Remarks on Mӧller mistaken famous paper from 1943 Jaykov Foukzon jaykovfoukzon@list.ru Center for Mathematical Sciences, Israel Institute of Technology,Haifa,Israel Abstract Einstein field equations was originally derived by Einstein in 1915 in respect with canonical formalism of Riemann geometry,i.e. by using the classical sufficiently smooth metric tensor, smooth Riemann curvature tensor, smooth Ricci tensor,smooth scalar curvature, etc.. However have soon been found singular solutions of the Einstein field equations with degenerate and singular metric tensor and singular Riemann curvature tensor. These degenerate and singular solutions of the Einstein field equations was formally accepted by main part of scientific community beyond rigorous canonical formalism of Riemannian geometry. 1.The breakdown of canonical formalism of Riemann geometry for the singular solutions of the Einstein field equations Einstein field equations was originally derived by Einstein in 1915 in respect with canonical formalism of Riemann geometry,i.e. by using the classical sufficiently smooth metric tensor, smooth Riemann curvature tensor, smooth Ricci tensor,smooth scalar curvature, etc.. However have soon been found singular solutions of the Einstein field equations with degenerate and singular metric tensor and singular Riemann curvature tensor. These degenerate and singular solutions of the Einstein field equations was formally accepted by main part of scientific community beyond rigorous canonical formalism of Riemannian geometry. 1.1.A. Einstein and N. famous paper from May 8,1935 In famous paper from May 8,1935 [1],(see [1],sec.1,p.74) A.Einstein originally emphasized that degenerate (singular) solutions of the Einstein field equations are problematic: "The first step to the general theory of relativity was to be found in the so-called "Principle of Equivalence": If in a space free from gravitation a reference system is uniformly accelerated, the reference system can be treated as being "at rest, " provided one interprets the condition of the space with respect to it as a homogeneous gravitational field. As is well known the latter is exactly described by the metric field ds2  dx1 2  dx2 2  dx3 2  2x12dx42 1. 1. 1 The g of this field satisfy in general the equations Rklm i  0, 1. 1. 2 and hence the equations Rkl  Rklm m  0, 1. 1. 3 The g corresponding to (1.1.1) are regular for all finite (i.e. nonzero) points of space-time. Nevertheless one cannot assert that Eqs.(1.1.3) are satisfied by (1.1.1) for all finite values of x1, . . . , x4. This is due to the fact that the determinant g of the g vanishes for x1  0. The contravariant g therefore become infinite and the tensors Rklmi and Rkl take on the form 0/0. 1. 1. 3 From the standpoint of Eqs.(1.1.3) the hyperplane x1  0 then represents a singularity of the field. We now ask whether the field law of gravitation (and later on the field law of gravitation and electricity) could not be modified in a natural way without essential change so that the solution (1.1) would satisfy the field equations for all finite points, i.e., also for x1  0. W. Mayer has called our attention to the fact that one can make Rklmi and Rkl into rational functions of the g, and their first two derivatives by multiplying them by suitable powers of g. It is easy to show that in g2Rkl there is no longer any denominator. If then we replace (1.1.3) by Rkl   g2Rkl  0, 1. 1. 3. a this system of equations is satisfied by (1.1.1) at all finite points. This amounts to introducing in place of the g the cofactors g  of the g in g in order to avoid the occurrence of denominators. One is therefore operating with tensor densities of a suitable weight instead of with tensors. In this way one succeeds in avoiding singularities of that special kind which is characterized by the vanishing of g. The solution (1.1.1) naturally has no deeper physical significance insofar as it extends into spatial infinity. It allows one to see however to what extent the regularization of the hypersurfaces g  0 leads to a theoretical representation of matter, regarded from the standpoint of the original theory. Thus, in the framework of the original theory one has the gravitational equations R ik  12 gikR  Tik, 1. 1. 4 where Tik is the tensor of mass or energy density. To interpret (1.1.1) in the framework of this theory we must approximate the line element by a slightly different one which avoids the singularity g  0. Accordingly we introduce a small constant  and let ds2  dx1 2  dx2 2  dx3 2  2x12  dx42 1. 1. 1. a the smaller  ( 0) is chosen, the nearer does this gravitational field come to that of (1.1.1). If one calculates from this the (fictitious) energy tensor Tik one obtains as nonvanishing components T22  T23   2 2x12   2 . 1. 1. 4. a We see then that the smaller one takes  the more is the tensor concentrated in the neighborhood of the hypersurface x1  0.From the standpoint of the original theory the solution (1.1.1) contains a singularity which corresponds to an energy or mass concentrated in the surface x1  0; from the standpoint of the modified theory, however, (1.1.1) is a solution of (1.1.3.a), free from singularities, which describes the "field-producing mass, " without requiring for this the introduction of any new field quantities. It is clear that all equations of the absolute differential calculus can be written in a form free from denominators, whereby the tensors are replaced by tensor densities of suitable weight. It is to be noted that in the case of the solution (1.1.1) the whole field consists of two equal halves, separated by the surface of symmetry x1  0, such that for the corresponding points x1, x2, x3, x4 and x1, x2, x3, x4 the gik are equal. As a result we find that, although we are permitting the determinant g to take on the value 0 x1  0, no change of sign of g and in general no change in the "inertial index" of the quadratic form (1.1.1) occurs. These features are of fundamental importance from the point of view of the physical interpretation, and will be encountered again in the solutions to be considered later." 1.2.Remarks on Mӧller ubnormal famous paper from 1943 Recall that the classical Cartan's structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. In order to study the mathematical properties of singularities, we need to study the geometry of manifolds endowed on the tangent bundle with a symmetric bilinear form which is allowed to become degenerate (singular). But if the fundamental tensor is allowed to be degenerate (singular), there are some obstructions in constructing the geometric objects normally associated to the fundamental tensor. Also, local orthonormal frames and co-frames no longer exist, as well as the metric connection and its curvature operator [2]. As an important example of the geometry with the fundamental tensor which is allowed to be degenerate, we consider now Mӧller's uniformly accelerated frame given by Mӧller's line element (1.2.1). Recall that Möller dealing with the following line element [3]: ds2  xdt2  dx2  dy2  dz2, 1. 2. 1 where x  a  gx2. Remark 1.2.1. Of course Mӧller's metric (1.2.1) degenerate at Mӧller horizon xhor  a/g. However in contrast with A.Einstein paper [1],in famous but ubnormal paper [3] Möller mistakenly argue that metric field (1.2.1) is an global vaccuum solution of the A.Einstein field equations (1.1.4), i.e. the g of this field for all values of t, x, y, z satisfy the equations R ik  12 gikR  0. 1. 2. 2 Remark 1.2.2. In physical literature this Möller's ubnormal mistake holds from Möller's time until nowadays. Remark 1.2.3. Note that formally corresponding to the Mӧller's metric (1.2.1) classical Levi-Civit'a connection is 44 1 x  a  gx,14 4 x  41 4 x  ga  gx1 1. 2. 3 and therefore classical Levi-Civit'a connection (1.2.3) of course is not available at Mӧller horizon since at horizon formal expressions (1.2.3) becomes infinity: 14 4  ag  41 4  ag  . 1. 2. 4 Remark 1.2.4.Note that Möller dealing with Einstein's field equations in the following form Gi k  R i k  12 i kR, 1. 2. 5 where R ik is the contracted Riemann-Christoffel tensor, formally calculated by canonical way by using classical Levi-Civit'a connection (1.2.3) and R  R ii By using the following ansatz ds2  Dxdt2  dx2  dy2  dz2, 1. 2. 6 Möller finally obtain G2 2  G3 3   1 2D D  D2 2D   D1/2  D1/2 . 1. 2. 7 where D  dDx/dx. Remark 1.2.5.From Eq.(1.2.7) Möller mistakenly obtain the following equation D1/2   0, 1. 2. 8 since it was mistakenly assumed that G22 and G33 for all values of t, x, y, z satisfy the equations G2 2x  G3 3x  0. 1. 2. 9 The equation (1.2.8) obviously has the following trivial general solution Gx  a  gx2. 1. 2. 10 Remark 1.2.6.Note that at horizon G22 and G33 ofcourse is not zero but becomes uncertainty since G2 2a/g  G3 3a/g   Da/g1/2  Da/g1/2  0 0 1. 2. 11 in acordance with (1.1.3) in A.Einstein paper [1]. Thus solution (1.2.10) obviously holds only except horizon xhor  a/g as A.Einstein emphasize in paper [1]. For better explanation see [4]. References [1] A. Einstein and N. Rosen,The Particle Problem in the General Theory of Relativity,Phys. Rev. 48, 73 – Published 1 July 1935 [2] D. N. Kupeli 1996 Singular Semi-Riemannian Geometry 1996 X 181 pp. Series: Mathematics and Its Applications 366 DOI: 10.1007/978-94-015-8761-7 [3] C. Mӧller, 1943 Danske Vid. Sel. Mat-Fys. Med. XX, No.19 [4] J Foukzon et al 2018 J. Phys.: Conf. Ser. 1141