© Michail A. Samokhvalov

Contact to the author: samochvalovma@ignd.tpu.ru

Tomsk Polytechnic University, Tomsk, Russia

A relativistic dynamics equation for the trajectory of motion of a proofmass in the central symmetrical gravitational field has been written and solved by four approximations with regard for the dependence of mass of a material body and the gravitational field [2]. Some differences from the solution of the classic equation of dynamics have been shown already at the second approximation, namely an additional small quantity appears in the energy integral. At the third and the fourth approximations the advance of the perihelion of the Mercury’s elliptic orbit has been found to be equal to per a century that is less than a value obtained by the GTR but higher than that one found by the STR. The analysis of the obtained equation of motion trajectory shows the instability of orbital motion of the proofmass in the central symmetrical gravitational field with gradual increase in dimensions and precession of the orbit.

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, (2)

.

This yields the equation:

. (4)

Application of Eq. (3) to determining the parameters of motion of the proofmass in the central symmetrical gravitational field by solution of the relativistic dynamics equation is a principal difference of this proposal from a similar solution to the task in the past (see, for example, [3]). Further Eq. (4) is analyzed according to the scheme used to determine parameters of motion of the proofmass in the central symmetrical field by using a classic equation of dynamics.

We expand Eq. (4) in x and y, assuming that the motion plane coincides with the coordinate plane XOY:

, (5)

, (6)

Taking into consideration that: ; ; ; , we will expand Eqs. (5), (6):

; (7)

. (8)

When we multiply Eqs. (7) and (8) by and , then sum them, and taking into consideration that , , , , , we will get:

. (9)

The equation (9) is equivalent to:

. (10)

Integration of Eq. (10) gives:

, (11)

where is the constant of integration.

It is seen that the module of velocity of the proofmass’s motion, like a classic task of gravitational interaction of two bodies [4], depends on neither the position of the coordinate axes nor the velocity attitude. The equation (11) can be rewritten as:

. (12)

Forming a convolution of Eq. (12) and doing necessary transformations we obtain an equation for , a so-called energy integral:

. (13)

This equation is valid for any direction of motion of the proofmass including its radial motion. The meaning of the constant of integration E becomes clear when we write Eq. (13) for a case when the proofmass is at infinity. In this case is equal to 1 and evaluating E, we will get:

. (14)

Eq. (14) implies that when the initial speed of the proofmass at infinity is equal to 0, E is equal to 1. And Eq. (13) is reduced to the equation described in the reference publication [2] when considering motion of the proofmass from infinity to a spherically symmetrical solid body. If the critical initial speed of the proofmass at infinity is equal to the velocity of light, . This implies that the constant of integration E is a function of the initial speed of the proofmass and varies depending on at what distance the proofmass was from the source of a spherically symmetrical gravitational field when it started moving.

Eq. (13) makes alterations of the gravity factor g defined earlier in [5] in the range of small values of r be easy-to-understand. When r → 0, goes to the velocity of light, and the rate of growth of absolute values of g gets lower reaching its maximum when , and then reaching 0 when r = 0.

Let us expand Eq. (2) into components:

; ; . (15)

Since we consider motion of the proofmass in the plane XOY, so z = 0, , N_x=0, N_y =0 and only one equation is left in (15):

. (16)

Let us determine the trajectory of motion of the proofmass and to do so let us come to polar coordinates of r and , where is an azimuth formed by the radius-vector r and the axis X . Taking into consideration that , , N_z = N , and m changes in accordance with (3), one can rewrite (16) for the polar coordinates system:

Let us also use polar coordinates for Eq. (13) that is necessary to determine the trajectory of motion of the proofmass considering that, and :

. (18)

It follows from Eq. (17) that:

; . (19)

If we substitute Eq. (19) in Eq. (18) and eliminate time we will obtain:

. (20)

Division of the left-hand and right-hand sides of Eq. (20) by yields:

. (21)

Let us rewrite Eq. (21) with a variable:

. (22)

Let us expand in Eq. (22) into the Maclaurin expansion and restrict ourselves to one, two and three members of the expansion. Thus Eq. (22) can be reduced to the following:

; (23)

; (24)

; (25)

where – is the constant of integration Eq. (23) is the first approximation of Eq. (22) for the case when the proofmass moves in negligible fields (at infinity or at negligible masses generating a superweak central symmetrical gravitational field). Comparing Eq. (23) with Eq. (27), let us write a solution of Eq. (28) for the case taking into consideration that when there are no gravitational fields, :

, (29)

where – is an initial speed of motion of the proofmass; – initial distance from the proofmass to the center of the solid body.

Eq. (29) is an equation of a straight line that represents the shortest distance from the gravitational field source at the angle.

Eq. (24) is the second approximation of Eq. (22) and characterizes motion of the proofmass in moderate central symmetrical gravitational fields. According to Eqs. (27), (28) this equation’s solution is as follows:

. (30)

When we divide the numerator and denominator of the right-hand side of Eq. (30) by , we will get:

. (31)

where p is a parameter; e is an eccentricity; u is an azimuth called a true anomaly in Astronomy.

Comparing Eqs. (32) and (31) gives:

; (33) ; (34) ; (35)

It is known for Eq. (32) that if the eccentricity is equal to 1, the conic obtained with it will be a parabola [4]. As seen from Eq. (34) it is possible when the following condition is met:

. (36)

In general N is not equal to 0, therefore the condition described in Eq. (36) will be met when E = 1. If the proofmass is at infinity, then, according to Eq. (14), E = 1 when. It follows that any proofmass starting its motion infinitely separated from the source of a central symmetrical gravitational field will generally travel in a parabola that fully coincides with a classic solution [4]. When a proofmass is in strong fields and at a finite distance from the field source, the initial speed of the proofmass must get increased to make its traveling in a parabola possible. In cases when is small, a central symmetrical gravitational field is moderate and the proofmass is near the field source, the constant E will be less than 1 and according to (34) the eccentricity e will also be less than 1, that result in elliptic motion of the proofmass.

In cases when is very large, E becomes greater than 1, and as it is seen from Eq. (34) the eccentricity is also greater than 1, and the proofmass will travel in a hyperbola. For example, a photon will travel in a hyperbola specifically in strong gravitational fields and at minimal distances from the field source. For such a hyperbolic motion one can write a formula to calculate a value of a semimajor axis à [4]:

Completing an analysis of the second approximation described by Eq. (24), let us rewrite Eq. (13) for a velocity squared of elliptic motion expanding into the Maclaurin expansion with regard to Eq. (37):

. (38)

The equation of velocity squared of elliptic motion for the second approximation (38) differs from a classic equation [4] by an additional summand. Taken as a whole the analysis of the second approximation has shown a qualitative coincidence of a pattern of motion of the proofmass in the central symmetrical gravitational field with a pattern of the proofmass’s motion defined by use of a classic equation of thermodynamics. Like in the classic case an absolute value of velocity at a given distance from the field’s source defines a value of the semimajor axis of the elliptic orbit. However, when considering the second approximation, parameters of motion of the proofmass are quantitatively different to a small extent from parameters defining the proofmass’s motion in a classic case.

It can also be shown in this approximation that the light travels in a hyperbola near the field source. Indeed, when we substitute parameters characterizing a photon’s motion perpendicular to the radius-vector to Eq. (31),, where is a mass of the photon at an infinite distance from the source of a gravitational field, is a minimum distance between a light beam and the source of a gravitational field, , we will get:

. (39)

This implies that the eccentricity cannot be less than or equal to 1, as far as in this case the equation is positive at any value of. Taking into consideration properties of the hyperbola [6], one can write the following equation for calculation of the angle of deviation of the light:

or (40)

Let us determine the angle of deviation of the light from a straight line when it travels near the Sun’s surface using Eq. (40) and taking into consideration that ,, , , where is a radius of the Sun; – is mass of the Sun.

.

This value of the angle of deviating the light beam by the Sun conforms with a classic value obtained by Newton and is two times less than that one obtained by the GTR.

The analysis of the second approximation of Eq. (22) makes it possible to ascertain that the results obtained in this approximation generally conform to results of a classic theory. And only there appears an additional square term in a so-called energy integral [4] in Eq. (38). Precession of the elliptic orbit in gravitational fields corresponding to this approximation is not observed since a relative parameter does not appear in an equation of the trajectory of motion. Now let us analyze the third approximation described by the equation (25). To make a solution of Eq. (25) easier let us simplify the equation and rewrite it as:

, (41)

where , and , but .

Differentiation of Eq.(41) gives:

. (42)

Having reduced a left-hand and a right-hand side of (42) and inserted into it we bring the equation to the form:

. (43)

This equation can be solved as follows [6]:

,

where R, S and are constants of integration. Going to y, we will get:

. (44)

Substituting the solution (44) in Eq. (41) and performing such operations as differentiation, simplification and reduction one can correlate the constants of integration R and S:

. (45)

It is possible to find these constants of integration R and S by going from the solution (44) of the equations of the third approximation (25), (41) to the solution (28) of the equations of the second approximation (24), (27) and by taking into consideration that A=1 for the second approximation. To do this let us rewrite Eq. (44) as:

. (46)

The solution (46) will be reduced to the solution (28) when S = 0. In fact it follows from Eq. (45) that. Substituting the found constants R and S in Eq. (46) we obtain Eq. (28). Now let us write a solution of Eqs. (41), (25) of the trajectory of the proofmass obtained in the third approximation as:

. (47)

And rewrite Eq. (47) for r as:

. (48)

When we substitute the coefficients in Eq. (48) we will obtain:

. (49)

The factor next to shows that precession of the perihelion is observed when the proofmass moves elliptically. To determine a value of the perihelion advance we will expand the equation for the case when the proofmass moves perpendicular to the radius-vector. In this case , and this formula can be written as:

. (50)

Using a well-known formula for the calculation of a value of the perihelion advance per a complete revolution around the gravitational field’s source [3] and taking into consideration Eq. (50) one can write:

. (51)

Now let us determine the perihelion advance per a revolution of the Mercury around the Sun using a formula (53) and taking into consideration that, , , , , where is a midradius of the Mercury’s orbit; – is a mean orbital velocity of the Mercury; is mass of the Sun. With the parameters mentioned above the perihelion advance or per a century that is significantly less than it has been obtained by the GTR, particularly itper a century and is two times greater that it has been determined by the STR [3] – particularly per a century.

Now we consider a photon’s motion perpendicular to the radius-vector in the third approximation when the distance is minimal to the center of mass.

Let us rewrite Eq. (51) taking into account that for this case and :

. (52)

In order to determine a type of trajectory of the photon’s motion along the Sun’s surface one is to calculate the eccentricity of e, taking into consideration that , , ,, where is a radius of the Sun; – is mass of the Sun:

. (53)

This implies that the photon moves in a hyperbola, the same as in the second approximation. Using a left-hand side of Eq. (40) and taking into consideration Eq. (53) one can find the angle of deviation of the photon’s trajectory near the Sun’s surface from its straight-line motion. The angle appears to be equal to, that, like in the second approximation, is two times less than that obtained by the GTR.

Now let us consider the fourth approximation described by Eq. (26). First we simplify it by introducing the following notations, , but leave the remaining coefficients like in the equation of the third approximation (41):

. (54)

This equation is similar to the equation obtained by Einstein for the trajectory of planetary motion [8] and is different from it only by polynomial coefficients in its right-hand side. The equation is solved as follows [6]:

. (55)

The integral (55) is performed by using a method by Einstein and it can be rewritten as:

. (56)

Further let us take roots and of the polynomial in the denominator (56). With a sufficient accuracy these roots are equal to roots of the polynomial. We can write these roots [7] using A, B and C:

, (57)

. (58)

Eq. (56) is put in the following form according to [8]:

. (59)

The integral of the right-hand side of Eq. (59) is tabulated [7], therefore, a solution of the equation of the fourth approximation (54) is as follows:

Let us show that the solution (60) is reduced to the solution (48) of the third approximation equation (41) when. Substituting in Eq. (60) and writing in a left-hand and right-hand side we will obtain:

. (61)

Let us take the square of both sides of the equation and, taking into consideration the already-known relations and , perform necessary transformations in Eq. (61):

. (62)

Substituting and from Eqs. (57), (58) and in Eq. (62), we obtain a mentioned-above solution (48) of the third approximation equation (41):

In order to calculate the perihelion advance of the elliptical orbit and the angle of deviation of the photon’s trajectory from a straight line in the central symmetrical gravitational field in the fourth approximation, one has to transform the solution (60). To do so we divide left-hand and right-hand sides of Eq. (60) by that yields:

, (63)

where , (64)

. (65)

Using Eqs. (63), (64), (65) one can write a formula for calculation of advance of the perihelion per a complete revolution of a planet in orbit:

, (66)

where is a value of the angle in the perihelion; – is a value of the anglein the aphelion; is a value of the angle in the perihelion; – is a value of the anglein the aphelion which can be found using the following equations:

, (67)

, (68)

, (69)

. (70)

In these equations and are distances to the source of the central symmetrical gravitational field in the perihelion and the aphelion. Taking into consideration that for elliptic motion , and [8], , Eq. (66) is transformed to the form:

, (71)

where . (72)

Using Eqs. (71) and (72) and expressions for coefficients A and D, we can find that a value of the perihelion advance of the Mercury’s orbit in the fourth approximation, like in the third approximation, is roughly equal to per a complete revolution or to per a century.

Let us write equations for calculation the angle of deviation of the trajectory of photon’s motion from a straight line. To do this we will use Eqs. (63), (64) and (65) taking into consideration that in the gravitational fields, :

, (73)

where , are the anglesand at an infinite separation of the photon from the Sun and can be calculated using the following equations:

, (74)

, (75)

Substituting the values obtained in before-mentioned expressions and parameter values in Eq. (74), (75) we can find that the angle of deviation of the trajectory of photon’s motion from a straight line when it travels near the Sun’s surface is roughly equal to that corresponds to the results obtained by calculations in the third and fourth approximations and is two times less than that obtained by the GTR.

In order to evaluate a solution of the standard equation (22) describing motion of the proofmass in central symmetrical gravitational fields let us substitute to it:

. (76)

We are to exclude the exponent from Eq. (76) and to do so we differentiate it with respect to and subtract Eq. (76) from the result obtained:

. (77)

Eq. (77) is a nonlinear equation of the second order. It is necessary first to take roots of the following characteristic equation [6] in order to solve it:

, (78)

where . Substituting the expression for in Eq. (78) yields:

.

This implies that and consequently [6], a special point (a rest point) of phase trajectories is an unstable focal point. Or clarifying it, the proofmass traveling around the gravitational field’s source will gradually move away it with simultaneous precession of the elliptic orbit. The analysis performed but not finished yet has shown that the equation obtained by taking into consideration the mass variation in the gravitational field, i.e. Eq. (22) of the trajectory of motion of the proofmass in the central symmetrical gravitational field, has been solved only in approximations. Areas of gravitational fields where these approximations are true can be evaluated only qualitatively. The first approximation that has a trivia solution as a straight-line trajectory of the proofmass is true for negligible gravitational fields. The second and third approximations which give trajectories of motion similar to classic trajectories – an ellipse, a parabola and hyperbola – are true for gravitational fields commeasurable to those of the solar system. However already in the second approximation a quadratic term appears in the energy integral as a small component to its classic expression. The fourth approximation whose equation is similar to the equation by Einstein [8] by its form has a solution that is true even for gravitational fields which are greater than those near the Sun’s surface.

A substantive problem of the Eq. (22)’s approximations considered in this article is a difference between the calculated value (equal to ) of the angle of deviation of the photon’s trajectory near the Sun’s surface from a straight line by the gravitational field and the value (equal to ) accepted as an observed one. However if one takes into consideration the variations of velocity of electromagnetic waves propagation determined by Einstein in his early works on the application of the Special Theory of Relativity to analysis of propagation of light in gravitational fields [9, 10], this problem is eliminated. Indeed, in his publication [10], Einstein determined the angle of deviation of a light beam traveling near the Sun’s surface to be equal to. He did it by using a Huygen’s principle and a relationship revealed between the light velocity and a gravitational potential , whereis velocity of light in the absence of gravitational fields in an unaccelerated reference system , is a gravitational potential. This deviation coincides in direction with the deviation revealed in this article. The angle of deviation amounts to, that is less than the angle accepted as an observed one. Here it should be noted that Einstein found a relationship between the light velocity and a gravitational potential in the first approximation. But if we use an exact relationship that was used by Einstein in his work [9] , i.e. the relationship between time in an accelerated reference system which is equivalent to the system with a gravitational field and time in a fixed reference system with an observes, one can write an exact relationship between the light velocity and a gravitational potential . Using this relationship let us write a corrected and more accurate equation for calculation of the angle of deviation of a light beam traveling perpendicular to the direction of a potential gradient. In this equation, are a radius and mass of the Sun and the rest parameters are like in a reference [10]:

.

Performing tabulated integration and substituting parameter values we will get , that gives a sought value equal to , which conforms to the value accepted as an observed one.

Thus, deviation of the trajectory of motion of the photon in a gravitational field is made up of two effects, particularly, attraction of the photon by the gravitational field’s source and the refraction of the trajectory in an inhomogeneous filed acting like an optical lens. It should be emphasized that according to the aforesaid equations, deceleration of propagation of electromagnetic waves by the gravitational field near the Sun’s surface are verified by experiments [11]. The value of advance of the perihelion of the Mercury’s orbit obtained when solving the third and fourth approximations of Eq. (22) is also less than that obtained by the GTR, however it is already significantly greater than that defined by the Special Theory of Relativity by Einstein. A preliminary analysis of the revealed equation of motion of a general proofmass in central symmetrical gravitational fields, of the equation in which a relationship between mass and a gravitational field has been used, has shown instability of celestial bodies and their clusters. Particularly the analysis has shown that orbits can possibly become larger in sufficiently strong fields.

References

1. Matveev A. N. Mechanics and the Theory of Relativity. Moscow: Vyschaya shkola, 1976.– 416 ps.
2. Samokhvalov M.A. The result of application of the relativistic law of dynamics to gravitational interaction of two bodies // E – print: http://www.physics-online.ru/php/paper.phtml?jrnid=null&paperid=267&option_lang=eng – 2009.
3.  Zommerfeld A. Structure of atom and spectra. Moscow: “Gos.izd.techn. teor.lit-ry”, v. 1, 1956.– 592 ps. Also see: http://eqworld.ipmnet.ru/ru/library/physics/quantum.htm
4. Subbotin M.F. Introduction to theoretical astronomy. Moscow: “Nauka”, 1968.– 800 ps. Also see: http://eqworld.ipmnet.ru/ru/library/mechanics/celestial.htm
5. Samokhvalov M.A. Basic relations for a central symmetrical field with regard to mass interdependence. Not published. 2009. See: http://www.physics-online.ru/php/paper.phtml?jrnid=null&paperid=277&option_lang=rus
6. Korn G., Korn T. Mathematics reference book for scientists and engineers . Moscow: Nauka, 1979,.– 832 ps. Also see: http://eqworld.ipmnet.ru/ru/library/mathematics/calculus.htm
7. Dwight G.B. Tables of integrals and other mathematical formalism. Moscow: Nauka, 1977.– 224 ps. Also see: http://eqworld.ipmnet.ru/ru/library/mathematics/handbooks.htm
8. Einstein A. Explanation of motion of the Mercury’s perihelion // in a collected volume “Scientific works, v.1”. Moscow: Nauka, 1965, pp. 439–447 Also see: http://eqworld.ipmnet.ru/ru/library/physics/relativity.htm
9. Einstein A. About a principle of relativity and its consequences // in a collected volume “Scientific works, v. 1”. Moscow: Nauka, 1965, pp. 65–114. Also see: http://eqworld.ipmnet.ru/ru/library/physics/relativity.htm
10. Einstein A. About the effect of gravity to the light propagation // in a collected volume “Scientific works, v.1”. Moscow: Nauka, 1965. pp. 165–173. Also see: http://eqworld.ipmnet.ru/ru/library/physics/relativity.htm
11. Rudenko V.N. Relativistic experiments in the gravitational field // Success of Physics, V. 126, Number 3, 1978, pp. 361 – 401. Also see: http://ufn.ru/ru/articles/1978/11/a/

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