Resolution of two fundamental issues in the dynamics of relativity and exposure of a real version of the emperor's new clothes Changbiao Wang∗ ShangGang Group, 10 Dover Road, North Haven, CT 06473, USA In this paper, we aim to resolve two fundamental issues in the dynamics of relativity: (i) Under what condition, the time-column space integrals of a Lorentz four-tensor constitute a Lorentz four- vector, and (ii) under what condition, the time-element space integral of a Lorentz four-vector is a Lorentz scalar; namely two \conservation laws", which are mispresented in traditional textbooks, and widely used in fundamental research, such as relativistic analysis of the momentum of light in a medium, and the proofs of the positive mass theorem in general relativity. To resolve issue (i), we have developed a generalized Lorentz four-vector theorem based on the principles of classical mathematical analysis, with a simplified analytic example given to illustrate how to transform a space integral from one inertial frame to another, and a strict mathematical derivation provided to confirm the effect of Lorentz contraction. We use this four-vector theorem to verify Møller’s theo- rem, and surprisingly find that Møller’s theorem is fundamentally wrong. We provide a corrected version of Møller’s theorem. We also use this four-vector theorem to analyze a plane light wave in a moving uniform medium, and find that the momentum and energy of Minkowski quasi-photon constitute a Lorentz four-vector and Planck constant is a Lorentz invariant. To resolve issue (ii), we have developed a generalized Lorentz scalar theorem. We use this theorem to verify the \invariant conservation law" in relativistic electrodynamics, and unexpectedly find that it is also fundamentally wrong. Thus the two \conservation laws" in traditional textbooks, which have magically attracted several generations of most outstanding scientists, turned out to be imaginary, just like the em- peror's new clothes; creating a scientific myth in the modern theoretical and mathematical physics: Believing is seeing. PACS numbers: 03.30.+p, 03.50.De, 04.20.-q, 11.30.-j Keywords: Conservation laws of relativity; Light momentum; Positive mass theorem; Gordon metric I. INTRODUCTION resolve the electrostatic field problem [6]. In contrast to Laue's theorem, Møller provided a four-vector theorem for a tensor that is required to be In the dynamics of relativity, the energy and momen- divergence-less, with a boundary condition imposed, but tum of a physical system is described by a Lorentz four- allowed to be time-dependent [3]. Møller’s theorem only tensor; such a tensor is usually called energy{momentum has a sufficient condition (instead of a sufficient and nec- tensor [1{3], stress tensor [4], stress{energy tensor [5, 6], essary condition) [13], but it is more attractive because or momentum{energy stress tensor [7]. If the tensor is the energy{momentum tensor for electromagnetic (EM) divergence-less, then the system is thought to be con- radiation fields varies with time [8{11]. It is widely rec- served [1{3], and it is a closed system [3]; thus the total ognized in the community that Møller’s theorem is abso- energy and momentum can be obtained by carrying out lutely rigorous so that this theorem has been widely used space integration of the time-column elements of the ten- in quantum electrodynamics [8] and relativistic analysis sor to constitute a Lorentz four-vector [1{5, 8{11]. of light momentum in a dielectric medium [9{11]. Mathematically speaking, if a tensor satisfies certain In this paper, we provide a generalized Lorentz four- conditions, the space integrals of the tensor's time- vector theorem for a tensor which is not required to be column elements can form a Lorentz four-vector. For the time-independent and divergence-less, and on which no sake of convenience, we call such a mathematical state- boundary conditions are imposed. This theorem has a ment \four-vector theorem". sufficient and necessary condition. We use this theo- Laue set up a four-vector theorem for a tensor that rem to verify Møller’s theorem, surprisingly finding that arXiv:1702.03200v24 [physics.gen-ph] 3 Dec 2020 is required to be time-independent [12]. Laue's theorem Møller’s theorem is fundamentally wrong. only provides a sufficient condition (instead of a sufficient Like the four-vector theorem, a Lorentz scalar theo- and necessary condition), and it cannot be used to judge rem is a mathematical statement that under what con- the Lorentz property of the energy and momentum of ditions, the time-element space integral of a four-vector electrostatic fields. In a recent study, Laue's theorem is is a Lorentz scalar. In Ref. [6], a scalar theorem for a improved to be a theorem that has a sufficient and nec- four-vector that is required to be time-independent is set essary condition, and it is successfully used to generally up, called \derivative von Laue's theorem", and it is suc- cessfully used to strictly resolve the invariance problem of total electric charge in relativistic electrodynamics. In this paper, we also provide a generalized Lorentz ∗Electronic address: changbiao [email protected] scalar theorem for a four-vector which is not required to 2 be time-independent and divergence-less, and on which Lorentz four-vector (Theorem 2), while the scalar theo- no boundary conditions are imposed. This theorem has rem provides a criterion to judge under what condition a sufficient and necessary condition. We use this scalar the space integral of the time-element of a four-vector is theorem to identify the validity of a well-known result in a Lorentz scalar (Theorem 3). The proofs of Theorem the dynamics of relativity that if a Lorentz four-vector 1 and Theorem 2 are very similar, and without loss of is divergence-less, then the time-element space integral generality, only the proof of Theorem 1 is given. of the four-vector is a Lorentz scalar [3, p. 168], namely Suppose that an inertial frame of X0Y 0Z0 moves uni- the \invariant conservation law" claimed by Weinberg [2, formly at βc relatively to the laboratory frame XYZ, p. 41]. We unexpectedly find that this widely-accepted where c is the vacuum light speed. The Lorentz trans- result is also fundamentally wrong. formation of time-space four-vector Xµ = (x; ct) is given The paper is organized as follows. In Sec. II, proofs are by [6, 7] given of Lorentz four-vector and scalar theorems based x0 = x + ξ(β · x)β − γβct; (1) on the principles of classical mathematical analysis, with 0 a simplified analytic example given to illustrate how to ct = γ(ct − β · x); (2) transform a space integral from one inertial frame to an- or conversely, given by other, and a strict mathematical derivation provided to 0 0 0 confirm the effect of Lorentz contraction. In Sec. III, x = x0 + ξ(β · x0)β − γβ ct0; (3) Møller’s theorem is proved to be incorrect, and a cor- ct = γ(ct0 − β0 · x0); (4) rected version of Møller’s theorem is provided. In Sec. IV, the \invariant conservation law" in relativistic electrody- where ξ ≡ (γ − 1)=β2 = γ2=(γ + 1), γ ≡ (1 − β2)−1=2, 0 ν namics is proved to be invalid; namely the current con- and β = −β. Note: Xµ = gµν X = (−x; ct), with µν tinuity equation r · J + @ρ/∂t = 0 cannot be taken as gµν = g = diag(−1; −1; −1; +1) the Minkowski metric. the charge conservation law in relativity. In Sec. V, some According to the definition of tensors [3, p.108], if remarks and conclusions are given. In Appendix A it Ωµν (x; t) is a Lorentz four-tensor given in XYZ, where is demonstrated why the hyperplane differential-element µ, ν = 1, 2, 3, and 4, with the index 4 correspond- four-vector, introduced to transform space integrals be- ing to time component, then in X0Y 0Z0 the tensor tween Lorentz inertial frames in textbooks, contradicts Ω0µν x = x(x0; t0); t = t(x0; t0) is obtained through the principles of mathematical analysis and the princi- \double" Lorentz transformation of Ωµν (x; t), given by ple of relativity. In Appendix B, as an application of Theorem 1 to Minkowski tensor for a plane light wave in 0µ 0ν 0µν @X @X λσ a moving uniform medium, the momentum{energy four- Ω (x; t) = Ω (x; t); (5) @Xλ @Xσ vector of the quasi-photon and the Lorentz invariance of 0µν 0 0 0 0 Planck constant are naturally derived. In Appendix C, Ω x = x(x ; t ); t = t(x ; t ) physical counterexamples of Thirring's claims are pro- @X0µ @X0ν vided. In Appendix D, an illustration is given of why = Ωλσ x = x(x0; t0); t = t(x0; t0) ; (6) @Xλ @Xσ the proofs of the positive mass theorem in general rela- tivity are based on a flawed theoretical framework. In where @X0µ=@Xλ and @X0ν =@Xσ are obtained from Appendix E, the covariance of Gordon optical metric is Lorentz transformation Eqs. (1) and (2), while x = questioned. x(x0; t0) and t = t(x0; t0) denote Lorentz transformation Eqs. (3) and (4), respectively. Eq. (5) is the expression of Ω0µν observed in XYZ, and Eq. (6) is the expression of II. LORENTZ FOUR-VECTOR THEOREMS Ω0µν observed in X0Y 0Z0. AND SCALAR THEOREM Theorem 1. Suppose that Θµν (x; t) is an integrable In this section, proofs are given of Lorentz four-vector Lorentz four-tensor, defined in the domain V in the lab- and scalar theorems, and a simplified analytic example oratory frame XYZ, where µ, ν = 1, 2, 3, and 4, with the is provided to illustrate how to transform a space inte- index 4 corresponding to time component, and V includ- gral from one inertial frame to another and to explain ing its boundary is at rest in XYZ, namely any x 2 V is why under time-space Lorentz transformation the effect independent of t.