Energy and Geometry: An Introduction to Deformed Special by Fabio Cardone

By Fabio Cardone

This e-book discusses intimately the mathematical features and actual functions of a brand new geometrical constitution of space-time. it really is in keeping with a generalization ("deformation") of the standard Minkowski area, supposedly endowed with a metric whose coefficients rely on the strength. power and Geometry: Geometrical Description of Interactions is acceptable for researchers, teachers and scholars in mathematical and theoretical physics.

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Additional resources for Energy and Geometry: An Introduction to Deformed Special Relativity

Sample text

Symmetrization of Deformed Boosts As in the case of standard SR, it is possible to symmetrize the expression of boosts in DSR by introducing suitable time coordinates. Let us first consider a deformed boost along x ˆi (i = 1, 2, 3); the symmetrization transformation (a “dimensionally homogenizing dilatocontraction”) of t is given by x0 ≡ ui t = c b0 (E) t; bi (E) xi ≡ xi . 47) The deformed metric tensor in the new “primed” coordinates, {xµ } = {x0 , x, y, z}, reads: ηµν (E) ESC on = ESC off = ηαβ (E) ∂xα ∂xβ = diag(b2i (E), −b21 (E), −b22 (E), −b23 (E)) ∂xµ ∂xν δµν [b2i (E)δµ0 − b21 (E)δµ1 − b22 (E)δµ2 − b23 (E)δµ3 ].

If ηµν is the deformed metric “seen” by the nth Fourier compo(n) nent of the evanescent wave, with metric coefficients bµ , we can build an effective metric tensor η¯µν for the evanescent wave as followsa : η¯µν (cn ) = n (n) |cn |2 ηµν . 39) reduces to the usual metric gµν . m. e. bµ = 1, µ = 0, 1, 2, 3, and therefore ηµν = gµν ∀n. 39) is analogous to the Cauchy stress tensor of a continuous medium. In fact, let us consider, in orthogonal Cartesian coordinates, an infinitesimal tetrahedron with edges parallel to the coordinate axes and the oblique face S opposite to the vertex O, origin of the Cartesian frame.

Total Internal Reflection 47 On the basis of the above analogy, we can also write the following expression for the wavevector along the z-direction inside the slab: k˜z2 = 2π λ 2 2 b0 b3 sin i sin ic − 2 . 31) and this case yields a superluminal speed of the light beam inside the slab, according to Eq. 10): uz = b0 c > c. 31) is also referred to as the superluminality condition (cf. 16)). By the formalism of the deformed Minkowski space-time, it is possible also to give an expression of the time-averaged Poynting vector in the z-direction for a monochromatic component of the light beam.

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