Galileo’s Ship

 It was Galileo who opened up the whole subject of ‘inertial frames’ and ‘relativity’, which has turned out to be of the utmost importance in physics. Nonetheless, he does not actually use the term ‘inertial frame’ or formulate a ‘Principle of Relativity’ as such.

Galileo wrote his Dialogue Concerning the Two World Systems, Ptolemaic and Copernican in 1616 to defend the revolutionary Copernican view that the Earth and the planets moved round the Sun. The Dialogue, modelled on Plato’s writings, takes the form of a three day long discussion where Salviati undertakes to explain and justify the heliocentric system to two friends, one of whom, Simplicius, advances various arguments against the heliocentric view. One of his strongest objections is, “If the Earth is moving, why do we not feel this movement?” Salviati’s reply is essentially this, “There are many other circumstances when we do not feel we are moving just so long as our motion is steady and in a straight line”.

Salviati asks his friends to conduct a ‘thought experiment’, ancestor of innumerable modern Gedanken Experimenten. They are to imagine themselves in “the main cabin below decks on some large ship” and this, given the construction at the time, meant there would have been no portholes so one would not be able to see out. The cabin serves as a floating laboratory and Galileo’s homespun apparatus includes “a large bowl of water with some fish in it”, “a bottle that empties drop by drop into a narrow-mouthed vessel beneath it”, a stick of incense, some flies and butterflies, a pair of scales and so on. The ship, presumably a galley, is moving steadily on a calm sea in a dead straight line. Galileo (via Salviati) claims that the occupants of the cabin would not be able to tell, without going up on deck to look, whether the ship was at rest or not. Objects will weigh just the same, drops of water from a tap will take the same time to fall to the ground, the flies and butterflies will fly around in much the same way, and so on — “You will discover not the least difference in all the effects named, nor could you tell from any of them whether the ship was moving or standing still” (Note 1).

Now, it should be said at once that this is not at all what one would expect, and not what Aristotle’s physics gave one to expect. One might well, for example, expect the flies and butterflies flying about to be impelled towards the back end of the cabin and even for human beings to feel a pull in this direction along with many other noticeable effects if the ship were in motion, effects that one would not perceive if the ship were safely in the dock.

What about if one conducted experiments on the open deck?  It is here that Galileo most nearly anticipates Newton’s treatment of motion and indeed Einstein himself. Salviati specifies that it is essential to decide whether a ‘body’ such as a fly or butterfly falls, or does not fall, within the confines of the system ‘ship + immediate environment’ ─ what we would call the ship’s ‘inertial frame’. Salviati concedes that flies and butterflies “separated from it [the ship] by a perceptible distance” would indeed be prevented from participating in the ship’s motion but this would simply be because of air resistance. “Keeping themselves near it, they would follow it without effort or hindrance, for the ship, being an unbroken structure, carries with it a part of the nearby air”. This mention of an ‘unbroken structure’ is the closest Galileo comes to the modern concept of an ‘inertial frame’ within which all bodies behave in the same way. As Salviati puts it, “The cause of all these correspondences of effects is the fact that the ship’s motion is common to all the things contained within it, and to the air also” (Dialogue p. 218 ).

Now, the claim that all bodies on and in the ship are and remain ‘in the same state of motion’ is, on the face of it, puzzling and counter-intuitive. For one might ask how an object ‘knows’ what ‘frame’ it belongs to and thus how to behave, especially since the limits of the frame are not necessarily, or even usually, physical barriers. Galileo does not seem to have conducted any actual experiments relating to moving ships himself, but other people at the time did conduct experiments on moving ships, dropping cannon balls, for example, from the top of a mast and noting where it hit the deck. According to Galileo’s line of argument, a heavy object should strike the deck very nearly at the foot of the mast if the ship continued moving forward at exactly the same speed in a straight line whereas the Aristotelians, on their side, expected the cannon ball to be shifted backwards from the foot of the mast by an appreciable distance. The issue  depended on which ‘structure’, to use Galileo’s term, a given object belonged to. For example, a cannonball dropped by a helicopter that happened to be flying over the ship at a particular moment, belongs to the helicopter ‘system’ and not to the system ‘ship’. In consequence, its trajectory would not be the same as that of a cannonball dropped from the top of a mast ─ unless the helicopter and ship were, by some fluke, travelling at an identical speed and in exactly the same direction.

By his observations and reflexions Galileo thus laid the foundations for the modern treatment of bodies in motion though this was not really his intention, or at any rate not at this stage in the argument. Newton was to capitalize on his predecessor’s observations by making a clearcut distinction between the velocity of a body which, other things being equal, a body retains indefinitely and a body’s acceleration which is always due to an outside force.

Families of Inertial Frames 

In the literature, ‘inertial frame’ has come to mean a ‘force-free frame’, that is, a set-up where a body inside some sort of a, usually box-like, container remains at rest unless interfered with or, if considered to be already in straight line constant motion, retains this motion indefinitely. But neither Galileo nor Newton used the term ‘inertial reference frame’ (German: Inertialsystem) which seems to have been coined by Ludwig Lange in 1885.

The peculiarity of inertial frames is, then, that they are, physically speaking, interchangeable and cannot be distinguished from one another ‘from the inside’. Mathematically speaking, ‘being an inertial frame’ is a ‘transitive’ relation : if A is an inertial frame and B is at rest or moves at constant speed in a straight line relative to A, then B is also an inertial frame. We have, then, a vast family of ‘frames’ within which objects allegedly behave in exactly the same way and which, when one  is inside such a frame, ‘feel’ no different from one another.

It is important to be clear that the concept of ‘inertial frame’ implies (1) that it is not possible to tell, from the inside, whether the ‘frame’ (such as Galileo’s cabin or Einstein’s railway coach) is at rest or in straight line constant motion, and (2) that it is not possible to distinguish between two or more frames, neither of which are considered to be stationary, provided their motion remains constant and in a straight line. These two cases are distinct: we might, for example, be able to tell whether we were moving or not but be unable to decide with precision what sort of motion we were in ─ to distinguish, for example, between two different straight-line motions at constant speed. As it happens, Galileo was really only concerned with the distinction between being ‘at rest’ and in constant straight-line motion, or rather with the alleged inability to make such a distinction from inside such a ‘frame’, since it was this inability which was relevant to his argument. But the lumping together of a whole host of different straight-line motions is actually a more important step conceptually though Galileo himself did not perhaps realize this.

So. Were Galileo in the cabin of a ship moving at a steady pace of, say, 10 knots, he would, so he claims, not be able to differentiate between what goes on inside such a cabin from what goes on in a similar cabin of a similar ship not moving at all or one moving at a speed of 2 or 20 or 200 or even 2,000 knots supposing this to be possible. Now, this is an extremely surprising fact (if it is indeed a fact) since Ship A and Ship B are not ‘in the same state of motion’ : one is travelling at a certain speed relative to dry land and the second at a quite different speed relative to the same land. One would, on the face of it, expect it to be possible to tell whether a ship were ‘in motion’ as opposed to being at rest, and, secondly, to be able to distinguish between two states of straight line constant motion with different speeds relative to the same fixed mass of land. Newton himself felt that it ought to be possible to distinguish between ‘absolute rest’ and ‘absolute motion’ but conceded that this seemed not to be possible in practice. He was obviously somewhat troubled by this point as well he might be.

 Galileo’s Ship is not a true Inertial Frame

 As a matter of fact, it would not only be possible but fairly easy today to tell whether we are at rest or in motion when, say, locked up without radio or TV communication in a windowless cabin of an ocean liner. All I would need to carry out the test successfully would be a heavy pendulum, a means to support it so that it can revolve freely, a good compass, and a certain amount of time. Foucault demonstrated that a heavy pendulum, suspended with the minimum possible friction from the bearings so that it can move freely in any direction, will appear to swing in a circle : the Science Museum in London and countless other places have working Foucault pendulums. The time taken to make a complete circuit depends on one’s latitude — or, more correctly, the time it takes the Earth to revolve around the pendulum depends on what we choose to call latitude. A Foucault pendulum suspended at the North Pole would, so we are assured, take 24 hours to make a full circuit and a similar one at the Equator would not change its direction of swing at all, within the margins of experimental error. By timing the swings carefully one could thus work out whether the ship was changing its latitude, i.e. moving ‘downwards’ in the direction of the South Pole, or ‘upwards’ in the direction of the North (geographical) pole. On the other hand, a ship at rest, whatever its latitude, would show no variation in the time of swing ─ again within the limits of scientific error.

However, suppose I noted no change in the period of the Foucault pendulum. I would now have to decide whether my ship, galley or ocean liner, was stationary relative to dry land or was moving at constant speed along a great circle of latitude. This is rather more difficult to determine but could be managed nonetheless even with home-made instruments. One could examine  the ‘dip’ of a compass needle which points downwards in regions above the Equator and upwards in regions south of the Equator ─ because the compass needle aligns itself according to the lines of force of the Earth’s magnetic field. Again, any change in the angle of dip would be noticeable and there would be changes as the ship moved nearer the magnetic south or north poles. Nor is this all. The magnetic ‘north pole’ differs appreciably from the geographical north pole and this discrepancy changes as we pursue a great circle path along a latitude : so-called isoclinics, lines drawn through places having the same angle of dip, are different from lines of latitude. There are also variations in g, the acceleration due to gravity at the Earth’s surface, because of the Earth’s slightly irregular shape, its ‘oblateness’ which makes the circumference of the Earth measured along the Equator markedly different from that measured along a great circle of longitude passing through the poles. And so, despite Galileo’s claim to the contrary, there would be slight differences in the weight of objects in the cabin at different moments if the ship were wandering about. Only if the Earth were a perfect sphere with the magnetic poles precisely aligned with the geographical poles, would such tests be inconclusive. But a perfect sphere does not exist in Nature and never will exist unless it is manufactured by humans or some other intelligent species.
Galileo’s claim is thus not strictly true : it is a typical case of an ‘ideal situation’ to which actual situations approximate but which they do not, and cannot, attain.

Einstein’s Generalizations

But, one might go on to argue, the discrepancies mentioned above only  arose because Galileo’s ship was constrained to move on a curved surface, that of the ocean : what about a spaceship in ‘empty space’?

The full Principle of Relativity, Galileian or early Einsteinian,  asserts that there is no way to distinguish from the inside between conditions inside a rocket stationary with respect to the Earth, and conditions inside one travelling at any permissible constant ‘speed’ in a straight line relative to the Earth. It is routinely asserted in textbooks on the Special Theory of Relativity that there would indeed be no way to distinguish the two cases provided one left gravity out of the picture.

Newton made Galileo’s idealized ship’s cabin into the arena where his laws of motion held sway. An object left to its own devices inside a recognizable container-like set-up (an inertial system) would either remain stationary or, if already moving relative to the real or imagined frame, would keep moving in a straight line at constant speed indefinitely. This is Newton’s First Law. Any deviation from this scenario would show that there was an outside force at work ─ and Newton, knowing nothing of interior chemical or nuclear forces, always assumed that any supposed force would necessarily come from the outside. Thus, Newton’s Second Law.

So, supposing I let go of a piece of wood I hold in my hand in this room, which I take as my inertial frame, what happens to it? Instead of remaining where it was when I had it in my hand, the piece of wood falls to the ground and its speed does not stay the same over the time of its trajectory but increases as it falls, i.e. is not constant. And if I throw a ball straight up into the air, not only does it not continue in a vertical line at constant speed but slows down and reverses direction while a shot fired in the air roughly northwards will be deflected markedly to the right because of the Earth’s rotation (if I am in the northern hemisphere). Neither this room nor the entire Earth are true inertial frames : if they were Newton’s laws would apply without any tinkering about. To make sense of the bizarre trajectories just mentioned it is necessary to introduce mysterious forces such as the gravitational pull of the Earth or the Coriolis ‘force’ produced by its rotation on its own axis.

As we know, Einstein’s theory of Special Relativity entirely neglects gravity, and introducing the latter eventually led on to the General Theory which is essentially a theory of Gravitation. Einstein’s aim, even in 1905, was quite different from Galileo’s. Whereas Galileo was principally concerned to establish the heliocentric theory and only introduced his ship thought-experiments to deal with objections, Einstein was concerned with identifying the places (‘frames’) where the ‘laws of physics’ would hold in their entirety, and by ‘laws’ he had in mind not only Newton’s laws of motion but also and above all Maxwell’s laws of electro-magnetism. Einstein’s thinking led him on to a search for a ‘true’ inertial frame as opposed to a merely stationary frame such as this room since the latter is certainly not a ‘force-free’ frame. Einstein, reputedly after speculating about what would happen to a construction worker falling from the scaffolding around a building, decided that a real or imaginary box falling freely under the influence of gravitation was a ‘true’ inertial   frame. Inside such a frame, not only would the ‘normal’ Newtonian laws governing mechanics hold good but the effects of gravity would be nullified and so could be legitimately left out of consideration. Such a ‘freely-falling frame’ would thus be the nearest thing to a spaceship marooned in the depths of space far away from the influence of any celestial body.

A freely falling frame is not a true inertial frame

So, would it in fact be impossible to distinguish from the inside between a box falling freely under the gravitational influence of the Earth and a spaceship marooned in empty space? The answer is, perhaps surprisingly, no. In a ‘freely falling’ lift dropping towards the Earth, or the centre of any other massive body, there would be so-called ‘tidal effects’ because the Earth’s gravitational field is not homogeneous (the same in all localities) and isotropic (the same in all directions). If one released a handful of ball-bearings or a basketful of apples in a freely falling lift, the ball-bearings or apples at the ‘horizontal’ extremities would curve slightly towards each other as they fell since their trajectories would be directed towards the centre of the Earth rather than straight downwards. Likewise, the top and bottom apples would not remain the same distance apart since the forces on them, dependent as they are on the distances of the two apples from the Earth’s centre of mass, would be different and this difference would increase as the falling lift accelerated.

It turns out, then, that, at the end of the day, Einstein’s freely falling lift is not a great deal better than Galileo’s ship ─ although both are good enough approximations to inertial frames, or rather are very good imitations of inertial frames. One can, of course, argue in Calculus manner that the strength of the Earth’s gravitational field will be the same over an ‘infinitesimally small region’ ─ though without going into further details about the actual size of such a region. Newton’s Laws in their purity and integrity are thus only strictly applicable to such ‘infinitesimal’ regions in which case there will inevitably be abrupt transitions, i.e. ‘accelerations’, as we move from one infinitesimal region to another. The trajectory of any free falling object will thus not be fluent and continuous but jerky at a small enough scale.

For that matter, it is by no means obvious that a spaceship marooned in the  middle of ‘empty’ space is a true ‘inertial frame’. According to Einstein’s General Theory of Relativity, Space-time is ‘warped’ or distorted by the presence of massive objects and this space-time curvature apparently extends over the whole of the universe ─ albeit with very different local effects. If the universe is to be considered a single entity, then strictly speaking there is nowhere inside it which is completely free of ‘curvature’, and so there is nowhere to situate a ‘true’ inertial frame.

What to Conclude?

 So where does all this leave us? Or, more specifically, what bearing does all this discussion have on the theory I am attempting to develop ?

In Ultimate Event Theory, the basic entities are not bodies but point-like ultimate events which, if they are strongly bonded together and keep repeating more or less identically, constitute what we view as objects. In its most simplistic form, the equivalent of an ‘object’ is a single ultimate event that repeats indefinitely, i.e. an event-chain, while several ‘laterally connected’ event-chains make up an event cluster. There is no such thing as continuous motion in UET and, if this is what we understand by motion, there is no motion. There is, however, succession and also causal linkage between successive ultimate events which belong to the ‘same’ event-chain.

Although I did not realize this until quite recently, one could say that the equivalent of an ‘inertial frame’ in UET is the basic ‘event-capsule’, a flexible though always finite region of the event Locality within which every ultimate event has occurrence. There is no question of the basic ‘building block’ in Eventrics ‘moving’ anywhere : it has occurrence at a particular spot, then disappears and, in some cases, re-appears in a similar (but not identical) spot a ksana (moment) later. One can then pass on to imagining a ‘rest event-chain’ made up of successive ultimate events sufficiently far removed from the influence of massive event-clusters for the latter to have no influence on what occurs. This is the equivalent, if you like, of the imaginary spaceship marooned in the midst of empty space.

So, where does one go from here? One thing to have come out of the endless discussions about inertial frames and their alleged indistinguishability (at least from the inside), is that the concept of ‘motion’ has little if any meaning if we are speaking of a single object whether this object or body is a boat, a particle, ocean liner or spaceship. We thus need at least two ‘objects’, one of which is traditionally seen as ‘embedded’ in the other more or less like an object in a box. In effect, Galileo’s galley is related to the enclosing dry land of the Mediterranean or, at the limit, to the Earth itself including its atmosphere. The important point is being able to relate an object which ‘moves’ to a larger, distinctive object that remains still, or is perceived to remain so.

In effect, then, we need a system composed of at least two very different ‘objects’, and the simplest such system in UET is a ‘dual event-system’ made up of just two event-chains, each of which is composed of a single ultimate event that repeats at every ksana. Now, although any talk of such a system ‘moving’ is only façon de parler , we can quite properly talk of such a system expanding, contracting or doing neither. If our viewpoint is event-chain A , we conceive event-chain B to be, for example, the one that is ‘moving further away’ at each ksana, while if we take the viewpoint of event-chain B, it is the other way round. The important point, however, is that the dual system is expanding if this distance increases, and by distance increasing we mean that there is a specified, finite number of ultimate events that could be ‘fitted into’ the space between the two chains at each ksana.

This is the broad schema that will be investigated in subsequent posts. How much of Galileo’s, Newton’s and early Einstein’s assumptions and observations do I propose to carry over as physical/philosophic baggage into UET?

To start with, what we can say in advance is that the actual distance (in terms of possible positions for ultimate events) between two event-chains does not seem to matter very much. Although Galileo, or Salviati, does not see fit to mention the point ─ he doubtless thinks it too ’obvious’ ─ it is notable that, whether the ship is in motion or not, the objects inside Galileo’s cabin do not change wherever the ship is, neglecting the effects of sun and wind, i.e. that position as such does not bring about changes in physical behaviour. This is not a trivial matter. It amounts to a ‘law’ or ‘principle’ that carries over into UET, namely that the Event Locality does not by itself seem to affect what goes on there, i.e. we have the equivalent of the principle of the ‘homogeneity’ and ‘isotropy’ of Space-time. As a contemporary author puts it : “The homogeneity of space means that all points in space are physically equivalent, i.e. a transportation of any object in space does not affect in any way the processes taking place in this object. The homogeneity of time must be understood as the physical indistinguishability of all instants of time for free objects. (By a free object we mean an object which is far from all surrounding objects so that their interaction can be neglected.)”  Saxena, Principles of Modern Physics  2.2)   

What about the equivalent of velocity? Everything we know about so-called ‘inertial systems’ in the Galileian sense suggests that, barring rather recondite magnetic and gravitational effects, the velocity of a system does not seem to matter very much, provided it is constant and in a straight line. Now, what this means in UET terms is that if successive members of two event-chains get increasingly separated along one spatial direction, this does not affect what goes on in each chain or cluster so long as this increase remains the same. What does affect what goes on in each chain is when the rate of increase or decrease changes : this not only means the system as a whole has changed, but that this change is reflected in each of the two members of the dual system. When travelling in a car or train we often have little idea of our speed but our bodies register immediately any abrupt substantial change of speed or direction, i.e. an acceleration.  This is, then, a feature to be carried over into UET since it is absolutely central to traditional physics.

Finally, that there is the question of there being a limit to the possible increase of distance between two event-chains. This principle is built into the basic assumptions of UET since everything in UET, except the extent of the Locality itself, has an upper and lower limit. Although there is apparently nothing to stop two event-chains which were once adjacent from becoming arbitrarily far apart at some subsequent ksana provided they do this by stages, there is a limit to how much a dual system can expand within the ‘space’ of a single ksana. This is the (now) well-known concept of there being an upper limit to the speed of all particles. Newton may have thought there had to be such a limit but if so he does not seem to have said so specifically : in Newtonian mechanics a body’s speed can, in principle, be increased without limit. In UET, although there is no continuous movement, there is a (discontinuous) ‘lateral space/time displacement rate’ and this, like everything else is limited. In contrast to orthodox Relativity theory, I originally attempted to make a distinction between such an unattainable upper limit, calling it c, and the highest attainable rate which would be one space less per ksana. This means one does not have the paradox of light actually attaining the limit and thus being massless (which it is in contemporary physics). However, this finicky separation between c s₀/t₀ and c* = (c – 1) s₀/t₀ (where s₀ and t₀ are ‘absolute’ spatial and temporal units) may well prove to be too much of a nuisance to be worth maintaining.  SH 21/11/14

 NOTES

 Note 1  This extract and following ones are taken from Drake’s translation of Dialogue concerning two world systems by Galileo Galilei (The Modern Library)