A DECISIVE TEST FOR THE GENERAL RELATIVITY A. B L O K L A N D Breda, Holland

(Received 14 March, 1971)

Abstract. A very accurate evaluation of the Schwarzschild constants is now possible from continuous observation of the spin axes of coupled gyroscopes, because the angular velocities of these axes can be more than 100000 times greater than the better known precession velocity of a single gyro in orbit around the Earth. This enormous increase in velocity is a consequence of a gravo-magnetic effect in gravitational interaction that resembles the electro-magnetic effect in electrical interaction.

Part h General Analysis of the Test A. INTRODUCTION There are a large n u m b e r of experimental facts that prove the reliability of predictions based on special relativity theory. General relativity, on the other hand, has led to remarkable little successful experimental research. The few predictions the theory does make a b o u t observable phenomena, that have been tested so far, require an almost impossible precision for any decisive measurement. We know o f only three experiments in the past in which a reasonable a m o u n t of success was achieved; namely, (1) In the total precession of the perihelion o f Mercury being 5600" of arc per century, there is an a m o u n t o f about 43" of arc per century that can be explained as a relativistic effect (Dicke, 1964). (2) The angle o f deflection o f a light ray which passes within a distance R o f the Sun is predicted by general relativity to be twice the angle predicted by Newtonian theory. (3) Electro magnetic signals are predicted to be retarded in a gravitational field, which retardation has been observed in radar signals reflected by Mercury. I n all three cases we can speak better o f a qualitative agreement with general relativity theory than o f a quantitative agreement, so that more reliable information is urgently needed.* (4) E S R O proposes, in view o f these facts, to launch a spacecraft that enables tracking o f a mass moving over a pure gravitational orbit. The essential point here is that the orbit has to present no perturbations due to n o n gravitational causes, such as solar radiation pressure and solar wind. One way to solve this problem is to use a drag free system, where a spherical ball is included in a hole inside the spacecraft, and submitted to the condition o f never touching the walls. (ESRO, 1970) * The observed frequency shift in the radiation of a specific source depending on the relative positions of observer and source in a gravitational field, (Pound and Rebka, 1960) is fully explained with the equivalence principle, so that the very accurate results constitute no base for an evaluation of the metric tensor values. (Weber, 1964) Astrophysics and Space Science 12 (1971) 219-242. All Rights Reserved Copyright 9 1971 by D. Reidel Publishing Company, Dordreeht-Holland

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A.BLOKLAND

(5) A totally different test is proposed by U.S. scientists, using a gyroscope in an earth orbiting satellite. Predicted is that the orbit motion effect will result in a precession of the axis of the gyro over an angle of about 7" of arc per year, whereas an earth rotation effect will result in 0.05" of arc per year. Since the gyroscope precession will be so extremely small, advanced technology much beyond the state of the art will be required. This technology is presently being developed by Stanford University. (6) In a short article (Blokland, 1970) the present author drew attention to an oscillatory effect of coupled gyroscopes in an earth orbiting satellite that offers a far better opportunity of establishing the dependence of the metric tensor components on the gravitational potential, because the angular velocities of the gyroscope axes are 4-5 mag. larger than those mentioned under (5). B. THEORETICAL ANALYSIS

It is a popular belief that the full beauty of relativity theory can only be revealed by the use of tensor relations. Whatever the truth may be in this respect, it may be well to remember that there is a beautiful skewsymmetric world-tensor in H and E, but that it is never used in practical problems regarding magnetism and electricity. Even in the analysis of the results of elementary particle collisions, as observed in bubble chambers, no tensors are used. We purposely mention these facts here as a motivation for stating two relations following from special relativity theory, and two comparable relations following from general relativity in a non tensor form, in order to make the discussion an open one for any interested reader. The origin and validity of the relations will be discussed in a third part that may be more interesting to the specialist in relativistic tensor calculus. From the special relativity theory follows that a negative electric charge Q~ moving in the spherically symmetric field of a stationary positive electric charge Q2 experiences a tangential acceleration ate d2~b dr dq~ ate = r ~ + 2 dt dt

Q1Q2 urut

7melr2 c2

,

(1.1)

where ate is expressed in polar coordinates, where u, is the radial velocity of Q1 in the field, and u t the tangential velocity, where c is the light velocity, where y = (1 - u2/c z) - 1/2 with u being the total relative velocity of Q1, where me~ is the rest mass of Q~, and where r is the distance between the charges, considered as point charges. It follows, further, that Q1 experiences a radial acceleration are d2r are=dtZ

r (dqS) 2 Q1Qe( \ d t ] =Tmeir 2 - 1 +

ur2~ ca ] ,

(1.2)

where the symbols have the same meaning as given above. Particularly the tangential acceleration is the direct cause of so called magnetic effects in the interaction between electric charges. Whereas the ancients saw no re-

A DECISIVE TEST FOR THE GENERAL RELATIVITY

221

lation between the characteristics of the material called 'elektron' and a 'magnetic stone', modern science (starting with Oersted in 1817) recognized an ever closer relation between the two, but only special relativity theory revealed that speaking about magnetism and magnetic fields (in a theoretical sense, at least) is nothing but an anachronism. Any so called magnetic effect is a result of relatively moving electric charges, and of nothing else. F r o m general relativity theory follows that a mass M 1 moving in the spherically symmetric field of a stationary mass ME experiences a tangential acceleration arm

M1M2 u,ut at,.- ~ 21 c2 ,

(1.3)

where 2~ is a constant or a function of r, for which we seek a definite value, where m i is the relativistic inertial value of the moving mass M~ (generally taken as to be numerically identical with the gravitational mass M1), and where the other relevant symbols have the same meaning as given above. It follows, further, that M~ experiences a radial acceleration arm at". -

M1M2( mlr 2

- 1 + 22

c2; '

(1.4)

where the symbols have the same meaning as given above. The acceleration values 'a' in these four relations have to be multiplied with a constant that is appropriate for the chosen units. Whereas there is overwhelming experimental evidence for the reliability of the first two relations, there is only very limited experimental evidence for the reliability of (1.3) and (1.4), in fact, only what has been mentioned under (1), (2) and (3) of the introduction. Assuming general relativity theory to be basically sound, however, there can be no doubt about the contributing components of the acceleration, although the values of 2~ and of 2 2 remain to a large extent unknown up till now. The characteristic difference between general relativity theory and Newtonian theory, that is open to investigation, is that in the first case we have a finite value for 21 and 22 (probably with a value of about 'two'), whereas in Newtonian theory they have a value zero. (Following from Kepler's second law.)* Having two pairs of almost identical relations we must now look for two physical objects in circumstances that make the relations applicable and the measuring results comparable. For (1.3) and (1.4) we decide on a thin rotating disc of neutral matter in a circular orbit around the Earth. It follows that we must take an identical disc of electrically non conductive material on which an evenly distributed negative charge is brought, * There is a certain danger of misunderstanding in speaking about magnetic forces in gravitational interaction that follow directly from (1.3) and (1.4), or even in speaking about pseudo magnetic forces, because they are very different in origin and effect from electro magnetic forces, although the relevant relations are almost identical. It might perhaps be better, therefore, to speak right from the outset about gravo-magnetic forces acting on electrically neutral matter.

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rotating about its axis and revolving in a circular orbit around a very pondorous positive charge, in order to apply in a comparable manner the relations (1.1) and (1.2). It is clear that a current in a super conductive wire loop is an entirely different configuration as practically nothing of the total inertial mass takes part in the rotation about the axis. We now consider an elementary charge dQ1 in an elementary mass dM~ (the chargemass ratio is of course many magnitudes larger than the ratio in an electron or in a proton!) with an inertial value drn~ = 7 dine1 at a radius rg from the axis of rotation of the disc at a time t. The radius r o forms an angle c~ with the orbital plane at this instant, and the axis of rotation of the disc forms an angle q5= mot with the radius r o connecting the center of the disc with the center of the field. I f the angular velocity of the disc is coo, and the orbital velocity of the center of the disc is uto = cooro, we then find a tangential velocity for dQ1 in relation to the field at this instant ut = Uto + cogro sinc~ c o s r

(1.5)

and a radial velocity u, = cogrg sin c~sin q~.

(1.6)

It is now immaterial from a mathematical point of view whether we apply (1.1) and (1.2) to the electrical case, or apply (1.3) and (1.4) to the gravitational case, if we remember the appropriate values of 21 and 22. However, being primarily interested in the behaviour of a rotating disc, we choose the gravitational case, and leave the better known electrical one in the background for easy reference. Substituting the values of (1.5) and (1.6) into (1.3) we find dM1 M2

at,. = d m i r 2 c 2 21

(Utocogrg sin e sin 4) +

2 2 sin 2 c~sin q5 cos qS), cogrg

(1.7)

and a radial acceleration a,,. with (1.4) at,. -

dM1 M z c2 2 2 9 2 sin 2 r d m i r~c 2 ( + 22co or g sin e

(1.8)

Integrating over the entire mass of the disc, we find the main component of the acceleration from (1.8), being a,.,,,1 = - M 1 M z / ( r 2 m , ) ,

(1.9)

that tends to keep the center of the disc in an exact circular orbit if this value is equal to rocoo2. The secondary accelerations have a value that depends on the position of dM1 at a time t so that small internal stresses result, over and above those caused by centrifugal forces, and apart from the stresses that always exist in a solid body. We will call the resulting mean accelerations for the disc as a whole at and at. In order to simplify the expressions somewhat we introduce the angular m o m e n t u m A of the disc, being A = Icog = (rn,r2.) coy.

(1.10)

A DECISIVE TEST FOR THE GENERAL RELATIVITY

at follows from (1.7). M1M2 ( 1A ) at - mir2c221 0 + ~ -mi- coo sinq~ cos~b .

223

The mean tangential acceleration

(1.11)

The mean (secondary) radial acceleration a r follows from (1.8).

M1M2 /1 ) ar- ~ 22~2 miAcogsin 2 ~b .

(1.12)

These expressions can be further simplified if we put M1/mi=1, and if we replace M2/c2 by the so called mass radius Rm of the central mass. Paying attention a moment to the electrical case in which we have well defined charges Q1 and Q2, and in which we have also an identical inertial mass mi, we could make a corresponding simplification

Q1Q2 mic2

R e

(1.13)

However, although it might be permissible in the case here under consideration, it should be remembered that in a non-circular orbit the velocity of Q1 is not constant, and, consequently, Q1/m~is not constant. This complication is absent in the gravitational case, because the force on M 1 is proportional to its relativistic inertial value. Left over is an important component of relation (1.7) that, admittedly, does not cause a tangential acceleration of the disc as a whole, but that does tend to turn the disc about the radial direction of to, thus exhibiting in this gravitational case the characteristic of a current loop in a magnetic field. Multiplying this component of arm with dt to get a velocity, and with dm~ and rg sinc~ to get an elementary angular momentum for the mass dM1 under consideration, we then integrate over the entire disc to find a radially directed angular momentum dA~

M1M2 miFoC

dA~ = .__-~_2~21 (89 sin (~Utodr) = R~m21 (89 sin q~ dq~).

(1.14)

Fo

This always radially directed angular momentum dA~ can be split in a component dA~ perpendicular to the momentary direction of A dA1 = (Rm/ro) 21 (89 sin 2 r d e ) ,

(1.15)

that does not change the magnitude of A, but that does change its direction, so that (1.15) gives the rate of precession of the disc along the orbit as a function of the angle r and in a component dA2 along the momentary direction of A dA2 =

(R,,/ro) 41 (89

sin ~b cos r d e ) ,

(1.16)

that gives the rate of change in angular momentum along the orbit, and, incidentally, the rate of change in electric current in the electrical case, because the current is directly proportional to the angular velocity of the charged disc.

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C. CONSERVATION OF ENERGY AND ADJUSTED ORBIT

An important aspect of our problem is that energy should be conserved in a closed system, which principle affords a good check on the results given above. Considering the comparatively very small velocities of the elementary masses in the disc in relation to the light velocity, we can express a change in total energy of the disc in a perfectly circular orbit as 1 2 d (flo~g) = Io) o dr9 o = o)o dA.

(1.17)

It follows from relation (1.16) that the rate of change in total energy of the disc along the orbit is then o)o dA2 = (Rm/ro) 21 (89

o sin q~ cos qb dqS).

(1.18)

Now, looking at relation (1.11), and multiplying a t with the inertial mass m i of the disc, we find a tangential force on the disc that feeds an elementary energy into the disc over a distance r o dq5 to be given by a~rniro dO = (Rm/ro) 21 (89162 sin q5 cos ~b dq~).

(1.19)

Whatever the actual value of 21 may turn out to be, we find from the equality of (1.18) and (1.19) that the change in total energy of the disc is accounted for. The tangential force acting on the disc is not used for an increase in orbital velocity, but for an increase (or decrease) in angular velocity of the disc. We find thus in this gravitational case also the characteristic tangential coupling, so well known from the interaction of electric fields, in which there is (principally and to a very small degree !) also a change in angular velocity of the central field if it is not perfectly rigid in space. We assumed at the outset an exact circular orbit for the center of the disc that presupposes a constant mean radial acceleration for the disc as a whole. However, in relation (1.12) we found a component that makes such a true circle impossible. A rotating disc must follow an elliptic orbit, but one with the interesting characteristic of having the center of the field right in the geometrical center of the ellipse. If we call the direction of the angular momentum A at the time zero the X-direction, and the direction perpendicular to the first one in the orbital plane the Y-direction, we then find that we must introduce a small radial correction rc together with a new constant angular velocity me, such that x=r ocoscoJ;

and

y=(r o-r~)sincoct;

(1.20)

where rc is so small in relation to r o that higher powers of rc are consistently neglected in the final results. Starting from the relations given in (1.20), the characteristic values of the orbit of the center of the disc follow easily. We will mark these values with a p, in order to tell them apart from previous values. rp

=

ro - rc

s i n 2 O~ct,

dpp = r162 - 89(r~/ro) sin2c%t,

(1.21) (1.22)

A DECISIVE TEST FOR THE GENERAL RELATIVITY

225

(1.23) (1.24) (1.25) (1.26)

utp = rp dCp/dt = coc (r o - r~ cos 2 co~t), Urp = drp/dt = - cocr~ sin 2coct, atp = 2 dr v dOv/dt 2 + rp d2r

2 = O,

arv = d2rp/dt 2 - rp (d(ov/dt) 2 = co2 ( _ ro + r~ sin 2 co~t).

The varying distance rp to the center of the field results in a varying force, which divided by rn i accounts for the following radial acceleration of the disc M1M2

M2

mir,

(-) 1 + 2 rc

sin2

coct

= _

09 c

(r ~ + 2r~

sin 2

COct) .

(1.27)

The spinning disc of neutral matter (or a comparable pure current loop in the form of a charged ring or disc) experiences an extra force in the field if compared with a non-spinning one that tends to give the spinning disc an extra radial acceleration. This force is, in fact, the primary cause for the orbit correction r~ so that from (1.12), (1.26), and (1.27) follows ( R , , / r 2) 22

- - coo sin 2 r mi

= 3co2r~ sin z coot.

(1.28)

Substituting for A the value given in (1.10), and neglecting the very small difference between r and coot in the final result, we can now solve for rc to obtain 2

2

r~ = -~22Rm r"co~ 2 2"

(1.29)

roco c

We conclude from this relation that only if the mean velocity of the mass in the disc is about equal to the orbital velocity of the disc may we find a radius correction in the order of about one millimeter. However, relation (1.29) does show how a direct check on the value of 22 is possible, for which it is necessary to compare the orbit of a suitably supported gyroscope with the orbit of the space craft enclosing the gyro. As a last theoretical measure we must check whether this orbit correction does not violate the principle of energy conservation in a closed system. The change in kinetic energy along the orbit (solely due to the orbit correction) is as a function of time d ( 89

= mirorcco~a sin 2coct.

(1.30)

The corresponding change in potential energy is d ( m i M 2 / r p ) / d t = -- m~ror~co 3~ sin 2coct.

(1.31)

The two effects compensate each other exactly so that the principle of energy conservation is not violated. D.

ROTATING GRAVITATIONAL FIELD

We are familiar with rotating electric fields, but in how far a rotating mass also creates a rotating gravitational field is still a complete mystery. The problem is ex-

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A.BLOKLAND

tremely involved, because even if we do know how the mass as a whole rotates, we generally know next to nothing about the movement of important sections of the mass, such as the core of the Sun or even of the Earth. Above all, we know nothing about the possible rotation of the field of a neutral elementary particle. As the Earth as a whole is certainly rotating against the physical background of the galaxies, we do have, however, an invaluable part of physical space close at hand to pursue practical investigations. In relation (1.15) we found that the precession is directly proportional to the dq5 of the field (being a well-known fact in case we deal with electric fields) so that the rate of precession does not depend on the angular velocity of the Earth as such, but only on the angular velocity of its field! Two space probes in equatorial orbits moving in opposite directions should show a clear difference in the observed precession values if there is such a thing as a rotating Earth field. However desirable a positive answer to the question may be, we will probably have to be careful in trying to attach undue significance to a particular value obtained. We have found that a spinning mass is influenced in a special manner so that we may safely conclude that any other spinning mass in - and around - the Earth will be influenced too, and probably in such a way as to counteract the original cause. As this has been going on for billions of years according to geologists, we may expect a situation that shows only a small angular velocity, unless we have a continually fluctuating state in which anything is possible. Considering the vast amounts of matter involved, and the rather weak coupling, these fluctuations m a y show periods of millions of years, and might even swing the effective field rotation into a negative one in relation to the rotation of the Earth. E. SUMMARYOF PART I (1) The geodetic path of a freely spinning disc of neutral matter in a static spherically symmetric gravitational field shows the same characteristics as the path of a pure current loop in a static spherically symmetric electric field. (2) The plane of the spinning disc in a virtually circular orbit around the Earth shows an oscillating precession according relation (1.15), amounting to

(dA/A)/d~? = 21 (R~/ro) 89sin 2 4 .

(1.32)

(3) The geodetic path of such a freely spinning disc can never be a true circle, but shows at least a radius correction rc according to relation (1.29). Part II: Observation of a Theoretically Existent Effect

A. OSCILLATINGGYROSCOPES We have already repeatedly referred in Part I to the electrical analogue of the gravitational effects we were investigating, because these electrical effects are better known, and are more striking in appearance due to the enormously much larger force-mass ratio. Coming to coupled gyroscopes we have again an analogue in an electric current

A DECISIVE TEST FOR THE GENERAL RELATIVITY

227

in a conductive ring, as opposed to a pure-current loop in which every elementary mass particle of a disc has the same velocity as its associated electric charge. If we submit an electric current in a superconductive ring to the same conditions as discussed in Part I, allowing an excess of charge in order to keep the ring in a circular orbit, we know that the ring will show only a negligible precession, but will rapidly align its plane with the orbital plane. This principle is applied in many electro technical products, but it follows also immediately from relation (1.15) if we realize that the couple on the ring remains the same whether we let the mass take part in the motion of the charge or not, and that in the latter case the couple is used to turn the ring about its Y-axis without experiencing an opposing effect of an already existing angular momentum in the X-direction, as happens in case we use a pure-current loop. The gravitational analogue of a current in a ring is a spinning disc of which the movement of its spin axis is restricted (without loss of energy !) to a plane that is perpendicular to dA1, being a plane through the spin axis of the disc and parallel to the Earth axis in case we use an equatorial orbit. It always comes as a bit of a surprise if we see the rotating disc of a demonstration gyro in a laboratory, and experience no customary gyroscopic reaction if the motion of the gimbals is suitably restricted. It follows again from relation (1.15) that the dA 1 is unaffected by the restriction of the gyro, but this angular momentum is not vectorially added to the A of the disc (being impossible), but now represents a rotation of the disc about its Y-axis, exactly like in the electrical analogue. Calling the resulting angular velocity coy, and the moment of inertia of the disc about its Y-direction Iy, we find from (1.15)

dA 1 = Iy dco, = 89 dco, = (Rm/ro) 2~ (89 sin 2 r d e ) .

(2.1)

From which we solve coy, using relation (1.10)

dcoy = (Rm/ro) 2~ (coo sin2 ~b d r

(2.2)

It is clear that the angular acceleration of (2.2) causes a deviation angle that can be seen with bare eyes already after one day in orbit, and this is the reason why the present author proposed this test in his Dutch publication. (Blokland, 1968) Specialists in the technical field (Huber, 1969) pointed out, however, that it was a physical impossibility to maintain the proposed plane with adequate accuracy, being within about l0 - ~ rad. The only possible solution seemed to be a second gyroscope that might maintain the direction with the required average precision over a certain length of time. But this stabilizing gyro would, naturally, show itself a precession, and therewith equalize the effect to be observed. Fortunately, the case is not as hopeless as it might seem to be at first sight, because the precession velocity varies along the orbit. If we take two gyro's that are rigidly connected with each other, but have their spin axes perpendicular to each other - both lying in the orbital plane - we then get the situation that the one gyro keeps the other one more or less in check during certain parts of the orbit so that we find an oscillating movement of the gyro's about a neutral position.

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A.BLOKLAND

It is not the average precession or the total precession after a certain time that is measured in this manner, but the difference between the rates of precession along the orbit. This, however, with an incomparable precision. B. PRECESSIONVELOCITYAND AMPLITUDEOF COUPLED GYRO'S We assume a Cartesian coordinate system with the X- and Y-direction lying in the orbital plane, so that the Z-direction is perpendicular to this plane. We call the first gyro the X-gyro, because its angular momentum points into the X-direction. We call the second gyro the Y-gyro for the identical reason. Both gyro's have the same mean precession velocity about the Z-direction COx = COt = ( d A a / A ) / d t = 88(Rm/ro) COo219

(2.3)

that follows again from relation (1.15). The X-gyro would show an additional precession velocity about the Z-direction if it were completely free moving COzx = (COo2iR,,/4ro) (2 sin 2 ~ - 1) = - (COo21Rm/4ro) cos2qS.

(2.4)

The Y-gyro would show an additional precession velocity under these circumstances for the simultaneous angle (~b +89 COzy = + (COo21Rm/4ro) cos 2~b.

(2.5)

ooba ao e II L Y 2~

IZ

Fig. 1

A DECISIVE TEST FOR THE GENERAL RELATIVITY

229

These velocities can be considered as being caused by internal couples (like we are used to with electric currents in magnetic fields) that tend to rotate the whole system of disc and gimbals about a particular axis. The internal couple of the X-gyro points into the Y-direction, which follows again from the direction of dA 1 out of relation (1.15), and has a value c0zxAx. The internal couple of the Y-gyro points into the X-direction, and has a value cozyAy.If we use the instrument as sketched in Figure 1 we must assume that the system will show additional angular velocities cox, coy, and coz at an arbitrary instant of time, that are superimposed on the angular velocities mentioned above, and that should be multiplied with A x and with Ay to represent additional internal couples. These internal couples are opposed by what might be called external couples, that are caused by the inertial masses around a given axis, which masses resist an acceleration. If we call Ix, Iy, and I~ the total moments of inertia of all the masses to be accelerated about a particular axis, we then can write down the three equations of equilibrium as + coyA~ - coxAy - I~ dco~/dt = O, + co~xAx + co~Ax + Iy dcoy/dt = O, + co~yAy + co~Ay - I~ d c o J d t = O.

(2.6) (2.7) (2.8)

We can now differentiate (2.6) over the time, and afterwards substitute the values found in (2.7) and (2.8), with the result -

-

A Ix + A2Iy coax - Azlx + Azlr- coz = d2coJ dtz ,

(2.9)

where cozyhas been substituted by -cozx, according to relations (2.4) and (2.5). Using our knowledge of cozx, and observing the relation between coz and d2coJdt 2, we can give coz the value co~ = Pl sin2coot + P2 cos 2co0t + P3 sin co: + P4 cosco:,

(2.10)

where we consistently neglect the mean steady precession of (2.3), and where co~ has the constant value co~ _ A~I~ + A zyIy ixlylz

(2.11)

We can now submit the system to the initial condition of having the additional angular velocities cox, coy, and co~ to be zero, from which follows that also dcoJdt is zero at the initial instant t = 0 , if we look at relation (2.6). We will now proceed to find the values of the constants Pl till P4. From relation (2.10) as it stands, and from the fact that co~=0 at the time t = 0 it follows that P* = - Pz.

(2.12)

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A.BLOKLAND

F r o m the fact that the first derivative of (2.10) is zero at the time t = 0 it follows that P3 = - Pl (2co0/coi).

(2.13)

I f we now add co2 times the value of (2.10) to its second derivative, we get c~

+ d2coJ dr2 = (co~ - 4co2) Pl sinZcoot + (co~ - 4coo2) P2 cosZcoot. (2.14)

I f we compare this value with the one found in (2.9) it follows that p~ must be zero, and consequently from (2.13) t h a t p 3 must be zero too. We can now use the value of co~ out of (2.4) in relation (2.9), and solve the value Pa out of (2.14) P2 =

- A~Ix + AZyIy -- coo21R,~/4ro I flyI: co~ - 4co2o

(2.15)

Considering the value of o)2 in this relation as defined in (2.11), and realizing that the I:`, Iy, and I~ will be kept as low as good engineering will permit, so that they may be from five to fifty times as large as the inertial moment I of the discs by themselves, as used in the values A:` and Ay of relation (2.11), we must conclude that cot is of the order of 10~ of cog, so that coo 2 is fully negligible in relation to o)2. Therefore, substituting the value of co2 in relation (2.15), we find for P2 the very accurate result 2

2

A fl~ - Ayly coo21R,~ P2-- 2 2 A fl:~ + A fly 4r o

(2.16)

It is rather clear that cot is a result of our choice in regard to the initial conditions, and that a very slight damping effect will suffice to let it die out soon, in any way. Assuming to have reached this condition, we are left with the single angular velocity 2coo. Combining the now known value of coz with the value of co~:` in relation (2.7), we get

2A~IyA:`

2

coo21R,~

2

AxI~ + Arly

- 4ro

COS2CO0t = Iy dcoy/dt,

(2.17)

and after integration 2

AyA:, 21Rm sin 2coot + Ka. coy - A2I:` + A2Iy 4ro

(2.18)

Likewise, we find for co:` 2

A:`Ay 21R,, sin2coot + K2. co:' = A~I:` + A2Iy 4r o

(2.19)

In our Figure 1 we have suggested a rather massive Y-gyro and a very light frame about the Y-direction, making Iy as small as possible. I f we succeed in keeping Iy under ten times the value of the X-gyro, and that we can give an angular velocity of about 6000 rad/s, which seems to be about the maximum we can expect at present, we then find an approximate value for the first term out of (2.18) of 1000.

A DECISIVE TEST FOR THE GENERAL RELATIVITY

231

I f we take about 95 min of time for one full revolution of the space craft around the Earth we have coo= 1.2 x 10 -a. Now, comparing the angular velocity coy out of (2.18) with the mean precession velocity out of (2.3) we find the remarkable figure

(2.20)

coy/cog = 800 000 sin 2coot.

We should remember, of course, that the mean angular velocity my has a cumulative effect, whereas coy only means an oscillation about a neutral position. The expected amplitude of the oscillating gyroscopes is, therefore, at least as important as the angular velocity. Using the value 0.45 cm for the mass radius R m of the Earth, and 6.5 x l0 s cm for the orbital radius r0, and further assuming that 21 has a value of about 2, we then can integrate relation (2.18) to find an angle 6 from

6 = 1000

2 x 0.45 4x6.5x10

- cos2coot s2x

1 . 2 x 10 -3

=-

1.45 x 10-4cos2coot. (2.21)

As the gyroscopes complete two oscillations per revolution of the space craft around the Earth we find that the axes of the gyro's (either the X-gyro or the Y-gyro, which ever has been chosen for the task) swings in about 24 min of time over an angle of 26, being about 1 min of arc, which is almost ten times the arc that is expected in Stanford after a full year. Or, with other words, the coupled gyro's give in 24 min of time the same result a single gyro would give after about 10 ~r ! A further tremendous advantage of the use of coupled gyroscopes is that the position of one gimbal in relation to another gimbal of the same instrument is observed, so that it is superfluous to observe an exact astronomical direction as necessary in the Stanford test.* C. PRACTICALEXECUTIONOF THE TEST We propose to discuss a design that probably will give satisfactory results, but there is no doubt in our mind that a fuller study may show considerable improvements over the proposal submitted here, so that any suggestion of interested readers is welcome. As the average deviation between the spin axes of the two rotors should be well within an angle of 10 -1~ rad, we believe that only precision ball bearings can fulfil this condition, where we have the advantage of averaging over periods of 24 min, and over more than 106 revolutions of the rotors. The frame on which the rotors are mounted should be symmetric about the Y-axis, and should be protected against varying temperature gradients in order to minimize deformation of this frame in space. I f we take rotors of currently used dimensions (Kranenborg, 1967) we find an angular m o m e n t u m A of some 500000 (g c m 2 / s ) . The

* The effect found above is about three to four magnitudes larger than mentioned in my first article in A. and Sp. S. This has been obtained by leaving out the third stabilizing gyroscope, which means that we have to incorporate a stabilizing force of another origin in the present instrument.

232

A.BLOKLAND

couple to accelerate the system is the according (2.4) and (2.7) maximum

o)zxAx -c~176

x=

4.2 x 10 -13 x 5 x 105 = 2.1

x 10 .7

dyne cm.

4r0

(2.22) It is fully clear that we can allow no bearing friction with such an extremely small couple, so that a natural solution is to support the frames by very thin, strong, and elastic torsion wires. These wires can at the same time fulfil the task of counteracting the instability of the system, which is caused by the fact that an increase in angular momentum of the X-gyro, for example, in the Z-direction is fully compensated by a decrease in angular momentum of the Y-gyro, which fact is expressed by the constants K1 and/(2 in the relations (2.18) and (2.19). It is exactly this fact that is the cause of the utter sensitivity of the coupled gyro's. In order to give an impression about the dimensions of these torsion wires we will assume a length of about 2 cm. The torque in such a wire should be about 10 .8 dyne cm after it is twisted over an angle of about 1 min of arc, in order to leave free play for the acceleration force mentioned in (2.22). If we take a material like steel with a gliding modulus G--8 x 1011 dyne/cm 2, we find (32Tl~1/,~( 3 2 x 10-8 x 2 )1/4 d=\-~-G~-~j = ~x8x 1011 x 3 x 1 0 -~ = a b o u t 2 x 10 - 4 c m . (2.23) This wire should be able to exert a longitudinal force of about 500 dyne easily, which may be considered ample for a floating suspension in absolute weightlessness. It may be superfluous under the given conditions to use a drag free space craft, so that a comparatively inexpensive space craft can be used. The calculated wires are, of course, unable to support the system during the time the space craft is brought in orbit, or during the switching on period of the gyro's. The frames have to rest in firm supports during these periods, but will be pulled free as soon as equilibrium is reached if the supports allow sufficient free play, but not too much in order to prevent over stressing of the wires. The present author deems it advisable to attach to frame 1x two rotating unbalances, of which the centrifugal forces will create a couple on frame 1~, having a period of about 5 min. Now, observing the direction of A~ (or rather frame Iy) in relation to frame lx we can expect the following values: (1) An oscillation with a period of about 48 min. (2) An oscillation with a period of about 5 min that gives a constant check on the free floating position of the frames. (3) A floating of perhaps up to one degree about a neutral position. (4) An irregular movement, or perhaps even a periodic movement (which is far worse) due to temperature variations, changes in the energy supply to the gyro's and such like.

A DECISIVE TEST FOR THE GENERAL RELATIVITY

233

There can be no doubt about it that the instrument is a very delicate one, but if we compare its requirements with those for measuring an angle of a few arc seconds per year it almost appears to be a minor task to measure an angle of one arc minute every 24 min of time, where the result can be checked over and over again, and where we have no need of an exact astronomical orientation. D. SUMMARY OF P A R T I I

(1) The spin axis of a gyroscope that is kept in check by a second gyroscope can show an angular velocity under the influence of the gravitational field of the Earth that is 100000 to 1000000 times greater than the precession velocity of a single free floating gyroscope under that same influence. (2) The oscillating motion of this spin axis covers in less than half an hour the same angle that a free floating gyroscope covers in about 10 yr. (3) The observation of this angle is greatly facilitated by the fact that no outside reference is required, such as a very exact astronomical direction, against which the precession angle of a free floating gyroscope is measured.

Part III: Tensor Application A. O R I G I N OF R E L A T I O N S

(1.1)

AND

(1.2)

Einstein discovered that the total energy content E of a mass m in relation to an inertial observer O is mc 2

mc

E - (1 - u2/c2) 1/2 - dr c d t ,

(3.1)

where m is an invariant quantity for every observer, called the rest-mass, where c is the light velocity, where u is the total velocity of tn in relation to O, where dt is the elementary time in O during which the mass was observed by O to cover the elementary distance dr between two space-time points, and where dr is the elementary time a local clock moving with the mass would have registered between these two points. A related discovery was that the three momentum components o f this same mass (multiplied by c) are for observer O at the same time mc PaC=(1-u2/c2)

dx a mc ~/2 dt - d r dx"

( a = 1, 2, 3).

(3.2)

The four quantities thus defined can be considered as the components of a four-vector, the energy m o m e n t u m vector pi. Now, to find the relations (1.1) and (1.2) we need, further, three postulates that are quite independent of special relativity, and that were, in fact, formulated much earlier. (1) The force on a charge in a static electric field is independent of the velocity of the charge, and the force has always the same direction as the field. (2) Energy is conserved in a closed system. (3) M o m e n t u m of a closed system is conserved in space.

234

A. BLOKLAND

F r o m (1) follows that a charge - Q1 with a rest mass mel at a distance r from the center of a static charge + Q 2 , where the charge - Q 1 has a radial velocity u r and a tangential velocity u t in relation to + Q 2 , experiences a radially directed force F, exactly like the force on a charge at rest in the field, being F r = - Q1Q2/r 2 .

(3.3)

F r o m (2) it follows that the gain in potential energy of the rest mass me~ over a distance d r is equal to the loss in kinetic energy, so that we find, f r o m (3.3) and (3.1), m e l 8 9 (t/2)

-- 0 1 Q 2

dE -

r2

dr

=

(1 --

U2/C2) ll/2"

(3.4)

F r o m (3) follows that the tangential m o m e n t u m remains unchanged under these conditions, so that we find, f r o m (3.2), me,

(1

--

mel 89 (u 2)

Ut

U2/C2) 1/2 dut "t- (1 -- U2/C2) ''/2 ~C = 0.

(3.5)

Solving du t out of (3.5), dividing the result by dt, and substituting the corresponding value f r o m (3.4), we find dut Q,Q2 UrUt (1 - u2/c2) */2 dt - at r 2 (22 mel

( = 1.1).

(3.6)

Having found relation (1.1), we n o w look again at relation (3.5), divide it by dt, and rewrite it as

(r

__

2

gl r

__

U2) at

= - - u t :1 d

(u, 2 +

u2)/dt

= -

(utu,a, + u2tat).

(3.7)

u2at to the left hand side, dividing the result by utur and then substituting the value found in (3.6), we find Bringing

Q~Q2 ( 1

-

( - - 1 -{- U2r/C2) " r2

u2/c2) ~/2 reel

a,

(3.8)

--

which is the relation cited in (1.2). B. ORIGIN OF RELATIONS (1.3) AND (1.4)

Einstein postulated that his field equations will be zero outside of the mass points creating the gravitational field. (Bergmann, 1946) Guv = R ~' - 89

= 0.

(3.9)

The solution for the line element in a static field with spherical s y m m e t r y following f r o m this postulate, as first found by Schwarzschild, can be written in the f o r m ds 2 = q dt 2

_

c-2

(q-1

dr 2 + r 2 (dO e + sin20 d~b2)),

(3.10)

where q = ( 1 - 2R,,/r) (McVittie, 1956). It is also k n o w n in the f o r m where q = e ~, in which q can be expended in powers of R,,,/r ( A m o n g others: ESRO, 1970).

A DECISIVE TEST FOR THE GENERAL RELATIVITY

235

The solution for the line element can also be expressed in isotropic coordinate values. (McVittie, 1956) ds 2 = q dt 2 - e - 2 q -1 (dr 2 + r 2 d 0 2 -1- r 2 sin20 d~b2).

(3.11)

In the E S R O Mission Definition (1970) we find the statement, that they consider (3.11) the more generalized form. We there find also a further definition of q and q - 1 as follows q = 1 - 2aR,,/r + 2 f l ( R m / r ) z + . . . . (3.12) q - 1 = 1 + 27Rm/r + . . . . (3.13) The object o f the E S R O mission is primarily to obtain more accurate values for c~, fi, and 7. A second objective is to find information on the quadrupole m o m e n t o f the sun, that would follow f r o m the observations o f Dicke and Goldenberg. A further possible f o r m for the expression of the line element, that is favoured by the present author, because it makes an effort to relate the expression to the position of an observer in ro, thus accounting fully for the results of Michelson and Morley on Earth, is ds 2 = ( q / q o ) 2 dt a - e - 2 (dr 2 + r 2 d02 + r 2 sin z 0 dq~2),

(3.14)

where (q/qo) = (1 - Rm/r)/(1 - Rm/ro). It should be clearly understood that every single one of these relations can be a correct solution, but also that all of them are jointly a correct solution, because they are all parameter relations in which t and r are not necessarily physical quantities, although they are closely related to the physical quantities 'observed time' and 'observed radial distance'. Obvious is that the parameters t and r out of (3.10) have another relation with 'observed time' and 'observed radial distance' than the parameters t and r out of the relations (3.11) and (3.14), but that it is possible to define a relation between the different parameters. Equally obvious is that every mathematical relation, cited above, has a physical meaning only in an instruction is given about the observational significance o f the parameters, and in this respect we find the E S R O Mission Definition quite explicit. It m a y be instructive, in this connection, to quote McVittie at some length on the interpretation o f a mathematical result (p. 87). "In order to interpret this result in the solar system it is first necessary to identify the coordinates (t, r, O, (o) with those used by astronomers." McVittie then goes on describing the methods of the astronomers, and the magnitude of an r-correction, after which he continues: "... the identification of r with the heliocentric distances computed by astronomers on the hypothesis of Euclidean geometry introduces an error no larger than one part in eight millions, which is far below the errors in the distances themselves. Similarly, the nearly Euclidean character of the geometry suggests that ~0..., and 0... are identifiable. Finally, the time coordinate t will be assumed to measure the time used in astronomical observatories, universal time for instance. These are purely pragmatic definitions of the coordinates (t, r, O, ~o)appropriate to, and made possible by, the analysis of a specific physical situation."

236

A.BLOKLAND

So far McVittie. Returning now to our subject of finding relations (1.3) and (1.4) we are first going to calculate the Christoffel symbols that follow from the metrics as defined in (3.10), (3.11) and (3.14). The results are given in the table below. In this table an accent indicates a first derivative over r, and q is a pure function of r only, but with the appropriate values mentioned in the discusssion above. Symbol

No.

(3.10

(3.11)

(3.14)

010)

(1)

89 c2

89 c2

qq'c2/qo2

(1013)

(2)

89

89

q'/q

(111)

(3)

--89

--89

0

(313)

(4)

--qr

--r +89

--r

(133)

(5)

1/r

1/r--89

1/r

Note that we have restricted our problem to the plane 0=89 so that dO is zero, and sin 2 0 is one. We are now going to use a further postulate from general relativity theory: 'The statement that a mass falls freely in a gravitational field is equivalent with the statement that the mass follows a geodesic in four-dimensional space'. The differential equations for a geodesic are d2xk

~ds+

( k ) dx/dxn - O. ~n ds -ds

(3.15)

Using Christoffel symbol No. (2) in (3.15), with (3.10), (3.11), and (3.14) respectively we find that (q dt/ds), (q dt/ds),

and

(q2 dt/ds)

are constant.

Using symbol No. (5) we find, in the same manner, (r 2 dqS/ds), (r 2 ddp/q ds), and (r a dqb/ds)

are constant.

(3.16)

(3.17)

We now divide the values of (3.17)by the corresponding values of (3.16) in order to find functions of dcHdt. Differentiating these values over t, we find from (3.10) that ( dr~t~t d2q~ (r/q) 2 d r + r dt 2

r d r )dq q dt ~ - = 0 "

(3.18)

With (3.11) and (3.14) we find that ( drd~ , (r/q 2) 2 dtt ~ + r dt 2

2rd~dq) q at ~ = o.

(3.19)

237

A DECISIVE TEST FOR TI-IE GENERAL RELATIVITY

In both relations we recognize in the first two terms between brackets a tangential acceleration as mentioned in (1.1). Giving q the appropriate values belonging to (3.10), (3.11), and (3.14) we find five distinct expressions for the tangential acceleration a t. (3.10): q = 1 -- 2R~/r at = (2/q) u,utRm/r 2 (3.20) (3.10): ESRO (3.11) : q = 1 - 2Rm/r (3.11): ESRO (3.14): q -

qo

urutRm/r 2 a t = (4/q) urutRm/r z at = (2e + 2y) u,utRm/r 2 at = (2e)

1 - R,,/r a t = (2/q) 1 - R~/r o

(3.21) (3.22) (3.23)

u,utRm/r z

(3.24)

We have herewith found the origin of the relation (1.3). And in a rather promising form, because 21 can have a wide range of values, so that a reliable observation is obviously of prime interest. We might, instead of 'promising', also consider these results a bit surprising, because relations (3.10), (3.11), and (3.14) were considered observationally almost indistinguishable (See McVittie, op. cit.), but they now prove to diverge strongly after only one further differentiation of a quotient, so that the lack of a precise definition of the relation between the parameters t and r, and the observational values time and radial distance leaves the theory very much in the blue. Illuminating is, in this connection, a comparison between (3.20) and (3.24). They are indistinguishable even with the oscillating gyro test. Yet, the consequences of everything that can be said about the Schwarzschild singularity are tremendous. With (3.10) we can write about neutron stars with light leaking tangentially from the Schwarzschild singularity, about black holes in space, etc. With (3.14) nothing of the kind occurs, because everything is related to a well-defined observer. As relation (1.4) proved to be observationally less important, we will go less in detail here. Again assuming dO to be zero, we divide (3.10) by ds 2, and solve dr/dt by multiplying dr/ds with ds/dt out of (3.16).

d r - q ( K(c2 ( K2"~1/2 at K 1 - q c2 -t- ~ ]] ,

(3.25)

where K 1 = q dt/ds, and K 2 = r 2 d(a/ds. Differentiating (3.25) over t, and subtracting r (dqS/dt) z, we find d2r

a,=dt z

dq~2 = c2Rm ( _ 2Rm 3u 2 r~ r2 \ 1 + r + qc 2

2u2~ c2 J

.

(3.26)

We have herewith found the origin of (1.4) also. As 2R,,/r and 2u2/c 2 have an exactly identical value in our gyro problem, we have left them out of(1.4) for simpIicity's sake. The value of 22 seems to be about three according to this relation, and not zero like in Newtonian theory, and neither one like in the electrical case. The present

238

A.BLOKLAND

author does not trust this value three, because a convincing physical explanation can be given for a value two, but not for a value three. What the full consequences are of an application of (3.26) as it stands may be a side issue in our present problem, but it might become a major issue if we investigated the fast moving particles in a nucleus, because we might find strong repulsion. C. PARALLEL DISPLACEMENT OF A VECTOR

In a previous publication mentioned before (Blokland, 1970) we approached the present problem with the question whether the direction of an angular momentum carried around the Earth in a circular orbit behaves like the direction of a parallel displaced vector in general relativity theory (De Sitter, 1916), and we now return to this part of the problem. It was pointed out to me (Peebles, 1970) that I used in my article a two-dimensional section of space time (r, ~b), but that in a displacement of an angular momentum both space and time are involved. The point is well taken, and certainly merits due consideration. We will therefore proceed to calculate the increments of a parallel displaced arbitrary four dimensional vector A * that transforms, consequently, like a proper elementary interval ds, by using the Christoffel symbols following from (3.10): namely, dA~ = dAa =

~ dr,

(3.27)

- ( k,13] 3 ~ A 1 dq5 = - ( l / r ) A 1 dqS,

(3.28)

\10]

dA l = - ( 1 ) A 0 ~

dt = - (89

d t - ( 1 ) A ~ A3dc~=-( 89 (3.29)

Remember in the choice of the symbols that d r is zero in this case. Now, replacing dq5 by co dt to make all relations a function of t, we then differentiate (3.29) with respect to t, and substitute the values of (3.27) and (3.28) in the result: we obtain d2A ~

dt 2

_ A 1,~ (~q ,a c 2 - qco2) = - A l c o 2 ( - Rm/r + (1 -- 2Rm/r)),

(3.30)

where q,2 =4R2/r 4, and caR,n/r 3= co2 in the circular case. It follows from (3.30) that we can give A t the value A t = A cos (1 - 3Rm/2r ) cot.

(3.31)

Substituting this value in (3.27) and (3.28), and then integrating, we find

A~ = (1 --

AR~/ra - sin (1 - 3Rm/2r ) cot, 2Rm/r ) (1 -- 3R~/2r) 09

A A 3 = -- r (1 - 3R,J2r) sin (1 - 3Rm/2r) cot,

(3.32) (3.33)

A DECISIVE TEST FOR THE GENERAL RELATIVITY

239

These relations are given as a function of a kind of universal time as used by McVittie and ESRO. We could also use the proper time of the gravitational center of the space craft, that possibly encloses the as yet physically undefined vector Ak This would mean a useless complication, because there is always a fixed ratio between universal time and the proper time of the center of the space craft in a circular orbit, that follows from the spherical symmetry of the field. It should be fully clear that one cannot speak about the proper time of a finite vector A ~ itself, because one of the characteristics of a curved space is that the proper time is a function of the coordinates. The maximum one can do about a finite extension in space and time is to speak about a mean proper time, but certainly never about 'a' proper time. We now come to another important consideration, that is apparently overlooked sometimes. The values A ~ A 1, and A a are not the vector components of A i, as little as dt, dr, and d~ are the vector components of the elementary interval (Is. I f A' represents, for example, a pure time interval (time-like vector) or a pure distance interval (space-like vector), we then deal with the same type of quantities that appear in an elementary proper time or in an elementary proper distance. The quantities A ~ A 1, and A 3 have then the dimension t, r, and q5 respectively. Exactly like in the elementary case, we find the components of the finite vector from cql/2 A o, q-i/2 )11 and rA 3. We will call these components A ti, A r, and A ~ respectively, and neglect higher powers of Rm/r,

A t ' = - (Rm/r)l/Za sin (1 - 3Rm/2r) r

(3.34)

where we have again replaced c2Rm/r 3 by co2; A r = (1 + Rm/r) A cos (1 - 3Rm/2r) (a,

(3.35)

A ~ = - (1 + 3Rm/2r) A sin (1 - 3Rm/2r) O.

(3.36)

It follows automatically from the postulated invariance of ds 2, but can be easily checked by using the above given values, that the absolute length of A ~ is invariant. (A') 2 = (At) z + (a4') 2 - (Aa) z = (1 + 2R,,/r) A z . (3.37) We could, of course, also have used the Christoffel symbols following from (3.11). This would have made no difference, here, as the results (3.34), (3.35), and (3.36) would have appeared in the end, although the intermediate relations are not exactly the same. Using, however, the Christoffel symbols following from (3.14), we get

a a = - (Rm/r)l/ZA sin (1 - R,,/2r) O, A" = A cos (1 - Rm/2r) c~, A ~ = - (1 + Rm/2r) A sin (1 - R,,/2r) c~, (Ai) z = A 2 = invariant.

(3.38) (3.39) (3.40) (3.41)

It is interesting to note that the values of (3.39) and (3.40) fully agree with those found in Part I, if we give 21 the value 2, and if we integrate the relations (1.15) and (1.16) over qS.

240

A.BLOKLAND

The mean precession is now nR,,/r per revolution around the Earth, instead of 3~R,,/r per revolution, as found earlier above. For those readers who are interested in vector diagrams the vector components of A *are set out in Figure 2 after the vector has covered an angle q~. The mean precession has been neglected in this diagram in order to show more clearly the oscillating motion of the vector, due to the different multiplication factors in A" and A*. The A ti component has been added in a relativistic manner to the sum of A r and A*.

A r ~ AOz. ~ ' - ~ < ....... /~-.. A*=-(I* J=~-m)Asin 0 -~

t'i\

\ \ Xl

t: L--~

~

|

- A t i : ~/-~T A s i n 0

/

.........................................

1

Fig. 2. Relations (3.34) - (3.37) with Rm =m.

D. PHYSICALMEANINGOF A i We have already spoken about A t as representing a finite extension in space and time, and this case is certainly covered by the general relativity postulate. We could take a homogeneous ball of matter (to prevent tidal effects) with a small hole drilled through the center, and two very accurate clocks fixed close to the exits of the hole. Observing the astronomical direction of the hole, having a fixed initial orientation, observing the length of the hole by some kind of interference method, and observing the time markings of the clocks, preferable from out of the center of the hole with a M6ssbauer effect, we then could find (in principle in any way) values for At, Ar, and A~b. We could then calculate the value of the finite interval, that should prove to remain invariant along the orbit if the general relativity postulate is correct. Note that we cannot speak about the proper time of the interval, but that we assume this proper time to be a linear function of the distance to the center, and measure it only at the extremities of the physical extension. We now have to ask again whether A ~can also be taken as a mathematical representation of an angular m o m e n t u m Ir o.

A DECISIVE TEST FOR THE GENERAL RELATIVITY

241

If we take (3.2) as a guide, where a momentum was multiplied by the light velocity in order to obtain a comparable energy, we are led, here, to multiply the initial angular momentum with half the initial angular velocity in order to obtain the kinetic energy of the disc as a recognizable part of the total energy. We then find with (3.14): 1 --tl zO9ooA = - l Ico2oo (Rm/r) ~/z sin (1 - Rm/2r) r

•2~'gO-~Ja~ -- - !r,.,2 2.trY, gO

COS (1

--

Rm/Zr) r

lo9 .o A r = - ~Ic%o ~ 2 (1 + Rm/2r ) sin(1

(3.42) (3.43)

Rm/2r ) r

(3.44)

Having the results of Part I available, where we investigated the energy balance more in detail, we now might recognize the invariance of (Ai) 2 as an illustration of the law of energy conservation in the closed system 'Central mass with revolving spinning disc'. Calling A t the 'Kinetic energy-angular momentum' vector, we have the further analogy with the 'Energy-momentum' vector P ~that in both cases the time component is not a direct observable, but a quantity that follows from observables, contrary to what we found about the direct observability of the time component in a finite interval. It is obvious that the invariance of (A~)2 does not mean that the angular momentum itself is invariant along the orbit, as one sometimes reads, as its change in magnitude follows plainly from (3.39) and (3.40). E. R E P R E S E N T I N G A N A N G U L A R

M O M E N T U M BY A TENSOR

To round off the subject, it should be mentioned that Corinaldesi and Papapetrou (1951) took the tensor S uv as a mathematical representation of an angular momentum. Marie-Antoinette Tonnelat treats this point of view and its implications at considerable length in her admirable work 'V&ifications exp6rimentales de la Relativit6 G6n6rales' (Tonnelat, 1964). The choice of the above mentioned tensor is, of course, quite logical, but a disappointing fact is that the ensuing relations prove to be unsolvable without additional (and necessarily rather arbitrary) assumptions, so that the results are by no means above suspicion. Corinaldesi-Papapetrou takes: S P ~ ( p = 1, 2, 3). Pirani (1956) takes SuouQ=O. Schiff (1960) explains the resulting difference. There emerges a steady precession again of 2re ( 3 R m / 2 r ) radians per revolution of the disc around the Earth, but no oscillatory effect is mentioned like we now found for a vector. Therefore, however fascinating the mathematical treatise may be, it looks as if the additional assumptions simplify the problem completely beyond recognition. The fuller treatment of Part I not only shows that there is such an oscillatory effect, but it shows also that the steady ( = m e a n ) precession is only one third of the value quoted above. F.

SUMMARY OF PART III

(1) The non-tensor relations (1.1) till (1.4) follow as a straightforward mathematical conclusion from special and general relativity theory.

242

A.BLOKLAND

(2) General relativity theory allows a large margin o f uncertainty in the values 21 and 22. (3) A four dimensional vector (and perhaps also a four dimensional tensor) can with due caution be used as a mathematical representation of a physical angular m o m e n t u m , and its kinetic energy. (4) M o r e observational data are required to enable a meaningfull interpretation o f general relativity theory. M o s t urgent is a binding relation between the parameters t and r, and the observational values time and radius for one well defined observer in space.

Acknowledgements I gratefully acknowledge the stimulating discussions with Ir H u b e r o f the T H Eindhoven (Dept. E R A ) a b o u t the theoretical and practical aspects o f the test. I also wish to t h a n k D r Kallenberg, lecturer in tensor calculus at the T H Delft, for his very kind help with some more abstruse aspects o f the subject.

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