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Maxwell’s theory of electromagnetism
Published in Edward J. Rothwell, Michael J. Cloud, Electromagnetics, 2018
Edward J. Rothwell, Michael J. Cloud
Several authors, notably Purcell [175] and Elliott [73], have used this approach. However, Jackson [105] has pointed out that many additional assumptions are required to deduce Maxwell’s equations beginning with Coulomb’s law. Feynman [81] is critical of the approach, pointing out that we must introduce a vector potential which adds to the scalar potential from electrostatics in order to produce an entity that transforms according to the laws of special relativity. In addition, the assumption of Lorentz invariance seems to involve circular reasoning since the Lorentz transformation was originally introduced to make Maxwell’s equations covariant. But Lucas and Hodgson [135] point out that the Lorentz transformation can be deduced from other fundamental principles (such as causality and the isotropy of space), and that the postulate of a vector potential is reasonable. Schwartz [198] gives a detailed derivation of Maxwell’s equations from Coulomb’s law, outlining the necessary assumptions.
Maxwell’s theory of electromagnetism
Published in Edward J. Rothwell, Michael J. Cloud, Electromagnetics, 2018
Edward J. Rothwell, Michael J. Cloud
Several authors, notably Purcell [155] and Elliott [56], have used this approach. However, Jackson [92] has pointed out that many additional assumptions are required to deduce Maxwell’s equations beginning with Coulomb’s law. Feynman [63] is critical of the approach, pointing out that we must introduce a vector potential that adds to the scalar potential from electrostatics in order to produce an entity that transforms according to the laws of special relativity. In addition, the assumption of Lorentz invariance seems to involve circular reasoning, since the Lorentz transformation was originally introduced to make Maxwell’s equations covariant. But Lucas and Hodgson [123] point out that the Lorentz transformation can be deduced from other fundamental principles (such as causality and the isotropy of space), and that the postulate of a vector potential is reasonable. Schwartz [172] gives a detailed derivation of Maxwell’s equations from Coulomb’s law, outlining the necessary assumptions.
Bohmian Quantum Gravity and Cosmology
Published in Xavier Oriols, Jordi Mompart, Applied Bohmian Mechanics, 2019
Nelson Pinto-Neto, Ward Struyve
Different choices of the shift vectors Ni correspond to different coordinates on the spatial hypersurfaces. The Bohmian dynamics does not depend on the choice of coordinates on the spatial hypersurfaces. That is, the dynamics is invariant under spatial diffeomorphisms. A convenient choice is to take the Ni = 0 [8]. The lapse function N > 0 determines the foliation. Lapse functions that differ only by a factor f(gt) (which only depends on the time t which labels the leaves of the foliation) determine the same foliation. The Bohmian dynamics does not depend on such different choices. Such a difference merely corresponds to a time-reparameterization. However, different lapse functions that differ by more than a factor f (t) generically yield different Bohmian dynamics [7, 8, 26, 27]. That is, if we consider the motion of an initial three-metric along a certain space-like hypersurface and let it evolve according to the dynamics given in Eq. (10.17) to a future space-like hypersurface, then the final three-metric will depend on the choice of lapse function or, in other words, on the choice of intermediate hypersurfaces. This was shown in detail in [26]. This is unlike general relativity, where there is foliation-independence. So in the Bohmian theory a particular choice of lapse function or foliation needs to be made. As such the theory is not generally covariant. This is akin to the situation in special relativity where the nonlocality (which is unavoidable for any empirically adequate quantum theory, due to Bell’s theorem) is hard to combine with Lorentz invariance. In that context, it is simpler to assume a preferred reference frame or foliation. The extent to which this extra space-time structure can be eliminated and the theory be made fully Lorentz invariant is discussed in [28]. One possibility is to let the foliation be determined in a covariant way by the wave function. Perhaps a similar approach can be taken in the case of quantum gravity [8], but so far no concrete examples have been considered.
Dislocation drag from phonon wind in an isotropic crystal at large velocities
Published in Philosophical Magazine, 2020
Daniel N. Blaschke, Emil Mottola, Dean L. Preston
In order to find the solutions of the linear elastic equations (15) for edge dislocations moving with uniform velocity (with , ), Eshelby decomposed the stationary edge dislocation (41) into its transverse and longitudinal parts as in (16), using the effective Lorentz invariance of the equations (19), and obtained [20]for an edge dislocation gliding in an isotropic elastic solid in the x direction with uniform velocity v, wherefollowing the standard notations of special relativity, whose definition of γ is related to that of Eshelby by . We consider below only gliding edge and screw dislocations, as dislocation climb is highly suppressed and hence can be neglected in the discussion of phonon wind [3].