How Do Objects Move?

It is a solid experiential fact that objects can move, and macroscopic objects appear to move continuously. However, direct experience does not tell us how objects move in reality. We cannot simply regard the appearance as the realistic picture. Today it is still a tough task to find how objects move and further understand the motion phenomena.

Here we try to find the realistic picture of motion from the familiar phenomena of motion. It is argued that the phenomena of inertial motion and spontaneous decay imply that motion is spontaneous. The spontaneity of motion requires that motion is essentially random and discontinuous. This conclusion is also supported by the theories of point set and measure in mathematics. However, the randomness and discontinuity of motion cannot emerge in continuous space and time. This is unnatural in logic and inconsistent with experience. We further analyze the motion in discrete space and time. It is shown that the discreteness of space and time not only leads to the existence of random discontinuous motion, but also can naturally release the randomness and discontinuity of motion as the experience reveals. Accordingly the real motion may be the random discontinuous motion in discrete space and time.

Zeno's Paradoxes and the 'at-at' Theory

It is standardly assumed that space and time intervals consist of extensionless points. The moving object is in one position at each instant during the course of motion. But it seems that such a natural assumption does not permit the existence of motion. The famous arrow paradox of Zeno provides an interesting argument (cf. Salmon 1970; Papa-Grimaldi 1996). It argues that at any instant a flying arrow cannot move for that would require the instant to have parts, and an instant is by definition a minimal and indivisible element of time. Since at each instant of its flight the arrow is at rest, and time is composed of such instants, the arrow never moves. The standard solution of the paradox rests on what is called the 'at-at' theory of motion. According to the theory, motion is merely a feature of being in different locations at different times, and that is that. As Russell (1903) wrote, "Motion consists merely in the occupation of different places at different times". So it is true that there is no motion during any instant. Motion has nothing at all to do with what happens during instants; it has instead to do with what happens between instants. If the object has the same location in the instants immediately neighboring, then we say it is at rest; otherwise it is in motion.

However, the 'at-at' theory does not tell us how the different points in space and time intrinsically correlate. In other words, this theory cannot explain dynamism as it never operates the synthesis which could intrinsically correlate different points in space and time. We all know that the transition of different positions is in fact accomplished, but how the transition from one position to another position has been accomplished remains a mystery in the existing theory. In the following, we will try to complete the 'at-at' theory of motion and find how objects move.

Is Motion Continuous?

How on earth do objects move? Most people may think motion is evidently continuous. This accords with our everyday experience of the motion of macroscopic objects. But is continuous motion the real motion?

We are only accustomed to continuous motion after all, and we cherish it so deeply. We have been taking for granted that continuous motion is the only possible form of motion, as well as the real motion. Indeed, the existence of continuous motion seems to be very natural. An object will hold its velocity if no influences are imposed on it, since there are no causes to change its velocity. Then the free object can only be at rest or move continuously with a constant velocity. In addition, a moving object is in one position at one instant, and it can only be in a neighboring position at next instant, since there are no causes to result in its sudden appearing in another nonadjacent position. In a similar way, an object cannot move from one position to another position without passing through the in-between positions either, since there is still no any cause to result in such a "jump". In a word, the existence of continuous motion is inevitable. If it is not the real motion, then which form of motion is the real motion? Whereas we never see and never learn of and even never dream of another form of motion, how could it be the real motion?

But what is continuous motion? How to confirm its existence? As we know, an object moving continuously from point 0 to point 1 in a line must pass through all points between 0 and 1. However, there are uncountable points between 0 and 1, say 1/2, 1/4, 1/8 etc. We cannot count up to them during a finite time interval. Then how can we know the object really passes through all these points? If we cannot know that, how can we confirm that the motion of object is continuous? There may exist some other ways to confirm the existence of continuous motion. For example, although we cannot directly verify that the object passes through all points between 0 and 1, we may confirm it through a plausible hypothesis. One of such hypotheses is that an object moving from one position to another position must pass through the middle position. However, even though the existence of continuous motion can be confirmed in terms of such hypothesis, how can we verify this hypothesis? It may be right for a large distance, but has it been verified for a very small distance? Since there exist uncountable distances, the above hypothesis cannot be verified either. In addition, even if the hypothesis has been confirmed by experiments, it can only confirm the dense property of the trajectory of object, and cannot confirm its continuity. For example, the trajectory which is only composed of rational points evidently satisfies the above hypothesis. Accordingly we cannot confirm the existence of continuous motion in terms of such a hypothesis either. In a word, continuous motion is only an assumption or a belief, which can never be confirmed. Moreover, the trajectory of continuous motion, if it exists, can only exist in the meaning of dense point set, since we can never measure a single point or a point set with zero measure in physics.

It appears that infinity prevents us from finding the real motion. In order to find the real motion, we must enter into the smaller and smaller space, even the infinitesimal space. Then how far can we walk along the logical road?

The Spontaneity of Motion

An object will continue to move after it is put into motion. This is the familiar phenomenon of inertial motion[1]. It is well summarized in Newton's first law of motion (i.e. the law of inertial motion) for macroscopic objects. According to the law, a free object can move or change its position, and no external causes are needed to sustain its motion. To our surprise, an in-depth analysis of such ordinary phenomena will lead us to find the real motion.

It seems that the inertial motion can be understood in the framework of classical mechanics which assumes objects move in a continuous way. A free object should hold its velocity, since there is no any cause leading to the change of its velocity. Thus the object must continuously move in a straight line with a constant speed as the law of inertial motion requires. If such an explanation is complete for the understanding of inertial motion, then the motion of a free object has no spontaneity. It just holds its previous state.

However, a further analysis will show that the motion of a free object may have spontaneity. It is a standard assumption that space and time intervals consist of extensionless points, and a moving object is in one position at each instant during the course of motion. Since there exists no motion for the moving object at one instant, the instantaneous state of the object contains no information about the position change of the object during a finite time interval or even an infinitesimal time interval. Thus the state of the object at one instant cannot determine the states of the object at other instants. In short, an object's instantaneous state does not imply, in virtue of logic and definition, any constraints on its instantaneous states at other times (cf. Albert 2000; Arntzenius 2000; Butterfield 2005). As a result, no causes determine the change of a free object's instantaneous state such as its position. Thus the object must change its position spontaneously during the course of inertial motion (cf. Gao 2001b, 2002a, 2003b).

We note that the velocity property, even if it exists, cannot determine the change of the instantaneous position of an object. On the one hand, velocity may not exist for some forms of motion such as Brown motion. Its valid definition requires that motion is continuous and the trajectory is differentiable relative to time. But it is still unclear whether or not the motion of objects is continuous, and thus the continuity of motion and the existence of velocity should not be a precondition when we analyze how objects move. On the other hand, even if velocity may exist for the motion of an object, it does not count as part of the instantaneous state of the object (cf. Albert 2000; Arntzenius 2000; Butterfield 2005). Thus velocity cannot determine the change of the instantaneous state such as position of the object. The orthodox definition of velocity is that the instantaneous velocity for an object is the limit of the object's average velocity as the time-interval around the point in question tends to zero. As a result, the orthodox velocity is local but temporally extrinsic (cf. Butterfield 2005). The object's instantaneous velocity at an instant codes a lot of information about what its velocity and location are at nearby times, which is given precisely by the limit definition of velocity. In a word, even if the orthodox velocity exists for the motion of an object, it is not an instantaneous intrinsic property of the object in essence, and thus velocity cannot determine the change of the instantaneous position of the object.

There exist two possible ways to avoid the spontaneity of motion. One way is to assume space and time consist of no smallest sized intervals such as points. Rather, space and time are infinitely divisible. The other way is to assume the state of an object at an instant does include a velocity. Such velocity is not defined in terms of the position development of an object during a time interval. Rather, it is a primitive intrinsic feature of an object at an instant, which causes the object to subsequently move in the direction in which the intrinsic velocity is pointing. There are some detailed discussions of these non-standard assumptions recently (cf. Tooley 1988; Vallentyne 1997; Albert 2000; Arntzenius 2000, 2003, 2004; Lewis 2001; Smith 2003; Floyd 2003; Butterfield 2005). If one of these assumptions is right, then the motion of objects will have no spontaneity, and the above explanation provided by classical mechanics may be complete for the understanding of the inertial phenomena. On the other hand, if motion is indeed spontaneous, then these assumptions will be wrong, i.e., space and time consist of smallest sized intervals, and there exists no intrinsic velocity that determines the change of the position of an object in each smallest sized time interval.

There are more direct evidences of the spontaneity of motion in the microscopic world. For example, the alpha particles can spontaneously move out from the radioactive isotopes without any external cause. Such a phenomenon is well known as radioactivity or spontaneous decay, which widely exists in the microscopic world. During the spontaneous decay process, the decay time of each radioactive atom in the substance is completely random. Such randomness also indicates that the decay process happens without causes, and it is spontaneous. In addition, there also exists spontaneous motion during the process of interaction between particles. According to quantum field theory, the interaction between particles is transferred by the transfer particles. Since there are no other transfer particles or interactions between the interacting particles and the transfer particles, the transfer particles must move spontaneously in the process.

The existence of the spontaneous motion without cause seems very counterintuitive. However, it may have a deeper logical basis (cf. Gao 2001b, 2002a, 2003b). If motions can exist only as a result of a certain cause such as interaction between particles, the particles would not be able to move without such interaction, but, on the other hand, the interaction cannot exist if there are no moving particles to transfer it. Thus either everything is immobile or there exist uncaused, spontaneous motions. In short, if the particles can not move in a spontaneous way, then all interactions will not exist, and all particles will also be resting. Furthermore, since the properties of a particle such as mass depend on its interaction with other particles, the particles will be devoid of any properties, and will not exist either. Thus it seems that objects must move spontaneously in order to exist. We may define the spontaneity of motion as a nature of object. Then such a nature can be regarded as the universal internal cause of the spontaneous motion of object. This kind of cause is independent of each concrete motion process.

Motion Is Discontinuous and Random

The evidences of spontaneous motion strongly favour the standard assumption that space and time consist of smallest sized intervals such as 0-sized points, and there exists no intrinsic velocity to determine the change of the position of an object in each smallest sized time interval. In such space and time, a moving object is in one position at one instant, and spontaneously moves to another position at another instant. Then how is the transition from one position to another position accomplished? Or how does the object move?

We first consider the motion of a free object. According to the above analysis, a free object can spontaneously change its position. The spontaneity of free motion means that no causes determine the position change of the free object. Here we give a summary of the no-causes argument. (1) There are no internal causes such as an intrinsic velocity to determine the position change of the free object. The object at one instant has no velocity to hold for determining the position of the object at the next instant. The object cannot hold its previous position either. It must move during the course of inertial motion. (2) There are no external causes such as the influences of other objects to determine the position change of the free object either. The free object moves without any external influence. Thus we conclude that no causes determine the position change of the free object.

Since no causes determine the position change of a free object, and the change without cause should be random, the position change of the free object will be random in nature. Whereas the change of position is random at any instant, the trajectory must be discontinuous everywhere. Accordingly the motion of free objects should be essentially discontinuous and random (cf. Gao 1993, 2001b, 2002a, 2003b). It should be stressed once again that the free object has no velocity to hold for determining the change of its instantaneous position. Thus the free object really does not know which direction to move along, and must move in a random and discontinuous way.

We then consider the motion of an object interacting with other objects. Can the interaction determine the position of the object at each instant and change the random discontinuous motion (RDM) to deterministic continuous motion (DCM)? The answer is negative. The essential reason lies in that a completely random process cannot be changed to a deterministic process in principle. If the interaction is not random, then it is evident that the motion of an object under such influence will still be random. If the interaction is also random, then since the combination of two random processes still leads to a random process, the motion of an object under such influence will also be random. Accordingly the motion of an object interacting with other objects is still discontinuous and random. Moreover, the mechanism of interaction may even require the existence of RDM. As we have argued, the interaction between particles is transferred by the transfer particles, and the transfer particles move spontaneously during the course of interaction. Since the spontaneous motion without cause should be discontinuous and random, the motion of the transfer particles will be discontinuous and random.

The existence of RDM may also be justifiable from a mathematical point of view (cf. Gao 1999c, 2000, 2001a, 2002a, 2006a). As we know, the motion state of an object is not the instantaneous state, but the infinitesimal interval state in continuous space and time. An infinitesimal time interval contains uncountable instants. This indicates that the motion state of an object should be described by a point set in space and time, in which the point represents the center of mass of the object at one instant. According to the point set theory in mathematics, the general point set in continuous space and time is a random discontinuous point set. Since we have no a priori reason to assume a special point set such as a continuous line for the motion state of an object, the point set describing the motion state should be a random discontinuous point set in space and time (see Figure 1). Thus the object will always move in a discontinuous and random way during an infinitesimal interval at any instant.

Figure 1 Discontinuous motion and continuous motion: It is really a wonder that so many points bind together to form a continuous curve in order.

To sum up, objects must move in a discontinuous and random way in the space and time, which consist of smallest sized intervals such as 0-sized points. In such space and time, continuous motion is impossible in logic. During the motion, the transition from one position to another position is discontinuous and random, and there is no correlation between the different points in space and time at all. This will complete the 'at-at' theory of motion.

Double-slit Experiment

Concerning the strange discontinuous motion, even those open-minded people may hardly accept it. This is very natural, since it contradicts our everyday experience of the motion of macroscopic objects. However, if you would like to take an objective attitude, you may also think motion is probably not continuous for very small objects, which cannot be directly observed by our naked eyes. Now let's come back to the domain of experience to see whether some phenomena or experiments have revealed the discontinuous motion.

Double-slit experiment may be one of the experiments which could agree with what you think, since it cannot be explained in terms of the assumption of continuous motion. In the double-slit experiment, a single particle such as an electron is emitted from the source one at a time, and then passes through the two slits to finally arrive at the screen. When a large number of particles reach the screen, they collectively form a double-slit interference pattern.

According to the assumption of continuous motion, the single particle can only pass through one of the two slits. One expects that the double-slit interference pattern should be the same as the direct mixture of two one-slit patterns, each of which is formed by opening each of the two slits independently. The reason is that the passing process of each particle in a double-slit experiment is exactly the same as that in one of the one-slit experiments. However, the results of experiment are that the interference patterns of the above two situations are very different. Thus a single particle must pass through both slits in the double-slit experiment, and its motion will be discontinuous.

Up to now, double-slit experiment has been accomplished for many kinds of microscopic particles such as electrons. Accordingly we find that the motion of small objects is really discontinuous.

Quantum Motion

We have been analyzing the motion of objects in continuous space and time, in which space and time consist of 0-sized points. However, the appearance of infinity in quantum field theory and singularity in general relativity has implied that space and time may be not continuous but discrete. In fact, it has been widely argued that the proper combination of quantum theory and general relativity may inevitably result in the discreteness of space and time. In this section, we will analyze the motion of objects in discrete space and time.

In discrete space and time, space and time consist of smallest finite-sized intervals, i.e., there exist a minimum time interval Tu and a minimum space interval Lu. As a result, a particle is no longer in one position at one instant (as in continuous space and time), but limited to a space unit Lu during a time unit Tu in discrete space and time. This defines the existent form of a particle in discrete space and time.

The analysis of motion in continuous space and time also applies to the motion in discrete space and time. In addition, the discreteness of space and time has more restrictions on the possible forms of motion. As we will see, the discreteness of space and time may also require the existence of RDM. Due to the limitation of discreteness of space and time, there are only two possible free motion states for continuous motion: one is rest state, the other is the motion state with a constant speed c=Lu/Tu. If the speed of an object is larger than c, the object will move more than a space unit during a time unit. Then moving a space unit will correspond to a time interval shorter than the time unit during such movement. This contradicts the above definition of discrete space and time. On the other hand, if the speed of an object is smaller than c, the object will move a space unit during a time interval longer than the time unit. Then the object will move a space interval shorter than the space unit during a time unit during such movement. This also contradicts the above definition of discrete space and time. Thus a free object can only be still or move with the constant speed c in discrete space and time. This result is evidently inconsistent with experience. A free object can move with a speed different from c in reality [2]. Thus if space and time are indeed discrete as defined above, the motion of objects must not be continuous, but be discontinuous and random. This means that an object moving from one space unit to another space unit must not pass the in-between space units. It is just in a space unit during a time unit, and is in another space unit during another time unit.

RDM can naturally exist in discrete space and time. In fact, it may exist only in discrete space and time. As we know, the discontinuity and randomness of motion is absorbed into the motion state defined during an infinitesimal time interval in continuous space and time. As a result, the evolution law of the motion state will be essentially a deterministic continuous equation. Then how can the randomness and discontinuity emerge? And how can the spontaneity of motion present itself? If space and time are continuous, then the inherent randomness of motion cannot be released. Since the 0-sized instants have no physical effects, the randomness and discontinuity of motion cannot emerge through detectable physical effects. This result seems very unnatural in logic, and contradicts one of our most basic scientific beliefs, the minimum ontology. According to the principle, existence should display itself. If a certain thing does exist, then it can be detected, whereas if a certain thing cannot be detected essentially, then it does not exist. Furthermore, if the randomness and discontinuity of motion cannot emerge, the spontaneity of motion cannot present itself either. This is also inconsistent with the evidences of the spontaneity of motion in the microscopic world. Certainly, we can assume there exist other possible sources of randomness, which revise the continuous evolution equation by adding a random evolution term. This can be consistent with the existing experience. However, the existence of RDM still contradicts the minimum ontology. In addition, assuming two different kinds of randomness may not satisfy the requirement of Occam's razor. By comparison, it is more natural and simpler to assume the inherent randomness of motion can emerge and generate the actual randomness and spontaneity of motion. Such process can happen in discrete space and time. There exists a minimum time interval Tu in discrete space and time. In contrast to the 0-sized instants, the minimum finite-sized intervals can have a physical effect. Concretely speaking, the object undergoing RDM stochastically stays in a space unit Lu during a time unit Tu, and such random stay can have a small random effect on the evolution of RDM due to the finite duration of the stay. Then during a longer time interval, such small random effect can continually accumulate to generate the detectable randomness and spontaneity of motion.

In a word, we show that space and time may be actually discrete, and the real motion of objects may be the RDM in discrete space and time. _____________________________________________________________________________________________________

[1] Note that pure inertial motion does not occur in nature. According to the existing physical theories, it can only occur at an infinite distance from all sources of gravity.

[2] It can be conceived that the free object moves with the speed c during some time units, and stays still during the other time units. The average speed of such motion can be different from the speed c, and thus such motion can be consistent with the existing experience. However, the speed change of the free object during such motion can hardly be explained. In addition, such motion will contain some kind of unnatural randomness (e.g. during each time unit the speed of the free object will assume c or zero in a random way), and the randomness has no logical basis either.