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USING ARTIFICIAL EARTH SATELLITES TO TEST THE GENERAL THEORY OF RELATIVITY
V. L. Ginzburg
The general theory of relativity belongs among the most fundamental physical constructions. In addition to its general-physical and methodological significance, this theory is the only reliable basis for the theoretical analysis of cosmological questions. Therefore the importance of an experimental test of the general theory of relativity appears obvious. However, the effects following from the theory that admit quantitative verification are very small, and, despite the fact that work in this direction has been going on for almost forty years, certain points here require further investigation and refinement.
The present state of the question of the experimental test of the general theory of relativity is discussed in article ¹. Here we shall confine ourselves only to a brief summary.
It follows from the theory that the perihelia of the planets should slowly rotate. For Mercury this effect is the largest and amounts to 43.03 seconds of arc per century. The observational data fully confirm this conclusion, since they lead to the value \(\Psi = 42'',56 \pm 0'',94\). For the Earth, according to the theory, \(\Psi = 3'',8\), while according to observational data \(\Psi = 4'',6 \pm 2'',7\). For the other planets there are no sufficiently reliable observations, and the effect of perihelion rotation may be regarded as well confirmed only for one case (Mercury).
The second effect following from the general theory of relativity is the deflection of light rays passing near the Sun. For a ray passing at the very edge of the solar disk, the deflection angle \(\alpha\), according to the theory, is equal to \(1'',75\). The corresponding observations, carried out during a whole series of solar eclipses beginning in 1919, have confirmed the fact of the existence of this effect. In this case the mean value of \(\alpha\) is \(1'',98 \pm 0'',12\). Thus, on this point, observations confirm the theory to an accuracy of about 10%. Undoubtedly it is desirable to achieve greater accuracy, and, most importantly, to test the dependence, following from the theory, of the angle of deflection of the rays on their distance from the center of the Sun*).
The third effect is the gravitational shift of spectral lines. Namely, if the source of radiation is at a point with gravitational potential \(\varphi_1\) and its proper frequency is \(\nu_1\) (that is, the frequency measured near the source), then on Earth there will be observed
*) According to the theory \(\alpha = \dfrac{C r_\odot}{R}\), where \(r_\odot\) is the radius of the Sun and \(R\) is the distance of the ray from the solar center. The values of \(\alpha\) given in the text correspond to the case when \(R = r_\odot\), i.e. are equal to the constant \(C\). The character of the dependence of \(\alpha\) on \(R\) has not yet been established from observations.
V. L. GINZBURG
radiation with frequency \(\nu_2\), moreover
\[ \frac{\Delta \nu}{\nu}=\frac{\nu_2-\nu_1}{\nu}=\frac{\varphi_1-\varphi_2}{c^2}, \tag{1} \]
where \(\varphi_2\) is the potential corresponding in the given case to the Earth's surface, \(c\) is the velocity of light in vacuum, and \(\nu \simeq \nu_1 \simeq \nu_2\), since the frequency shift \(\Delta \nu\) is usually very small.
When observing on Earth the radiation of some star or of the Sun, one may in practice put \(\varphi_2=0\) and \(\varphi_1=-\dfrac{\varkappa M}{c^2 r_0}\), where \(\varkappa=6.670 \times 10^{-8}\ \mathrm{dyn}\cdot\mathrm{cm}^2\cdot\mathrm{g}^{-2}\), \(M\) is the mass of the star, and \(r_0\) is the radius of its photosphere. In the case of the Sun we hence obtain the value
\[ \frac{\Delta \nu}{\nu}=-\frac{\varkappa M_{\odot}}{c^2 r_{\odot}}=-2.12\cdot 10^{-6}. \tag{2} \]
For some white dwarfs the value of \(\dfrac{\Delta \nu}{\nu}\) is 10–30 times greater than for the Sun. However, in this case, owing to the absence of sufficiently accurate data on the mass \(M\) and radius \(r_0\) of the star, it is difficult to test the theory quantitatively. For the Sun, the quantitative test of the theory is also made difficult by the existence, in addition to the relativistic effect, of certain other sources of frequency shift of spectral lines (see \(^{1,2}\)). As a result, although the effect of the gravitational shift of spectral lines may be regarded as established both for white dwarfs and for the Sun, a quantitative check of formula (1) has not been carried out.
Thus, the effects of the general theory of relativity have been detected, and the theory as a whole has successfully withstood comparison with observational data. Nevertheless, the fundamental character of the theory does not permit us to be satisfied with the results achieved, and its further testing is, of course, desirable.
The astronomical methods used up to now have not yet been exhausted. Along with this, however, it is desirable to find new ways that would make it possible to test the theory more quickly and with greater accuracy. Certain possibilities in this respect are opened up by the use of artificial Earth satellites.
The perigee of an artificial Earth satellite will rotate in a manner similar to the perihelia of planets moving around the Sun. In this case, as it turns out (see \(^{3}\), and also \(^{1}\)), this rotation is considerably greater even than for Mercury, and for close satellites reaches approximately \(1500''\) per century. The effect is the greater, the closer the satellite is to the Earth; at a distance of 3 radii from the center of the Earth it is approximately ten times smaller than that indicated, while for the Moon the rotation of the perigee is already negligible (it amounts to \(0''.06\) per century *). The possible accuracy of determining the trajectory of a satellite is considerably higher than the accuracy of determining the elements of Mercury’s orbit, and it would apparently be possible to notice the relativistic effect on satellites without particular difficulty. But, on the other hand, in this case there are also great difficulties for comparing the theory with observations. The point is that the trajectory of a satellite will not be strictly elliptical even if relativistic effects are completely neglected, since the motion of the satellite is influenced by air resistance in the ionosphere, the nonspherical distribution of mass on the Earth, and the poss—
* This refers to the shift of the perigee caused by the influence of the Earth. Under the influence of the Sun’s field the perigee of earthly satellites turns through an angle of \(1''.9\) per century (in \(^{1}\), because of an error encountered in the literature, the incorrect value \(7''.6\) is given).
disturbing action of other celestial bodies (in particular, the Moon). Although the latter factor is easy to take into account, the same cannot be said of the first and especially of the second. Therefore it is still not clear to what extent and how soon it will be possible to compute in advance the trajectory of a satellite with an accuracy sufficient to reveal the relativistic effect. Since, however, we have no grounds for doubting such a possibility, we shall point out one more effect of the general theory of relativity which, in principle, could be detected by studying a satellite’s trajectory.
This concerns an additional displacement of the satellite’s perigee, as well as the motion of the nodes of its orbit, caused by the rotation of the Earth (see \(^{4}\) and, in more detail, \(^{1}\)). This rather interesting fourth effect of the general theory of relativity for Earth satellites reaches \(50''\) per century, i.e., is of the same order as the entire relativistic effect for Mercury (the relativistic displacement of Mercury’s perihelion, caused by the rotation of the Sun, is only \(0''.01\) per century, which lies far beyond the limits of measurement accuracy). The relativistic “rotation effect” in question may thus amount to as much as \(1/30\) of the total relativistic effect of the displacement of satellite perigees. Therefore, if the accuracy of determining this displacement is sufficiently high, it will also be possible to separate out the “rotation effect.”
An artificial satellite may also be used to measure the gravitational shift of frequency. From formula (1) it is clear that, when radiation from a sufficiently distant source is received on the Earth, for which \(|\varphi_1| \ll |\varphi_2|\), we have:
\[ \frac{\Delta \nu}{\nu} = -\frac{\varphi_2}{c^2} = -\frac{\varkappa M_\oplus}{c^2 r_\oplus} = 7\cdot 10^{-10}, \tag{3} \]
where \(r_\oplus = 6.37\cdot 10^8\ \text{cm}\) is the radius of the Earth and \(M_\oplus = 5.98\cdot 10^{27}\ \text{g}\) is its mass.
The frequency shift (3) is a “violet” one, whereas in observing solar radiation on the Earth the gravitational shift is “red,” i.e. \(\Delta \nu < 0\) (see (2)). In optics there can at present be no question of measuring a relative frequency shift of order \(10^{-9}\), but in the radio range this is quite realistic (the stability of atomic clocks and of a molecular generator, corresponding to values
\[ \frac{\Delta \nu}{\nu}\sim 10^{-9}\div 10^{-10}, \]
has already been achieved).
In this connection it also seems possible to use artificial satellites to measure the gravitational frequency shift in the radio range \(^{1,2,4,5}\). Unfortunately, for nearby satellites, which apparently will be launched at first, this shift is considerably smaller than the limiting value (3). Thus, for a satellite at height \(h\) above the Earth, with \(h \ll r_\oplus = 6.37\cdot 10^8\ \text{cm}\),
\[ \frac{\Delta \nu}{\nu} = \frac{gh\left(1-h/r_\oplus\right)}{c^2} = 1.09\cdot 10^{-18}\left(1-\frac{h}{r_\oplus}\right)h \tag{4} \]
and, for \(h = 8\cdot 10^7 = 800\ \text{km}\),
\[ \frac{\Delta \nu}{\nu}=7.6\cdot 10^{-11}. \]
Further, in addition to the gravitational frequency shift, there is also its shift due to the Doppler effect of first and second order. As a result, the total frequency shift is equal to
\[ \left(\frac{\Delta \nu}{\nu}\right)_t = 1+\frac{v_1}{c}\cos\theta - \frac{v_1^2}{2c^2}\left(1-2\cos^2\theta\right) + \frac{\varphi_1-\varphi_2}{c^2}, \tag{5} \]
where \(v_1\) is the satellite’s velocity and \(\theta\) is the angle between its velocity and the line of sight. In a circular orbit \(v_1^2=-\varphi_1\), since \(\varphi_1\) is the potential of the terrestrial
field in the satellite’s orbit. Hence it is clear that for close satellites, when \(|\varphi_1-\varphi_2|\ll|\varphi_2|\), even the quadratic Doppler effect is greater than the gravitational frequency shift, to say nothing of the linear Doppler effect, i.e., the effect of first order in \(v/c\). True, the linear effect vanishes for \(\theta=\pi/2\), but for a close satellite the angle \(\theta\) is close to \(\pi/2\) only for a very short interval of time, and it is not easy to “render harmless” the linear effect. On the other hand, this can be done if one measures not the frequency, but the difference between the readings of terrestrial clocks and clocks on the satellite (see the end of § 2 and \(^{5}\)). The ratio of this difference \(\Delta\tau\) to the time interval \(\tau\) itself in the case of a circular orbit is equal to
\[ \frac{\Delta\tau}{\tau} = \frac{\tau_1-\tau_2}{\tau} = \frac{\varphi_1-\varphi_2}{c^2} - \frac{v_1^2}{2c^3} = \frac{\frac{3}{2}\varphi_1-\varphi_2}{c^2} = \frac{\varkappa M_{\oplus}}{c^2 r_{\oplus}} \left( 1-\frac{3}{2}\frac{r_{\oplus}}{r_{\oplus}+h} \right). \tag{6} \]
Of course, relation (6) does not exceed the value (3), but, owing to the fact that the linear Doppler effect is excluded*), the measurement of \(\Delta\tau\) may prove more convenient and easier than the measurement of \(\Delta\nu\). In both cases, in order to obtain the maximum gravitational effect of the change in frequency or of the difference in times, distant satellites must be used. In this case the role of the Doppler effect, obviously, decreases, and the separation of the gravitational frequency shift is facilitated. Consequently, at least for distant satellites, when \(h\gg r_{\oplus}\), the measurement of the relativistic (gravitational) frequency shift appears feasible. Perhaps it will also be possible for close satellites.
Thus, the use of artificial Earth satellites opens certain, very tempting prospects for a further test of the general theory of relativity**).
References
- V. L. Ginzburg, UFN 59, 11 (1956); Forschr. d. Physik 5, 16 (1957); collection Einstein and Contemporary Physics, p. 93, Gostekhizdat (1956).
- V. L. Ginzburg, DAN 97, 617 (1954).
- L. La Paz, Publ. Astron. Soc. Pacific 66, 13 (1954).
- V. L. Ginzburg, ZhETF 30, 213 (1956).
- S. F. Singer, Phys. Rev. 104, 11 (1956). See also V. Hoffman, Phys. Rev. 106, 358 (1957).
- W. A. Baum, Publ. Astron. Soc. Pacific 68, 118 (1956).
*) As indicated in \(^{5}\), if the errors in the measurement of time are of a purely random character, then the accuracy of determining \(\dfrac{\Delta\tau}{\tau}\) increases with increasing \(\tau\), which can be taken advantage of in this case.
**) In addition to the possibilities listed, one may also point to the determination, using artificial satellites, of the brightness of the Metagalaxy \(^{6}\). From the surface of the Earth this brightness is difficult to measure because of the night sky glow, which interferes with observations. Determining the brightness of the Metagalaxy is of great cosmological significance and is connected with the general theory of relativity, on which theoretical investigations in cosmology are based.