Experimental Verification of the General Theory of Relativity*)

V. L. Ginzburg

Contents

Introduction ........................................................................ 11
§ 1. Motion of the perihelia of planets and their satellites ............ 12
§ 2. Gravitational shift of spectral lines ................................. 22
§ 3. Deflection of light rays passing near the Sun ....................... 32
§ 4. Significance of Einstein’s general theory of relativity for physics and astronomy .................................................................. 35
Conclusion .......................................................................... 47
Cited literature ................................................................... 47

Introduction

The general theory of relativity, which is the greatest scientific achievement created by the genius of Albert Einstein, is above all a theory of the gravitational field, generalizing Newton’s law of universal gravitation. At the same time, the relation between Newtonian gravitation theory, or gravistatics, and the general theory of relativity is, within known limits, analogous to the relation between electrostatics and electrodynamics. The need for such a generalization of Newton’s law of gravitation became entirely clear and, at the same time, fully matured only after the creation and success of the special theory of relativity, which is based on the principle of the finite speed of propagation of any signals and is not in agreement with Newton’s theory of universal gravitation, with its action at a distance, equivalent to an infinite speed of propagation of gravitational disturbances.

From this alone it is already evident how important the construction of the general theory of relativity was for theoretical physics, even independently of the question of the role of this theory in understanding particular

*) Expanded presentation of a report delivered on November 30, 1955, at a session of the Division of Physical and Mathematical Sciences of the Academy of Sciences of the USSR.

physical and astronomical phenomena. At the same time, the general theory of relativity, like any other scientific theory, on the one hand rests on definite experimental facts and, on the other hand, is constructed on the basis of known assumptions that to one degree or another have a hypothetical character. Therefore the genuine triumph of the general theory of relativity cannot but be connected with the experimental discovery of new effects predicted by the theory. In this respect it is quite natural that the question of the experimental verification of the general theory of relativity has attracted attention from the very time of its creation, i.e. for about 40 years already. This problem, owing to the smallness of the corresponding effects that allow a quantitative comparison of theory with experiment, has not lost its relevance up to the present time, although it has also found itself in the shade in connection with the stormy successes of atomic and nuclear physics.

The purpose of the present article is a relatively brief account of the current state of the question of the experimental verification of the general theory of relativity; moreover, the discussion will deal mainly with the three effects indicated by Einstein: the motion of the perihelia of planets, the deflection of light rays passing near the Sun, and the gravitational displacement of spectral lines (the last effect is usually called the gravitational red shift). In addition, bearing in mind readers little acquainted with this field, we shall touch on some related questions, with the aim of making clearer the significance of the general theory of relativity for physics and astronomy.

§ 1. THE MOTION OF THE PERIHELIA OF PLANETS AND THEIR SATELLITES

The content of one of Kepler’s well-known laws consists in the fact that planets move along ellipses, in one of whose foci the Sun is located. This result, established with very high accuracy by observations, follows from the laws of Newtonian mechanics for the motion of a planet in a field with potential \(\varphi=-\dfrac{\mathrm{const}}{r}\), or, specifically for the Sun, with the potential:

\[ \varphi_{\odot}(r)=-\frac{\varkappa M_{\odot}}{r}, \tag{1} \]

where \(r\) is the distance from the center of the Sun, the gravitational constant \(\varkappa=6.670\cdot 10^{-8}\ \mathrm{dyne}\cdot \mathrm{cm}^{2}\cdot \mathrm{g}^{-2}\), and the mass of the Sun \(M_{\odot}=1.991\cdot 10^{33}\ \mathrm{g}\).

In the general theory of relativity the motion of a particle (planet), even in a given centrally symmetric field, is, generally speaking, rather complicated. However, in the case of the solar system the situation is substantially simplified, since the gravitational field may be regarded as weak and the effects of the general theory of relativity ...

relativity are very small. Indeed, from the foundations of the theory it follows*) that the gravitational field is weak if

\[ \frac{|\varphi|}{c^2}\ll 1. \tag{2} \]

At the same time, on the surface of the Sun (radius \(r_\odot=6.963\cdot 10^{10}\ \text{cm}\))

\[ \frac{|\varphi|}{c^2}=\frac{\chi M_\odot}{r_\odot c^2}=\frac{\rho_\odot}{r_\odot}=2.12\cdot 10^{-6},\qquad \rho_\odot=1.47\cdot 10^5\ \text{cm}, \tag{3} \]

where \(\rho=\dfrac{\chi M}{c^2}\) is the gravitational radius of a body of mass \(M\); in the Earth’s orbit \(\dfrac{|\varphi|}{c^2}=2\cdot 10^{-8}\). For the Earth \(M_\oplus=5.98\cdot 10^{27}\ \text{g}\), the mean radius \(r_\oplus=6.37\cdot 10^8\ \text{cm}\), \(\rho_\oplus=0.43\ \text{cm}\), and, taking into account only the terrestrial field, on the surface of the Earth

\[ \frac{|\varphi|}{c^2}=\frac{\chi M_\oplus}{r_\oplus c^2}=7\cdot 10^{-10}. \]

Condition (2) can also be written in the form

\[ \frac{v^2}{c^2}\ll 1, \tag{4} \]

where \(v\) is the velocity of a body in a gravitational field; the transition from (2) to (4) is made by virtue of the fact that, by the virial theorem, on an orbit, averaged over time, \(|\varphi|=v^2\).

The possibility indicated above of treating the influence of the effects of general relativity on the motion of planets as a small perturbation is connected precisely with the fact that these effects are determined by the parameter \(\dfrac{v^2}{c^2}\sim\dfrac{|\varphi|}{c^2}\), which is very small. In this connection, as was clarified by Einstein in 1915\(^{4,5}\), the change in the motion of a planet in comparison with the classical one consists in the fact that the elliptical orbit very slowly rotates in its plane in the direction of motion (rotation) of the planet itself**). As a result

*) See, for example, books \(^{1,2,3}\); below, for results available in \(^{1,2,3}\) and other courses of the theory of relativity, we shall generally not give references to the literature.

**) We are speaking of the so-called secular perturbations, i.e. perturbations accumulating with time. In addition, there may exist periodic perturbations of the elements of the orbit which, however, are so small that they are of no interest. Let us note, moreover, that in the case of two bodies of comparable mass (double stars) formula (5), obtained\(^{4,5}\) for a body (planet) whose mass is negligibly small in comparison with the mass of the Sun, also remains valid\(^{10}\). In this case, \(M\) should be understood as the sum of the masses of both stars, while the parameters \(a\) and \(e\) refer to the classical orbit corresponding to the relative motion of both bodies. Relativistic corrections for binary stars, or for such a system as the Earth and the Moon, are, however, of no interest (in the latter case the corrections are very small, as is clear from Table II; in the case of binary stars the relativistic effect is very difficult to isolate, since there are also possible effects connected with tidal phenomena on both stars, which cannot be regarded as either rigid or spherical).

the perihelion of the planet (the vertex of the ellipse nearest to the Sun) shifts, and in one revolution the angular displacement of the perihelion, in radians, is equal to:

\[ \varepsilon = \frac{24\cdot \pi^3 a^3}{c^2 T^2(1-e^2)} = \frac{6\pi \varkappa M_{\odot}}{c^2 a(1-e^2)}, \tag{5} \]

where \(a\) is the semimajor axis of the ellipse (orbit), \(e=\dfrac{\sqrt{a^2-b^2}}{a}\) is the eccentricity of the orbit (\(b\) is the semiminor axis), and \(T\) is the period of revolution of the planet; in passing to the second expression, Kepler’s third law was used:

\[ a^3=\frac{\varkappa M_{\odot}}{4\pi^2}T^2. \]

Expression (5) is, of course, in agreement with the assertion made above that the effect is of order

\[ \frac{v^2}{c^2}\sim \frac{|\varphi|}{c^2}\sim \frac{\varkappa M_{\odot}}{ac^2} \]

(since for the planets of the solar system \(e\ll 1\), and the effect of general relativity is more properly characterized by the quantity \(\dfrac{\varepsilon}{2\pi}\), i.e. by the ratio of the angular displacement to the angle described by the planet in one revolution, we see from (5) that the effect is approximately equal to \(\dfrac{3v^2}{c^2}\)). Let us also note that the motion of the perihelion of a planetary orbit occurs not only from the point of view of general relativity, but also within the framework of special relativity. In the latter case the motion of a planet is determined\(^2\) by the equation

\[ \frac{d}{dt}\frac{m_0\mathbf{v}}{\sqrt{1-\dfrac{v^2}{c^2}}} = -m_0\operatorname{grad}\varphi = -\frac{\varkappa m_0 M_{\odot}}{r^3}\mathbf{r}. \]

From this equation it is immediately clear that the difference of the motion from the classical one is determined by the parameter \(\dfrac{v^2}{c^2}\). As a result of the calculation\(^2\) it turns out that the motion of the planet has the same character as in general relativity, but the rotation of the perihelion is 6 times smaller than according to formula (5). Special relativity leads to an incorrect result because the dependence of inertial mass on velocity is taken into account in it, but the gravitational field, instead of the tensor \(g_{ik}\), is described, as in the classical theory, by the potential \(\varphi\) (moreover, in the equation of motion written above the heavy mass is taken equal to the rest mass \(m_0\))*.

*) As shown by a calculation carried out by V. V. Zhukov, if in the equation of motion the heavy mass is taken equal to

\[ \frac{m_0}{\sqrt{1-\dfrac{v^2}{c^2}}}, \]

then the rotation of the perihelion in such a theory of relativity turns out to be three times smaller than according to formula (5).

According to formula (5), the perihelion of a planet turns over a century through an angle (in arc seconds):

\[ \Psi=\frac{5\pi^2 a^2Y}{24c^2T^3(1-e^2)} =8.35\cdot 10^{-19}\frac{a^3}{T^3(1-e^2)} =\frac{3.34\cdot 10^{33}}{a^{5/2}(1-e^2)}, \tag{6} \]

where \(Y=365.25\) is the number of days in a year, the period of revolution \(T\) is measured in days, and the semi-major axis \(a\) in centimeters. The values of \(\Psi\) and \(e\Psi\) for several planets are given in Table I.

Table I

Angle of rotation of the perihelia of planets in arc seconds per century

Planet	\(\Psi\) by formula (6)	\(e\Psi\)	\(\Psi\) from observations
Mercury	\(43'',03\pm0'',03\)	\(8'',847\)	\(42'',56\pm0'',94\)
Venus	\(8'',63\)	\(0'',059\)	—
Earth	\(3'',8\pm0'',0\)	\(0'',064\)	\(4'',6\pm2'',7\)
Mars	\(1'',35\)	\(0'',126\)	—
Jupiter	\(0'',06\)	\(0'',003\)	—

The accuracy indicated in the table for the theoretical value of \(\Psi\) for Mercury is determined by the accuracy with which the quantities entering into (6) are known. The accuracy of measuring the effect under discussion is not determined by the value of \(\Psi\) alone, but also depends on the eccentricity of the orbit. The latter is already clear from the fact that, for a strictly circular orbit, it is altogether impossible to speak of the motion of the perihelion or of the perihelion itself. Therefore, in the third column of Table I the value \(e\Psi\) is given—the product of \(\Psi\) by the eccentricity of the orbit \(e\), characterizing the accuracy with which the effect can be determined from observations.\(^*\)

Even for Mercury, for which the effect is the greatest in comparison with all the planets, it amounts to only \(43''\) per century. Moreover, the relativistic effect is not the only cause of the motion of the planet’s perihelion: as is well known, such motion also occurs under the influence of perturbations from

\(^*\) In §7, as a parameter characterizing the accuracy of observing the effect, not \(e\Psi\) is used, but a somewhat different quantity, which more fully takes into account the conditions of observation of the orbit. However, we shall not discuss this question here in detail (see, however, Table II) and shall confine ourselves to introducing the parameter \(e\Psi\), since for Mercury and the Earth the introduction of another parameter is not especially essential.

of the other planets. For Mercury, the perturbing effect amounts to \(532''\), i.e. is 12.5 times larger than the relativistic effect; for the Earth, the perturbations from the other planets and the Moon lead to a displacement of the perihelion by \(154''\), whereas the effect of the general theory of relativity is only \(3.8''\). It is therefore clear that isolating the relativistic effect is a rather difficult problem, which can be solved only as a result of careful calculations (especially important are calculations of the perturbations from individual planets; for more detail see \({}^{6}\)) and of the use of observations of Mercury’s motion over more than 150 years. As a result it turns out that allowance for all perturbations cannot explain the observed motion of the perihelion, and the remaining displacement of the perihelion (see the last column of Table 1) is in excellent agreement with the result obtained in the general theory of relativity (the latter applies to Mercury, since for the Earth, owing to the smallness of the effect, one can rather only say that the theory does not contradict experiment).

Thus, one may say that on this point the experimental facts speak in favor of the general theory of relativity in the most definite and, moreover, quantitative way*). Nevertheless, further study of relativistic deviations from the Newtonian theory of the motion of planets and their satellites is of unquestionable interest. Suffice it to say that the relativistic effect has been reliably established only for one object (Mercury), while its isolation against the background of other perturbations, although carried out quite reliably, is a rather complicated operation. In this respect two possibilities are of interest. The first of them\({}^{7}\) consists in observations of the motion of the perihelion of the minor planet Icarus, for which the mean distance from the Sun is the smallest, and the eccentricity the largest among the other minor planets (for Icarus \(a = 1.077\) astronomical units \(\simeq 1.6 \cdot 10^{13}\) cm, \(e = 0.8265\), and \(T = 408.67\) days). In this case \(\Psi = 10''.05\) and \(e\Psi = 8''.3\), but a more detailed analysis shows\({}^{7}\) that the possible accuracy of determining \(\Psi\) is 4–5 times higher than for Mercury. However, measuring \(\Psi\) will require decades of astronomical observations, and the perturbations of the orbit of Icarus caused by the planets must be taken into account with sufficient accuracy (according to \({}^{7}\), such an allowance seems possible). Therefore, of greater interest is the study of relativistic perturbations in the case of planetary satellites and, in particular, artificial satellites of the Earth\({}^{8,11}\).

*) It should be noted that a certain displacement of Mercury’s perihelion, unexplained by planetary perturbations, was discovered already by Le Verrier in the middle of the last century. In Einstein’s paper\({}^{4}\), in which formula (5) was derived, it is pointed out that the remaining unexplained displacement of Mercury’s perihelion, according to astronomical data, is \(45'' \pm 5''\). From Table 1 it is clear what great accuracy has been achieved at the present time.

Relativistic perturbations in the motion of a satellite of a planet are composed of perturbations caused by the planet itself and by the Sun. The first of these perturbations, \(\Psi_1\), is determined by formulas (5) and (6), where, of course, \(a\), \(e\), and \(T\) must be understood as the parameters of the satellite’s orbit, and the mass \(M_{\odot}\) must be replaced by the mass of the planet. The effect of the general theory of relativity associated with the field of the Sun consists in the motion of the perihelion (perigee) of the satellite, and also in the motion of the nodes of the satellite’s orbit. In this case the corresponding displacement of the perigee, \(\Psi_2\), is determined2 by expressions (5) and (6), multiplied by 2 (in this case, of course, no replacement of quantities need be made in formulas (5) and (6), and the parameters of the orbit of the planet under consideration enter into them); as for the displacement of the nodes, it is described by expressions (5) and (6), divided by 2.

For satellites of the Earth (in angular seconds per century)

\[ \Psi_1 = 8.35 \cdot 10^{-19}\,\frac{a^2}{T^3(1-e^2)} = \frac{1.74 \cdot 10^{25}}{a^{5/2}(1-e^2)} , \tag{7} \]

where the period of revolution of the satellite \(T\) is measured in days (sidereal days) and the semimajor axis of its orbit \(a\) in centimeters (in (7), specifically for the Earth, only the last expression applies, of course). The quantity \(\Psi_2\), i.e. the displacement of the satellite’s perigee associated with the field of the Sun, is equal for satellites of the Earth to \(7''.6\) per century.

For the Moon (mean distance from the Earth \(r = 3.844 \cdot 10^{10}\ \mathrm{cm}\)) \(\Psi_1 = 0''.06\), but for satellites close to the Earth the value of \(\Psi_1\) can already be relatively very large. At the same time, the discovery of such small natural satellites is not excluded,* and, most importantly, in the coming years, according to press reports, artificial satellites of the Earth will be created. A satellite moving at the very surface of the Earth (this, of course, is an unreal case because of air resistance) would have to revolve around it in 1.41 hours \(= 5.88 \cdot 10^{-2}\) days, while a satellite at an altitude, for example, of \(400\ \mathrm{km}\) above the Earth revolves in 1.54 hours. The displacements \(\Psi_1\) for these, as well as for some other satellites1, are given in Table II on p. 18 (the displacement \(\Psi_2 = 7''.6\) is not taken into account; for comparison, the values of \(\Psi\) for Mercury from Table I are also given).

It follows from Table II that the values of \(\Psi_1\) and \(e\Psi_1\) for artificial satellites of the Earth may exceed the corresponding values for Mercury by tens of times, to say nothing of the other planets. The attainable accuracy of measuring the effect for satellites is, apparently, still relatively greater than follows from the values of the parameter \(e\Psi\). This is seen from the last column of Table II,

* Let us note that already for the known satellites of a number of planets the displacement \(\Psi_1\) is very large; thus, for example, for Jupiter’s satellite V it amounts to about \(2200''\) per century.

Table II

Displacements of the perigees of Earth satellites (in arc seconds per century)

Satellite	Mean distance to the center of the Earth (in cm)	\(e\)	\(\Psi_1\)	\(e\Psi_1\)	\(\alpha\)
C1	\(r_{\oplus}=6{,}367\cdot 10^8\)	0	\(1700''\)	—	—
C2	\(r_{\oplus}+4\cdot 10^7=6{,}77\cdot 10^8\)	0,01	\(1450''\)	\(14'',5\)	—
C3	\(17\cdot 10^8\)	0,06	\(146'',0\)	\(8'',75\)	\(\geq 2'',6\)
C4	\(17\cdot 10^8\)	0,40	\(194'',6\)	\(77'',7\)	\(\geq 30''\)
C5	\(7{,}2\cdot 10^8\)	0,02	\(1250'',5\)	\(25'',0\)	\(\geq 75''\)
C6	\(10\cdot 10^8\)	0,25	\(586'',6\)	\(146'',6\)	\(\geq 485''\)
Moon	\(3{,}844\cdot 10^{10}\)	0,06	\(0'',06\)	\(3{,}6\cdot 10^{-3}\)	—
Mercury	—	0,2	\(43'',03\)	\(8'',847\)	\(3'',1\)

where the value of the parameter \(\alpha\), used in 7,8 instead of the parameter \(e\Psi\) to characterize the possible accuracy of measurements of the displacement of the perigees of satellites and planets, is given. If the parameter \(\alpha\) is used, it becomes clear that for satellite C6, in a year of observations, an accuracy greater than that for Mercury over a century can be achieved. It may be that the use of radiomethods suitable in the case of artificial satellites will lead to still more favorable results.*)

But the study of the motion of satellites of planets and, in particular, artificial satellites of the Earth is of interest from the point of view of general relativity by no means only for the purpose of verifying formula (5). The point is that the use of satellites opens up the fundamental possibility of testing yet another effect of general relativity which has hitherto remained entirely unconfirmed experimentally. We have in mind the influence on the motion of a satellite of the rotation of the planet about its axis.**) As is well known, in Newtonian theory the gravitational field of a rotating body is exactly the same irrespective of whether this body is at rest or is rotating about its axis.

*) Here and below we completely abstract from a number of technical difficulties with which the use of satellites is connected. In addition, in order to identify relativistic perturbations, it is of course necessary to take into account, with sufficient accuracy, other perturbations (the influence of the Moon and of the nonsphericity of the Earth, the influence of gas resistance in the ionosphere).

**) In addition, the use of satellites opens up certain possibilities for testing the effect of the gravitational displacement of spectral lines, which will be discussed in § 2.

In general relativity this is no longer so, and the rotation, for example, of a sphere about its axis changes the gravitational field it creates. The essence of this effect becomes clear if one recalls that general relativity is such a generalization of Newton’s theory of gravitation which, in a certain respect, is analogous to the transition from electrostatics to electrodynamics (this has already been mentioned in the introduction). In electrostatics a stationary charged sphere creates only a Coulomb electric field, but when the sphere rotates about its axis there also appears a magnetic field, produced by the corresponding current. In exactly the same way the rotation of a gravitating sphere creates an additional gravitational field analogous to the magnetic field of currents. In the case of interest to us, when the gravitational field is weak, this additional field of the sphere in a quasi-Euclidean coordinate system*) is equal to

\[ \mathbf{g}=-\frac{2\chi}{c^{3}r^{3}}[\mathbf{I}\mathbf{r}],\qquad \mathbf{I}=\int[\mathbf{r}',\mu\mathbf{v}']\,dV, \tag{8} \]

where \(g_\alpha=-\dfrac{g_{\alpha 0}}{g_{00}}\approx g_{\alpha 0}\), \(\mu\) is the mass density at the point \(\mathbf{r}'\), moving with velocity \(\mathbf{v}'\), \(r\) is the distance from the center of the sphere to the point of observation, and \(r'\) is the distance from the center of the sphere to its points. If \(\mu=\mathrm{const}\), then the vector \(\mathbf{I}\), directed along the axis of rotation, is in absolute value equal to \(I=\dfrac{2}{5}Mr_0^2\omega\), where \(M\) is the mass of the sphere, \(r_0\) its radius, and \(\omega\) the angular velocity of rotation. In the case of bodies of nonspherical form, formula (8) is applicable not everywhere outside the body, but only at distances \(r\gg r_0\) (see\(^3\) § 100). The maximum value of \(|\mathbf{g}|\) is attained at the equator and is equal to

\[ |\mathbf{g}|=\frac{4\chi Mr_0\omega}{5r_0c^{3}}\sim \frac{|\varphi_0|}{c^{2}}\cdot\frac{v_0}{c}, \]

where \(\varphi_0\) is the Newtonian gravitational potential and \(v_0\) is the velocity of rotation at the surface of the sphere (at the equator). At the same time, in a weak field

\[ g_{00}=-1-\frac{2\varphi}{c^{2}}. \]

The influence of the effects of general relativity on the motion of bodies in a weak gravitational field can be represented by introducing into the classical equation of motion in a field with gravitational potential \(\varphi\), i.e. into the equation \(m\,d\mathbf{v}/dt=-m\,\operatorname{grad}\varphi=\mathbf{f}_{\mathrm{cl}}\), a certain additional force \(\mathbf{f}'\).

In the case of a static gravitational field, when \(g_{\alpha 0}=0\),

\[ f'=f'_1\sim \frac{\varphi}{c^{2}}f_{\mathrm{cl}}\sim \frac{\chi mM}{r^{2}}\cdot\frac{v^{2}}{c^{4}}\sim \frac{\chi mM}{r^{2}}\cdot\frac{\chi M}{c^{2}r} \]

(the field

*) What is meant is a coordinate system in which the deviations of the metric tensor \(g_{ik}\) from the Galilean values \(g_{00}=-1\), \(g_{\alpha\beta}=\delta_{\alpha\beta}\), and \(g_{0\alpha}=0\) \((\alpha=1,2,3)\) are everywhere small.

sphere). If, however, \(g_{\alpha 0}\ne 0\), as occurs when the source of the field rotates, an additional force also appears (see\(^3\) § 88)

\[ \mathbf{f}'_2=mc[\mathbf{v}\operatorname{rot}\mathbf{g}], \tag{9} \]

which is analogous to the Coriolis force appearing in a coordinate system rotating with angular velocity \(\Omega=\dfrac{c}{2}\operatorname{rot}\mathbf{g}\). In view of the estimate of \(|\mathbf{g}|\) given above, it is clear that

\[ f'_2 \lessgtr \frac{\varkappa mM}{r^2}\cdot\frac{vv_0}{c^2}\cdot\frac{r_0}{r} \lessgtr \frac{\varkappa mMvv_0}{r_0^2c^2} \quad\text{and}\quad \frac{f'_2}{f_1}\lessgtr \frac{v_0r_0}{vr}, \]

where \(v_0\) is the velocity of rotation of the surface of the sphere (at the equator) and \(v\) is the velocity of the body under consideration, on which the gravitational field acts. For \(r=r_0\)

\[ v=v_{\max}=\sqrt{\frac{\varkappa M}{r_0}} \quad\text{and}\quad \frac{f'_2}{f_1}\lessgtr \frac{v_0}{\sqrt{\dfrac{\varkappa M}{r_0}}}. \]

In the case of revolution around the Sun the velocity of the body reaches the value \(v_{\max}=4.36\cdot 10^7\ \text{cm/sec}\), while for revolution around the Earth \(v_{\max}=7.9\cdot 10^5\). At the same time, for the Sun the period of rotation about its axis is \(\tau\simeq 28\) days and \(v_0\simeq 2\cdot 10^5\), while for the Earth \(\tau=24\) hours and \(v_0=4.6\cdot 10^4\). Hence it is evident that the influence of the rotation of the Sun and the Earth on their satellites can prove to be only one or two orders of magnitude smaller than the relativistic effect which occurs in the absence of rotation. A quantitative account of the rotation effect was already carried out long ago in work\(^9\); it was shown there that the corresponding displacement of the perihelion of a planet (satellite) per century, in seconds of arc, is equal to*)

\[ \Psi_{\mathrm{v}}= -\frac{\varkappa^2 r_0^2 Y}{9c^2\tau T^2(1-e^2)^{3/2}}, \tag{10} \]

where \(r_0\) is the radius and \(\tau\) the period of rotation (in days) of the sphere producing the field, and the remaining notations are the same as in the case of the for—

*) In\(^9\) it is stated that formula (10) is valid only for \(a\gg r_0\), i.e. when the radius of the orbit is considerably greater than the radius of the sphere. In fact, for a sphere this formula is valid for any \(a\) (of course \(a\ge r_0\)), since formula (8) in the case of a sphere is applicable not only far away but also near the rotating body. We also note that the influence of the rotation of the sphere, besides the precession (displacement) of the perihelion, leads to a precession of the nodes of the orbit of the planet (satellite), and the angle of rotation of the nodes is half as large and has the opposite sign as that given by formula (10).

...formula (6). In (10) it is assumed, for simplicity, that the plane of the orbit coincides with the equatorial plane of the rotating body, and that the rotation of the satellite and of the sphere takes place in one and the same direction; in the general case, when the angle between these planes is equal to \(i\), in (10), according to\(^9\), an additional factor \(\left(1-3\sin^2 \dfrac{i}{2}\right)\) appears. The effects determined by formulas (6) and (10) must simply be added, with the negative sign in (10) indicating that, when the rotation of the Sun (planet) is taken into account, the resulting relativistic displacement of the perihelion of the planet (satellite) decreases.

According to (6) and (10),

\[ \Delta=\frac{|\Psi_{\mathrm{v}}|}{\Psi}=\frac{8}{15}\left(\frac{r_0}{a}\right)^2 \frac{T}{\tau(1-e^2)^{1/2}}. \tag{11} \]

In the case of Mercury (\(a=0.39\) astr. units, \(r_0=5.8\cdot10^{12}\) cm, \(T=0.241\) yr), \(\Delta\simeq2.5\cdot10^{-4}\) and \(\Psi_{\mathrm{v}}=-0''.01\), at a time when the accuracy of measuring the rotation of the perihelion is of order \(1''\) (see Table I). For the satellite Jupiter V, \(\Psi_{\mathrm{v}}=-3'.46''\) and \(\Psi=36'.37''\), but the observational data do not possess, according to\(^9\), sufficient accuracy for isolating relativistic effects. Therefore, as far as we know, after work\(^9\) there has been no return to the discussion of the “rotation effect,” which leads to the displacement \(\Psi_{\mathrm{v}}\). At the same time this effect is very interesting, which makes it appropriate to draw attention to the new possibilities\(^ {11}\) opened up by the use of artificial Earth satellites\(^*\). Thus, for the satellite C1 moving very near the Earth (see Table II),

\[ \left. \begin{aligned} T_1&=1.41\ \text{hours},\qquad \Delta=3.14\cdot10^{-2},\\ \Psi_{\mathrm{v}}&=-53'', \end{aligned} \right\} \tag{12} \]

i.e., the “rotation effect” is greater than the entire relativistic effect for Mercury (as was emphasized, the satellite C1 in practice cannot be used, but it is given as an example because, for a number of other realizable satellites, values of \(\Delta\) and \(\Psi_{\mathrm{v}}\) of the same order of magnitude are obtained; for example, for the satellite C2, \(\Delta\simeq3\cdot10^{-2}\) and \(\Psi_{\mathrm{v}}=-43''\)). Thus, attaining the relative accuracy of determining the displacement of the perihelion

\(^*\) As regards the use of the satellites of planets (Jupiter, Saturn)\(^9\) for checking the “rotation effect,” it should be noted that the author has no data permitting him to express an opinion on this question. At the same time, it would be highly desirable for astronomers to carry out the corresponding analysis on the basis of modern data and possibilities.

Let us note in this connection that the possibility is not excluded of discovering small close natural satellites of the Earth (for example, captured meteors). If such satellites are discovered, they can, of course, also be used for checking the general theory of relativity.

of an artificial satellite by an order of magnitude exceeding the accuracy of the corresponding determination for Mercury, will make it possible, in only a year of observations, to detect the relativistic “rotation effect” for the Earth. The solution of this problem should attract attention.

§ 2. GRAVITATIONAL SHIFT OF SPECTRAL LINES

According to the general theory of relativity, the proper (true) time \(\tau\) at some fixed point is related to the coordinate time \(t=\dfrac{x_0}{c}\) by the relation

\[ \tau=\frac{1}{c}\int \sqrt{-g_{00}}\,dx_0 . \tag{13} \]

In a constant gravitational field the frequency of light, measured in coordinate (world) time, is the same along a light ray and, consequently, the experimentally determined frequency \(\nu=\dfrac{1}{\Delta\tau}\) (\(\Delta\tau\) is the period of oscillation measured in proper time) is not the same at different points: the ratio of the frequencies \(\nu_2\) and \(\nu_1\) at points 2 and 1 is equal to \(\dfrac{\nu_2}{\nu_1}=\sqrt{\dfrac{g_{00}(1)}{g_{00}(2)}}\). In a weak field, to within terms of higher order, \(g_{00}=-1-\dfrac{2\varphi}{c^2}\), and with the same accuracy

\[ \frac{\Delta\nu}{\nu_1} = \frac{\nu_2-\nu_1}{\nu_1} = \frac{\varphi_1-\varphi_2}{c^2}. \tag{14} \]

If on the Earth one observes the spectrum emitted by atoms on the Sun or on the stars, then the potential at the Earth \(\varphi_2\) may be neglected, and

\[ \frac{\Delta\nu}{\nu} = -\frac{\Delta\lambda}{\lambda} = \frac{\varphi_1}{c^2} = -\frac{\chi M}{c^2 r}, \tag{15} \]

where \(r\) is the radius of the emitting layer (photosphere), \(\lambda=\dfrac{c}{\nu}\) is the wavelength and, in view of the smallness of the effect, \(\nu\) may be understood as the frequency emitted by the atom in the absence of a gravitational field (i.e., practically, on Earth). For the Sun (when \(r=r_{\odot}\))

\[ \frac{\Delta\nu}{\nu} = -\,2.12\cdot 10^{-6}. \tag{16} \]

In the case of (15)—(16) \(\Delta\nu<0\), i.e. the spectral lines are shifted toward the red. Therefore the effect under discussion is usually called the red shift of spectral lines. But, of course, the shift may also have the opposite sign. For example, when one observes on Earth radiation from a very distant source located at a point with zero potential, the shift of the lines occurs

toward the violet side, with

\[ \frac{\Delta \nu}{\nu}=\frac{\varkappa M \delta}{c^2 r\delta}=7\cdot 10^{-10}. \tag{17} \]

Since both cases (16) and (17) are discussed below, we shall call the frequency shift in a gravitational field the gravitational shift of spectral lines.

The simple formula (14), which in practice is almost the only one that has to be used (if cosmological questions are not touched upon), can be obtained in an elementary way from the principle of equivalence; this was first done by Einstein\(^{12,13}\) in 1907, and then in 1911, even before he had created the completed theory of the gravitational field.\(^{5}\) The same result is obtained on the basis of quantum concepts, assuming that a quantum has not only inertial but also gravitational mass \(m=\dfrac{h\nu}{c^2}\). Then, when moving in a gravitational field, the quantum does work

\[ m(\varphi_1-\varphi_2)=\frac{h\nu}{c^2}(\varphi_1-\varphi_2), \]

which can occur only at the expense of a change in frequency. Hence

\[ h\Delta \nu=\frac{h\nu}{c^2}(\varphi_1-\varphi_2), \]

i.e., we obtain formula (14).

Attempts to detect the gravitational shift of spectral lines in the spectra of the Sun and stars have been going on for more than 40 years. The difficulty of the problem is connected with the relative smallness of the effect and with the presence of a Doppler shift of the lines, due to the motion both of the stars themselves and of their photospheres. The significance of the Doppler shift becomes especially clear if the gravitational shift is measured in effective velocities*):

\[ v_{\mathrm{eff}}=c\,\frac{\Delta \nu}{\nu}=\frac{\varphi_1-\varphi_2}{c}. \tag{18} \]

Then for the Sun \(v_{\mathrm{eff}}=-0.636\ \mathrm{km/sec}\), and for the Earth (see (17)) \(v_{\mathrm{eff}}=21\ \mathrm{cm/sec}\). At the same time the radial velocities of stars reach tens of \(\mathrm{km/sec}\), and the velocity of streams in the solar photosphere is of the same order as \(v_{\mathrm{eff}}\).

Without dwelling on individual works and, in particular, on the early investigations of the gravitational shift (see\(^{14}\)), let us briefly characterize the present state of affairs. The gravitational shift has been established with certainty in white dwarfs, where \(v_{\mathrm{eff}}\) reaches several tens of \(\mathrm{km/sec}\) (this is explained by the fact that, for a mass \(M\sim M_{\odot}\), the radius of white dwarfs is \(r\sim 10^{-2}r_{\odot}\)). At the same time, insufficiently accurate knowledge of the radii of these stars

*) It is obvious that \(v_{\mathrm{eff}}\) is the radial velocity of the source (relative to the Earth) that leads to the same Doppler shift of the spectral lines as is caused by the gravitational effect.

hinders a reliable quantitative comparison of theory with experiment. If, nevertheless, one uses the best available data, treating them in the most natural way, then, according to a kind communication from P. P. Parenago, for the white dwarfs Sirius B \((v_{\mathrm{eff}} \simeq -20\ \mathrm{km/sec})\) and \(o^2\) Eridani the experimental value agrees well with the theoretical one. The statistical analysis of the redshift for a number of hot stars leads to the same conclusion (this is the so-called \(K\)-effect)\(^{15}\). A recent attempt to revise these conclusions\(^{16}\) meets with serious objections (see, for example,\(^{17,18}\)) and will not be discussed here in greater detail.

Despite the fact that the data concerning stellar spectra are favorable to the theory, they, as has been said, cannot be regarded as fully conclusive in quantitative terms. In this connection, the detection of a redshift in the spectrum of the Sun is of great importance. In this case the effect is very small (thus, for \(\lambda = 6100\ \text{Å}\), \(\Delta \lambda = 1.29 \cdot 10^{-2}\ \text{Å}\)), but this negative circumstance is more than compensated by the possibility of carrying out observations with very sophisticated solar instruments possessing high resolving power. As a result, the accuracy of determining the centers of lines in the solar spectrum proves quite sufficient for detecting the gravitational effect. This is clearly seen from the figure, in which data are plotted on the shift of lines in the spectrum of the Sun (absorption lines are meant) in the region \(\lambda = 6100\ \text{Å}\), as a function of the position of the emitting point on the solar disk \((\theta\) is the angle between the direction of observation and the solar radius directed to the point at which the line of sight intersects the solar surface; for the center of the solar disk, evidently, \(\theta = 0\), and for the edge of the disk \(\theta = \frac{\pi}{2}\)). In the figure, borrowed from\(^{16}\), the black points correspond to the observations of Adam\(^{19}\) and the circles to the observations of Freundlich et al.\(^{20}\). The measurements lead to a paradoxical result at first sight: the displacement of the lines is not the same for different points of the disk, and only at its very edge, according to the best available data\(^{19}\), is the displacement almost exactly equal to that predicted by the general theory of relativity. Such a picture will not, however, seem so strange if one recalls that the layers of the solar atmosphere responsible for the formation of spectral lines (the reversing layer) are in continuous motion of a convective character. It is precisely these motions that explain solar granulation, and, as is easy to see, they must lead to some additional violet shift of spectral lines. In fact, the hotter masses of gas move in the reversing layer along the radius away from the Sun, while the colder ones move toward the Sun. Therefore, if, solely for simplicity, we speak not of absorption lines but of emission lines, then immediately

EXPERIMENTAL VERIFICATION OF THE GENERAL THEORY OF RELATIVITY

it is clear that the line as a whole will, because of the motion of the gas, be shifted toward the violet side*). Moreover, if the velocities of the streams are purely radial, the violet shift will be proportional to \(\cos \vartheta\), i.e., maximal at the center of the disk and equal to zero at its edge. This latter circumstance, as well as taking into account other factors essential in the emission of light in the solar atmosphere, makes it possible to conclude that

near the edge of the disk the motion of the emitting layers may be disregarded, and only the gravitational effect should be observed (we abstract here from pressure effects; see below). This is indeed observed experimentally. At the same time, the use of the available data on solar granulation makes it possible to explain quite naturally the presence of an additional violet shift over the remaining part of the disk. Thus, the data on the shift of spectral lines on the Sun may be regarded as confirming the conclusions of the general theory of relativity. However, from the point of view of strict

*) In the limiting case, when the radiation of the “cold” gas moving toward the Sun may be completely neglected, the shift of the lines is determined simply by the velocity of motion of the hot gas.

It is not difficult to see that the same conclusions are obtained in the case of absorption lines.

V. L. GINZBURG

there is still insufficient available data for a quantitative verification of formulas (14)—(15) for the gravitational shift of spectral lines. This is due to the fact that the shift of lines on the Sun is also caused by the pressure effect (i.e., ultimately, by the interaction between atoms), which, generally speaking, is not small in comparison with the gravitational effect. At the same time, allowing for the influence of pressure (see, for example, \(^{19}\)) is associated with certain assumptions and therefore so far could not be carried out with complete confidence*). The same can be said about taking into account the shift of lines caused by granulation. Therefore, further work is still needed for a rigorous quantitative verification of formula (14).

In addition to the study of the spectra of the Sun and stars, certain very attractive prospects in this respect are opened up by the development of radiophysics and, in particular, radiospectroscopy. In optics the smallest noticeable relative frequency shift does not exceed

\[ \left(\frac{\Delta \nu}{\nu}\right)_{\min} \gtrsim 10^{-7}. \]

In radiophysics the situation is already different—in principle, shifts by small fractions of a hertz can be observed there, and thus, for \(\nu \sim 10^{10}\) \(\left(\lambda=\frac{c}{\nu}\sim 3\ \text{cm}\right)\), even a value

\[ \left(\frac{\Delta \nu}{\nu}\right)\sim 10^{-13} \]

cannot be regarded as a limiting one. On the other hand, the development of frequency-stabilization techniques and, in particular, the transition to molecular stabilization (atomic clocks) open up known possibilities for achieving stability of the source frequency

\[ \frac{\Delta \nu}{\nu} \gtrsim 10^{-12}\div 10^{-13} \]

(see \(^{21,22,23}\)). But if it proves possible to measure frequency shifts with such accuracy, then the effect of the gravitational shift of spectral lines can be observed \(^{21}\) even within the Earth. Indeed, on Earth one can quite well place a receiver and a transmitter at heights differing, say, by \(3\ \text{km}=3\cdot 10^{5}\ \text{cm}\). Then, according to (14),

\[ \frac{\Delta \nu}{\nu}=\frac{gh}{c^{2}}=1.09\cdot 10^{-18}h\simeq 3\cdot 10^{-13} \tag{19} \]

and, with sensitivity \(\frac{\Delta \nu}{\nu}\sim 10^{-13}\), the effect can be measured (in formula (19), \(g=981\ \text{cm}/\text{sec}^{2}\) is the acceleration of gravity). As far as we know, practical realization of frequency measurements with such accuracy is still far off, but values

\[ \frac{\Delta \nu}{\nu}\gtrsim 10^{-10} \]

have already been achieved \(^{23}\).

*) We note that the figure gives the directly measured red shift of the lines, i.e., the pressure effect has not been taken into account. If this is done, then, according to \(^{19}\), at the center of the disk \(\Delta\lambda=0.9\cdot 10^{-3}\ \text{\AA}\) instead of \(5\cdot 10^{-3}\ \text{\AA}\), but near the edge of the disk the correction associated with the pressure effect is insignificant.

EXPERIMENTAL TEST OF THE GENERAL THEORY OF RELATIVITY

In this connection, the possibility of detecting the gravitational shift with the use of an artificial satellite of the Earth is of interest \(^{11,17}\). For a very distant satellite, the gravitational violet shift of the frequency of its radiation, observed on the Earth, is determined by expression (17), i.e., it reaches the value \(\Delta\nu/\nu \sim 10^{-9}\). If, however, the satellite is at a height \(h\) above the Earth, then

\[ \frac{\Delta\nu}{\nu} = \frac{\chi M_{\oplus}}{c^{2}} \left( \frac{1}{r_{\oplus}}-\frac{1}{r_{\oplus}+h} \right) \approx \frac{\chi M_{\oplus}h\left(1-\dfrac{h}{r_{\oplus}}\right)} {c^{2}r_{\oplus}^{2}} = \frac{gh\left(1-\dfrac{h}{r_{\oplus}}\right)} {c^{2}} = 1.09\cdot 10^{-18} \left(1-\frac{h}{r_{\oplus}}\right), \tag{19a} \]

where the last two expressions refer to the case \(h \ll r_{\oplus}=6.367\times 10^{8}\).

For \(h=800\) km, \(\Delta\nu/\nu=7.6\cdot 10^{-11}\), and the effect can probably already be measured in the near future. These measurements, however, are substantially complicated by the motion of the satellite, which leads to a large Doppler frequency shift; moreover, it is also necessary to take into account the quadratic Doppler effect, which is of order \(v^{2}/c^{2}\), where \(v\) is the speed of motion of the satellite; the complete gravitational effect (17) is also of order \(v^{2}/c^{2}\) \(\left(\chi M_{\oplus}/r_{\oplus}=v^{2}/c^{2}\right.\), where \(v\) is the speed of a satellite moving at the very surface of the Earth\(\left.\right)\), while the effect (19a) for \(h\ll r\) is already substantially smaller. As for the first-order Doppler effect, for a satellite close to the Earth it is

\[ \frac{c}{v}\sim \frac{3\cdot 10^{10}}{10^{6}}=3\cdot 10^{4} \]

times greater than the complete gravitational effect, unless the angle between the velocity and the line of sight \(\theta\) is close to \(\pi/2\) (for \(\theta=\pi/2\) the first-order effect, as is known, vanishes).

Thus there arises the necessity of obtaining a formula for \(\Delta\nu/\nu\) taking into account both the gravitational shift and the Doppler effect. Already from elementary considerations it is clear that, with accuracy up to terms \(v^{2}/c^{2}\), for which alone formula (14) is suitable, the Dopple-

rovian and gravitational shifts will simply add, i.e.,

\[ \frac{\Delta \nu}{\nu} = \frac{\sqrt{1-\frac{v^2(t_1)}{c^2}}} {1-\frac{v(t_1)\cos\theta(t_1)}{c}} + \frac{\varphi_1(t_1)-\varphi_2}{c^2} = \]

\[ = 1+\frac{v}{c}\cos\theta -\frac{v^2}{2c^2}\left(1-2\cos^2\theta\right) +\frac{\varphi_1-\varphi_2}{c^2}, \tag{20} \]

where \(v\) is the velocity of the source (satellite) relative to the Earth at the moment of emission of the light \(t_1\), \(\theta\) is the angle between the velocity and the line of sight, and the well-known formula for the Doppler effect in the special theory of relativity

\[ \frac{\Delta \nu}{\nu} = \frac{\sqrt{1-\frac{v^2}{c^2}}} {1-\frac{v}{c}\cos\theta} \]

has been written out in full, although it must be used with accuracy up to terms \(\sim \frac{v^2}{c^2}\) (this has also been explicitly taken into account in the last of the written expressions*). The author has had, however, to encounter certain known doubts about the applicability of formula (20) to the case where the source is undergoing accelerated motion, as is the case for a source on a satellite. The satellite “falls freely,” and in a freely falling reference system there is no gravitational field and, consequently, no gravitational frequency shift—such, insofar as could be understood, are the considerations giving grounds for the doubts mentioned. Here, however, there is a simple misunderstanding. In a local inertial reference system associated with the satellite, no frequency shift will indeed be observed if not only the emitter but also the receiver are at rest relative to this system. In the experiments in question, however, the receiver is on the Earth, i.e. in the system associated with the satellite it moves with acceleration in the direction toward the satellite. Therefore, if at the moment of emission of the signal the relative velocity of the satellite and the Earth was equal to zero, then at the moment when the signal reaches the Earth the velocity of the latter will be equal to

\[ v=\frac{gh}{c}, \]

where \(\frac{h}{c}\) is the time of propagation of light over the distance \(h\), and \(g\) is the accel—

* For simplicity, in (20) the difference of the refractive index from unity in the region of the ionosphere in which the satellite moves is not taken into account (a difference which may prove appreciable in the radio range).

of the force of gravity (i.e., the acceleration of the Earth in the system connected with the satellite; for simplicity we regard the gravitational field as homogeneous); because of the Doppler effect, the presence of such a velocity will lead to the fact that the receiver will receive a radiation frequency shifted by

\[ \frac{\Delta \nu}{\nu}=\frac{v}{c}=\frac{gh}{c^{2}}, \]

i.e., one obtains precisely the gravitational frequency shift (19). Arguments of this kind, quite analogous to those used by Einstein in the initial analysis of the question of the gravitational shift,\(^{13}\) leave no doubt that the gravitational shift does not depend on the acceleration of the satellite. Other considerations also lead to the same conclusion, for example quantum ones (a quantum “works” while propagating in a gravitational field and at the same time its momentum \(\frac{h\nu}{c}\), and hence also the frequency \(\nu\), do not depend on the acceleration of the emitting atom)\(*\). Finally, the general expression for the frequency shift in a gravitational field, with moving source and receiver, can be obtained by directly using the invariance of the interval \(ds=\sqrt{-g_{ik}dx_i dx_k}\) (see, for example, \(^{24}\) § 116). Let the source be at the point \(x_{1\alpha}(t_1)\), and the receiver at the point \(x_{2\alpha}(t_2)\), where \(t_1\) and \(t_2\) are the instants of emission and reception of some light signal, with \(t_2=f(t_1)\). The elements of proper time for the source and the receiver are respectively

\[ c\,d\tau_1=\left[-g_{ik}\frac{dx_i}{dt}\frac{dx_k}{dt}\right]^{1/2}_{x_1(t_1)}dt_1, \]

\[ c\,d\tau_2=\left[-g_{ik}\frac{dx_i}{dt}\frac{dx_k}{dt}\right]^{1/2}_{x_2(t_2)}dt_2; \]

further, the ratio of the frequencies is

\[ \frac{\nu_2}{\nu_1}=\frac{d\tau_1}{d\tau_2} \quad\text{and}\quad dt_2=\frac{\partial f}{\partial t_1}\,dt_1. \]

Hence

\[ \frac{\nu_2(t_2)}{\nu_1(t_1)} = \frac{ \left[ -g_{00} -g_{\alpha\beta}\frac{dx_\alpha}{c\,dt}\frac{dx_\beta}{c\,dt} -|g_{0\alpha}|\frac{dx_\alpha}{c\,dt} \right]^{1/2}_{x_{11}(t_1)} }{ \dfrac{\partial f(t_1)}{\partial t_1} \left[ -g_{00} -g_{\alpha\beta}\frac{dx_\alpha}{c\,dt}\frac{dx_\beta}{c\,dt} -g_{0\alpha}\frac{dx_\alpha}{c\,dt} \right]^{1/2}_{x_{21}(t_2)} }, \tag{21} \]

where \(\frac{dx_\alpha}{dt}\) is the component of the velocity along the axis \(x_\alpha\). The trajectory of the ray and the coor-

\(*\) As A. L. Zelmanov rightly noted, if free fall of the emitter led to the elimination of the gravitational shift, this latter effect would never be observed in astrophysical conditions, since the emitting atoms in stellar photospheres always fall freely onto the star.

the coordinate velocity of light is determined from the condition \(ds=0\), whence one can find the function \(t_2=f(t_1)\). In Euclidean space, if point 2 is at rest,

\[ t_2=t_1+\frac{r_{12}(t_1)}{c}, \]

where \(r_{12}(t_1)\) is the distance between points 1 and 2 at the moment \(t_1\) (we speak of \(t_1\) and \(t_2\) as the moments of emission and reception of the signal, or, better, of the leading front of the signal, but all the arguments are also applicable to continuous emission of light)*). Since

\[ \frac{dt_2}{dt_1}=\frac{\partial f}{\partial t_1} =1-\frac{v_1(t_1)\cos\theta(t_1)}{c}, \]

from (21) we at once obtain the usual relativistic formula for the Doppler effect, but with the values of \(v_1\) and \(\theta\) referred to the moment of emission \(t_1\). For a stationary source and receiver situated in a constant gravitational field, \(\partial f/\partial t_1=1\), and the formula given above is obtained

\[ \frac{\nu_2}{\nu_1}=\sqrt{\frac{g_{00}(1)}{g_{00}(2)}}, \]

i.e., for a weak field, formula (14). If, however, the source moves in a weak field, with \(\frac{v_1^2}{c^2}\ll 1\), we obtain formula (20), since the influence of the curvature of the rays and of the change in the coordinate velocity of light in the gravitational field affects the value of \(\partial f/\partial t_1\) only in terms of higher order. With the aid of formula (21) one can also consider more complicated cases and, in particular, take into account the motion of the receiver. When observing a near satellite from the Earth there is as yet no necessity for this, if by \(v\) in (20) one understands the velocity of the source relative to the receiver at rest on the Earth. For the latter reason, the linear Doppler effect associated with the rotation of the Earth—

*) The frequency of radiation at a given moment \(t\) is, of course, a concept having no meaning, and in (20)—(21) by \(\nu(t)\) is meant the frequency corresponding to the maximum of the spectral line which is obtained upon reception of the radiation during some time interval \(\Delta t\) near the moment \(t\). In order that the line be sufficiently narrow, the velocity \(v\) and other quantities (i.e., \(\cos\theta\) and \(\varphi_1\)) must change little over the period of emission

\[ \Delta\tau=\frac{1}{\nu}\ll\Delta t . \]

For a satellite, the quantity

\[ \frac{d}{dt}\left(\frac{v}{c}\cos\theta\right)\cdot\Delta\tau \]

is of order

\[ \frac{v^2\Delta\tau}{cr} \]

because of the change of the angle \(\theta\) (here \(r\) is the distance to the satellite), and of order

\[ \frac{g\Delta\tau}{c} \]

because of the change of the velocity \(v\). Since

\[ \frac{v^2\Delta\tau}{cr}\sim 10^{-16} \quad\text{and}\quad \frac{g\Delta\tau}{c}\sim 3\cdot 10^{-18} \quad(\text{for } \Delta\tau\sim 10^{-10}), \]

whereas

\[ \frac{v^2}{c^2}\sim 6\cdot 10^{-10}, \]

there is no doubt about the possibility of choosing an interval \(\Delta t\gg\Delta\tau\) during which the change in

\[ \frac{\Delta\nu}{\nu} \]

may be neglected. Moreover, in the experiment it will be not \(\Delta\nu(t)\) that is measured, but the change in the phase difference between the incoming radiation and the radiation of the terrestrial standard, so that no spectral decomposition in explicit form need be carried out.

is taken into account, whereas the quadratic effect (or, in a system connected with the rotating Earth, the influence of rotation on \(g_{ik}\)) is of the order

\[ \frac{v_0^2}{c^2}\sim 2\cdot 10^{-12}, \]

where \(v_0=4.6\cdot 10^4\) is the velocity of the Earth’s surface at the equator. At the same time, for a satellite at a height \(h=800\ \text{km}\) from the Earth,

\[ \frac{v^2}{c^2}\simeq 6\cdot 10^{-10} \]

and

\[ \frac{gh}{c^2}\sim 10^{-10}, \]

i.e. the term of order \(\frac{v_0^2}{c^2}\) may be neglected. In this example (\(h=800\ \text{km}\)), on the other hand, the quadratic Doppler effect is already four times greater than the gravitational effect, whence it is clear that separating out the gravitational frequency shift is far from a simple task, even if the frequency itself can be measured with sufficient accuracy.

In this connection it is clear that, for measuring the gravitational frequency shift, it is advantageous to use distant satellites, in passing to which the Doppler effect decreases and the gravitational shift increases. Especially convenient in this respect is a satellite stationary relative to the Earth’s surface, i.e. one making one revolution in 24 hours in the equatorial plane (such a satellite will be at a distance \(6.6r_{\oplus}\) from the center of the Earth and have a velocity \(v\simeq 3\cdot 10^5\); the quadratic Doppler effect in this case is

\[ \frac{v^2}{2c^2}=5\cdot 10^{-11} \]

and the gravitational shift

\[ \frac{\Delta\nu}{\nu}=6\cdot 10^{-10} \]
).

The difference between the proper time measured by clocks and the coordinate time can be detected not only by the frequency shift, but also in the following way. Let clocks 1 and 2 at the moment \(t=0\) be at rest on the Earth (gravitational potential \(\varphi_2\)), and then clocks 1 begin to move with velocity \(v_1(t)\) in a region with potential \(\varphi_1\) (i.e. clocks 1 are placed on an artificial Earth satellite). Then, if at the moment \(t\) (in coordinate time) clocks 1 are again at rest and have returned to point 2 (i.e. clocks 1 have been brought down from the satellite to the Earth), they will show a different time from clocks 2. Namely, clocks 2 will show the time

\[ \tau_2=\left(1+\frac{\varphi_2}{c^2}\right)t, \]

whereas clocks 1 the time

\[ \tau_1=\left(1+\frac{\varphi_1}{c^2}-\frac{v_1^2}{2c^2}\right)t, \]

where, for simplicity, we assume that the satellite moves throughout with velocity \(v_1\) in a region with potential \(\varphi_1\) (i.e. the orbit is considered as ...).

circular, and the time of descent and ascent is neglected). From the expressions given for $\tau_1$ and $\tau_2$, excluding the auxiliary coordinate time $t$, with the adopted accuracy we obtain:

\[ \frac{\tau_2-\tau_1}{\tau_1}\simeq \frac{\tau_2-\tau_1}{\tau_2} = \frac{\varphi_2-\varphi_1}{c^2} + \frac{v_1^2}{2c^2}. \tag{22} \]

If in this expression one neglects the difference $\varphi_2-\varphi_1$, which is permissible for a satellite sufficiently close to the Earth, then one obtains the well-known result for the lagging of “traveling clocks,” indicated by Einstein already in his fundamental paper on the special theory of relativity[^25].

For a satellite close to the Earth,

\[ \frac{\tau_2-\tau_1}{\tau_2}\sim 2\cdot 10^{-10}, \]

and over a year the lag of the clock will be $\tau_2-\tau_1\sim 6\cdot 10^{-3}$ sec. It is not excluded, in principle, that observation of such an integral effect will also prove possible.

§ 3. DEFLECTION OF LIGHT RAYS PASSING NEAR THE SUN

In a gravitational field the “coordinate velocity of light” $c'$, determined from the condition $ds=0$, depends on $g_{ik}$, and in a weak field

\[ c'=c\left(1+\frac{2\varphi}{c^2}\right), \]

where $c=3\cdot 10^{10}$ is the speed of light in the absence of a field. Therefore, as is immediately clear from Huygens’ principle, or simply by analogy with the case of propagation of light in an inhomogeneous refracting medium, in an inhomogeneous gravitational field light rays will be bent. Specifically, for rays passing near the Sun, the bending will be directed toward the Sun, and the stellar field near the Sun photographed during a total solar eclipse will be deformed in comparison with the same field photographed in the absence of the Sun (the latter can be done, say, half a year after the eclipse). This effect, like the gravitational shift of frequencies, was predicted by Einstein already in his first papers on the theory of gravitation[^12],[^13], and for the deflection of a ray passing at a distance $R$ from the center of the Sun the expression was obtained[^13]

\[ \alpha=\frac{|2\varkappa M_{\odot}|}{c^2 R} = 4.24\cdot 10^{-6}\frac{r_{\odot}}{R} = 0'',87\,\frac{r_{\odot}}{R}. \tag{23} \]

Subsequently, after the creation of the general theory of relativity, it became clear[^4],[^5] that the effect of deflection of rays must be twice as גרויס

EXPERIMENTAL VERIFICATION OF THE GENERAL THEORY OF RELATIVITY

greater*) and thus, according to the theory,

\[ \alpha=\frac{4\chi M_\odot}{c^2 R}=8.48\cdot 10^{-6}\frac{r_\odot}{R}=1''.75\frac{r_\odot}{R}, \tag{24} \]

i.e. the deflection of a ray reaches, at the solar limb, 1.75 seconds of arc (a more exact value is \(1''.745\)).

It is curious to note that expression (23) for the deflection of light rays was obtained as early as 1801 by Soldner (his paper is reprinted in \(^{20}\)) on the basis of ideas about light corpuscles and classical mechanics. This result, strange at first glance, is obtained very simply. An elementary calculation, leading to Rutherford’s well-known formula for the scattering of \(\alpha\)-particles by nuclei, shows that a particle of mass \(m\), moving in a field with potential energy \(-\dfrac{\beta}{r}\), is deflected through an angle \(\alpha\) determined by the formula

\[ R_\infty^2=\frac{\beta^2}{m^2 v_\infty^4}\operatorname{ctg}^2\frac{\alpha}{2}, \]

where \(R_\infty\) is the impact parameter and \(v_\infty\) is the velocity of the particle at infinity. If

\[ \frac{m v_\infty^2}{2}\gg \frac{\beta}{R_\infty}, \]

then the deflection is small (i.e. \(\operatorname{tg}\dfrac{\alpha}{2}\simeq\dfrac{\alpha}{2}\)); along the entire trajectory \(v\simeq v_\infty\), and, moreover, \(R\simeq R_\infty\), where \(R\) is the closest distan-

*) Value (23) was obtained \(^{13}\) on the basis of the expression for the speed of light

\[ c'=c\left(1+\frac{\varphi}{c^2}\right), \]

which results when the influence of the gravitational field only on the course of time is taken into account. In the complete theory, the change in the spatial metric is also taken into account (the non-Euclidean character of space in a gravitational field). In this case, in a weak static gravitational field,

\[ ds^2=\left(1+\frac{2\varphi}{c^2}\right)c^2dt^2-\left(1-\frac{2\varphi}{c^2}\right)dr^2, \]

whence for a light ray (i.e. under the condition \(ds=0\)) one obtains the expression indicated in the text

\[ c'=c\left(1+\frac{2\varphi}{c^2}\right), \]

leading to (23). To avoid misunderstandings it is necessary to emphasize that the “coordinate speed of light” \(c'=\dfrac{dr}{dt}\) (under the condition \(ds=0\)) has no direct physical meaning, since the speed of light measured experimentally in a small neighborhood of any point is always equal to \(\dfrac{dl}{d\tau}=c=3\cdot 10^{10}\), where \(dl\) is the spatial distance and \(d\tau\) is the interval of proper time.

...between the particles and the scattering center. As a result,

\[ \alpha \approx \frac{2\beta}{m v^2 R} \]

or, for the gravitational field of the Sun,

\[ \alpha=\frac{2\varkappa M_{\odot}}{v^2 R}, \]

where the mass of the particle \(m\) has dropped out owing to the assumption of the equality of inertial and gravitational masses. Putting \(v=c\) in the formula obtained, we also obtain expression (23). The possibility of obtaining this expression without any assumptions about the mass of light corpuscles is connected, obviously, with the equality of gravitational and inertial masses, and also with the application of the classical law of motion up to the velocity \(v=c\).

The experimental verification of formula (24) was carried out during a whole series of total solar eclipses, beginning in 1919. A summary of the corresponding results is given in Table III, where all

Table III

Deflection of light rays in the field of the Sun (theoretical value \(\alpha_{\max}=1'',75\))

Observers	Year	Angle of deflection	Reference to literature
Crommelin and Davidson	1919	\(1'',98\pm0'',18\)	14
Eddington and Cottingham	1919	\(1'',61\pm0'',45\)	14
Campbell and Trumpler	1922	\(1'',78\pm0'',17\)	14
Dowdell and Kennedy	1922	\(1'',77\)	27
Freundlich	1929	\(2'',24\)	27
Mikhailov	1936	\(2'',73\pm0'',31\)	28
Van Biesbroeck	1947	\(2'',01\pm0'',27\)	29
Van Biesbroeck	1952	\(1'',70\pm0'',10\)	30
Mean		\(1'',98\pm0'',12\)
Mean without the largest value		\(1'',87\pm0'',08\)

values are referred to the edge of the solar disk. We do not indicate the sign of the deflection, since in all cases it coincides with the theoretical one (the ray is “attracted” to the Sun). The error of the mean values indicated at the bottom of Table III is simply the root-mean-square error without taking into account the accuracy of the mean values \(\alpha\) obtained in the individual works. The basic error of the observations, which, apparently...

because of this, may exceed the indicated errors, is associated with the inaccuracy of taking refraction in the Earth’s atmosphere into account or, more precisely, the difference in refraction during the eclipse and on the control day. From this point of view Mikhailov’s measurements^28, which give the greatest deviation from the theoretical value, were carried out under unfavorable conditions (during the eclipse the air temperature was \(+23.6^\circ\mathrm{C}\), and on the control day \(21^\circ\mathrm{C}\); as a result the differential refraction reached \(0''.85\)). The latest observations^29,30 lead to values of \(\alpha\) that agree with the theoretical ones to within the experimental error. The same applies to the data of Campbell and Trumpler (see Table III), which, according to a kind communication from A. A. Mikhailov, are the most reliable of those known. Hence, as also follows from all of Table III, the prediction of the general theory of relativity concerning the deflection of light rays in the field of the Sun has been confirmed: an effect has been found which is certainly larger than the “classical” value (23) and agrees with the theoretical prediction to an accuracy of approximately 10% *).

The acquisition of more accurate experimental data is, of course, of interest; unfortunately, in this respect there appears to be no way to radically improve the accuracy, since radio methods (for example, observation of cosmic radio emission), owing to their relatively low angular resolving power, are quite unsuitable here.

§ 4. THE SIGNIFICANCE OF EINSTEIN’S GENERAL THEORY OF RELATIVITY FOR PHYSICS AND ASTRONOMY

The effects of the general theory of relativity discussed above are very small, and their significance, taken by themselves, is not great in astronomy. This circumstance may lead to an underestimation of the role of the general theory of relativity. We therefore did not consider it superfluous to make below a few remarks on the significance of the general theory of relativity for physics and astronomy.

The general theory of relativity is, above all, a relativistic theory of gravitation, a theory of the gravitational field. Only with the creation of this theory did the edifice of non-quantum physics acquire a certain completeness, since the Newtonian theory of universal

*) It should be noted that in processing the experimental data one starts from the law
\[ \alpha=\frac{Cr_\odot}{R} \]
and determines the constant \(C=\alpha_{\max}\), which is also given in Table III. As for checking the dependence of \(\alpha\) itself on \(R\), which is also necessary for a complete comparison of theory with experiment, the available data are still insufficient for this purpose. We note, however, that the influence of the Sun’s rotation on the deflection of rays can be neglected, since this effect is smaller than the gravitational deflection (24) by an amount of order
\[ \frac{v_0}{c}\sim 10^{-5}, \]
where \(v_0\sim 2\cdot 10^5\) is the velocity of the solar surface at the equator.

gravitation, while preserving action at a distance, is not in accord with the foundations of field theory and the special theory of relativity (this has already been discussed in the introduction). At the same time, the transition to the theory of the gravitational field, although analogous to the transition from electrostatics to electrodynamics, is connected with considerably deeper generalizations and, in particular, made it possible to reflect in the very foundation of the theory the long-known fact of the equality of inertial and gravitational mass. Space and time, which before the creation of the general theory of relativity had been regarded as independent of matter and unconnected with it, became in this theory inseparable from matter. The metaphysical Newtonian absolute space and absolute time disappeared from physics; in place of “emptiness,” or space “in itself,” there arose the gravitational field \(g_{ik}\). Celestial bodies (and, in fact, any bodies), the electromagnetic and other fields determine the character of the gravitational field and at the same time the space-time metric. In the presence of masses—and they are always present—the geometry of space is already non-Euclidean and depends on the magnitude, distribution, and motion of these masses. Therefore the question of the geometry of our real space (as distinct, of course, from the geometry of the innumerable multitude of conceivable spaces of any number of dimensions) becomes a physical question. Thus, in the general theory of relativity there found their reflection and, within certain limits, their completion the ideas on the connection of geometry with physics and with experiment, ideas going back to Lobachevsky, Gauss, and Riemann.

Here we must confine ourselves to these few words of a general character, since an exposition of the fundamental questions of space-time, geometry, and field theory in their connection with the general theory of relativity goes far beyond the scope of the present article. We shall therefore dwell briefly on only three more particular points: the methodological significance of the general theory of relativity from the standpoint of theoretical physics, the role of the general theory of relativity in atomic physics and in the theory of elementary particles, and, finally, its significance for astronomy and, in particular, for cosmology.

The methodological significance of the general theory of relativity is connected mainly with the fact that it is a very perfect field theory (what is meant is a non-quantum field theory). The latter is manifested not only in the possibility of freely using a broad class of the most diverse coordinates, but first of all in the following fact: the equations of motion of the masses producing the field in the general theory of relativity follow from the field equations themselves*).

*) In electrodynamics, as is known, this is not so: from Maxwell’s equations there follows the continuity equation, but the equations of motion of charges are independent. As a result, for example, the field of two charges at rest satisfies Maxwell’s equations, although it is incompatible with the equations of motion, since because of their interaction the charges cannot remain at rest.

EXPERIMENTAL VERIFICATION OF THE GENERAL THEORY OF RELATIVITY

This is explained by the fact that from the equations of the gravitational field there automatically follows the equality to zero of the covariant divergence of the energy-momentum tensor \(T_j^k\), i.e. the equality \(T_{i;k}^k=0\) (in Galilean coordinates this equality has the primary form: \(\dfrac{\partial T_{ik}}{\partial x_k}=0\)). But the equality \(T_{i;k}^k=0\) is nothing other than the law of conservation of the energy and momentum of “matter,” i.e. of substance and the electromagnetic field, and thus the equations of motion of particles of substance and Maxwell’s equations, containing points and charges, follow from the equation for the gravitational field*).

Further, in the general theory of relativity the energy-momentum tensor, when obtained from the variational principle, automatically turns out to be symmetric (see, for example, \(^{3}\), § 93). This makes it convenient to obtain the expression for \(T_{ik}\) also in the absence of a gravitational field, using the same method as in the general theory of relativity (otherwise the tensor \(T_{ik}\) may turn out to be nonsymmetric and must be additionally symmetrized; see \(^{3}\), § 31).

An analogous device (the introduction of arbitrary \(g_{ik}\) with a transition to the Galilean metric at the very end) makes it possible to derive the equations of motion from the field equations in linear field theories \(^{34}\). Incidentally, the nonlinear character of the equations for the gravitational field \(g_{ik}\) is also an essential feature of these equations in comparison, say, with Maxwell’s theory. In this connection, the experience accumulated in the study of the equations for \(g_{ik}\) proves very valuable when working with other nonlinear equations.

Without risking dwelling further on methodological questions**), we shall emphasize only the great influence which the general theory of relativity has had on the development of geometry and tensor analysis.

*) The result just cited is contained, in essence, already in Einstein’s fundamental work \(^{5}\), where, nevertheless, the equation of motion of a material point in a gravitational field (the geodesic equation) is introduced independently. Subsequently it was established with complete definiteness that the “postulate of the geodesic” is superfluous (see, for example, \(^{31}\)), and also, starting from the equations of the gravitational field, the equations of motion of a system of bodies were derived in various approximations (\(^{32, 32a, 33}\); see also \(^{2}\), Ch. XV).

**) As a quite elementary example of the use of the “principle of equivalence” for methodological purposes, let us point to the following problem. In considering electron-inertia experiments with metals, it is necessary to write the Schrödinger equation for the wave function of electrons in a uniformly accelerated coordinate system; although this presents no difficulty, it requires known calculations, since a gradient transformation \(^{35}\) has to be performed on the \(\Psi\)-function. At the same time the answer is immediately clear \(^{36}\) on the basis of the “principle of equivalence,” accord-

V. L. GINZBURG

Let us turn to the question of the significance of the general theory of relativity for atomic and nuclear physics and the theory of elementary particles. As is well known, gravitational forces are negligibly small in comparison with the electric forces acting in atoms, and also with the electric and nuclear forces between particles in atomic nuclei. Thus, the energy of the electrostatic interaction of two particles with charges \(e\) and masses \(m\) is equal to \(U_{\mathrm{el}}=\dfrac{e^2}{r}\), while the energy of their gravitational interaction is equal to \(U_{\mathrm{gr}}=-\dfrac{\chi m^2}{r}\), whence the ratio of the energies or forces of the two types, in absolute value, is equal to

\[ \frac{U_{\mathrm{gr}}}{U_{\mathrm{el}}}=\frac{\chi m^2}{e^2}= \begin{cases} 2.4\cdot 10^{-43} & \text{(for electrons)},\\ 8.1\cdot 10^{-37} & \text{(for protons)}. \end{cases} \tag{25} \]

The same can be expressed by noting that the gravitational radii of the electron and proton

\[ \rho_e=\frac{\chi m_e^2}{c^2}=6.75\cdot 10^{-56}\ \text{cm} \]

and

\[ \rho_p=\frac{\chi m_p^2}{c^2}=1.24\cdot 10^{-52}\ \text{cm} \]

are incomparably smaller than their classical electromagnetic radii

\[ r_e=\frac{e^2}{m_e c^2}=2.8\cdot 10^{-13}\ \text{cm} \]

and

\[ r_p=\frac{e^2}{m_p c^2}=1.5\cdot 10^{-16}\ \text{cm}. \]

Because of the smallness of gravitational forces, they can be completely disregarded both in atoms and in nuclei; thus, for atomic and nuclear physics, the theory of gravitation has no direct significance.* The same can be said—

\[ \text{———————} \]

—of motion of a particle in a frame of reference with acceleration \(a\), in exactly the same way as in a gravitational field with potential \(\varphi=-az\) (the \(z\)-axis is directed along \(a\)); therefore the Schrödinger equation in an accelerated frame differs from the usual one only by the addition of the term \(maz\), i.e., of the potential energy of a particle of mass \(m\) in a field with potential \(az\).

*) In other words, one may use Galilean coordinates and the special theory of relativity valid in these coordinates, or, at small velocities, the corresponding nonrelativistic theory. What has been said also applies to the quantum domain, where, as applied to atoms and nuclei, space–time may be regarded as Galilean. Let us note, incidentally, that in analyzing thought experiments carried out when discussing the foundations of nonrelativistic quantum mechanics, there was nevertheless a case in which it was necessary to take into account\(^{37}\) the change in the rate of clocks in a gravitational field.

...one might assert also with regard to the theories of elementary particles, i.e., in the question of the structure and properties of the electron, proton, neutron, etc. In general this is indeed so: gravitational forces, even at very small distances, are negligible in comparison with electromagnetic ones, but nevertheless no categorical assertions can be made here. The point is that in the case of an elementary charged particle it is unclear what distances are essential, and at distances of the order of the gravitational radius of the particle one can no longer use the Newtonian approximation

\[ U_{\mathrm{gr}}=-\frac{\varkappa m^{2}}{r}, \]

which leads to (25). Moreover, at very small distances, when the fields are extremely large, the equations of gravitation usually used in the general theory of relativity may prove unsuitable. All this at one time, soon after the creation of the general theory of relativity, gave rise to well-known hopes that this theory and its generalizations would prove fruitful in solving the problem of the structure of elementary particles.

As a result, beginning with Einstein’s work of 1919 (see \(^{38}\)), a whole trend in theoretical physics arose, usually called an attempt to construct unified field theories\(^*\). However, along this path no results have yet been achieved that have direct physical significance (what is meant are new conclusions consistent with experiment, or the explanation of any known facts, and not hopes and expectations, of which there have been many). Therefore we shall not dwell here on unified field theories, some of which are distinguished by great mathematical elegance, and shall confine ourselves to references to the corresponding literature (see \(^{1}\), Chap. V, \(^{2}\), Part III, \(^{39}\), Appendix II, and \(^{40}\)). It must be thought that the failures of unified field theories are not accidental, but are connected with the fact that, within the framework of these theories, attempts are made to solve the problem of elementary particles entirely without invoking quantum concepts or, at any rate, on a classical foundation.

Meanwhile, quantum effects set in at distances of the order

\[ \frac{\hbar}{m_e c}=3.85\cdot 10^{-11} \]

for the electron and

\[ \frac{\hbar}{m_p c}=2.1\cdot 10^{-14} \]

for protons and neutrons, i.e., at distances substantially exceeding even the classical electromagnetic radii of these particles, not to mention their gravitational radii. Therefore it appears highly probable that the problem of elementary particles can be solved only within the framework of quantum theory, and not by starting from a classical (nonquantum) field theory.

\(^*\) The term “unified field theory” is connected with the fact that in theories of this type two fields appear (the gravitational and electromagnetic), which one tries to connect with each other as closely as possible, usually treating them as a certain unified field with the corresponding number of components.

What has been said does not yet mean that taking gravitational effects into account in the theory of elementary particles must necessarily be unnatural, since it appears possible to consider these effects also in quantum field theory. Moreover, in the quantum theory of the electron and in quantum electrodynamics there are even weightier grounds for bringing in gravitational effects than in the classical theory. The point is that in classical electrodynamics the self-energy of a charged sphere of radius \(r_e\) (the “classical model” of the electron) is of the order of \(\dfrac{e^2}{r_e}\), and, if the mass of the electron is electromagnetic,

\[ \frac{e^2}{r_e} = mc^2, \]

whence one obtains the expression for the “classical radius” of the electron

\[ r_e = \frac{e^2}{m_e c^2}. \]

In quantum field theory there occur such effects as vacuum polarization and zero-point oscillations of the field, as a result of which the electron’s own electromagnetic energy is of the order of

\[ m_e c^2 \cdot \frac{e^2}{\hbar c}\cdot \ln \frac{\hbar}{m_e c r'_e}, \]

where \(\hbar = 1.05\cdot 10^{-27}\) is Planck’s constant (divided by \(2\pi\)) and \(r'_e\) is the “quantum radius” of the electron (for details see, for example, \(^{41}\)). Equating the indicated self-energy to the rest energy \(m_e c^2\), we obtain for \(r'_e\) the expression

\[ r'_e \sim \frac{\hbar}{m_e c}\, e^{-\frac{\hbar c}{e^2}} \sim 10^{-70}\ \text{cm}. \tag{26} \]

Since \(r'_e\) is smaller than the gravitational radius \(\rho_e \sim 10^{-55}\ \text{cm}\), it is clear that the entire discussion carried out without taking the gravitational field into account is illegitimate, and one might think that gravitation is essential for the theory of the electron (see, in particular, \(^{42,43}\)). This conclusion, however, is by no means indisputable, since the electron interacts not only with electromagnetic and gravitational fields, but also with meson fields. In this case the interaction with the meson field is considerably stronger than the gravitational one, owing to which gravitational effects would seem, after all, to be quite insignificant (see, for example, \(^{44}\)). On the other hand, it is not excluded, although it seems to us extremely improbable, that the meson effects somehow compensate one another or prove to be “ineffective” in calculating the self-energy, thereby leaving room for the action of gravitation.

Even if not gravitation itself, the methods and apparatus of the general theory of relativity are of interest for the theory of elementary particles from yet another point of view. At present the opinion is becoming more and more generally accepted that the solution of the fundamental problems of the theory of elementary particles, this principal task of theoretical physics, will require a radical revision of our ideas about space and time in the region of microscopic scales—

bov \(l\sim \dfrac{\hbar}{m_p c}\sim 10^{-14}\ \mathrm{cm}\) and \(t\sim \dfrac{l}{c}\sim 10^{-25}\ \mathrm{sec}\). Such a program appears natural, since up to now, in the domain of microphenomena, spatio-temporal representations, concepts, and images borrowed from macroscopic physics have been used without restriction, yet they may prove inadequate to reality in the transition to quite different scales. Along what path development will proceed here is entirely unclear, but it seems possible that the introduction of appropriately generalized methods of the general theory of relativity will prove especially fruitful in this respect, since the spatio-temporal representations associated with this theory are the richest of those known*).

Thus, the general theory of relativity, understood sufficiently broadly, turns out to be connected with modern research in the theory of elementary particles, but only the future will show how deep and essential these connections are.

The situation is quite different in astronomy and, in particular, in cosmology, where the outstanding role of the general theory of relativity is beyond doubt. This is, of course, explained simply by the fact that in the Universe gravitational forces play the principal role. True, within the limits of the solar system and on stars the gravitational field is weak (see condition (2)), as a result of which relativistic effects, though perceptible, are nevertheless relatively small (even on the surface of the white dwarf Sirius B \(\dfrac{|\varphi|}{c^2}\sim 10^{-4}\)**). But when large regions of the Universe are considered the situation changes. The most powerful modern telescope (mirror diameter \(5\ \mathrm{m}\)) makes it possible to obtain certain information about regions separated from us by \(\sim 2\cdot 10^9\) light-years and makes it possible to study the red shift in the spectra of extragalactic nebulae (see below) out to distances \(\sim 10^9\) light-years \(=10^{27}\ \mathrm{cm}\). Further, the mean density of matter \(\mu_0\) in regions containing many nebulae, according to the available data, is approximately the same throughout the entire part of the universe accessible to observation***), and \(\mu_0\sim 10^{-28}\div 10^{-29}\ \mathrm{g/cm^3}\). Hence it is clear that already in the region of space now accessible to observation one must, generally speaking, use the general theory of relativity, since for such a region with radius

*) In this connection see, for example, \(^{46}\).

**) The exception is constituted by “neutron cores” (central regions of a star consisting of neutrons), which appear in certain stellar models and possess radii so small that the gravitational field is strong and the treatment must, generally speaking, be carried out on the basis of the general theory of relativity \(^{46}\). The question of whether neutron cores exist in any existing stars has not yet been clarified.

***) The assertion that the mean density of matter in the universe is everywhere the same is sometimes called the “cosmological principle.”

$R \sim 2 \cdot 10^{27}\ \text{cm}$ and density $\mu_0 \sim 5 \cdot 10^{-29}\ \text{g}/\text{cm}^3$, the gravitational radius

\[ \rho=\frac{\varkappa M}{c^2}\sim \frac{4\varkappa \mu_0 R^3}{c^2}\sim 10^{26} \]

is comparable with the radius of the region and, consequently, the gravitational field is not weak.

Thus, in analyzing cosmological questions, i.e., questions concerning the structure of the Universe on large scales, it is necessary to rely on the general theory of relativity.

Not being able to set forth here even the foundations of relativistic cosmology,* we shall confine ourselves to only a few remarks. Besides the homogeneity of the distribution of the mean density of matter in space (see above), the principal experimental cosmological fact is the red shift of lines in the spectra of extragalactic nebulae, which is proportional to the distance to the nebula (Hubble’s law, 1929):

\[ \frac{\Delta\lambda}{\lambda}=\frac{\lambda-\lambda_0}{\lambda_0}=\frac{v_{\mathrm{eff}}}{c}=\frac{H}{c}\,r, \tag{27} \]

where $\lambda_0$ and $\lambda$ are the wavelengths of some spectral line, respectively, for a terrestrial source and in the spectrum of the nebula; $v_{\mathrm{eff}}$ is the effective velocity of the nebula (the radial velocity that it would have if the shift of the lines were explained by the Doppler effect), $r$ is the distance from the Earth to the nebula, and $H$ is the Hubble constant. According to present-day data,${}^{50}$ $H=6.47\cdot 10^{-18}\ \text{sec}^{-1}$, i.e.,

\[ \frac{1}{H}=4.9\cdot 10^9\ \text{years}, \]

or, according to${}^{51}$,

\[ \frac{1}{H}=5.4\cdot 10^9\ \text{years} \]

with an accuracy of $\pm 20\%$ (only a few years ago a value of $\dfrac{1}{H}$ smaller by a factor of 2–3 was given, which was connected with an incorrect determination of the distances to the nebulae).

The cosmological red shift has been measured out to distances of $\sim 10^9$ light-years, where it corresponds to the colossal velocity

\[ v_{\mathrm{eff}}\cong 6\cdot 10^9\ \frac{\text{cm}}{\text{sec}}=\frac{c}{5}. \]

Attempts have repeatedly been made to explain this red shift by some processes occurring with light on its way from the nebula to the Earth (“aging” of photons). However, all these attempts have led to no results, and it may be asserted that the cosmological red shift

* The application of the general theory of relativity to the investigation of the cosmological problem was begun by Einstein${}^{47}$ and then continued by a whole series of authors, among whom the most substantial results were obtained by A. A. Friedmann.${}^{48}$ The significance of A. A. Friedmann’s work is already evident from the fact that almost the entire exposition of relativistic cosmology in his book${}^{39}$ Einstein calls “nothing other than an account of Friedmann’s ideas.” One may become acquainted with relativistic cosmology from the following sources: ${}^{39}$ appendix I,${}^{3}$ §§ 102–105,${}^{24,49,50}$.

cannot be explained by any effects known in physics that occur during the propagation of light. Therefore it seems highly probable that the red shift is due to the Doppler effect and indicates a real “recession” of the nebulae*). It follows from this that the region of the Universe observed by us is in a distinctly nonstationary state, and that approximately 5 billion years ago, when the nebulae had not yet “managed to scatter,” the physical conditions in the indicated region were substantially different from those obtaining at the present time. This conclusion does not contradict the data on the solar system, since its age does not exceed 2–5 billion years (for example, the age of meteorites is \(2 \div 4 \cdot 10^9\) years \({}^{56}\)). On the other hand, the existence in the past of physical conditions different from the present ones makes it possible to approach the question of the formation of the elements and to understand why naturally radioactive substances have survived to the present time, despite the fact that their lifetime usually does not exceed several billion years (for uranium the half-life is \(T = 4.5 \cdot 10^9\) years). True, there are attempts to explain the origin of the elements without connection with cosmology, but this question is not the place to discuss here, especially since the sole purpose of what has been said is to point to the existence of facts and arguments that convincingly speak in favor of the hypothesis of the nonstationarity of the known part of the Universe. Further verification of this supposition is, of course, necessary and is being carried out; but it seems completely inadmissible to reject it on the basis of some a priori considerations or on the grounds that nonstationary cosmology has sometimes been used for fideistic conclusions having nothing in common with its essence. To reject ideas about the nonstationarity of the Universe only because these ideas are used for unscientific conclusions is as little justified as, for example, denying quantum mechanics because “conclusions” about “free will,” etc., having no relation to it, have been drawn from it.

A major success of the general theory of relativity in the light of what has been said is the following circumstance. If one considers a space uniformly filled with matter, then the solution of the equations of the general theory of relativity, into which no

*) The supposition of the Doppler character of the cosmological red shift admits experimental verification \({}^{50,52,53}\), connected, first of all, with observation of deviations from the linear law (27) and with the Doppler change in the duration of various processes in remote nebulae, for example, a change in the period of Cepheid oscillations and in the time of the bright glow of supernovae. Unfortunately, progress in the field of extremely difficult experimental investigations of the cosmological red shift is proceeding very slowly; nevertheless, further successes may now be expected here in connection with the use of the 5-meter telescope \({}^{54}\) and the involvement of radio-astronomical data \({}^{53,55}\).

no changes automatically turns out to be nonstationary and is in agreement with observations (under the interpretation of the cosmological redshift as the “recession” of nebulae*).

It is not surprising that this fact attracts special attention in relativistic cosmology.

Thus, in cosmology the general theory of relativity is not only widely used, but has already led to great success, despite the fact that in this field, in essence, only the first steps have been taken. There can be no doubt that further progress here is possible only on the basis of the general theory of relativity.

Since this article was written only a few months after the life path of the great creator of the theory of relativity came to an end, I should like, in conclusion, to say a few words about the distinctive features of his scientific creativity.

The first feature, well known to all, consists in the boldness and depth of Einstein’s ideas, which brought about a genuine revolution in physics. How long ago this was recognized can be seen, for example, from the fact that as early as 1910 M. Planck called Einstein “the Copernicus of the twentieth century.”

The second feature is the breadth and variety of his interests, and his almost simultaneous work in many areas of physics. Thus, in 1905 alone there appeared the special theory of relativity, the hypothesis of light quanta, and the theory of Brownian motion. Or another example: in 1907 Einstein constructed the first quantum theory of the heat capacity of solids, worked on the theory of Brownian motion, and began the construction of the general theory of relativity. In addition to the universally known works on the theory of relativity, quantum theory, and the theory of Brownian motion, Einstein is responsible for: the theory of light scattering in liquids, the foundations of the theory of sound dispersion in gases, the idea of gyromagnetic experiments, and a whole series of other investigations.

*) For more details and more precision, see the literature cited; in particular, see §§ 102–105, where certain reservations are made concerning the possibility of comparing the nonstationary solutions in question with reality. Nonstationary solutions (they were first found by Friedman) have, corresponding to the beginning of expansion, special points at which the energy density is infinite. This point, which gives rise to objections of various kinds, in Einstein’s authoritative opinion (see 39, Appendix I), testifies only to the limited applicability of the equations of the general theory of relativity at extraordinarily large energy densities (the quantum considerations also point to this). Thus, there are no grounds for being “afraid” of singularities and therefore objecting to the application of relativistic cosmological solutions in regions with very large, but finite, energy density—which is, in fact, precisely what is at issue.

EXPERIMENTAL VERIFICATION OF THE GENERAL THEORY OF RELATIVITY

The third feature is the exceptional completeness of Einstein’s works. From the history of physics many examples are known in which the content of major scientific discoveries was not fully realized by their authors, when the most important new results at first remained only partially understood. Particularly close to our own time as illustrations of what has been said may be the first works of de Broglie, Heisenberg, and Schrödinger on quantum mechanics (for example, even in the clearest of the aforementioned works by Schrödinger, the meaning of the \(\Psi\)-function for which Schrödinger established his wave equation remained unclear). The situation is different with Einstein’s works. In the work on the special theory of relativity \(^{25}\) this theory is immediately set forth in completed form; everything in the article is correct and, reading it 50 years after its publication, one sees that even now one can become acquainted with and study the theory of relativity from it; one sees how little has been added in these questions by later investigations. Completeness and a classical style of exposition distinguish the majority of Einstein’s other works as well, although, of course, he did sometimes err and follow an incorrect path, as Einstein himself emphasized more than once (see, for example, \(^{57}\), p. 285). What has been said applies fully also to the general theory of relativity. In the work of 1916 (see \(^{5}\)) this theory is set forth by its author almost in the same way as it is presented in most courses at the present time. Despite the 40 years that have passed, in this work essentially nothing needs to be changed, and the additions that could be made have very little relative weight (we are no longer speaking of the fact that these “additions” to the theory belong essentially to Einstein himself*).

Here one should make only one further remark.

The basic physical proposition in the general theory of relativity is the “principle of equivalence,” based on the experimentally established equality of inertial and gravitational mass and leading to the description of the gravitational field by means of the metric tensor \(g_{ik}\). At the same time, in constructing the theory another proposition also played a very large role, one bearing primarily a methodological and heuristic character, namely the “principle of general covariance,” or the “principle of general relativity,” as

*) The additions meant here reduce mainly to several clarifications \(^{58,59}\) and to the investigation of questions on gravitational waves \(^{60}\) and on obtaining the equations of motion from the field equations (see above). As for the application of the general theory of relativity to cosmology and attempts to construct a unified field theory, here one has to speak of new directions, and not of a completion of the theory. At the same time, as has already been emphasized earlier, the unified field theory, with which Einstein was chiefly occupied in the last period of his life, did not prove fruitful.

it is often called Einstein’s*). The use of the latter name for the principle of general covariance also led to the fact that the theory of the gravitational field, constructed on the basis of the special theory of relativity and the principle of equivalence, was called the general theory of relativity. In his 1916 work^5 Einstein may have somewhat overestimated the significance of the principle of general covariance and, in any case, did not emphasize that this principle by itself is not physical. However, soon afterward, in response to certain critical remarks,^61 Einstein fully agreed that the principle of general covariance has mainly heuristic significance.^59 And only the combination of the principle of general covariance with the principle of equivalence, by virtue of which the gravitational field is described only by the quantities \(g_{ik}\), proves to be the key to constructing the equations of the theory of the gravitational field. But this is precisely how Einstein actually proceeded, as a result of which the subsequent discussion of the foundations of the general theory of relativity led, on the whole, only to a clarification of certain concepts and formulations that were of secondary importance, but not to a change in the essence of the matter**).

There is no doubt that Einstein’s general theory of relativity will remain through the ages as one of the greatest achievements of human thought.

*) In the 1916 work this principle is formulated as follows:^5 “The laws of physics must be so constructed that they are valid for arbitrarily moving coordinate systems.” In the same work, as well as later, this statement was usually used in the form of the broader requirement that “the laws of nature be covariant with respect to arbitrary continuous transformations of the coordinates.” Einstein also calls this principle of general covariance the general principle of relativity, as it seems to us, to a considerable extent because the special principle of relativity was formulated by him in a related form (for example, in^5 the special principle or postulate of relativity is called the following assertion: “If the coordinate system \(K\) is chosen so that, with respect to it, the physical laws are valid in their simplest form, then the same laws are valid as applied to any other system \(K'\), which is in uniform translational motion with respect to \(K\)”).

**) In view of all that has been said, the author cannot agree at all with the opinion of V. A. Fok, who says that “the theory of gravitation was incorrectly understood by its author.”^63 At the same time, of course, in the modern presentation of the theory of relativity it is by no means necessary in everything to adhere to Einstein’s exposition, and separate propositions of the theory should naturally be illuminated somewhat differently, or even substantially otherwise, than was done in some of Einstein’s early works.

Among the latter propositions is the “principle of the relativity of inertia,” or “Mach’s principle,” to which Einstein at one time attached great importance (according to this principle, the field \(g_{ik}\) must be completely determined by matter).^47,^59 The equations of the gravitational field used do not satisfy this principle and, as Einstein himself later emphasized (see, for example,^39 Appendix I), there are now no grounds for changing these equations in the spirit of the principle of the relativity of inertia, against which a number of objections can be raised.

CONCLUSION

The results of the experimental test of the general theory of relativity may be summarized as follows (we leave cosmological questions aside here; see § 4).

The data concerning the motion of Mercury’s perihelion are in excellent quantitative agreement with the theory (see Table I). In the case of the other planets the relativistic effect is so small that it is hardly of interest for testing the theory (nevertheless, from the example of the Earth it is clear that known possibilities exist here). The motion of the perihelion should, on the other hand, be very much larger for artificial satellites of the Earth (see Table II). In this case, moreover, it seems possible in principle to detect still another relativistic effect—the influence of the Earth’s rotation on the motion of a satellite.

The available data on the gravitational displacement of spectral lines testify to the existence of this effect. At the same time, owing to a number of complicating circumstances, the existing astrophysical material is still insufficient to speak of a confident quantitative verification of the predictions of the theory. In addition to the progress that is possible in this area as a result of further astrophysical investigations, successes connected with the application of radiophysical methods and the use of artificial Earth satellites may be expected here.

Observations during total solar eclipses have led to the detection of the deflection of light rays passing near the Sun, and the predictions of the theory have been confirmed within the limits of the attained accuracy of the experiment, which is approximately 10%.

Thus, on the whole, the general theory of relativity has well withstood experimental testing, and at present there are no grounds for doubting its validity in application to the corresponding range of phenomena.

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