V.6 No 1

65

The problem of physical time in today physics

6. Mach’s problem of pail

6.1. Origin of the problem

The problem of pail rotating relative to the resting stars was raised by Mach and used by relativists in postulating the equivalence of inertial and non-inertial reference frames, and as the corollary of this postulate – in postulating the concept of relative acceleration: “The kinematic equivalence of two coordinate systems factually is not limited by the case when both systems K and K' move about each other uniformly and directly. From the kinematic point, this equivalence also well works, for example, if one frame uniformly rotates about another. So it seems necessary to generalise the existing theory of relativity so that to lift from it the seemingly untrue preference of uniform and direct motions over relative motions of other kinds. Each who is aware of the subject has to feel necessity of such extension of theory. True, it first seems that such extension of theory of relativity is impossible by physical reasons; namely – let K denotes the coordinate system in the Galilee – Newton meaning and K'  is the coordinate system uniformly orbiting about K. Then the centrifugal forces affect the masses resting in K' , while they do not affect the masses resting in K. Already Newton saw in it the proof that we have to think the orbiting of K'  as an ‘absolute’, and that we may not choose K'  as the ‘resting’ system equally to K. But this argument is not substantiated, and E. Mach showed it. The existence of centrifugal forces not certainly is caused by the very frame K'  . With the same success we can attribute them to the mean orbital motion of the remote weighty masses in the vicinity of frame relative to K' , with it we think K' resting. If Newtonian laws of mechanics disallow such interpretation, we can rather think it to be the defects of these theories. On the other hand, the following important argument says pro relativistic treatment. The centrifugal forces that affect some body at such conditions will be determined by exactly same world constant as the gravity field affecting it, so we have no means to distinct ‘the field of centrifugal forces’ from the gravity field. Thus, this what we measure as the weight of some body on the Earth’s surface factually is the result of joined affection of fields of both kinds which we cannot separate” [54, p. 327–328]. At the same time, as Poincare pointed, “it is not enough to confine ourselves to a simple comparison of formulas that agree with each other only due to a happy occasion; we need these formulas to be, so to say, penetrating into each other. Our mind will not be satisfied until we cannot believe that we have found the cause of this agreement so well that seemingly could predict it. However, we can imagine this issue from another point, best of all comparing. Imagine an astronomer living before Copernicus and pondering on the Ptolemy system; he would notice that for all planets one of two circles, epicycle either deferent, is the main circle that is passed at one and the same time. As it cannot be occasionally, then some mysterious interrelation exists between all planets. But Copernicus, changing only the coordinate axes that still were thought resting, cancelled this seemed interrelation at once; each planet describes only one circle, and periods of orbiting become independent of each other (until Kepler established the relation between them that was thought cancelled)” [55, p. 435–436].

Not in vain Poincare gave an example of Copernicus’ model, as the issue of substitution of centrifugal forces of inertia by the attraction of celestial bodies surrounding the Earth goes back to Aristotle; it became especially sharp at times of Copernicus, when he succeeded to prove the heliocentricity of the Earth orbiting. It was Galilee who stated the quintessence of this view of Aristotle’s followers in his monologue of Simplicio: “we can explain enough the phenomena of which you said; in particular, this last (predominance of east winds over those west – Authors) we can explain enough out of applying to the moving Earth, with only sky moving, doing not introducing something new except the opposite to that what you are putting forward. The peripatetic schools assume that the element of fire, as well as a considerable part of air, are involved into the diurnal rotation and touch with the firmament of lunar sphere that contains them like a vessel. And so, doing not digressing from your ways, I would like to get your admittance that the quantity of air participating in this motion reaches the tops of highest mountains and that it would reach the Earth, should the surmounts like just these mounts did not prevent it. This agrees with what you are saying; namely, you say that the air surrounded by the tops of mountains rotates because of uneven surface of moving Earth; we, on the contrary, say that the whole element of air is circularly moved by the motion of sky, except the part located lower than the tops and which is braked by the roughness of resting Earth. And when you are saying that should we have this unevenness lifted, it would take away the ability of air to be involved, we would be able to say that, should we lift this roughness, the whole air would go on moving. So, as the surface of vast seas is even and smooth, the motion of winds permanently blowing from the east goes on above them; it is more noticeable in the parts adjacent to the equator and between the tropics, where the sky moves faster; and as far as this motion of sky is able to involve with itself the whole amount of free air, we have good grounds to say that it imparts the same motion to the moving water which is flowable and is not bound to the resting Earth. We can state it even more surely that, as you admitted, such motion has to be very small comparing with its cause; this last, going diurnally the whole earthly sphere, passes many hundreds of miles hourly, especially near the equator, while the speed of flow in the open sea does not exceed several miles per hour. Thus, to float west will appear more convenient and fast not only due to the permanent east wind but also due to the flow of waters. Such flow can indirectly cause the ebb and flow, dependently on different location of the land shores. When the water strikes them, it can go back in the reverse motion, as we see on the example of flow of many rivers; when flowing among rough banks, the water encounters some ledge or a hollow that formed below, it produces a whirl and clearly flows back. So it seems to me that for the phenomena from which you deduce the motion of Earth, and take this last as the cause of firsts, we can find quite convincing cause, remaining the Earth resting and returning the motion to the sky” [39, p. 133–134].

The same, the issue of equivalence of inertial direct motion and of uniform orbiting goes back also to the times of Aristotle who postulated orbiting as the primordial and eternal motion: “orbiting necessarily has to be primordial (in the ontological hierarchy of different kinds of motion, not in the meaning of preceding in time – comment of the author of translation). Actually, the consummate in its nature is primordial in relation to the non-consummate. While a circle is something consummate, which we can say of no one direct line, neither of an infinite line (or [should it were finite] it would have the end and limit), neither of whatever finite line (as they all are not drawn to their end, since we can elongate any)” [56, p. 268]. With it, “any body moving up or down necessarily has to have lightness or weight, or them both together (only not in relation to the same [body]…). However, the orbiting body cannot have lightness or weight, as neither with nor against its nature it can move to or from the centre. The direct motion cannot be for it in agreement with nature, because each of simple bodies [according to the initial premise] has only one [natural] motion… And as far as the whole and its part in natural motion move in one direction (for example, the whole globe and its little lump), it follows from it that, first, it has no lightness either weight (in the opposite case it would be unable, according to its own nature, to move to or from the centre [which is impossible]), and second, it cannot do its motion in space, being drifted up or down, as it can move otherwise [than orbiting] neither in agreement nor opposing its nature, and this concerns both to it [as the whole] and to its parts, as in relation both to the whole and its part, the same consideration is in power” [56, p. 269–270].

Taking into account that at Aristotle’s times and even at times of Galilee, the concept of inertia still has not been formed in its appearance formulated by Newton, we can see from this citation that Aristotle factually admitted the orbital motion around the gravitating centre to be fully inertial in the today meaning of this term, as he stated that in such motion the body does not show such properties that it shows in an accelerated motion up or down. Important that Galilee in his “Dialogues” [39], disproving the Aristotle’s theory in many aspects, factually agreed with Aristotle that it is impossible to reveal the uniform orbiting in the frame involved in this motion: “Thus, the meaning of objection of this philosopher is, whether the principle by whose power we move together with Earth will be outer either inner, we anyway would have to feel it; as we do not, it is neither, so we do not move, so the Earth does not move. I say, it can be either and we will not feel something. The fact that it can be outer is proven more than enough by the experiment with the boat; I say, more than enough, as, having a possibility at any time to make it moving, as well as to stop it and to observe more thoroughly, whether this difference causes any tangible feeling by which we could get to know and notice, whether it is in motion or not, we still have not found a possibility of such identification; what is surprising that this circumstance remains unknowable for us concerning the Earth which can move us permanently, while we cannot make an experiment putting it to the state of rest? Hopefully, you, Signor Simplicio, thousand times floated on boats from Padua and can honestly admit that you never felt yourself involved in such motion, except cases when the boat, having run aground or encountering some obstacle, stopped, and you with other passengers, being taken unawares, risked to fall. The Earth also would have to encounter some obstacle which would stop it, because, I can assure you, then you would notice the impulse which is inside you, as it would throw you to the stars. True, via another sensation but accompanying by reasoning, you can note the boat’s motion, namely, via vision, looking at trees and buildings on the shore: separated from the boat, they seem inversely moving; if you want to make sure in the Earth’s motion through such experiment, I would advice you to look at stars which due to this seem you to be moving in another direction. Furthermore, the astonishment that we do not feel such principle if it is our internal hardly has grounds, because, if we do not feel the alike principle coming from the outwards and often interrupted, then what are the grounds on which we would have to feel it, should it were inside us unchanged and continuous?” [39, p. 191–192]. “And this occurs because the motion which the ship passes to the mast, it passes also to you and your eye; thus, you need not at all to move this last to look at the top of mast, due to which it seems to you resting. [The ray of vision goes from the eye to the mast the same as the rope stretched between two points of ship; but hundreds of ropes fixed at different points will remain at the same places, will the ship move or rest.] Now apply this reasoning to the Earth’s rotation and to a stone that is on the top of tower. Here you cannot distinct its motions, as the motion which you should follow you do equally with it and with Earth, and you need not move your eye. And when you join to it the motion down that belongs exceptionally to it, not to you, and that merges with that orbiting, then the orbiting part being common to the stone and eye continues to be intangible, and the only tangible remains the direct motion, as to follow it, you have to move your eye downwards” [39, p. 188].

Only Newton with his genius gave to the orbiting the rank of accelerated, irrespectively of, whether it was caused by the affection of lumped either distributed force – in particular, gravity or magnetic field:

Of this sort is gravity, by which bodies tend to the centre of the earth; magnetism, by which iron tends to the load-stone; and that force, whatever it is, by which the planets are perpetually drawn aside from the rectilinear motions, which otherwise they would pursue, and made to revolve in curvilinear orbits. A stone whirled about in a sling, endeavours to recede from the hand that turns it; and by that endeavour, distends the sling, and that with so much the greater force, as it is revolved with the greater velocity, and as soon as ever it is let go, flies away. That force which opposes itself to this endeavour, and by which the sling perpetually draws back the stone towards the hand, and retains it in its orbit, because it is directed to the hand as the centre of the orbit, I call the centripetal force. And the thing is to be understood of all bodies, revolved in any orbits. They all endeavour to recede from the centres of their orbits; and were it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call centripetal, would fly off in right lines, with a uniform motion. A projectile, if it was not for the force of gravity, would not deviate towards the earth, but would go off from it in a right line, and that with an uniform motion, if the resistance of the air was taken away. It is by its gravity that it is drawn aside perpetually from its rectilinear course, and made to deviate towards the earth more or less, according to the force of its gravity, and the velocity of its motion. The less its gravity is, for the quantity of its matter, or the greater the velocity with which it is projected, the less will it deviate from a rectilinear course, and the farther it will go. If a leaden ball, projected from the top of a mountain by the force of gunpowder with a given velocity, and in a direction parallel to the horizon, is carried in a curve line to the distance of two miles before it falls to the ground; the same, if the resistance of the air were taken away, with a double or decuple velocity, would fly twice or ten times as far. And by increasing the velocity, we may at pleasure increase the distance to which it might be projected, and diminish the curvature of the line, which it might describe, till at last it should fall at the distance of 10, 30, or 90 degrees, or even might go quite round the whole earth before it falls; or lastly, so that it might never fall to the earth, but go forward into the celestial spaces, and proceed in its motion in infinitum. And after the same manner that a projectile, by the force of gravity, may be made to revolve in an orbit, and go round the whole earth, the moon also, either by the force of gravity, if it is endued with gravity, or by any other force, that impels it towards the earth, may be perpetually drawn aside towards the earth, out of the rectilinear way, which by its innate force it would pursue; and would be made to revolve in the orbit which it now describes; nor could the moon without some such force, be retained in its orbit. If this force was too small, it would not sufficiently turn the moon out of a rectilinear course: if it was too great, it would turn it too much, and draw down the moon from its orbit towards the earth. It is necessary, that the force be of a just quantity” [2, p. 26–27].

In this citation Newton, to substantiate the centripetal force, repeats Galilean reasoning about the sling (Galilean split for throwing stones), and of the ball shot horizontally. Agreeing on one hand with Galilee that “Increasing the speed, we can, by our wish, increase the distance of flight and decrease the curvature of line along which the ball moves”, Newton on the other hand, grounding on new treatment of physical processes based on the concept of centrifugal and centripetal forces, concludes that at definite speed the ball can leave the Earth, while Galilee rejected it: “Like this, should a cannon put horizontally was on the top of tower and shot in parallel to the horizon, then, dependently on more or less charge, the ball before falling would fly a thousand cubits, or four thousands, or six thousands, or, finally, ten thousands etc., however, the flights would take the time intervals equal to each other, and each would be equal to the time which the ball would spend to path from the muzzle of cannon to the earth, should it felt from there without any other pulse perpendicularly down. It seems surprising that in the short time necessary for a steep falling to the earth from the height, say, of hundred cubits a ball driven by fire can fly sometimes four hundreds, sometimes thousand, sometimes four thousands and even hundred thousands cubits, so that with all shots made in parallel to horizon the ball keeps in the air equal intervals of time” [39, p. 124–125]. “… The recoil motion to do not ensue, the sections of space in which the thrown body has to fall to join the wheel necessarily has to be so brief and negligible that, however slow, even infinitely decelerated would be the fall of moving body, it has, none the less, to be enough to return the body. So it is necessary to find such decrease of these sections which not only would go to infinity but to a such infinity which would exceed the double infinity of decrease of the speed of the body falling down. But how can one value decrease more than another which decreases to infinity twice?” [39, p. 156].

The same Newton, to prove the accelerated motion, uses the example with the change of water surface in accelerated motion which Galilee used before him and Mach after him and which then took the shape of problem of Machian pail. But if Galilee used this example with water to prove the accelerated motion of wooden barge with drinking water or terrestrial tides, due to the non-uniform motion of barge or of Earth, Newton used the properties of water to prove the accelerated motion of water in the pail when revolving and proved the presence of centrifugal force. Though in Galilee’s works we see yet unshaped understanding of affection of some radial force that arises just in the revolved pail – true, not relative to the axis of pail, as Newton’s, but as Mach’s (or rather, Mach’s as Galilee’s) – on a long rope: “I would like to add and even more to support it, so that to show even more visually how much true is that heavy bodies spinning about resting centre, gain a pulse for motion away from centre, even though they had a penchant to go in direction inherent to them by their nature. Let us fasten to the end of rope a pail filled with water, and another end strongly take in hand; then, making of the rope and arm a half of diameter with the centre in the humeral joint, let us make this vessel quickly orbiting so that it described a circle which will be once parallel to the horizon, once vertical in reference to it, or having whatever inclination; in no one of these cases the water will not pour from the vessel, and he who revolves it will always feel the stretch of rope and the force directed away from the shoulder; and if we make holes in the bottom of pail, we will see the water sprinkle out irrespectively to the sky, sides or earth; and if we change the water for pebbles and revolve them in the same way, we will feel from them the same stretch of rope; finally, we see how children throw pebbles far describing a circle by a cane into whose butt the stone is embedded. These all are proofs of the truth of the statement that the orbiting passes to the moving body a pulse directed to the circle, when the motion is fast” [39, p. 156].

Thus, despite a brief analysis, we see that from Aristotle to Newton the understanding of properties and revelations of accelerated motion gradually transformed. Started from stating the circular motion as primordial and so inertial, through the proof that Earth moves about Sun, it has led Newton to rejection of circular motion as inertial. But Newton did not complete the proof, confining himself to select the inertial motion only as direct and uniform. Some issues related to the likeness of outward revelations in the rectilinear and circular motions he did not touch. This unfinished proof made Mach to revive a transformed idea by Aristotle – to take the force of inertia as the affection of celestial spheres. And Einstein in a transformed, again, appearance has put it into his conception of general relativity as an equivalence of inertial and non-inertial reference frames.

We can understand the reason, why Newton did not finish his proof: the concept of field processes then only started, thanks to just Newton’s discoveries. In absence of experimental and theoretical basis he naturally was unable to cover the whole variety of this issue. Noting it, we here will fill in some lacking chains, basing on the knowledge that the mankind gained after Newton. We will begin this study, considering, how much legally first Mach, then relativists substituted the centrifugal force of inertia by the external affection of gravitating masses.