Deformed surfaces in holographic interferometry. Similar aspects in general gravitational fields

W Schumann

doi:10.3176/proc.2008.1.04

Abstract

In the first part some principles of the large deformation analysis in holographic Interferometry are briefly outlined. This may also give a link to the second part here. Modifications of the set-up at the reconstruction should recover the previously invisible fringes. The spacing and the contrast of them are characterized by the fringe and visibility vectors. The relevant derivative of the path difference involves the polar decomposition of the deformation gradient into strain and rotation and the image aberration implies further changes of geodesic curvature and of surface curvatures. In the second part these considerations lead then to similar aspects for hypersurfaces, before all to an interpretation by two virtual deformations for the Schwarzschild-solution of the gravitation. That is further usefull for non-spherical gravitational fields, for the invariants there and for the TOV-relation between pressure and density. The null-geodesics or light rays can also be interpreted by these virtual deformations. An approach towards the Kerr-solution for rotating stars is added. As for the linearisation a connection is outlined which confirms the non-existence of gravitational waves.

Highlights

Many authors [e.g. 2] have studied the recovering of fringes
The calculus there may serve as an introduction for a link to the principal part afterwards
Modifications of the set-up at the reconstruction should recover the previously invisible fringes. Their spacing and the contrast are characterized by the fringe and visibility vectors

Summary

The Christoffel symbols certain condition

A wave equation (?) for the Ricci tensor in vacuum:. , λ ψαβ = 0. The development of the projector on the curved space becomes N ′ ≈ N +ψ i ⊗ ni + ni ⊗ψ i −ψ i ⊗ψ i + (ψ i ⋅ψ j )(n j ⊗ ni ) The derivative of this projector reads according to Eq (3), ∇n′ ⊗ N ′ = B′i ⊗ ni′ + Bi′ ⊗ n′i ]T. This leads to two exterior curvature tensors, here we have B′i ≠ Bi′ ≠ Bi′T , see in particular Section 4.4, Eqs (A38) and (A39). R = Bi′T B′i − (Bi′ ⋅ N ′)B′iT is in vacuum (see sections 4.1, Eq (A6), 4.2, Eq (A16), and 4.6, Eqs (A58)–(A60))

Components of the exterior curvatures and the Ricci tensor

Theorem of energy impulse

Three illustrative steps

The role of the surrounding complex space