# Metric theory of relativity

Metric theory of relativity (MTR) is the theory that describes the transformation of physical quantities in the laws of motion, including the laws of mechanics, electrodynamics and the covariant theory of gravitation, based on the spatial-temporal relations in arbitrary reference frames. The special cases of MTR are the special theory of relativity (STR) and the extended special theory of relativity (ESTR) in the limit of the weak field and the inertial reference frames, as well as the general theory of relativity (GTR) in the part concerning the transformation of physical quantities from one frame to another. The reference systems in which the rate of flow of time as the rate of similar movements differ due to shift along the scale dimension also are included in the scope of the MTR.

## Creation of MTR

The analysis of the concept of relativity in physics and the need of transformation of physical quantities between different reference frames led first to the Galilean transformations in mechanics, and then to the Lorentz transformations in STR, already suitable for mechanical and for electromagnetic quantities. As it is shown in the Lorentz-invariant theory of gravitation (LITG), in the inertial reference frames gravitational quantities satisfy the Lorentz transformations.

The transition from the relativity in STR to the relativity of arbitrary reference frames was done in the beginning of GTR development. The feature of GTR is that the transformation of quantities from one frame to another is done using the corresponding 4-dimensional tensor transformation function, the components of which are the partial derivatives relating the time and the coordinates of both reference frames. GTR is a metric theory, so that in the transformations of the physical quantities the metric tensor of the curved spacetime is involved.

The next step was development of the extended special theory of relativity (ESTR), which showed that the axiom of STR about the constancy of the speed of light in all reference frames can be derived from a different set of initial axioms.  In this case in ESTR the same formulas are obtained as in STR.

MTR is described in the works of Sergey Fedosin.  The characteristic feature of MTR is that it introduces the concept of various wave representations of phenomena. The events can be registered not only with the help of electromagnetic, but also other waves, such as gravitational waves.  If these waves have different propagation speeds, it should be taken into account in all the formulas of transformations of physical quantities from one frame to another.

According to MTR, the constancy of the speed of light, its independence on the speed of the light source and on the observer's velocity is the consequence of the procedure accepted in SRT of measuring the spatial and temporal parameters, such as the distance between the points in space and time intervals between the events. Synchronization of clocks at different points and measurements of the length in SRT are carried out remotely by the waves and always imply that the wave returns to the point from which it was originally sent as a signal. The wave passes a closed path in space, so that no matter how the absolute velocity of the wave changes relative to the moving source, being averaged on the way back and forth the wave speed equals the speed of light in the medium at rest, in which the source is moving.

In fact, the wave speed in different parts of the path is not equal relative to the source. To determine the absolute velocity of the wave relative to the moving source it is necessary to consider non-closed paths during the propagation of waves. These are such well-known experiments on measuring with the help of Doppler effect the temperature of the cosmic microwave background radiation coming to Earth from different directions, and showing the absolute velocity of the Earth’s motion relative to this radiation, about 600 km/s in direction of constellation Leo. 

It is possible to use a geostationary satellite, which emits waves in one direction towards the telescope on the ground, with a fixed distance between the transmitter and the receiver. The Earth with this satellite revolves around the Sun, the Solar system revolves around the galactic center, and the Galaxy itself is moving relative to an isotropic reference frame in which the speed of light is the same in all directions. In the observation of the satellite we should consider the effect of aberration, which is dependent of the absolute velocity of the Earth relative to the isotropic frame. Experiments show the orbital velocity of the Earth around the Sun and the absolute velocity of the Solar system, about 600 km/s, the speed and direction of which is sufficiently close to the axis of the dipole microwave background radiation. 

Another example is the experiments of Stefan Marinov on measuring the speed of light from lasers with the help of rotating discs with small holes in them.  In these experiments, the measured speed of light varied depending on the diurnal cycle of the Earth's rotation and the corresponding change of position measuring system in space and change the direction of light propagation. For the absolute velocity of the Earth, there were obtained about 360 km/s.

The analysis of the Michelson–Morley experiment based on the ideas of MTR brought Fedosin to the following dependence of the absolute speed of light in the interferometer: 

$~c_{\varphi }={\frac {c{\sqrt {1-V^{2}/c^{2}}}}{1-{\frac {V}{c}}\cos \varphi }},$ where $~\varphi$ is the angle between the direction of the light beam velocity and the direction of the velocity $~V$ of motion relative to the interferometer of the isotropic reference frame in which the speed of light is equal in all directions. The velocity $~V$ is not equal to zero, because the interferometer moves with the Earth relative to the isotropic reference frame, and there is expected the aether wind. For comparison, Marinov under the same conditions of motion for the speed of light found the following: 

$~c_{\varphi }={\frac {c}{1-{\frac {V}{c}}\cos \varphi }}$ .

In acoustic Michelson-Morley experiment,  the ultrasonic range finder was used to measure the distances and there were sound waves in the work instead of electromagnetic waves. The range finder was mounted on a moving car and the incoming air has the ability to change the speed of sound as it propagates back and forth. The experimental results can be interpreted in the same way as in the special theory of relativity, that is, involving the effect of reducing the longitudinal dimensions and time dilation. Both effects are the result of measurements using the reflected wave returning to the starting point.

## The postulates

MTR includes five postulates (assumptions accepted without proving):

1. The properties of the used spacetime manifold in this or that reference frame depend on the properties of the test bodies or waves, with the help of which the spacetime measurements are carried out in this frame reference. The most important property of test bodies or waves is their propagation speed $~c$ , as it is used in the formulas as the measure of velocities of other bodies and of delay of information in remote measurements.
2. The geometrical properties of spacetime are fixed by the corresponding mathematical object, which is the function of spacetime coordinates of the reference frame. For a wide range of reference frames the suitable mathematical object is the non-degenerate four-dimensional symmetric metric tensor of second rank $~g_{ik}$ , the components of which are the scalar products of unit vectors of the coordinate axes of the chosen reference frame. The tensor $~g_{ik}$ allows us to find any invariants associated with 4-vectors and tensors.
3. The square of the interval $~ds^{2}$ between two close events, understood as the contraction of the metric tensor $~g_{ik}$ with the product of differentials of the coordinates $~dx^{i}\ dx^{k},$ is the invariant, the measure of the proper dynamic time $~\tau$ of the moving particle, and does not depend on the choice of the reference frame: $~ds^{2}=c^{2}(d\tau )^{2}=g_{ik}\ dx^{i}\ dx^{k}=g'_{ik}\ dx^{'i}\ dx^{'k}=ds^{'2}.$ The square of the interval $~ds^{2}$ for two close events is zero, if these events are associated with the propagation of test bodies or waves, by means of which the spacetime measurements and recording the metric are carried out.
4. The physical properties of ordinary matter and the fields, including the gravitational field, in the given reference frame are specified by the corresponding stress-energy tensors. There is a mathematical function of the metric tensor $~g_{ik}$ , which is found by certain rules and is proportional to the total stress-energy tensor of the fields in the given reference frame. In the simplest case this function is the tensor, which is the left side of the equation for the metric: $~R_{ik}-{\frac {1}{4}}g_{ik}R={\frac {8\pi G\beta }{c^{4}}}T_{ik},$ where $~R_{ik}={R^{n}}_{ink}$ is the Ricci tensor, which is a trace of the Riemann curvature tensor, $~R=R_{ik}g^{ik}$ is the scalar curvature, $~g^{ik}$ is the metric tensor with contravariant indices, $~\beta$ is the coefficient to be determined in the comparison with the experiment (in GTR this coefficient is equal to 1), $~G$ is the gravitational constant. For the stress-energy tensor $~T_{ik}$ of the general field can be written:  $~T^{ik}=k_{1}W^{ik}+k_{2}U^{ik}+k_{3}B^{ik}+k_{4}P^{ik}+k_{5}Q^{ik}+k_{6}L^{ik}+k_{7}A^{ik}+cross\quad terms,$ where $~k_{1}{,}k_{2}{,}k_{3}{,}k_{4}{,}k_{5}{,}k_{6}{,}k_{7}$ are some coefficients, $~W^{ik}$ is the electromagnetic stress-energy tensor, $~U^{ik}$ is the gravitational stress-energy tensor, $~B^{ik}$ is the acceleration stress-energy tensor, $~P^{ik}$ is the pressure stress-energy tensor, $~Q^{ik}$ is the dissipation stress-energy tensor, $~L^{ik}$ is the stress-energy tensor of the strong interaction field, $~A^{ik}$ is the stress-energy tensor of the weak interaction field. The equation for the metric realizes connection between the geometrical properties of the used spacetime manifold, on the one hand, and the physical properties of the available matter and the acting fields, on the other hand. The covariant derivative acting on the both sides of the equation for the metric, makes them vanish. This fixes the properties of the tensor in the left side of the equation for the metric, and at the same time specifies the equation of motion of the matter under the influence of the fields.
5. There are additional conditions which specify the required for calculations amount of relations for displacements and rotations of the compared reference frames, the velocities of their motion relative to each other, taking into account the symmetry properties of the reference frames.

Comparison of the postulates of MTR with the postulates of GTR shows that the latter are a special case of the postulates of MTR.  

## The consequences

The postulates of MTR are intended to summarize all the possible ways by which in the framework of metric theories of spacetime the relations between different reference frames can be realized and the corresponding physical quantities can be found.

For the inertial reference frames, by definition moving at a constant velocity, the metric tensor is constant and independent on the coordinates and time. Under the influence of forces and fields acting on the reference frame accelerations arise in it and the frame becomes non-inertial. At the same time the curvature of motion of any test bodies and the waves relative to this reference frame takes place, since the action of forces (fields) on the reference frame and the test bodies or the waves as a rule is not equal because of the differences in their velocities and accelerations. In this case the velocity of test bodies and the waves in the accelerated reference frame becomes the function of time, coordinates, and the internal parameters of the reference frame such as mass and the sizes of the frame. The curvature of motion of the test bodies or the wave, by means of which the measurements are made, is perceived by the observer in the accelerated reference frame as the curvature of spacetime. It can be described as the dependence of the metric tensor on the coordinates and time.

On the basis of axiom 3 of MTR we can compare different reference frames with well-known metric tensors with each other. Thus, the comparison of two different inertial reference frames allows us to obtain the formula of velocities addition and the Lorentz transformations for the coordinates and time in STR. This is sufficient to develop the special theory of relativity. From the perspective of the four-dimensional vector-tensor formalism, from the equation of the squared interval, expressed through the time and coordinates of two different reference frames, we can determine the matrix for the transformation of any 4-vector (tensor) from one frame into the corresponding 4-vector (tensor) of another reference frame. And according to axiom 5 the additional conditions should be introduced, which are necessary to derive transformations of the coordinates and time from one frame to another. The typical condition is that the velocity of the reference frames relative to each other, is up to sign the same in each frame.

Just as in GTR, according to axiom 4 the effective spacetime curvature is determined by all the mass-energy sources, including the mass-energy of different types of fields which take place in the frame (because they first of all affect the propagation of test bodies and waves). Accordingly, in each reference frame from the equations for the metric we can find its metric tensor. Since the test bodies and the waves, used for spacetime measurements, can differ significantly from each other, then according to axioms 1 and 2 the metric tensor becomes the function of the properties of test objects or waves. For example, the metric can be presented as depending on the electric charge of test bodies, or on the speed of the wave. In contrast to GTR, in MTR the gravitational field of the body is also the source of mass-energy in determining the metric.

Axiom 3 allows us from the condition the equality of the interval to zero to find the dependence of the velocity of test bodies and the waves, used for spacetime measurements, relative to the accelerated reference frame, depending on the coordinates and the time of the reference frame. Since the axiom 4 implies the possibility of determining the metric tensor through the known stress-energy tensor of the frame, it leads to the principle of local equivalence of the energy-momentum: "In the accelerated reference frame the metric depends locally not on the type of the acting force, causing this acceleration, but on the configuration of this force in the spacetime of the reference frame, determined by the stress-energy tensor".

This principle is the generalization of Einstein equivalence principle according to which based on the equality of the gravitational and the inertial masses the acceleration in the gravitational field can be equated to the acceleration from the inertial force with respect to their influence on the physical processes. In the principle of the local equivalence of the energy-momentum it is emphasized that the curvature of spacetime is not simply the embodiment of the gravitational field, as it is considered in GTR. The effective curvature of spacetime is rather the consequence of any fields and forces available in the reference frame, and the metric itself exists even in the absence of gravitation. Besides it is shown that to describe the equivalent phenomena we should use not the equality of acting forces or accelerations, as in GTR in the equivalence principle, but the equality of the form of stress-energy tensors, present in the compared frames and leading to the same form of the metric tensor.

In MTR the gravitational field is only one of the types of fields existing in nature and causing the change of the metric in comparison with its form in the inertial reference frames. In MTR it is also possible to make the spacetime measurements in this or that reference frame with the help of the proper test bodies or waves. This implies that for mathematical determination of the metric in this representation it is necessary to express the effective stress-energy tensor in terms of the characteristics of the given test bodies or waves. For example, for the electromagnetic wave the main characteristics are its speed in the vacuum, the equality of the rest mass, charge and magnetic moment of the wave to zero. The momentum and the angular momentum of the wave are neglected in the measurements carried out in the frame.

The most important conclusion of MTR is that the dependence of the metric on the type of the used test bodies and waves means the absence of the unified spacetime for each reference frame. Thus the reduction of the physical system only to one chosen geometry of spacetime is incomplete for the description of the system – the geometry can not completely replace the physics of the phenomena. Therefore the geometry of GTR, based on the measurements by means of electromagnetic waves, can not be considered sufficient for complete description of physical systems. For example, once the particles in the system obtain charges or spins, then the previously found in GTR metric for uncharged particles becomes unsuitable for describing the free motion of charged and rotating particles.

## The effects of MTR

Since STR and GTR are part of MTR, then for MTR there are classical relativistic effects, which include:

1. time dilation in moving reference frames (the absolute effect).
2. gravitational time dilation in the reference frames in the gravitational field (the absolute effect).
3. reduction of linear sizes in moving reference frames (the apparent effect).
4. reduction of sizes along the gradient of the gravitational field (the apparent effect for the external observer).
5. change of the speed of light in the non-uniform gravitational field.
6. gravitational redshift of the wavelength.
7. precession of the perihelions of planets.
8. additional deflection of waves and test particles near massive bodies due to the impact of the gravitational field on the spacetime metric and its difference from the Euclidean form.

Improvement of measurement techniques and the use of satellites allowed us to measure the number of orbital and spin effects predicted earlier, such as spin and orbital precession in the Lense-Thirring effect, geodetic precession, etc.

The calculation of the metric inside the uniformly accelerated reference frame leads to new effects. In such a reference frame the rate of clock becomes dependent on the location of the clock and on the duration of the acceleration, which changes the velocity.  In the accelerated reference frame it is possible that the time runs faster than in the inertial reference frame. Besides, there is visual extension of bodies along the direction of acceleration, shortening of the transverse dimensions and deformation of the body shape. With this in mind, it becomes possible to describe the so-called twin paradox more accurately.