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

Abstract. In the introductory part (Section 1) some aspects of the large deformation analysis in holographic interferometry are briefly outlined. The calculus there may also 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. 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 the geodesic curvature and of surface curvatures. In the principal part (sections 2, 3, 4) these considerations lead then to similar aspects for hypersurfaces, above all to an interpretation of gravitation by two virtual deformations for the Schwarzschild solution. This is further useful for nonspherical 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 to linearization, a connection is outlined, which confirms the non-existence of gravitational waves if they are described by pure geometrical considerations of the field equations. Detailed equations for calculations are presented in Section 4.


DERIVATIVES OF THE OPTICAL PATH DIFFERENCE, STRAIN, ROTATION, CHANGES OF CURVATURE, FRINGE AND VISIBILITY VECTORS
The basic expression in holographic interferometry is the optical path difference ( ) .D

λν = ⋅ − = u k h
Here u is the displacement, h and k are unit vectors, λ is the wave length, and ν the fringe order.In the case of a large deformation, when two modified holograms are used [ 1 ], the exact expression becomes where , L L′ denote the distances from the image points P, P′ to a point K of fringe localization (see Fig. 1).The phases at P, P′ are T S T T (2 ) ( ) L L p q q p q q ϕ π λ Many authors [e.g. 2 ] have studied the recovering of fringes.The contrast depends on the derivative of , D and the spacing leads to the strains.Therefore the differential p q q q ′ ′ ′ − − − + − is primary.We insert now some elements with convenient notations for the geometry and deformation of 2D-surfaces in the 3D-space.In the principal part these concepts will be generalized to 4D-hypersurfaces, embedded in an 8D-space or in a 4D-complex space.We write in particular for the distance S L in Fig.
where only the semi-projection For small values, a strain tensor , γ an inclination vector , ψ a p-rotation scalar , Ω and the permutation tensor holds with factors 0 , v 0 , E the stress tensor , τ and an involution ( ) .
− … E E Next, the equation of a geodesic curve, relative to the arc , s can be written as because the osculating plane contains the unit normal .
n More generally, for any curve and its image we obtain with where | N marks a projection of the middle factor in a triadic.Finally, gives the change of surface curvature forwards Again in holography: The image P < > is defined by P d 0 θ = (or Eq. ( 9)) of P 2 ( ) p q p q θ π λ = − − + for the rays through the aperture.Thus we find with θ = leads also to Eq. ( 9).Therefore we get from neighbouring rays 2 d ( ) 0, N r we apply Eq. ( 5) so that the total term 2 ˆd ( ) is cancelled because of Eq. ( 9).We use also the affine connection shows the curvature of the non-spherical wavefront and the astigmatic interval R .< > The ray aberration reads KV Q M k gives with ( 1) n n The fringe vector [ 3 ] (indicating the spacing) and the visibility vector [ 4 ] (marking the distance of the homologous rays) are therefore

ASPECTS OF DEFORMATION AT SPHERICAL AND NONSPHERICAL GRAVITATIONAL FIELDS, GRAVITATIONAL LENS, AND ROTATING BODIES
This section is only indirectly related to the previous subject.An extension should illustrate Eqs (2)- (7).Equation (7) gives the curvature ′ B of a deformed surface 2 3 .

⊂ » A
For a hypersurface , n which are at the moment supposed to be real.For the complex version see Section 3, Eqs (40)-(43).If we use these vectors, the Riemann-Christoffel tensor can be written (see also Section 4.1, Eqs (A1)-(A5)) as according to Eq. ( 3) and ( 1) n n ( ) )).The bracket T indicates a transposition of the factors 2 and 4. The Ricci tensor is the contraction of , R alternatively ( ) .
For a spherical gravitational field first one uses the Schwarzschild radius as well as the known vase-like surface [ 6 ].Second, as for the time-radial terms in Eq. ( 13), we introduce a vector 2 cos , M w The factor cos 2 cos r M η ϖ χ ψ = is not relevant in Eq. ( 17).Note also that we have The field equations and the theorem of energy-impulse are with 4 In this static case the principal components of the energy-impulse tensor T are 0 where ( ) p r denotes the 'pressure'.We introduce now the curvature 1  1 The Ricci tensor 4D ( ) However, as we have a connection between ω and ϖ (Eq.( 22)), another connection between the curvatures (Eq.( 23)) and, for a given ( ), r ρ a linear differential equation for 1 ϖ (Eq.( 24)): (sin 2 ) .
where * E is the 2D-permutation tensor normal to * .k With the involution we have here the Gauss curvature For the general gravitational lens with 1, ω ϖ = = we use the equation of a geodesic curve A type of Eqs ( 5) and ( 6) gives then the backwards deformation into the flat space.We have similar to Eqs (25)-(28) four parts The image equation is The surrounding field of a rotating star for instance is nonspherical.In the rotating system there, we may write for the scalar of the inertial force For small Ω we have Let us now look at the dynamic case from another standpoint.We mention that the general Lorentz transformation requires second order spinors ,

S S
(41) The covariant spinors read ;

SS
(i not summed) and .
Further, if we use once more an involution ( ) T − … E E with the symplectic matrix , E we get the conjugates and i.e. a product: ) .

d x
But the fundamental form Eq. ( 13) has incidentally reversed signs, implying alters signs: in three matrices from 4 M but in one only of the last two from 4 .M Accordingly, (1 ) , , 0, 0, After this preparation, again in general, n ( , i k summed from 1 to 4) denotes here a modified projector for which the transpose is equal to the conjugate: . T = N N We have also , where I is the 'identity' in a complex space 4 4 4

⊕ =
M M C (see Section 4.5 and Eqs (A44)).We write then the development ⋅ n with small 'inclination vectors' or spinors in the flat space 4 : Therefore the development of the projector on the curved space becomes The derivative of this projector reads according to Eq. ( 3 This leads to two exterior curvature tensors, here we have , B B see also in particular Section 4.4, Eqs (A38) and (A39) We can also use a base , The Ricci tensor ( ) is in vacuum (see also sections 4.1, Eq. (A6), 4.2, Eq. (A16), and 4.6, Eqs (A58)-(A60)) This is not a wave equation.Therefore gravitational waves, based on the incorrect Eqs (39), cannot exist (see also [ 12 ]).The contradiction between Eqs (48) and (39) originates from the semi-exterior part , ).
We could also develop the complementary projector where , is then obtained from the 'one-third-exterior triadic' curvatures

Riemann-Christoffel tensor
Some remarks concern Eq. ( 12) in the general dynamic case where spinors i i ′ ′ ≠ n n and a modified projector reads (pay attention to the non-commutative products and the transposition of the factors 2 and 3 by the sign ] ) .
n so that we get in accordance with Eqs ( 46) and ( 47) ] .
The second derivatives of the projector become afterwards The bracket ] T indicates here a transposition of the factors 3 and 4, the double bracket T indicates, as previously, a transposition of the factors 2 and 4. Referring to the 2 4 8 × = terms in Eqs (A3) and (A4), we see on one hand that the 1st, 3rd, 6th, and 7th quarter-exterior terms vanish all together if the full projection N N is applied.One term disappears by the first ( ) left (applied to the 2nd factor), two by the fourth | ( ) T ′ ′ = … N N right (applied to the 3rd factor), and one by the third ′ N right (applied to the 4th factor).The second ′ N left is actually not necessary, but it serves to accentuate the interior character of these 4th order tensors.In the difference appearing in the Riemann-Christoffel tensor only four terms remain therefore: The Ricci tensor R is the 'outside' contraction of , the last skewsymmetric factor in Eq. (A5) vanishes after this contraction.Then R can be written in two simple conjugate expressions (the real components R R

Components of the exterior curvatures and the Ricci tensor
The derivatives of the spinors Using the auxiliary relations n , Now as for the development in case of a small curvature, we get with i , , , so that the Ricci tensor of Eq. ( 48) becomes in fact ( ) In general we have also ( ) The components of Eq. (A17) correspond to the usual representation α β from 0 to 3, i from 1 to 4) expresses the vanishing Ricci tensor in the space

Theorem of energy impulse
It is well known that Eq. ( 20) is compatible with Eqs (18) and ( 19), because of some integrability (Bianchi) equations.Here we give an alternative explanation, which refers to the present notation by exterior spinors.We use the rules Y X valid for any 2nd order tensors , X .Y First, the derivative of the trace [sin , cos , 0, 0] , ˆγ The exterior curvature tensor 1 n N according to Eq. ( 46), is therefore By comparison with Section 2 we find now 2 tan .
γ ω ζ = In the same way, but in addition with 2 n , 0 The expressions (A38) and (A39) lead then with 1 , ′ B 2 ′ B and Eqs (A6) and (A33) to a relation for the Ricci tensor R similar to Eq. (21).
We can either write two matrix representations or a complex representation:

The problem of linearization, an alternative development with details
In order to point out explicitly a sort of kinematic meaning of Eq. ( 48) and also of the statement The tensor 0 R may be called the image in the flat space of the true Ricci tensor on the curved space 0 0 0 n n ( ) ( ) .
Because Eq. (48) represents already an approximation after the forgoing development of the projector, R has two semi-exterior parts of higher order.We arrive therefore at the following conclusion: The field Eq. ( 19) becomes in vacuum 0.

= R
We would then get from Eqs (A58) and (A59) and from Eq. (A52) simultaneously two differential equations: Nh with the normal projector = − ⊗ N I n n [note that for any dyadic ⊗ an expanded version of the paper published in Speckle06 (Slangen, P. & Cerruti, C., eds), Proceedings of SPIE, Vol.6341.Nîmes, 2006.The layout has been slightly modified and Section 4, an extended appendix, has been added.

Fig. 1 .
Fig. 1.Recording of a large surface deformation.Modification at the reconstruction to recover fringes.
and the symmetric dilatation , U defined by the Cauchy-Green tensor .T = F F UU At the surface the decomposition is with a rotation n Q

>
The projector n = − ⊗ K N k k refers to the radial unit vector ( , ).
tion [ 8 ]).In a nonspherical field we have , The vector k differs from * k by an angle , α which must be determined by the conditions of vanishing mixed terms.The inclination ψ appears then between * the details this gives finally for the equatorial plane , (ln ) (

i n first in the complementary space 4 Mn
instead of the previous unit vectors .iThis implies a triadic connection S to real unit 'vectors' .

4 [
With a small parameter ψ of inclination we write tentatively B NThe integrability equation can be written with the 4D-permutation tensor E in the form (b) The derivative of the spinor in the static spherical case, according to that in front of Eq. (14)to the key relation and in addition to i . (A52) in another manner.If we apply now the operator ∆ onto Eq. (A51) we obtain with Eq. (A57) possible; therefore gravitational waves, described by pure geometrical considerations, cannot exist.As for the (incorrect) Eq. (39), note that the operator , (A58) and (A60) the equivalent operator n n∇ ⋅∇ = ∆ is applied on the complete tensor γ including its base.Moreover, one has , generalized displacement (spinor) v does unfortunately not intervene in Eq. (39).The commonly used condition this equation, is stated as a choice of simplifying special coordinates.However, this statement disguises the fact of the additional restrictive kinematic relation Eq. (A52).
) Analogously to the nonlinear kinematic equations in the 3D-shell theory, we define here a modified strain tensor M We assume further that | | w is smooth of first order small, whereas | | v is smooth of second order small.