von Laue’s theorem and its applications

What is it about?

Based on the principles of classical mathematical analysis, this paper (10.1139/cjp-2015-0198) proves two theorems. (1) von Laue’s theorem, which provides a criterion for a Lorentz four-vector, is strictly proved in detail for its sufficient and necessary condition (SNC). This SNC version of Laue’s theorem is used to show that the total (Abraham=Minkowski) electromagnetic momentum and energy for electrostatic field cannot constitute a Lorentz four-vector. (2) A derivative von Laue’s theorem, which provides a criterion for a Lorentz invariant, is proved. It is used to rigorously prove the Lorentz invariance of total electric charge.

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Why is it important?

(1) In the theory of relativity, the momentum and energy of a closed physical system can be described by a four tensor, usually called momentum–energy tensor. It is well established in traditional textbooks [for example, C.W. Misner, et al, Gravitation, W.H. Freeman, San Francisco, 1973, p443] that if the tensor is divergence free, then the total momentum and energy of the system can be obtained by carrying out integration of the time-column elements of the tensor to form a constant four-vector, which is usually called conservation law. This conservation law is the basis of Positive Mass Theorem in general relativity. Unfortunately, this textbook-level conservation law is incorrect. This paper (10.1139/cjp-2015-0198) provides a counterexample, EM stress–energy tensor for electrostatic field, which directly disproves this well-established conservation law, as shown in the section of “4. Application of Laue’s theorem to charged metal sphere in free space”. This is a conceptually challenging result. (2) Invariance of total charge is the cornerstone in electrodynamics. In traditional textbook of Weinberg [Gravitation and cosmology, John Wiley & Sons, New York, 1972, p41], a standard proof is provided, stating that the divergence-free property of any current density four-vector defines a Lorentz invariant total charge, which has been accepted as a well-established result. Unfortunately, Weinberg's proof contains a conceptual flaw as it requires satisfying a certain kind of infinite boundary conditions; however, at the physical level, invariance of charge is independent of specific boundaries. No matter whether the boundaries are finite or infinite, the invariance of charge should always hold. Therefore, Weinberg’s proof is incorrect; a simple counterexample is the localized wave four-vector for a plane light wave in free space. This paper (10.1139/cjp-2015-0198) has rigorously proved the invariance of charge by use of the derivative von Laue’s theorem, which indicates that the divergence-free property of current density four-vector is irrelevant to the invariance of charge, as shown in the section of “8. Derivative von Laue’s theorem for a Lorentz invariant”. This is also a conceptually challenging result. Putting it bluntly, physically, Weinberg’s standard textbook proof [Gravitation and cosmology, John Wiley & Sons, New York, 1972, p41] conflates the local charge conservation law and the Lorentz invariance of total charge, which are completely different concepts. The divergence-free property of the current density four-vector (current continuity equation) means the charge conservation law that holds in all inertial frames, while the invariance of charge means that the total charge has the same value observed in any frames; they (divergence-free property of current four-vector and invariance of charge) have nothing to do with each other. Thus it is impossible to derive the invariance of total charge from the divergence-free property of current four-vector.

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