View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server General relativity as a complete theory and its new consequences ∗† Zahid Zakir Institute of Noosphere, 11a Sayram, Tashkent, 700170 Uzbekistan [email protected] Abstract It is shown that a local extension of distances in a gravitational field leads to a global topological difference between the Schwarzschild coordinates and the physical coordinates of local inertial frame. In term of a physical time and a physical radius the falling in Schwarzschild field particles and photons elastically reflect from a gravitational radius. It is shown that the Schwarzschild frame is complete, an interior region of gravitational radius is unphysical, a horizon and a singularity not appear. As result the general relativity really predicts an existence of heavy barionic and quark stars and quasars instead of black holes. The high redshifts of quasars can be explained mainly as a gravitational redshift. A model of oscillating Universe without a cosmological singularity is proposed. A new type of cosmological redshift is predicted which can partly solve the dark matter problem. In the theories of strings and branes the singularities of black hole solutions disappear in term of physical variables. A renormalization in quantum field theory also is derived as a natural consequence of general relativity prescriptions. 1 Introduction The modern standard treatment of general relativity leads to an intrinsic incompleteness of the theory because of the appearance of horizons and singularities in the black holes and cosmology. Is this a necessary consequence of first principles or the theory can be reformulated as a complete one without changing the first principles? Up to now the predicted black holes are not convincingly revealed, but at the same time more than 13200 quasars are discovered with very high red shifts (z<5.8) which can not be explained in the standard treatment of general relativity without assumptions about fantastic masses, luminosities and fantastically small sizes and large distances. Why so many difficulties appear in an identification of few candidates to predicted objects, while the objects which difficultly explained in the theory so easily discovered by astronomers every week during last 35 year? In the existing standard treatments of general relativity a main attention was drawn to the metrics and coordinates as some formal mathematical tools and little attention is given to their physical meaning and measurability. In this paper it is argued, that the general relativity becomes a complete theory after an adequate to first principles interpretation of the results of spatial and temporal measurements. We describe a new picture of the particle’s motion in the Schwarzschild field with a reflection of falling particles from the source. We show that the process is fully reversible in time, the Schwarzschild’s frame becomes complete, the horizons and singularities disappear. The correctly interpreted general relativity not predicts a collapse and black holes, but it naturally explains quasars with the arbitrary high redshifts as relativistic stars or galaxy nucleus with ordinary masses, luminosities, sizes and distances. It is shown that the general relativity not contains a cosmological singularity and it leads to the oscillating cosmological models. A new type of cosmological redshift is predicted as result of ”potential difference” between scale parameter at a time of emission and at the present time with respect to a ”center of mass” of Universe in fourth dimension. The problem of dark matter probably can be naturally solved taking into account this additional redshift. Another surprise of the new treatment is a derivation of renormalization procedures in quantum field theory from the first principles of general relativity. ∗To be published in: "Z.Zakir. Structure of space-time. Part II, paper 18, I.N., Tashkent, 2000." c Institute of Noosphere, 2000. y 1 2 SCALES AND CLOCKS IN A GRAVITATIONAL FIELD 2 In the paper the corrected version of the new formulation is presented, the preliminary properties of which was published in my previous papers [3] . More detail considerations and discussions will be presented in [4] and in the forthcoming publications. 2 Scales and clocks in a gravitational field In spacetime with metric gik the arbitrary curvilinear coordinate systems can be used and a spacetime 2 i k i i i0 i0 interval ds(g) = gikdx dx is invariant under the coordinate transformations dx =(∂x /∂x ) dx .Buta definite coordinate system has a meaning from a physical point of view only after describing a· behavior of scales and clocks on a physical frame, on the basis of which these coordinates can be measured. In a static gravitational field four types of special frames can be introduced which we describe below in a form which we will use in this paper. 2.1 Rest frame at infinity (I-frame) For the static local source it is natural to choice a Galilean frame at infinity which is at rest with respect to this source (I-frame) with flat euclidean spacetime interval between two events: ds2 = c2dt2 dR2 R2dΩ2 (1) (I) − − where dΩ2 = dθ2 +sin2 θ dϕ2 is the line element on 2-sphere. At finite distances from the· source the curvilinear coordinates xi with gravitationally deformed scale and time units can be expressed in terms of the I-frame coordinates Xa =(t, R, θ, ϕ) with the scale and time units same as on I-frame: dxi =(∂xi/∂Xa)dX a andwehaveforthespacetimeinterval: 2 (0) a b ds(g) = gab (X)dX dX (2) Here ∂xi ∂xi g(0)(X)=g (3) ab ik ∂Xa ∂Xb is a true remainder metric of spacetime with a nonzero curvature tensor which (on a rest frame of the source) describes the true gravitational field. In such consideration the gravitation represents as a difference of the spacetime geometry with respect to the I-frame geometry. This true remainder metric can not be transformed to the trivial flat metric ηab = diag(1, 1, 1, 1) without transformation of I-frame coordinates Xa to another coordinates X 0a which means that− the− scale− and time units then becomes different from the I-frame units. 2.2 Local inertial rest frame (L-frame) Let we introduce a local inertial frame which begins free falling from a given point of the gravitational field at finite distance from the source (L-frame). A scale and a clock of L-frame at given point are momentary at rest with respect to the source (and to the I-frame). On L-frame the gravitational field absent (only locally and during a small time interval) and the scale, clock and comoving objects are at weightlessness.Bythe coordinate transformations dxi =(∂xi/∂Xα)dX α with property: ∂xi ∂xi g = η (4) ik ∂Xα ∂Xβ αβ we obtain L-frame line element as: 2 α β ds(L) = ηαβdX dX (5) We consider a special form of this line element as: ds2 = c2dτ 2 dr2 R2dΩ2 (6) (L) − − where dr(R)anddτ(R) are spatial and temporal intervals which measured by L-frame scales and clocks and a length of orbit is l =2πR. Are the scale and time standards of the L-frame in gravitational field 2 SCALES AND CLOCKS IN A GRAVITATIONAL FIELD 3 identical with the scale and time standards of the I-frame, and can we identify dr(R)withdR and dτ(R) with dt? L- and I-frames are at rest with respect to each other and on both frames scales and clocks are at weightlessness. Nevertheless we can see that the scale and time standards on L- and I-frames are different since they are situated at the different points of space with different curvatures and that moreover 2 2 ds(L) <ds(I). The L-frame coordinates have another scale and time units than I-frame with the relations: ∂Xb dXb = dX α = hb (X)dX α (7) ∂Xα α We have in the gravitational field a conserved energy of the particle E which is equal to a rest energy E = m c2 at infinity (R ) and for the falling from infinity particle can be written as: 0 0 →∞ 1 rg/R E = m c2 − = m c2 (8) 0 1 v2/c2 0 s − Here v = dr/dτ is a local velocity and therefore: dr v r = = g (9) cdτ c R r 2 where rg =2GM/c is the gravitational radius, G is Newton constant, M is a mass of the source. If the spacetime interval on L-frame we represent in term of I-frame coordinates, we get the ordinary Schwarzschild 1=2 1=2 solution with a contraction of time intervals dτ = γ dt and an extension of space distances dr = γ− dR, where γ =1 rg/R. We have on L-frame: − 2 2 2 1 2 2 2 ds = γc dt γ− dR R dΩ (10) (L) − − 1=2 It is very important for our purposes that dr = γ− dR > dR since the gravitational extension of the radial distances on L-frame with respect to I-frame distances. The I-frame observer at infinity receive 1=2 photons emitted by L-frame atom with a gravitational redshift z =∆λ/λ = γ− 1. 0 − 2.3 Freely falling from infinity frame (F-frame) The velocities of scales and clocks of this frame are increasing with respect to L-frame observers due to energy conservation at falling and the L-frame observers really can measure the potential of the gravitational field by the measuring these velocities. The scales, clocks and comoving objects on F-frame are also at the weightlessness but the space and time units of F-frame contain Lorentz contractions with respect to L-frame standards.