If we now substitute into the above formulas the values of the post-Newtonian parameters of the field theory of gravitation (16.37) it is easy to see that the motion of the center of mass of an extended body with a spherically symmetric distribution of matter does not proceed along a geodesic of Riemannian space--time. If the extended body does not possess spherically symmetry, then in the expression for the difference of accelerations of its center of mass and the test body there appear additional terms caused by the presence of multipole moments of the mass of the extended body, and the motion of such an arbitrary extended body will not proceed along a geodesic of Riemannian space--time for any metric theory of gravitation possessing conservation laws of the energy--momentum of matter and the gravitational field taken together. The magnitude of the concrete value of the difference of the accelerations of the test body and the center of mass of the extended body will depend both on the post-Newtonian parameters of the theory and on the magnitude of the multipole moments of theorems of the extended body as well as on the character of its motion. Thus, in any metric theory of gravitation possessing conservation laws of the energy-momentum of matter and the gravitational field taken together the center of mass of an arbitrary extended body in its motion along the orbit performs small oscillations relative to a support geodesic of Riemannian space--time. The frequency of these oscillations ~ / ~ F ~ 3 is a small quantity of the order of g, while the amplitude Ib~I/a 2 is a small quantity of the order s2L. 31.

Motion of the Sun--Earth System

To compare our results with the results of Will's investigation [9] we apply the general formulas obtained to the concrete post-Newtonian sun-earth system~ This system is characterized by the following quantities. I. The averaged value of the gravitational potential the earth's orbit [5]

of the sun in a neighborhood

of the

U @ ~ I O -8. 2.

The average value of the proper gravitational potential of the earth

[5]

O| 3.

The average velocity of the earth in its orbit about the sun [5]

V| 4.

-4 C.

The ratio of the mass of the sun to the mass of the earth M@

N

~3.10 5.

The r a t i o of the p r o p e r g r a v i t a t i o n a l

5. e n e r g y of the e a r t h to i t s

~| 6.

The radius of the earth

-I0.

cm.

The frequency of rotation of the earth about its axis

co| 2 7 . 1 0 ~ 8.

t o t a l energy [8]

[5] R|

7.

[5]

tad

see

The average distance between the sun and the earth

R ~ 1.5.10 :~cm9.

The density of matter

in a neighborhood of the center of m a s s

(o. 0 - 1 3 10. The eccentricity of the earth's orbit e |

1834

cm 3

of the earth

[4]

It is also necessary to note that for all bodies of the solar system [5] the magnitude of shear stresses is considerably less than the magnitude of the isotropic pressure, and hence all bodies of the solar system may be assumed to consist of an ideal fluid. It follows from these estimates that the characteristic parameter of smallness c for describing the motion of the sun-earth system must be taken equal to s ~ 10 -5 . Then the quantities O@, UQ,~| will have magnitudes of order 2 . The velocity of the earth in a reference system connected ~, while the velocity

with the center of mass of the sun--earth system of the sun is a quantity of order 2 :

is a quantity

of order

M|

v(D ~ ----M--~Gv@ ~ 3.10-~o c. The mean value of the gravitational

potential

M| is a quantity

of order

of the earth

in a neighborhood

of the sun

R|

:O@ --U-" 10-~0|

~3.

For generality we assume that the velocity of the solar system relative to a hypothetical universal rest system w ~ is a quantity of order g. Moreover, following Will, we consider the sun a massive point body. Under these conditions the general formulas obtained above simplify considerably. In this case from expression (30.1) for the averaged acceleration of the center of mass of the earth we have

a~----

n~ 1-}-~'v~+(a,--(~:-}-~,+5~w+3-}-V--4J3)~|

M@ (sl .~) As we have seen, the separation of the characteristics of bodies on the right side of (28.4) and (30.1) was not possible in the general case, and hence it was not possible for us to introduce the tensor of passive gravitational mass. In the case of the sun-earth system such a separation is possible. Indeed, in the present case the magnitude of the reduced multipole moments of the solar mass can be neglected as a result of which the definition (27.5) of the tensor of passive mass of a body takes the form

my"~Me M|

-

-

R~

Thus, the expression for the acceleration of the center in the quasi-Newtonian form presented above.

of mass

of the earth must be written

For this we must transform the right side of relation (31.1) so that it not contain terms decreasing faster than I/R 2 with increase in the distance between the bodies. Using the orbital equation r = P [ l + e ~ c o s ~ ] -1 , we find the relation between the radial and transverse components of the earth's velocity: Vr

Since

the quantity

Then the equations take the form

e~NO(e)

of motion

e@sin

1 dr

, for the square

of the center

of the velocity

of mass of the earth

V|2 T

of the earth we have

in Newtonian

approximation

M| [1 -{- 0 (e|

----R' "

1835

From this it follows that

M| = v~ q- 0 (s~e| Considering this relation, we can introduce the tensor of passive gravitational mass of the earth in correspondence with the expression (27.5):

M@ 3 ~Z2 (nfy.Ojg)2

|

+

+

If we now set w ~ = 0, ~w = 0 in expression

(31.2)

-

+ 89

+o

(31.2), we obtain

M@ - 2

+ 1)

-

+

+ o

(31.3)

We now compare this expression with the expression (27.2) obtained by Will. the principle difference between these expressions is the absence of the terms ~ in expression

(27.2).

Now consideration of the terms

~2

and ~ S

We see that and ~ S ~

is necessary,

since

they have post-Newtonian order of smallness ~s 2 and exceed the remaining post-Newtonian corrections in expressions (27.2) and (31.3) by more than one order of magnitude. Since data on laser ranging of the moon with consideration of other experiments has made it possible to determine with sufficient accuracy the values of all the post-Newtonian parameters contained in the expression (31.2), we can estimate the deviation from one of the ratios ofthe passive mass to the inertial mass. Indeed, results of experiments [8, 12] on the laser ranging of the moon with consideration of other experiments have shown that

Substituting estimate

(31.4) into expression

m~ _-- _ M@

(31.3), we obtain

y~O I1 -- 3v~ -}-0 (10-'o)]- 4 v ~ v ~ q - 0 ~e (10-~o).

(31.5)

Thus, in any metric theory of gravitation satisfying conditions (31.4) and possessing conservation laws of the energy--momentum of matter and gravitational field taken together in the post-Newtonian approximation the ratio of passive gravitational mass to inertial mass by expression (31.5) is not equal to one, differing from it by a quantity approximately equal to 10 -8 in contrast to assertions regarding the equality of the passive gravitational and inertial masses of the earth in the post-Newtonian approximation made in [8, 12] on the basis of data on the laser ranging of the moon. This interpretation of the results of experiments on the laser ranging of the moon presented by the authors of [8, 12] is incorrect, since their assertions are based essentially on the formula (27.2) for the passive gravitational mass obtained by Will. As we have shown in this chapter, such a definition of passive gravitational mass is entirely incorrect and has no physical meaning. Nevertheless, it is not the terms violating equality of passive gravitational and inertial masses of the earth which are the reason for its nongeodesicmotion. As follows from the expression (30.2), the averaged difference of the accelerations of a point body and the center of mass of the earth does not contain terms of the form 7 ~ Q ~ / R 2 , 7 P I Q ~ / R 2 This makes it possible to assert that the difference of the passive mass of the earth from its inertial mass is not the reason for the nongeodesic motion of its center of mass. In order to estimate the quantities characterizing the oscillatory motion of the center of mass of the earth relative to a support geodesic, we use formulas (30.7)-(30.9), setting 1836

there all parameters

except Y = @ = I equal

to zero.

It should be noted that the estimates obtained here will have approximate character, since to obtain their exact value a more detailed analysis with consideration of the internal structure of the earth is required. If we assume the earth to be a homogeneous, spherically symmetric, nonrotating ball, then it follows from the formulas obtained at the end of the preceding section that its center of mass will perform oscillations relative to a support geodesic with period

T=

2s~ _ = . ~ 3 . 3 - t 0 a see, (55 ,rain)

and amplitude

_,. M@ 2 [ A t=-~-~- R | The r o t a t i o n

of the earth

leads

to considerable 1

--~

2

+---*

10 -4 em. increase , M@R

I A l = ~ R e [o~eve] - r - - U Since

m|

-5

rad/sec,

for the amplitude

in the amplitude

of the oscillations:

(31 6)

2

Re"

of oscillations

in this case we have

[ At~4.6.10 -2 cm ~e2R| S i n c e the axis of rotation of the earth makes an angle ~ ~ 66033 ' with the plane of the ecliptic, this amplitude will be subject to seasonal variations, varying from a value ~4.6. --2 * I0 cm (wlnter and summer) to a value 4.2"10 -2 cm (at the time of the vernal and autumnal equinoxes). The ratio of the difference of the accelerations of the center of mass of the earth and a test body to the magnitude of the acceleration in this case is

8a ~ 1 M| Rz _-~ ~ - - ~ - 2 M@ R e l[m|174 1i~10-7"

(31.7)

If we note that the earth is not a ball but a spheroid, then in the expression for the amplitude of oscillations there appear additional terms caused by the presence of multipole moments of the mass of the earth which, however, yield corrections only to the amplitude

(31.6). We point out, for example, that for the ratio of the difference of the accelerations of the center of mass of the earth and a test body to the magnitude of the earth's acceleration among other corrections to expression (31.7) we obtain the correction

la=[NJ3V@-'~'-~ where J3 is the coefficient appearing as a factor in front of Pa (cos 0) in the expansion of Newtonian potential of the earth in spherical harmonics. According to Allen's data, we have fa,~,--2.5.10 -s. From this it follows

that the magnitude

of this correction 16ac~ I

is

lO_lo ~ e~.

Thus, in any metric theory of gravitation possessing conservation laws of the energy-momentum of matter and gravitational field taken together the motion of the center of mass of an extended body does not proceed along a geodesic of Riemannian space--time. Application of the general formulas obtained to the sun--earth system and use of results of experiments on the laser ranging of the moon with consideration of other experiments have shown to high accuracy O(I0 -l~ that the ratio of the passive mass of the earth to its inertial mass is not equal to one, differing from it by a quantity approximately equal to

1837

10-8 . The earth in its motion along its orbit performs oscillatory motion relative to a support geodesic with period of the order of I h and amplitude not less than 10 -3 cm. Although this amplitude is a small quantity, it has post-Newtonian order of smallness A ~ g 2 R @ , and hence the deviation of the motion of the center of mass from geodesic motion can be detected in an appropriate experiment having post-Newtonian degree of accuracy. In the present chapter we showed for simplicity only one of the experimental consequences of the deviation of the motion of the center of mass of the earth from the motion along a geodesic of Riemannian space--time. However, the nongeodesic nature of the motion of the center of mass of the earth will, of course, lead to observable consequences in other post-Newtonian experiments, in particular, in experiments carried out with test bodies on the surface of the earth. Therefore, theoretical analysis of various gravimetric experiments and their subsequent expedient formulation will make it possible to further refine the numerical value of the post-Newtonian parameters. LITERATURE CITED I. 2. 3.

4. 5. 6. 7. 8. 9. 10.

11. 12.

1838

K. W. Allen, Astrophysical Quantities [Russian translation], Mir, Moscow (1977). V. I. Denisov, A. A. Logunov, and M. A. Mestvirishvili, "Does an extended body move along geodesics of Riemannian space--time?," Teor. Mat. Fiz., 47, No. I, 3-37 (1981). V. I. Denisov, A. A. Logunov, and N. A. Mestvirishvili, "The motion of extended bodies in an arbitrary metric theory of gravitation," in: Reports of the Fifth Gravitational Conference [in Russian], Moscow State Univ. (1981), p. 82. V. N. Zharkov, The Internal Structure of the Earth and Planets [in Russian], Nauka, Moscow (]978). C. W. Misner, K. Thorn, and J. Wheeler, Gravitation, W. H. Freeman (1973). V. A. Fock, The Theory of Space, Time, and Gravitation, Pergamon (1964). K. Nordtvedt, Jr., "Equivalence principle for massive bodies," Phys. Rev., 169, No. 5, 1014-1025 (1968). I. Shapiro, C. C. Counselman, and R. W. King, "Verification of the principle of equivalence for massive bodies," Phys. Rev. Lett., 36, No. 11, 555-558 (1976). C. M. Will, "Parametrized post-Newtonian hydrodynamics and the Nordtvedt effect," Astrophys. J., 163, No. 3, 611-628 (1971). C. M. Will, "The theoretical tools of experimental gravitation," Proc. of Course 56 of the International School of Physics "Enrico Fermi," Academic Press, New York (1974), pp. 1-110. C. M. Will and K. Nordtvedt, Jr., "Preferred-frame theories and an extended PPN formalism," Astrophys. J., 177, No. 3, 757-774 (1972). J. G. Williams, R. H. Dicke, P. L. Bender, C. O. Alley, W. E. Carter, D. G. Currie, D. H. Eckhard, J. E. Failer, W. M. Kaula, J. D. Mulholland, H. H. Plotkin, S. K. Poultney, P. J. Shelus, E. C. Silverberg, W. S. Sinclair, M. A. Slade, and D. T. Wilkinson, "New test of the equivalence principle from lunar laser ranging," Phys. Rev. Lett., 36, No. 11, 551-554 (1976).