Abstract
AbstractMaxwell's equations are transformed from a Cartesian geometry to a Riemannian geometry. A geodetic path in the Riemannian geometry is defined as the raypath on which the electromagnetic energy efficiently travels through the medium. Consistent with the spatial behavior of the Poynting vector, the metric tensor is required to be functionally dependent on the refractive index of the medium. A symmetric nonorthogonal transformation is introduced, in which the metric is a function of an electromagnetic tension. This so‐called refractional tension determines the curvature of the geodetic line. To verify the geodetic propagation paths and wavefronts, a spherical object with a refractive index not equal to one is considered. A full 3‐D numerical simulation based on a contrast‐source integral equation for the electric field vector is used. These experiments corroborate that the geodesics support the actual wavefronts. This result has consequences for the explanation of the light bending around the Sun. Next to Einstein's gravitational tension there is room for an additional refractional tension. In fact, the total potential interaction energy controls the bending of the light. It is shown that this extended model is in excellent agreement with historical electromagnetic deflection measurements.
Highlights
The transmission of electromagnetic energy along rays has been of interest in our community
A full 3-D numerical simulation based on a contrast-source integral equation for the electric field vector is used
We argue that the geodetic path in the Riemannian geometry determines the raypath on which the electromagnetic energy efficiently travels through the medium
Summary
The transmission of electromagnetic energy along rays has been of interest in our community. We argue that the geodetic path in the Riemannian geometry determines the raypath on which the electromagnetic energy efficiently travels through the medium. The metric is a function of an electromagnetic potential, which gives rise to a tension that determines the curvature of the geodetic line This tension is present in vacuum outside the object. The geodetic lines are not equivalent to the raypaths in optics The latter paths follow from a high-frequency approximation of Maxwell's equations. It is obvious that any solution of the Maxwell equations in a Riemannian geometry needs the specification of the refraction index and the impedance in whole space. Both material parameters occur in the curl operators. In view of (11) and (12), we choose the metric tensor gij to be a function of the refraction index n only
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