Unformatted text preview: INERTIAL AND GRAVITATIONAL MASS 443 ANNALS OF PHYSICS: 26, 442517 (1964) The Equivalence of Inertial and Passive Gravitational P. G.
Palrner Physical
ROLL, Mass * R. KROTKOV, t AND R. H. DICKE Laboratory, Princeton University, Pl'inceton, New Jersey Torsion balance measurements of the difference in ratios of gravitational to inertial mass for different materials have been carried out, confirming to higher precision the null results obtained 60 years ago by Eotvos and assumed by Einstein as the Principle of Equivalence upon which the General Theory of Relativity is founded. If the parameter 'leA, B) is defined as
'1(A, B) = [(M/rn)A  (M/rnhl/!.z[M/m).{ + (M/rn)B], where M and rn represent the passive gravitational and inertial masses respectively of materials A and B, then the results from the most sensitive torsion balance used enable us to conclude with 95% confidence that I '1(Au, AI) I < 3 X 1011 Stated more exactly, the various measurements of '1, obtained from the gravitational acceleration toward the sun, gave a substantially Gaussian distribution with mean value 'l(Au, AI) = (1.3 x 1.0) X lOlI. The probable error quoted for the mean is based upon the observed scatter in results from individual data runs, assuming a Gaussian distribution. The importance of the Eotvos experiment to contemporary gravitational theories is discussed, and the earlier measurements of Eotvos and J. Renner are examined critically. The torsion balance and associated equipment used in the present experiment are described in detail, along with the considerations involved in their design. Methods of data analysis are also discussed extensively and tables of individual results are presented. I. INTRODUCTION It is possible to ascribe three types of mass to a body (1); inertial, passive gravitational, and active gravitational. By active gravitational mass one means a measure of the source strength of a body for generating a gravitational field, and by the passive gravitational mass one means a measure of the gravitational force acting on a body in a given gravitational field. If the inertial and passive masses are always proportional, independent of composition; the gravitational accelerations will be compositionindependent. The first crude experiments that demonstrated the uniqueness of the gravitational acceleration are probably lost in antiquity. While Galileo may not have * This work has been supported by research contracts with the Office of Naval Research,
the U. S. }.It.tomic Energy Commission, and the ~!&tional Science Foundation. t Present address: Sloane Physics Laboratory, Yale University, New Haven, Connecticut.
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"f dropped iron and wooden balls from the leaning tower of Pisa, he was certainly aware that the gravitational acceleration of a body is substantially independent of the material of which the body is composed. Later, a pendulum experiment which showed the equivalence of inertial and passive gravitational mass with an accuracy of a part in 103 was performed by Newton. Bessel (2) in 1827 improved this null result to an accuracy of 2 X 105 H. H. Potter (3) in 1923, also using the pendulum technique, was able to im6 prove the precision to 3 X 10 The most accJrate of the classic experiments were those of Baron Roland v. Eatvas, perfornwd with a torsion balance. His first series (4), published in 1890, established the constancy of the gravitational acceleration with a precision of .5 X 108. Later, with D. Pekar and E. Fekete 9 (5), he improved this to 3 X 10 Although this work first appeared in 1909 as an essay which won the Benecke Trust Fund award of the University of Gattingen, it was not published until 1922, three years after the death of Eatvas. These later remarkable experiments, using only classical techniques, showed that a wide variety of substances, including such exotic materials as "Schlangenholz" and magnalium, fell with the same acceleration. It is of interest to note that as recently as 1935 one of Eatvas' torsion balances was used with little modification by J. Renner (6) to repeat Eatvas' old experiment, substituting for snake wood the equally exotic material "Batavian glass drops." Perhaps the first question to occur to a physicist is, "Why is there any necessity of increasing the precision of the Eatvas experiment? Is not a precision of 9 3 X 10 enough?" What this question usually implies is the feeling that this experiment and others establish the ~quivalence principle, and that Einstein's "general relativity," being based on this principle, is almost certainly correct. If general relativity is beyond question, the experiment is without merit, for the null result of the Eatvas experiment is a result which can be derived from "general relativity." It is well known that Einstein's principle of equivalence, the "strong principle" (7), is such a rigid constraint on gra vitatibnal theory that "general relativity" is almost uniquely determined by this principle, together with the principle of general covariance. Far from being the most general of relativity theories, it is actually a strongly restricted special case. According to Synge (8), the presence of the word relativity in the narhe is something of a misnomer. The theory is more properly described as a. theory of the dynamies of a Riemannian spacetime, rather than a relativistic theory in the sense of the concept of space introduced by G. Berkeley and E. Mach. As will be discussed below, it is a mistake to regard Einstein's equivalence principle as directly established by the Eatvas experiment. Actually, what is directly supported is a wea.ker forIn of the principle, This '(,veak equivalence principle" can be used as a basis for theories more general than "general relativity." Perhaps the best way to answer the question posed above, as to the impor. .. 444 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 445 tance of the Eotvos experiment, is to exhibit its unique role as a basis for rela constants are fixed. It should be noted, however, that the Eotv6s experiment tivistic theories. This has been discussed previously in a number of publications. and the improved version to be reported here do not set useful limits on the Here we simply summarize several important points: variability of the weak coupling "constant" or the gravitational coupling "constant." 1. It is well known that Einstein's "general relativity" is based on his assumption, sometimes called the "strong equivalence principle" (7). This is the as4. The strong equivalence principle requires that the gravitational interaction sumption that in an electromagnetically shielded laboratory, freely falling and between two small bodies, closely spaced, should be independent of the distribunonrotating, the laws of physics, including their numerical content, are inde tion of distant matter surrounding the bodies. This may not be consistent with pendent of the location of the laboratory. In such a laboratory all particles, free the requirements of Mach's principle, according to which the gravitational of nongravitational forces, move without acceleration. Hence, more generally, acceleration depends upon the distribution of the total mass of the universe in any arbitrary coordinate system all particles move with the same acceleration. _about the point in question. These two basic principles are not necessarily inIt should be remarked that the uniqueness of the trajectory of a particle compatible. For example, it is possible that, with the application of suitable moving gravitationally is essential to the geometrical interpretation of the space boundary conditions, only certain distributions of distant'matter may be pertime orbit as a geodesic of the geometry. Thus, the fact that the gravitational mitted. However, it is also possible that only the weak form of the equivalence field tensor is interpreted as the metric tensor of the geometry, rather than as a principle is valid. Unfortunately, the Eotvos experiment is of no assistance iii. prosaic force field, is due in part to the unique gravitational acceleration. deciding between these two alternatives. It must be noted that it is here assumed that the "partiele" is so small and has 5. Lee and Yang have suggested the existence of a gaugeinvariant vector so little spin angular momentum that the "gravitational tidal forces" or "gravi field, similar to electromagnetism, associated with baryon conservation (9). tational gradient" effects on the particle are negligible. These forces may be They have also shown that the E6tvos experiment sets drastic limits to the defined as those associated with nonzero second derivatives of metric tensor strength of an interaction with such a field. components in a locally Minkowskian coordinate system. It is an experimental 6. It has frequently been suggested that antimatter may fall up, not down. problem associated with the Eotvos experiment to measure accelerations under L. Schiff (10) has made use of the null result of the E6tvos experiment in an ingenious argument to suggest that it falls down, not up. conditions for which such forces are negligible. 2. A weaker form of the equivalence principle has been stated (7). It requires It is clear from the above that if one believes that general relativity is estabthe uniqueness of the gravitational acceleration, at least to the accuracy of the lished beyond question by its elegance, beauty, and the three famous experiEotvos experilnent, while permitting the numerical content of physical laws to mental checks, then the Eotv6s experiment has no point, for its null result can vary from point to point. It provides a broader base upon which gravitational be derived from the theory. However, if gravitational theory is to be based Oil theories can be constructed. experiment, the null result of the Eotvos experiment is probably the single most It may be noted that a massless scalar field has no place in "general relativity" important experimental result available to us. As such, it is important that its under the "strong equivalence principle," for the value of the scalar varying froIn accuracy be improved, if possible. point to point would contribute a variable' element as part of the numerical content of physical laws. In similar fashion a seQ<?ndlong range tensor field. II. EOT\'OS' APPARATUS would be impossible, for it would generate a scalar by contraction. However, the It IS a good general experimental rule that the most precise experiments are existence of both of these fields might be possible under the "weak equivalence null experiments, and the E6tv6s experiment is no exception. With E6tv6s' principle." a~para.tus, consisting of a Cavendish balance (more exactly Boy's modification) 3. Under the "weak ~quivalence princi.ple" particle masses and coupling Withdlffere~It materials at opposite ends of the beam, the centrifugal forces on constants co~ld vary, b~mg pe~~aps functIOns .of a 9 scalar fiel~. :r:o~ever, the the two weI~hts d~e t~ the earth's rotation are balanced against a component Eotvos experIment and Its preCISIOnof 3 parts m 10 sets drastIC limIts to such ~ft~e earth s gravitatIOnal field. If the ratio of passive gravitational mass to possible variations (7). These limit~ are improv~~ su~stantially by the experi Illert.mlmass ~hould diffe.r from one weight to the other, there would be a torque ment r~ported here. Thus, to a consIderabl~ preCISIOn,It may be ~oncluded that ten~mg to tWiSt the torSIOn balance. A rotation of the whole apparatus through the ratIOS of the masses of elementary partICles to each other are mdependent of 180 would then reverse the sense of this twist. position in 'pace.thoe, and also that the ,trongand eleotmmagoetie eoupling At a latitude of 45, the ho","ntal eomponent of the ea<th'e een"if"" l a,. I 446 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 447 celeration is only 1.6 cm/sec2 If there were an anomaly of 1 part in 1011 in the inertialgravitational mass ratio, there would appear an anomalous turning force of at most 1.6 X 1011 dyn/gm on one of the weights. For purposes of design considerations we shall have in mind a standard gravitational anomaly of 1 11 part in lOll; namely, an added anomalous gravitational force, 10 of the normal gravitational force, acting on one of the bodies. In a Galileo freefall experiment, the anomalous force would be much greater, 108 dyn/gm, but it would now be necessary to measure the acceleration to l' part in 1011. In a more sophisticated experiment, two 'weights might be dropped together and their relative accelerations compared. (The extraneous effect of gas drag could be eliminated by dropping them inside a freely falling container.) In a drop of 5 meters taking 1 sec, an anomalous acceleration of one of the' 9 weights of 108 cm/sec2 would lead to a relative displacement of only 5 X' 10 c between the two weights. A great advantage of the E6tv6s technique is that it: permits an anomalous force, though much smaller, to act for a far longer periodi of time and accumulate a bigger displacement. An interesting experimental possibility would be to put an apparatus in a artificial satellite. In this case, the advantage of the large force anomaly coul be combined with that of a long observation time. It is interesting to calculate the maximum rotation to be obtained in Eotvosl apparatus from an anomaly of 1011 His apparatus employed weights of 30 gml had a beam length of 40 em, and the suspension wire had a torsion constant 0 11 0.5 (dyn cm)l. The resulting maximum rotation from an anomaly of 10 1.9 X 108 rad. E6tv6s observed rotations with the traditional instrument 0 that day, a telescope and scale. A rotation of this size represents 1/20,000 0 ERTICAL ROTATIONAL AIR SPAf:;E ADJUSTMENT ADJUSTMENT SUSPENSION WIRE MIRROR TELESCO~E WEIGHT
I c:=:::> c=== \!I ; .. WEIGHT IIj r ~ n "J ~ FIG; 1. Reproduction of a drawing of a single torsion balance used by Eatvas for some of the smallest division on his scale. 's measurements (11). The scale below the drawing is one meter in length. Perhaps the most serious objection which can be raised to the E6tv6s exper.i ment is the lack of a suitable control. There is no way of turning off the centrifui gal force field of the earth. Hence, there is no overall zero check upon the pell' . formance of the torsion balance with some specific set of weights in place. T While E6tv6s made use of ac~eleration toward the sun for some of his meas avoid this difficulty, it is possible to use the a~celeration of the a~paratus towar ~nts,. these results we~e of mferior accuracy and his experiment depen~:~ the sun. At 6 a.m. and 6 p.m., and for a torslOn balance beam m a northsout rll1~anly upon the centnfugal force field (5). His principal apparatus is "h . direction, an anomalous gravita~ional pull upon one .of the .weightsat an e~d ~ FIg. 1, reproduced from an old article (11). It may be noted that one ~~e~~~~ the beam would produce a turnmg force. The resultmg tWIst would be penodi as suspend~d lower than the other. The resulting elimination of the tw f Id with a 24hr period. While the horizontal component of acceleration towar ~mmetryId aXISd' the apparatus increased the number of basic types., of gl'aVI,aof .at 2 lonal fi. ( . the sun is at most only 0.6 em/sec, about threeeighths of the rotational ce e gra I~nts tIdal effects) to which the apparatus was sensitive a d trifug~l a~celeration, it occurs wi~h a 24hr period.',and hence the me~~ureme. d no worthwh~le purpose. It .is difficult to see why this was done, as it' OI~~V contaIns Its own zero check. ThIS type of expenment has the addltlOnal a ade the. expenment more dIfficult, requiring twice as many observ t' . 0 vantage that a rotation of the apparatus is unnecessary, eliminating the resultin observatIOns 90 apa:t) as would have been required by an apparatus \~i~,~ns 0 disturbance of the delic.ate balance, for the rotation is generated automaticallf ... ~Of?ld symmetry aXIs (180 rota~ions). The only reason that we can see f. ~ o and gently by the rotatmg earth. IS IS that the apparatus was deSIgned for making geophysical measurements 448 ROLL, KROTKOV, AND DICKE INERTIAL AND GRAVITATIONAL MASS 449 of gravitational gradients, and was used only incidentally for this imp9rtant basic experiment. . In the apparatus shown in Fig. 1, a platinum weight was inserted into one end of the torsion beam, while the weight suspended from the opposite end could be changed. The observing telescope was supported by a bracket at the right of the apparatus and carried a short scale on top of it. The telescope had a 90 bend in its optical axis, and the figure shows the open end of the eyepiece. This arrangement required the observer to place himself approximately a meter from the torsion fiber, on a line making an angle of about 45 with the torsion beam. The distance from the torsion fiber to the scale was 62 em, and the variable weight was suspended 21.2 em below the torsion beam, 40 em long. In order to make an observation, the whole apparatus was rotated about its vertical axis and then left undisturbed for one hour or more until air damped the oscillations, after which the scale was read. Many of the measurements of E6tv6s, Pekar, and Feteke (5), and all of those of Renner (6), were made with a socalled "double gravity variometer." 'This device was nothing more than two torsion balances of the type shown in Fig. 1, mounted i~ the same housing with torsion beams approximately parallel and the suspended variable weights at opposite ends. Table I summarizes the results reported by Eotv6s, Pekar, and Feteke (5), and by Renner (6). Since E6tvos and his colleagues did not publish their data 'and details of their calculations, we are not certain how to evaluate the standard deviations which accompany their measurements (see the third column of Table I). However, assuming that the measurements they made were independent and normally distributed, it is possible to estimate the implied accuracy of a single position measurement on the torsion balance scale from the number of measurements made and the quoted standard deviation of the final result. Such estimates have been made and are all consistent with the reading _ accuracy claimed by E6tv6s et al., 7'20 of the smallest division on the scale used, provided this was the only significant source of error. Similar estimates of the scale reading accuracies for the measurements of Renner (6), however, are not consistent with his claimed 7'20 scale division reading accuracy. This inconsistency has been at least partially resolved through the courtesy of Dr. Renner himself, who recently made available to us copies of his original data and calculations for the measurements on brass and paraffin. A careful examination of these records has revealed that his method of calculating differences in position readings of the torsion balance yielded sets of values which were not statistically independent. For instance, his method utilized each of 24 independent north and 24 independent south position readings a number of times, and in various combinations, to obtain 90 values of t.he northsouth difference. The rms fluctuation of these differences about their MEASUREMENTS IN. PASSIVE BY EOTVOS TABLE I et al. (5) AND RENNER (6)
MATERIALS OF THE DIFFERENCE FOR VARIOUS GRAVITATIONAL TO INERTIAL MASS RATIOS Materials
A B TJ (A,B) ::I: standard deviation of the mean Magnalium Snakewood Copper Ag2SO. + FeSO. glass and brass vials + Water brass vial Cu'SO. crystals + brass vial CuSO. solution + brass vial Asbestos + brass vial Talc brass + + Platinum Batavian glass drops Ground Batavian ...
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 Spring '09
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 Inertia, Mass, General Relativity, Equivalence Principle, torsion balance, Dr. Renner, Eotvos

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