Abstract:The notion of a two-point susceptibility kernel used to describe linear electromagnetic responses of dispersive continuous media in non-relativistic phenomena is generalized to accommodate the constraints required of a causal formulation in spacetimes with background gravitational fields. In particular the concepts of spatial material inhomogeneity and temporal non-stationarity are formulated within a fully covariant spacetime framework. This framework is illustrated by re-casting the Maxwell-Vlasov equations … Show more
“…Since the variations are purely local, this makes no statement about the global conservation of charge in non trivial spacetimes. It should also be noted that most Lagrangian formulations of electromagnetism implicitly assume a model for H. For example the Maxwell vacuum where Λ contains the term Λ EM = 1 2 dA ∧ dA, or a model of a simple non-dispersive "antediluvian" medium 7 where Λ EM = 1 2 dA ∧ Z(dA) and Z is a constitutive tensor [32]. It would also be interesting to attempt to construct Lagrangians which do not imply a well defined excitation 2-form.…”
A careful examination of the fundamentals of electromagnetic theory shows that due to the underlying mathematical assumptions required for Stokes' Theorem, global charge conservation cannot be guaranteed in topologically non-trivial spacetimes. However, in order to break the charge conservation mechanism we must also allow the electromagnetic excitation fields D, H to possess a gauge freedom, just as the electromagnetic scalar and vector potentials ϕ and A do. This has implications for the treatment of electromagnetism in spacetimes where black holes both form and then evaporate, as well as extending the possibilities for treating vacuum polarisation. Using this gauge freedom of D, H we also propose an alternative to the accepted notion that a charge passing through a wormhole necessarily leads to an additional (effective) charge on the wormhole's mouth. * https://orcid.org/0000-0003-1597-6084; j.gratus@lancaster.ac.uk † https://orcid.org/0000-0001-5744-8146; Dr.Paul.Kinsler@physics.org ‡ https://orcid.org/0000-0003-0643-7169; m.mccall@imperial.ac.uk arXiv:1904.04103v1 [gr-qc] 8 Apr 2019 III IV L i g h t c o n e J f = 0 J b = 0 J f = 0 J b = 0 J f = 0 J b = 0 J f = 0 J b = 0 J b = 0 J f = 0 FIG. 5: Domains in a spacetime M where J f (blue) and J b (red) may be non-zero; note in particular that the supports of J f and J b do not intersect. For completeness, we show several different regions where the various possible combinations of zero and non-zero J f and J b hold, although alternative (and simpler) scenarios are possible. Note that Region I matches the cone shown on fig. 4, and Region IV encompasses both a section above the light line, as well as below.
“…Since the variations are purely local, this makes no statement about the global conservation of charge in non trivial spacetimes. It should also be noted that most Lagrangian formulations of electromagnetism implicitly assume a model for H. For example the Maxwell vacuum where Λ contains the term Λ EM = 1 2 dA ∧ dA, or a model of a simple non-dispersive "antediluvian" medium 7 where Λ EM = 1 2 dA ∧ Z(dA) and Z is a constitutive tensor [32]. It would also be interesting to attempt to construct Lagrangians which do not imply a well defined excitation 2-form.…”
A careful examination of the fundamentals of electromagnetic theory shows that due to the underlying mathematical assumptions required for Stokes' Theorem, global charge conservation cannot be guaranteed in topologically non-trivial spacetimes. However, in order to break the charge conservation mechanism we must also allow the electromagnetic excitation fields D, H to possess a gauge freedom, just as the electromagnetic scalar and vector potentials ϕ and A do. This has implications for the treatment of electromagnetism in spacetimes where black holes both form and then evaporate, as well as extending the possibilities for treating vacuum polarisation. Using this gauge freedom of D, H we also propose an alternative to the accepted notion that a charge passing through a wormhole necessarily leads to an additional (effective) charge on the wormhole's mouth. * https://orcid.org/0000-0003-1597-6084; j.gratus@lancaster.ac.uk † https://orcid.org/0000-0001-5744-8146; Dr.Paul.Kinsler@physics.org ‡ https://orcid.org/0000-0003-0643-7169; m.mccall@imperial.ac.uk arXiv:1904.04103v1 [gr-qc] 8 Apr 2019 III IV L i g h t c o n e J f = 0 J b = 0 J f = 0 J b = 0 J f = 0 J b = 0 J f = 0 J b = 0 J b = 0 J f = 0 FIG. 5: Domains in a spacetime M where J f (blue) and J b (red) may be non-zero; note in particular that the supports of J f and J b do not intersect. For completeness, we show several different regions where the various possible combinations of zero and non-zero J f and J b hold, although alternative (and simpler) scenarios are possible. Note that Region I matches the cone shown on fig. 4, and Region IV encompasses both a section above the light line, as well as below.
“…Then, as illustrated in fig. 8(a) (32) This states that the total charge in Σ at time t 1 equals the charge in Σ at time t 0 plus any charge that enters (or leaves) via S II and S I . We cannot let S I go to infinity as then it would disappear from the right hand side of (32) and Stokes' theorem will no longer apply.…”
A careful examination of the fundamentals of electromagnetic theory shows that due to the underlying mathematical assumptions required for Stokes' Theorem, global charge conservation cannot be guaranteed in topologically non-trivial spacetimes. However, in order to break the charge conservation mechanism we must also allow the electromagnetic excitation fields D, H to possess a gauge freedom, just as the electromagnetic scalar and vector potentials $\varphi$ and A do. This has implications for the treatment of electromagnetism in spacetimes where black holes both form and then evaporate, as well as extending the possibilities for treating vacuum polarisation. Using this gauge freedom of D, H we also propose an alternative to the accepted notion that a charge passing through a wormhole necessarily leads to an additional (effective) charge on the wormhole's mouth.
“…have a position z dependence), as well as spatially dispersive (i.e. have a wavevector k dependence), which is challenging as k and z are Fourier conjugate variables [17,20,21]. We start with the undamped hydrodynamic Lorentz model which is spatially dispersive and homogenous.…”
Section: Inhomogeneous and Spatially Dispersive 1d Modelmentioning
confidence: 99%
“…Note that this profile shape requires much longer cell lengths than the others, as a result of its a, q Mathieu parameters. (20). For each of the three profiles, Γ1 = 0.5406647836mm for all three shapes.…”
Section: The Wire Shape Needed To Match a Mathieu Equation: R(z)mentioning
We show how spatial dispersion can be used as a mechanism to customize the longitudinal profiles of electric fields inside modulated wire media, using a fast and remarkably accurate 1D inhomogeneous model. This customization gives fine control of the sub-wavelength behaviour of the field, as has been achieved recently for transverse fields in simpler slotted-slab media. Our scheme avoids any necessity to run a long series of computationally intensive 3D simulations of specific structures, in order to iteratively converge (or brute-force search) to an empirical 'best-performance' design according to an abstract figure-of-merit. Instead, if supplied with an 'ideal waveform' profile, we could now calculate how to construct it directly. Notably, and unlike most work on photonic crystal structures, our focus is specifically on the field profiles because of their potential utility, rather than other issues such as band-gap control, and/or transmission and reflection coefficients.
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