason is given: First, (in the 1911 paper) Einstein derives equation 11.2 for the energy shift; then he starts ”all over again” and derives the frequency shift (equation 11.5). It is no accident, I am sure, that he did not derive only one of these equations and from there go to the other one with the help of U = hν. He had had something to do with U = hν. It cannot have slipped his mind; the quantum theory never slipped his mind. However, it was Einstein’s style forever to avoid the quantum theory if he could help it - as in the case of the energy and frequency shift ([32], p. 197). It seems that Einstein’s avoidance has become the norm amongst specialists in GTR. We on the other hand do not pretend to be able to derive Planck’s quantum postulate, we just accept it as a basic principle. Then the question becomes unavoidable: ”Why do clocks slow down?” Because of their velocity? Because of the curvature? Because of gravity? Because of the quantum architecture of matter? Why? If we inverse the logics of the previous derivation and start with the end results Ug = hνc and Ug = (1/γ)U0 , we can interpret the decrease of gravitational energy and the slowing down of quantum clocks as coinciding events. Thus the slowing down of

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moving quantum clocks and the slowing down of quantum clocks who are deeper in a gravitational field can be related to one single physical circumstance: the decrease of the gravitational energy of the quantum particle used for time measurement in a clock quantum-mechanism. A moving quantum-clock has less Ug , because Ug = (1/γ)U0 , and thus also a lower clock frequency, because Ug = hνc . A quantum-clock which is positioned lower in a g-field has less Ug and thus also a lower quantum-clock frequency, because Ug = hνc . Both phenomena, the first in STR and considered to be purely kinematic in its origin, the second in GTR and considered to be part of space-time deformation, are in this way related to one and the same circumstance in the Mie-de Broglie QG. In our interpretation, the gravitational red shift has to be interpreted as a direct effect of Quantum Gravity and not as something caused by the curvature of the metric. So Mie-de Broglie QG and Einstein’s GTR are competing theories. At the same time, Mie-de Broglie QG adds a dynamics to Einstein’s purely kinematical STR. Our conclusions based on equation (35) and equation (36) are in accordance with the results of Yarman in his 2004 paper [30]. In this paper, presenting a summary of years of research on the subject, Yarman wrote : A wave-like clock [=a quantum-clock] in a gravitational field, retards via quantum mechanics, due to the mass deficiency it develops in there, and this, as much as the binding energy it displays in the gravitational field;... and Note that, according to our approach, the classical gravitational red shift and a related mass decrease, occur to be concomitant quantum mechanical effects. 6. Implications for the Unification program 6.1. Wave Mechanics and Einstein’s GTR are non-unifiable As we already stated, in all cases where Tµν = Vµ Gν is symmetric and ug = −Vµ Gµ we have 1 ug = 2 ui . (37) γ To get Einstein’s mg = mi , we must further assume v · g = 0, which, because of gi = (1/c2 )ui v and ui 6= 0, implies v = 0. So according to Laue’s tensor dynamics, the principle of equivalence mg = mi is only valid for completely static situations. In 1992 John Norton came to the same conclusion [15]. The stress energy tensors Tµν that may be used in the GTR must satisfy two criteria, ∂µ Tµν = 0 and Nµν = 0. The principle of equivalence mg = mi is an additional demand that further restricts the set of allowed stress energy tensors in GTR. But in order to have a wave-length λi , particles must move. Stationary quantum states, described by the Schr¨odinger Equation, are about standing waves, so with moving particles. Without movement of the particles involved, we have νi = νc , so no wave-length is needed to harmonize the phases. If we have νi = νc we can multiply it by h to get hνi = hνc or Ui = Uc . Because we identified clock energy as gravitational energy this implies Ui = Ug and mi = mg . We conclude that the principle of equivalence as an axiom of

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GTR excludes wave-lengths and thus wave-mechanics or Quantum Mechanics. Because their basic principles are mutually exclusive, the validity of the principle of equivalence for the one and the existence of particles with wave-lengths for the other, we proclaim that GTR and QM are non-unifiable. The General Theory of Relativity as a theory of gravity seems much to restricted to be unifiable with the wavelength postulate of Louis de Broglie. I emphasized the word restricted because I do not claim GTR to be a wrong theory. In those regions of reality where the axioms of GTR are valid, the theory of GTR, being bases on those axioms, is valid. What I claim is that in the regions where de Broglie’s wavelength postulate can be successfully applied, Einstein’s restrictions put on his stress-energy tensors are such that they cannot be fulfilled and so GTR cannot be applied. 6.2. A new theory of gravity is needed. Einstein’s GTR cannot be integrated with de Broglie’s quantum postulates. If we choose de Broglie’s phase harmony as fundamental, a new theory of gravity is needed. Einstein’s principle of equivalence is only valid for static masses and cannot be applied to moving objects. Because a particle on a geodesic is not really moving, GTR’s space curvature has been applied to moving objects. But it can only work for so called ”test particles” or point objects with a negligible mass or for objects with to small a motion to make a difference. For fast moving objects with considerable mass, thus influencing the local g-field, we do need a new theory. This theory of gravity must be in accordance with 1 Ug = 2 Ui . (38) γ If, in contrast to Einstein, we directly apply Planck’s postulate U = hν, the condition for quantum gravitational phase harmony becomes 1 νg = 2 νi . (39) γ By identifying gravitational frequency as clock frequency we arrive at de Broglie’s 1923 result. In the words of de Broglie: ... the phase agreement is realized if one has 1 ν1 = 2 ν. γ a condition that is clearly satisfied by the definitions of ν1 and ν. The demonstration of this important result rests uniquely on the principle of special relativity and on the correctness of the quantum relationship as much for the fixed observer as for the moving observer. In the original text, de Broglie used the notations ν for the inertial frequency νi and ν1 for the internal periodic phenomenon or the clock frequency νc [8]. In the notation we use this gives 1 νc = 2 νi (40) γ as the condition for de Broglie’s 1923 Harmony of the Phases.

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Mie’s theory of gravity is compatible with de Broglie’s phase harmony and it’s basic assumptions can be proven in relativistic tensor dynamics. But Mie was not successful in predicting the cosmological experimental facts. Yarman’s theory of gravity is compatible with de Broglie’s phase harmony and it does predict the crucial cosmological experimental facts [31]. Yarman’s theory of gravity seems to a high degree unifiable with Mie’s theory of gravity. So the Mie-Yarman theory of gravity might be the candidate needed. As a scalar theory of gravity, Yarman’s approach should be translated into the framework of relativistic tensor dynamics in order to fully grasp its potentials in the domain of QG. 7. Conclusion We started this paper with the observation that our Mie-de Broglie theory of quantum gravity had a restricted value due to the fact that it was disconnected from modern physics. This isolation was due to the circumstance that both Mie’s and de Broglie’s theories have become ”losing theories” in the history of physics. But with the use of von Laue’s relativistic tensor dynamics we managed to connect the Mie-de Broglie theory of QG to the matter input side of the General Theory of Relativity. We derived Mie’s relations and the connection of Mie to de Broglie from the relations ug = −ρ0 Uµ Uµ = u0 and U0 = hν0 . So the basic assumptions of relativistic tensor dynamics, as they are common in modern GTR, together with Planck’s quantum postulate, lead to de Broglie’s postulates as they are given in his first paper and in his thesis. De Broglie’s work cleared the way for Schr¨odinger to find his basic equation and thus to formulate a wave-mechanical version of Quantum Mechanics. Since then, the unification of Einstein’s Equations with the Schr¨odinger Equation has been a major challenge for modern physics. Up until the present day all attempts to unify GTR with QM failed. By connecting de Broglie’s postulates to relativistic tensor dynamics, we were able to show that one of the basic axioms of the General Theory of Relativity, the principle of equivalence mg = mi , is incompatible with the existence of wave-lengths λi in Quantum Mechanics. We come to the conclusion that GTR and QM are non-unifiable. If we choose the combination of relativistic tensor dynamics and Planck’s postulate as fundamental, a combination which leads to de Broglie’s phase harmony, a new theory of gravity is needed. The Mie-Yarman theory of gravity seems to qualify as such, but it has to be modified into the language of tensor dynamics in order to fully judge its potentials. Acknowledgments The author would like to thank Tolga Yarman, Alexander L. Kholmetskii and Michael C. Duffy for their support and encouragement.

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