A THEORY OF GRAVITATION BASED ON SCALE-ROTATION-REFLECTION

TRANSFORMATIONS (SRR-THEORY)

G.M.Telezhko

The paper deals with the agreement of the theory of gravitation, gravitational potential being represented as a 4*4 matrix of scale-and-rotation transformations, improper rotation transformations included, with the observable phenomena: gravitational red shift, radio-location signal time delay while locating planets, declination of ultra-relativistic particles and Hubble's effect.

CONTENTS

1. Electromagnetic reference frames and gravitation potential

2. Agreement of SRR-theory with observable phenomena

2.1. Effective co-ordinate transformations and the dependence of the speed of light as observed at a distance on gravitation potential

2.2. Time delay of radio-location signals

2.3. Transformations of wave 4-vectors and gravitational red shift. Group and phase velocities

2.4. Transformations ofљ 4-momentum. Newtonian law of gravitational attraction, declination of ultra relativistic particles, Hubble's effect

3. Disputable issues

4. Conclusion

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1. ELECTROMAGNETIC REFERENCE FRAMES AND GRAVITATIONAL POTENTIAL

Creation of macroscopic physical co-ordinate systems using them for observation of physical events seems to be possible due to long-distance forces inevitably accompanied by gravitational counter-action revealing itself as gravity and/or inertia. Absence of forces opposing gravitation in Nature would, probably, turn the Universe into a fog of dust-like matter, a variety of 'blind' particles incessantly and freely falling somewhere. There would be no point events of General Relativity Theory (no collisions, etc.), no frames of reference of any kind - neither Euclidean, nor Riemannian - in this Universe.

For constructing of co-ordinate systems we use (explicitly or implicitly) electromagnetic forces, in most cases. Those can be intermolecular forces providing rigidity of rods - length standards, constancy of length of pendulum threads or elasticity of springs for generating periodical movements - time standards. We can also use electromagnetic interaction of light with matter permitting radio-location measurements of distances, provided that the above mentioned standards of time or electromagnetic spectral standards are available for measuring of the time required for passing of the radio-location signal there and back. However, it is known that measurements of speed of light in any direction are ambiguous because of the fundamental impossibility of checking simultaneity of equal readings of clocks located at different places (at the initial and final points of the path of the light signal). It is postulated in Relativity Theory that speed of light is a standard isotropic value, although, in principle, factors influencing speed of light and the readings of devices for measuring of lengths and time intervals in such a way that speed-of-light variations would remain unnoticeable are not excluded. This seems acceptable because rigidity of rods, elasticity of springs and other characteristics of materials, that permit to create devices for measuring distance and time, are caused by electromagnetic interaction between molecules of the materials, i.e., depend on propagation of virtual photons - carriers of electromagnetic interaction. It appears that we measure speed of light with devices, whose parameters depend on the value of the speed.

Let us regard different electromagnetic reference frames from Mach's hypothesis viewpoint. Variations (spatial and temporal) of background gravitation potential components are the factor that influences both speed of light and measuring instruments, so that changes of speed of light after a transition from one frame to another moving with respect to the first one can not be detected, the variations being caused by the global change of relative velocities of all bodies in the Universe after the said transition (there are many papers dealing with deformations of real rods and clocks in gravitational field, see [1]). If one maintains one and the same value of the components of gravitational potential, passing from one reference frame to another (e.g., makes the potential diagonal), then he must correspondingly vary the phases of wave-functions of all the particles participating in gravitational interaction, photons included.

The difference between the vector components of the gravitational potentials of the two observers causes difference between the readings of clocks in the two frames of reference that depends on the clocks' location and 'disguises' the relative light speed anisotropy in the moving frame, observable from the remote stationary frame. To make the entire Universe move differently with respect to the two observers (i.e., to make one of the observers move with respect to the other in the same equi-potential area of the Universe) proves to be the simplest and most effective way of creating non-zero difference between the vector components of the total potential, the components dependent on the velocities of external field sources.

Then Lorentz's transformations can be interpreted as gauge transformations:

k'_i - k_i = (a_ij - d_ij) k_j,љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (1)

where k_i = j/x_i, k'_i = j/x'_i are wave 4-vectors of a particle viewed from the 'stationary' frame of reference and from the moving one (primed) correspondingly;

a_ij are coefficients of Lorentz's transformations, playing the role of components of potential into which the primed frame is placed, from the non-primed frame's viewpoint;

4-vector k_j in the right-hand part of (1) represents the particle's gravitational charge placed into this potential;

d_ij - the unit symmetric tensor (Kronecker's symbol), playing the role of potential, chosen in each of the reference frames as the gauge potential.

(Here beneath we will only use Cartesian co-ordinate systems with the imaginary-number time axes, hence lower indexes of tensor components will be used).

Transformations (1) (those of Lorentz rotation) "disguise" the anisotropy of speed of light with respect to the primed frame, which is observable, naturally, from the immovable frame. Besides, four-dimensional deformations of wave functions (1) mean re-determination of simultaneity and the units of length and time and cause disagreement between observers, one moving with respect to another (or being in potentials with different vector components), about the angles between the normal to the wave front of a particle and spatial axes of the corresponding electromagnetic co-ordinate systems (relativistic aberration). These effects are being caused by the difference between the vector components of the transformation matrix, and, independent of the way the difference is created, the effects can be eliminated by change of the velocity of one of the observers.

With regard to static field, it causes changes of the scales of instruments for measuring length and time, the instruments being posed at points with different diagonal components of the potential and 'hides' isotropic variations of the speed of light value with the transition of a local observer to another point with different potential, while a remote observer would see, that the speed of light value varies reciprocal proportionally to the square of the scaling factor. We will suppose that diagonal components' variation is uniform (e.g., inside a sphere, its radius changing) and causes isotropic variation of the measured at a distance speed of light. This supposition results in a less general form of the effective metrical tensor, but still holds the test on the standard gravitational phenomena.

The transformations (1) show the influence of two factors on the description of one and the same phenomenon: one is the change of the vector part of the potential at the transition from one reference frame to another, the other is Galileean 'drift' of one of the frames with respect to the other. The latter of the two factors will be excluded, as unessential for our consideration, by an appropriate substitution. Within Mach's approach, although it originates from the Relativity theory, proofs of one's movement relativity and the issue of correct determination of inertial and non-inertial reference frames lose their sense. They are replaced with the problem of correct calculation of the field crated by the ambiguous distribution of gravitational sources in the Universe, their movements not being well known as well. (These issues had occupied the mind of late K.V.Anisovich [4,5]). We will be, however, interested in matching the variant of a Mach-oriented field theory with the observable gravitational phenomena rather than in looking for a way of finding the exact value of the potential created by the total number of bodies in the Universe.

The following example will show what we mean by the two above-said factors. It seems obvious that if a train passes by a platform with a velocity V, then all that is at rest with respect to the train, is observed as moving with respect to the platform with the same velocity V. Gravitation has nothing to do with that. However, if there is a passenger walking along the train with velocity W then an observer on the platform would see him walking with the relative velocity (with respect to the train) (W+V)/(1+VW/c²) - V, which does not equal W. Differences of this kind that are not directly caused by the change of distance between observers are caused by the difference in "tuning" of particle's wave functions when the differing, from the two observer's viewpoints, corresponding components of the gravitational potential are transformed to one and the same standard set of components d_ij in each of the frames of reference.

The consideration of relativistic effects as caused by gauge invariance means that one takes the connection between potentials and probability amplitudes for primary, and their geometrical interpretation - for secondary. Every time we mention co-ordinate transformations we will have this connection in mind. Surely, it remains a matter of agreement what to consider primary and what - secondary.

In the suggested in [2] variant of the theory gravitational field was represented geometrically, in the general case, in the form of the spatial and temporal dependence of the angles of rotation of a 4-dimensional electromagnetic reference frame, improper rotations included, and scales along all its axes. In paper [3] the gauge interpretation of the relativity principle was considered in detail (in paper [2] the focus of the attention was on the correspondence of the variant of the theory to Einstein's requirements to a theory satisfying Mach's principle).

In this theory of gravitation (here referred to as SRR-theory) many relativistic phenomena (relativistic Dopler effect, aberration, etc.) appear to be gravitational analogies of Aharonov-Bohm effect, and the field of inertia forces in non-inertial reference frames is analogous to vortex electrical field -dA_m/dt (m = 1,..,3) in the theory of electromagnetism and is considered, as well as the field produced by the changing vector component of the electromagnetic potential A_m, to be a relativistic field of true gravitational forces.

2. AGREEMENT OF SRR-THEORY WITH OBSERVABLE PHENOMENA

2.1. Effective co-ordinate transformations and the dependence of speed of light as measured at a distance on gravitational potential

Let us exclude Galileean drift from Lorentz's transformations:

љљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (2)

substituting dx'_n = dX'_n- v_n dt' (x' - the co-ordinate with respect to the origin of non-primed reference frame moving with the 3-velocity v_n= -ic a_0n/ a₀₀, n = 1, 2, 3; j = 0,..., 3, as viewed from the primed frame), and combine these transformations with scale transformations: a_ij R H'_ij º Ha_ij. Thus we will find the connection between the results of measurements of the same length performed by observers located at the areas with different constant values of gravitational potential components, while Lorentz's transformation give the connection between the difference of the corresponding co-ordinates of certain events as measured by the observers moving one with respect to another. (It is worth saying that the value vdt', removed from (2), corresponds to the movement of an instant accompanying inertial reference frame, so that when a_0n are variable, we incessantly pass from the current accompanying frame to another with the constant velocity v differing from the previous one by an infinitely small value).

We will obtain that locally measured elementary lengths dx_i are connected with elements dx'_j (dx'₀ = icdt') of the extrapolated Cartesian co-ordinate network of a relatively resting remote observer as follows:

љљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (3)љљљ

љљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (3Б)

where the primed values correspond to the remote observer's viewpoint, dx₀љ ºљ icdt, H'_ij H'_kjљ ºљ H² d_ik, and when x'_n are constant, x_m are also constant, as it should be when two observers are at rest with respect to each other.

Square of 4-interval dx_i dx_i can be represented as the sum of squares of modified co-ordinate differentials dx"_j = H'_ji dx_i /H² (dx_i = H'_ji dx"_j):

љ (4)

where dx"_jљ ºљ (ic/H² dt', dx'_n + H'_n0 / H'₀₀ ic/H² dt'), m, n = 1, 2, 3; i, j = 0,..., 3.

Local (dx_i) and modified (dx"_j) co-ordinates are connected with orthogonal transformations, i.e. dx'_i are co-ordinates providing the standard isotropic value of the speed of light. The local observer is moving with respect to the origin of this co-ordinate system, as follows from the form of dx"_n in the primed co-ordinate system of the remote observer, with the velocity icH'_n0/(H²H'₀₀). Independence of dx"₀ and dx'_n means that the clocks of the double-primed system seem synchronised to the remote observer. The term, proportional to time, has the following sense in the modified co-ordinate system. In order to transport a test body from the remote observer to the local one, located at the potential with non-zero vector component, one must apply a transversal non-gravitational force (e.g., an electromagnetic one) to the body, which force would cause the transversal speed component increase, if the vector component were zero (i.e., if the potential were diagonal: Hd_ij) along the way. If the potential were diagonal, the double-primed co-ordinate system would be an immovable co-ordinate system with synchronous clock. We also will have in mind, that the term H'_n0H'_0j/H'₀₀ in (3) is caused by the compensation of the movement of the double-primed system, i.e. is instantaneously fixed, and its differentiating makes zero.

The transformations (3, 3a) are not orthogonal, therefore the speed-of-light value depends both on co-ordinates and direction. From (3, 3a) one can find the dependence of projections c'_n (the remote observer's viewpoint on the speed-of-light value) on projections c_m of the locally measured speed of light (c_mc_m = c²), substituting components of the potential of the place where the measurement is performed, from the point of view of the remote observer:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (5)

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (5a)

modulus of remote light speed being expressed as:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (5b)

where e'_n is a unity vector parallel to light's direction, dx_i dx_i = 0.

If the field is static (H'_ij = H d_ij), (5a, 5b) give c' = c/H². If light moves orthogonal to the vector H'_0n (H'_0n e_n = 0), they give c' = c/(HH'₀₀), which means that the radial speed of light in a rotating reference frame is c' = c ×љ ×, where w is the angular velocity, R is the distance between the observer in the centre of rotationљ and the observed point. If light moves orthogonal to an inertial frame, we will have, correspondingly, c' = c× ×, where v - is the frame's velocity. In the inertia field on a rotating platform the tangent speed of light (i.e. when the speed-of-light vector is parallel to the vector H'_0n, so that H'_0n e_n = ), from the viewpoint of the observer in the centre, equals c+wR (H'_ij is a matrix of Lorentz's rotation corresponding to v=wR). The anisotropy of speed of light at the area with non-zero H'_0n coincides with that observed by the immovable observer (c'_n º dx'_n /dt') with respect to a moving body.

Besides, equation (5a), applied to relative velocity w'_n in general (i.e. not exclusively to relative speed of light c'_n), is equivalent to the classical rule of addition of velocities. For the example with a passengerљљ walkingљ inљ a trainљ we would obtain from (5a) w'= (W+V)/(1+VW/c²) - V = W (1 - V²/c²)/(1 + WV/c²) by virtue of substitutions: H = 1, H'_0n = - H'_n0 = (V/ic)/ , H₀₀ = H₁₁ =љ 1/.

2.2. Time delay of a radio-location signal

Let us find additional delay of a radio-location signal reflected from Mercury, using (5a, 5b). We have for static field H'₀₀ = H'_nn = H; H'_n0 = 0.

Actual time spent by the signal travelling there and back is:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љ(6)

where c'(H) is the signal velocity as observed from the Earth, c'(H) = c/H²;

H² @ (1+F_Newt)² @ 1+2F_Newt is square of the sum of the background potential at the origin (always fixed equal to d_ij) and Newtonian contribution of the Sun F_Newt = gM/(R(l') c²)<<1; g is gravitational constant, M - the Sun's mass, R(l') - the distance between the point of the signal's path and the Sun's centre;

dl' is length of an elementary part of the path measured using units of the reference frame origin located on the Earth.

The second term in (6) is the value of the additional time delay and, in the case of location of Mercury, equals 220 mcs coinciding with experimental data.

2.3. Wave 4-vector transformations and gravitational red shift. Group and phase velocities

Knowing the rule of "gravitational" co-ordinate transformations in electromagnetic co-ordinate systems, one can obtain the rule of wave vector transformations (the same result would be obtained if one excludes Galileean 'drift' from (1) and complements (1) with scale transformations):

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (7)

When a source of radiation sinks into an area with greater gravitation potential the observed frequency of radiation n'ºicj/x'₀ would decrease being reciprocal proportional to the scalar potential (n'=n/H'₀₀). This gives the gravitational red-shift value and also predicts the effect analogous to the transverse Dopler effect for sources of radiation located at areas with non-zero value of the vector component of the potential (i.e. when ½H'₀₀½ > ½H½). The expression for j/x'_n describes the effect analogous to the longitudinal Dopler effect.

From (3) one can obtain the remote 3-velocity v'_m vs local 3-velocity v_n, and from (7) - the remote phase velocity (j/x'₀)/(j/x'_n) vs local phase velocity. Postulating that the local group and phase velocities are parallel we come to the conclusion that the angle between the remote group and phase velocities is observed as a non-zero value depending on the vector component H'_m0, and this is the manifestation of gauge invariance: any change of the potential causes the change of the wave function phase distribution in space and time. It is seen from (7) that the wave 3-vector in the primed frame corresponds to the test body velocity in the double primed frame (see the note to (4)). The physical sense of the wave vector rotation implies that particle's group velocity constancy, the particle moving through an area with varying vector components, demands applying a non-gravitational force to the particle (e.g., electromagnetic force), manifesting in rotation of the particle's wave function.

Changes of the angle between the group and phase velocities can be demonstrated using the example of light propagation in the reference frame of a rotating disk. Let a cylinder shaped light wave be sent from the centre of the disk at the initial moment of time (or a particle with a uniform distribution of probable directions of movement within 360^п).љ While the photons (or the probable locations of the particle) propagate along paths that are straight in an inertial reference frame the circle rotates so that in the circle's reference frame the paths are curved. Then, from the viewpoint of a rotating observer at the centre of each element of the wave front coincides with an element of the cylindrical surface, the axis of the surface passing through the circle's centre (the wave remains cylinder as well as for the inertial observer resting with respect to the centre), and from the point of view of the peripheral observer is turned with respect to the local element of the concentric cylinder (aberration caused by the disagreement of the central and peripheral observers about the issue of simultaneity).

As to the photons' group velocities, according both to the viewpoints of the central and peripheral observers on the disk, they are not orthogonal to the elements of the cylinder surface they pass through, because their velocities have the tangent component. Therefore the central observer 'sees' the wave 3-vector of every photon turned with respect to its group velocity, while the two 3-vectors remain parallel for any local observer.

We can also say that, from the point of view of the central observer, the time (t) axis of the local peripheral observer (or the world line of a resting, with respect to him, particle) is not orthogonal to the 3-hyper-plane t=const (is not parallel to the normal to the 3-hyper-plane of the wave front of the resting particle).

2.4. 4-momentum transformations. Newtonian law of attraction, declination of ultra-relativistic particles, Hubble's effect

4-momentum P'_i º љtransformations are analogous to those of wave 4-vector (7):

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (8)

.

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (8Б)

We can define a modified 4-momentum P"_j, representing square of modulus of local 4-momentum as the sum of squares:

љљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (9)

where P"_i º(HP'₀ - ; ) is the modified 4-momentum equal to the orthogonal transformed locally measured 4-momentum P_i; P₀ ºiE/c. A body possesses the modified momentum (differing from the locally observed one and that in the primed frame both) from the point of view of the double primed frame, introduced in (4).

The components of a 4-force, accelerating a test body and changing its energy in gravitational field are as follows:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (10)

Here P_i is the 4-momentum of a freely moving body in the local instantly resting and freely falling reference frame, independent of time and co-ordinates.

In the particular case of the field with a diagonal matrix H'_ij = H d_ij at each point we can get from (10) the rate of the 4-momentum change:

љљљљљљљљљљљљљљљљљ љ(11)

The first term in (11) means that there is a 4-acceleration of the test body, proportional to the observed velocity of the test body, in the case when the potential varies in time (e.g. if the surrounding the body area of the Universe expands or contracts). No matter what sign H has, in the case of expansion 'island-like' areas of the Universe are likely to be gravitationally unstable, because the sign of the velocity change coincides with that of the velocity, and if the component of the force is greater than attraction (the second term in (11)), then the radial velocities of peripheral bodies in this area would increase (the red shift would increase in time for each of the bodies, i.e. the greater the distance between the centre and a body the greater the velocity - thus we predict Hubble's effect). In [10] we analysed the geophysical data concerning the deceleration of Earth's rotation (from one and a half billion years ago up to nowadays, 6 values) that had been within reach to us, which could be interpreted as the effect of two phenomena: the tidal deceleration and the Earth's radius increase, and it appeared that the range of estimations of the-rate-of-the-radius-increase-to-the-radius ratio approximately equalled the range of estimations of Hubble's constant (50 - 100 km/(sec Mps)). The dispersion of values of the ratio of the radial velocity of a body to the radius can be caused by difference of the rate of the potential H changes in different areas of the Universe. Besides, from (10) one can deduce, that accelerations of cosmic objects depend on the velocities of the nearest more massive objects (the term proportional to H'_n0 dH/dt exists in the formula for F'_n implicitly).

In the case of contraction its rate decreases, because the sign of the velocity change is opposite to that of the velocity, and the velocities of peripheral bodies would decrease (if attraction is comparatively small). The above said means that the first term 'prefers' expansion to contraction. One can suppose that this effect (that modulus of H tends to decrease) is to some extent responsible for creation of local 'time arrows'.

The second term is responsible for Newtonian attraction when W'_mP'_m = 0, -W'₀ P'₀ = -ic mic = mc²:љ F'_n = 1/H mc² H/x'_n = mc² (-g M/(R²c²)) (for a spherically symmetric body), - and produces the effect of additional declination of ultra-relativistic particles at tangent paths near gravitational sources (with a 3-velocity, approaching that of light, the term W'_mP'_m asymptotically approaches -W'₀P'₀; in the expression for the latter case in [2] the following signification was used: H₀₀ º H'₀₀, H₂₂ º H'₂₂/H²).

The example (11) shows that particularities of the action of gravitational field on test bodies can be explained suggesting that the potential influences the space-like and time-like components of physical quantities differently.

4. DISPUTABLE ISSUES

Some issues concern the specifics of geometrical representation of the SRR-theory.

The example (6) shows that it doesn't matter if we regard light as moving with the locally standard and isotropic velocity in Riemannian space or as moving with a variable velocity in Euclidean space. 'Local truth' about constancy of the speed of light in Riemannian space is equal to 'remote truth' about its variability in Euclidean one. The choice of the viewpoint is, as it seems, a matter of agreement: we know that curved space-time is not used for the description of propagation of light in optically non-uniform media, but is postulated for the space with co-ordinate dependent gravitational potential, on the contrary.

We have chosen the analogy with varying refraction coefficient in the space with Cartesian co-ordination of its points.

The observability of curvature of space is often demonstrated using the difference between results of angle measurements on a sphere and on a plane. Let a two-dimensional observer moving upon a sphere and measuring length in steps go from the equator to one of the poles (making A steps), turns by 90њ, reaches the equator passing along the other meridian (making б steps more), turns once again to return to the starting point (having made б steps more). This stepping observer will think that he has walked around a triangle the sum of angles being 270њ (the perimeter being 3б steps), from which he would make a conclusion that he is on a convex surface with the constant, probably, radius of curvature R= 2б/p, directed in the third dimension. Let a resting with respect to the sphere two-dimensional observer watch the walking one using light propagating along geodesic lines - arcs of the largest circles of the sphere. This resting observer would think that the farther a light signal moves the slower it moves, its velocity being zero at the horizon of events of the sphere. As concerns the stepping observer, he is sure that the local speed of light remains unchangeable. Therefore the immovable observer will see that the first observer's steps getting shorter as cos j becoming zero at j=90њ, with the increase of distance along the first meridian (which is a straight line from the resting observer's viewpoint), and increasing as cos j with j decreasing from 90њ to 0њ along the equatorial part (the last part of the travel, also straight from the resting observer's point of view). Adding varying step lengths the resting observer will see that the first and the third parts of the first observer's travel are equal to the sphere's radius R. While the walking observer moves along the second meridian (after he turned at 90њ the first time), which is arc-shaped (where the distance between the observers is constant and equals R), the steps seem to have their standard size, and the length of the second part of the travel equals therefore pR/2. Thus the immovable observer thinks that the stepping one has walked around a flat figure with the varying speed, the figure being a circular sector with the central angle equal to 90њ (the observers do not disagree about the sum of the figure's angles) and the perimeter equal to 2R+ pR/2 (which is characteristic of the flat sector with the 90њ central angle).

The both viewpoints on curvature of this two-dimensional surface do not cause contradictions for observers existing in the same dimensionality. Similarly, since the dependence of the speed of light on co-ordinates and direction (which is caused by gravitational field and manifests itself in the changes of length and time standards because of changes of the conditions of balance of electromagnetic and gravitational (including inertia) forces in macroscopic measuring devices) can be interpreted as a specific deformation of space-time, the brilliant geometrical model of the General Relativity (which was placed by A.A.Logunov into pseudo-Euclidean space-time) can serve as a mathematical image of the field.

Still we can hardly agree with A.A.Logunov [6] that only the invariant speed of light is 'physical', and all the others are 'co-ordinate' velocities. The observed from the Earth dependence of speed of light moving to Mercury and back on gravitational potential can be observed (because the delay of the radio-location signal with respect to the expected time of arrival is observable) by any other observer: it is an objective fact, and, following Logunov, this dependence should be called physical. Of course, a description of this dependence can imply co-ordinate arbitrariness also, but this is not a relevant issue. The same could be said about a light signal travelling along a rotating circle: all observers agree that a signal moving in the direction of rotation returns to the source later than an analogous signal moving in the opposite direction. Thus both anisotropy and variability of the speed of light along finite paths are no less physical interpretations of experimental data (the measured value of time spent on the way there and back), than the co-ordinate dependence of metric tensor components in pseudo-Euclidean space-time.

Square of an interval in the SRR-theory is expressed the following way (it is obtained from (3), the sign is chosen, as it is often done, coinciding with that of (cdt')²):

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (12)

Hence we get the components of the metric tensor: g₀₀ = 1/H'₀₀²; g_n0 = g_0n = -H'_n0/H'₀₀; g_nm = -H² d_nm. (It is worth reminding that we use Cartesian co-ordinate systems with imaginary time co-ordinates x'₀ºict' which permits usage of lower indexes only). Since the components H'_ij describe all possible rotations, rotations with reflection included, and isotropic scale transformations, the components of the metric tensor in (12) are represented in the most general way in the SRR theory.

It remains unclear if the zero values of non-diagonal components g_nm of the metric tensor (a consequence of isotropy of space scales) form an unnecessary limitation of the SRR-theory. Yet, up to now there was no necessity to consider anisotropic scale variations.

A large group of problems concerns field equations, the nature of the quantum of gravitational field and the cosmology with anti-gravitating anti-matter; these issues are not discussed in the present paper (in [2] we had mentioned that a universe with anti-gravitating anti-matter suffers gravitational polarisation).

5. CONCLUSIONS

The gauge interpretation of the special principle of relativity means that local observations of gravitational phenomena do not depend on components of the gravitational potential created by all the external bodies of the Universe if the surrounding area is equi-potential: i.e. uniform movement of the Universe with respect to any observer and its orientation as a whole do not influence the locally observed gravitational phenomena in the areas where these bodies provide independence of the components of the gravitational potential, created by them, on time and co-ordinates. It does not even matter how the change of the potential components can be achieved, either by a uniform change of movements and angular co-ordinates of the sources or using a set of engines of the observer's laboratory. Thus we replace the thesis about the invariance of descriptions of physical phenomena with respect to the 4-co-ordinate transformations by the thesis about the invariance of locally observed phenomena with respect to gauge transformations of the gravitational field.

A non-uniform movement of the external field sources influences the locally observed phenomena because of the field (analogous to the vortex component of electric field) acting on the bodies participating in these phenomena. More generally, considering the co-ordinate dependence of the gravitational potential, we can say that the general principle of relativity means, from this point of view, that similar phenomena go on similarly being subject to the same distribution of gravitational forces under equal initial conditions. The equivalence principle then means that a gravitational field similar to that described by the gradient A₀/x_m (electrostatic field) and a gravitational field similar to that described by the derivative -dA_m/dt cannot be told one from another locally using test charges only (the field action does not depend on the way the field is being created, either by a static source or by a number of sources moving with a relative acceleration), and when a body freely falls these fields compensate each other in its reference frame.

E.Wigner stated [7] that space-time, possessing symmetry with respect to 4-translations and 4-rotations, does not possess symmetry with respect to scale transformations, because microscopic and macroscopic events look substantially different, i.e. using observations one can determine the 'magnification' of the observed picture. Using the gauge approach we are not concerned with the properties of space, we just talk about the invariance of field phenomena with respect to potentials of the same field; here we deal with the invariance of gravitational phenomena with respect to potential H'_ij, and the scaling factor H, in particular, and say nothing about phenomena, caused by other fields. For example, we are not surprised at the possibility of observing background electrostatic potentials (e.g., if the Solar system were surrounded with a charged sphere) detecting the difference between accelerations of free falling of different electric charges [2]. In a Mach-oriented theory we are not surprised at the possibility of observing background gravitational potential detecting the difference between accelerations of charged particles with different masses (e.g., a proton and a positron) moving in one and the same electromagnetic field. (The SRR-theory satisfies Mach's principle due to non-zero values of the diagonal components of the potential at the origin of a reference frame: any accelerated movement of particles near the origin would result in appearing a non-zero value of the derivative of the vector part of the potential by time, and, consequently, in appearing an inertia force acting on the particles). Similarly, we are not surprised at the observability of scale detecting changes of gravitational phenomena when other fields start to manifest (in atomic scale, for example) or, on the contrary, examining fields which act differently at different values of H.

Zero values of the diagonal components of potential at some distant point would mean that remote measurements of the speed of light and lengths would give infinitely great values, and those of time intervals - zero values (the tempos of electromagnetic phenomena and their characteristic sizes, as observed at a distance, seem increasing infinitely). Bodies at remote areas with zero diagonal components would seem to have no inertial mass. The SRR-theory supposes existence of negative values of the diagonal components of potential (caused by anti-matter), which mathematically correspond to improper rotations of a reference frame [8,9]. If it is true, then the exotic areas with total potential equalling zero are not infinitely far, and discovering of peculiarities characteristic of these areas would confirm the hypothesis about the negative sign of the potential crated by antimatter. With tending H to zero some interactions should manifest themselves, those that would forbid infinite distortions of the electromagnetic scale, scattering of photons, perhaps (producing 'particle - anti-particle' pairs, in that number). Among the candidates, i.e. objects that are located in such areas, probably, we could mention pulsars, Roentgen sources ('hasty' objects), maybe, quasars.

After all, adjusting of the potential at the origin of any reference frame that makes gravitation potential become Kronecker's symbol leads to the following:

- the equality of the non-diagonal components of potential to zero means that we define the proper state of an electromagnetic reference frame as a state of rest with 'never turned and not reflected' spatial axes (the observer's face looks always forward, his head is above the rest part of the body, his wristwatch is on the left arm), the time flowing from the past to the future;

- the equality of the diagonal components of potential to 1 means a uniform and standard scale along the axes of the reference frame built using the standard light velocity and 'always correct' proper time.

The original suppositions for this paper were the following: Lorentz transformations (transformations of 4-dimentional rotation) are valid for moving electromagnetic reference frames; there exists a non-zero background gravitational potential, created by bodies of the Universe (Mach's principle is valid), the potential manifesting in scale transformations of 4-co-ordinates in Cartesian electromagnetic co-ordinates under which time intervals and lengths change reciprocal proportional to each other. The latter is a heuristic consequence of the known relativistic expressions for the proper time and proper length of a moving body. The equation for the force (11) can be constructed, for example, substituting the known Lagrangian [11] into Euler-Lagrange equation.

Thus the known variety of observable gravitational phenomena (only orbit precession calculations are absent in this paper) has been described using minimum ideas of the Special Relativity theory, validity of Mach's principle being accepted, without using the power of the General Relativity and without numerous additional suppositions.

REFERENCESљљљљљљљљљљљљљљљ

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