Absorption of the Force of Gravity

Q. Majorana, On gravitation. Theoretical and Experimental Researches. Phil. Mag. 39, p. 488. (1920).

In connection with the very great interest aroused in recent years by the general principle of relativity and the associated theory of gravitation, new experimental investigations have lately begun to appear whose purpose is to reconsider our basic ideas about the force of gravity. Among these works is the paper under review by Q. Majorana, who has specially studied the question of the absorption of the force of gravity by matter.

The theoretical investigations of Q. Majorana reduce to the following. Let \(\Phi\) be the flux of the force of gravity in the solid angle \(d\omega\), let \(dm\) be the mass producing the gravitational force, and, finally, let \(k\) be the gravitational constant; then

\[ \Phi = k \frac{dm\, d\omega}{4\pi}, \]

according to Newton’s fundamental law.

Next, let \(x\) be the thickness of the absorbing medium through which the flux \(\Phi\) passes, \(H\) the coefficient of absorption, and \(\delta_v\) the quantity which Majorana calls the “true density”; then, according to Majorana, the relation must hold:

\[ \Phi = k \frac{dm\, d\omega}{4\pi} e^{-Hx} = k \frac{dm\, d\omega}{4\pi} e^{-h\delta_v x}. \]

Majorana applies this expression for \(\Phi\) to a sphere with constant “true” density \(\delta_v\) and radius \(R\). By means of simple calculations he finds that the total flux of force issuing from this sphere will be:

\[ F = k\pi \delta_v R^3 \left[ \frac{1}{p} - \frac{1}{2p^3} + e^{-2p}\left(\frac{1}{p^2}+\frac{1}{2p^3}\right) \right], \qquad \text{where } p=RH. \]

Let \(M_a\) be that “apparent” mass of the sphere to which, according to our usual conceptions, the action of gravitation should be ascribed; then \(F=kM_a\), and

\[ M_a = \pi \delta_v R^3 \left[ \frac{1}{p} - \frac{1}{2p^3} + e^{-2p}\left(\frac{1}{p^2}+\frac{1}{2p^3}\right) \right]. \]

Put

\[ \psi = \frac{3}{4} \left[ \frac{1}{p} - \frac{1}{2p^3} + e^{-2p}\left(\frac{1}{p^2}+\frac{1}{2p^3}\right) \right] \]

and call

\[ M_t = \frac{4}{3}\pi R^3 \delta_v \]

the “true” mass of the sphere; we have: \(M_a = M_t\psi\); \(\delta_a = \delta_v\psi\).

For \(p=HR=0\), i.e. for \(H=0\) or \(R=0\), we have \(\psi=1\). As \(p\) increases, i.e. as the absorption or the radius of the sphere on which the apparent mass depends increases, \(\psi\) rapidly decreases.

From \(p=RH=R\delta_a \dfrac{h}{\psi}\) it follows that

\[ h = p\psi \frac{1}{R\delta_a}. \]

\(R\) and \(\delta_a\) are known for particular cases (for example, for the sun).

As \(\delta_v\) increases, the quantity \(\psi\) decreases; in the limit, for \(\delta_v=\infty\), \(p=\infty\), \(\psi=0\); the quantity \(h\) has a limiting value, which from the data for the sun \((R=6.95\cdot 10^{10}\) and \(\delta_a=1.41)\) is derived as equal to \(h=7.64\,10^{-12}\).

The constant \(h\) Majorana regards as a “universal constant.” With this the theoretical investigations of Majorana are limited.

Let there now be a small sphere with true mass \(m_v\), and let it be placed in a large sphere of radius \(r\) and density \(\delta\); the relation between the “apparent” mass and the “true” one will be

\[ m_a=m_v e^{-h\delta r}=m_v(1-\delta_v hr+\ldots\ldots) \]

Approximately,

\[ \varepsilon=m_v-m_a=m_v h\delta r \]

If one takes \(m_v=1\), \(\delta=13.60\) (mercury), \(r=10\), and \(h=7.64\cdot 10^{-12}\), then

\[ \varepsilon=1.4\cdot 10^{-7}\ \mathrm{gr}. \]

This difference in weight Majorana considers it possible to determine experimentally. To a sensitive balance, on one side of the beam, a lead sphere weighing 1274 kilos is suspended on a long thread. This sphere may be surrounded on all sides by mercury, the total weight of which is 104 kilos, and in such a way that the mercury touches neither the sphere nor the thread supporting it. The sphere is weighed surrounded by mercury and also without mercury.

In this case a difference in weight was observed equal to

\[ \varepsilon=0.00209\pm 0.00007\ \mathrm{mgr}. \]

The sensitivity corresponding to such an insignificant weight—with a comparatively large total load on the balance—is attained by making the observations with a mirror and scale at a distance of twelve meters from the mirror (!). Since the zero position of the scale is not constant, the curves giving the equilibrium points for the balance in the presence of mercury and without it (if they are plotted as functions of time) are arranged so that they are at a definite distance from one another. This distance gives \(\varepsilon\), and there can be no doubt that, in the presence of mercury, the equilibrium position of the balance is indeed different from that without mercury.

Majorana investigates all possible causes which might have produced this difference in weight. The result of this investigation is not given in Phil. Mag. It is briefly summarized in the following table:

TABLE.

\[ \text{Actual deviation }\varepsilon=+0.00098\pm 0.00016 \]

Majorana also lists other causes which, at first glance, might affect the result of the experiment, namely:

1) perturbations of a mechanical character, 2) perturbations of a caloric character, 3) radiometric effects, 4) magnetic effects, 5) electrostatic effects, 6) electromagnetic effects.

But Majorana believes that they cannot noticeably influence the result of the experiment. Since in the article under review no explanations whatever are given on this point, it is difficult to draw any conclusion as to the correct—

...of the reliability of his conclusions. There is no doubt, however, that Zeeman, comparing Eötvös’s inertial and gravitational masses by the method and dealing with observations of approximately the same precision as Majorana, found in some cases so strong an influence of the magnetic field that it completely masked the true course of things.

Since in Majorana’s experiment

\[ \varepsilon = 9.8 \cdot 10^{-7}\ gr.;\quad m_w = 1278\ gr.;\quad \delta = 13.60;\quad \gamma = 8.40 \]

then

\[ h = 6.73 \cdot 10^{-12} \]

which agrees very well with the theoretical number given above.

Applied to the Sun this gives, if one sets \(\delta_2 = 1.41\), a value for the true density approximately three times greater: \(\delta_w = 4.27\).

The excessively brief account of the work in Phil. Mag. does not allow a strictly critical analysis of the results obtained. The extraordinarily high sensitivity of the apparatus and the perhaps too good agreement of the calculated and observed values for \(h\) compel one to express the wish that Majorana’s planned repetitions of these experiments on a broader basis be published with sufficient detail not only in exclusively specialized publications, but also in widely circulated physics journals. There is no doubt that the result obtained by Majorana, if only it is confirmed, will have enormous theoretical significance.

V. Frederiks.