Modern physics has confirmed most dramatically one of the basic ideas of Eastern mysticism; that all the concepts we use to describe nature are limited, that they are not features of reality, as we tend to believe, but creations of the mind; parts of the map, not of the territory. Whenever we expand the realm of our experience, the limitations of our rational mind become apparent and we have to modify, or even abandon, some of our concepts.
Our notions of space and time figure prominently on our map of reality. They serve to order things and events in our environment and are therefore of paramount importance not only in our everyday life, but also in our attempts to understand nature through science and philosophy. There is no law of physics which does not require the concepts of space and time for its formulation. The profound modification of these basic concepts brought about by relativity theory was therefore one of the greatest revolutions in the history of science.
Classical physics was based on the notion both of an absolute, three-dimensional space, independent of the material objects it contains, and obeying the laws of Euclidean geometry, and of time as a separate dimension which again is absolute and flows at an even rate, independent of the material world. In the West, these notions of space and time were so deeply rooted in the minds of philosophers and scientists that they were taken as true and unquestioned properties of nature.
The belief that geometry is inherent in nature, rather than part of the framework we use to describe nature, has its origin in Greek thought. Demonstrative geometry was the central feature of Greek mathematics and had a profound influence on Greek philosophy. Its method of starting from unquestioned axioms, and deriving theorems from these by deductive reasoning, became characteristic of Greek philosophical thought; geometry was therefore at the very centre of all intellectual activities and formed the basis of philosophical training. The gate of Plato’s Academy in Athens is said to have borne the inscription, ‘You are not allowed to enter here, unless you know geometry.’ The Greeks believed that their mathematical theorems were expressions of eternal and exact truths about the real world, and that geometrical shapes were manifestations of absolute beauty. Geometry was considered to be the perfect combination of logic and beauty and was thus believed to be of divine origin. Hence Plato’s dictum, ‘God is a geometer.’
Since geometry was seen as the revelation of God, it was obvious to the Greeks that the heavens should exhibit perfect geometrical shapes. This meant that the heavenly bodies had to move in circles. To present the picture as being even more geometrical they were thought to be fixed to a series of concentric crystalline spheres which moved as a whole, with the Earth at the centre.
In subsequent centuries, Greek geometry continued to exert a strong influence on Western philosophy and science. Euclid’s Elements was a standard textbook in European schools until the beginning of this century, and Euclidean geometry was taken to be the true nature of space for more than two thousand years. It took an Einstein to make scientists and philosophers realize that geometry is not inherent in nature, but is imposed upon it by the mind. In the words of Henry Margenau,
The central recognition of the theory of relativity is that geometry … is a construct of the intellect. Only when this discovery is accepted can the mind feel free to tamper with the time-honoured notions of space and time, to survey the range of possibilities available for defining them, and to select that formulation which agrees with observation.1
Eastern philosophy, unlike that of the Creeks, has always maintained that space and time are constructs of the mind. The Eastern mystics treated them like all other intellectual concepts; as relative, limited, and illusory. In a Buddhist text, for example, we find the words,
It was taught by the Buddha, oh Monks, that … the past, the future, physical space, … and individuals are nothing but names, forms of thought, words of common usage, merely superficial realities.2
Thus in the Far East, geometry never attained the status it had in ancient Greece, although this does not mean that the Indians and Chinese had little knowledge of it. They used it extensively in building altars of precise geometrical shapes, in measuring the land and mapping out the heavens, but never to determine abstract and eternal truths. This philosophical attitude is also reflected in the fact that ancient Eastern science generally did not find it necessary to fit nature into a scheme of straight lines and perfect circles. Joseph Needham’s remarks about Chinese astronomy are very interesting in this connection:
The Chinese [astronomers] did not feel the need for [geometrical] forms of explanation—the component organisms in the universal organism followed their Tao each according to its own nature, and their motions could be dealt with in the essentially ‘non-representational’ form of algebra. The Chinese were thus free from that obsession of European astronomers for the circle as the most perfect figure, … nor did they experience the medieval prison of the crystalline spheres.3
Thus the ancient Eastern philosophers and scientists already had the attitude which is so basic to relativity theory—that our notions of geometry are not absolute and unchangeable properties of nature, but intellectual constructions. In the words of Ashvaghosha,
Be it clearly understood that space is nothing but a mode of particularisation and that it has no real existence of its own … Space exists only in relation to our particularising consciousness.4
The same applies to our idea of time. The Eastern mystics link the notions of both space and time to particular states of consciousness. Being able to go beyond the ordinary state through meditation, they have realized that the conventional notions of space and time are not the ultimate truth. The refined notions of space and time resulting from their mystical experiences appear to be in many ways similar to the notions of modern physics, as exemplified by the theory of relativity.
What, then, is this new view of space and time which emerged from relativity theory? It is based on the discovery that all space and time measurements are relative. The relativity of spatial specifications was, of course, nothing new. It was well known before Einstein that the position of an object in space can only be defined relative to some other object. This is usually done with the help of three coordinates and the point from which the coordinates are measured may be called the location of the ‘observer’.
To illustrate the relativity of such coordinates, imagine two observers floating in space and observing an umbrella, as drawn opposite. Observer A sees the umbrella to his left and slightly inclined, so that the upper end is nearer to him. Observer B, on the other hand, sees the umbrella to his right and in such a way that the upper end is farther away. By extending this two-dimensional example to three dimensions, it becomes clear that all spatial specifications—such as ‘left’, ‘right’, ‘up’, ‘down’, ‘oblique’, etc.—depend on the position of the observer and are thus relative. This was known long before relativity theory. As far as time is concerned, however, the situation in classical physics was entirely different. The temporal order of two events was assumed to be independent of any observer. Specifications referring to time—such as ‘before’, ‘after’ or ‘simultaneous’—were thought to have an absolute meaning independent of any coordinate system.
Einstein recognized that temporal specifications, too, are relative and depend on the observer. In everyday life, the impression that we can arrange the events around us in a unique time sequence is created by the fact that the velocity of light—186,000 miles per second—is so high, compared to any other velocity we experience, that we can assume we are observing events at the instant they are occurring. This, however, is incorrect. Light needs some time to travel from the event to the observer. Normally, this time is so short that the propagation of light can be considered to be instantaneous; but when the observer moves with a high velocity with respect to the observed phenomena, the time span between the occurrence of an event and its observation plays a crucial role in establishing a sequence of events. Einstein realized that in such a case, observers moving at different velocities will order events differently in time.* Two events which are seen as occurring simultaneously by one observer may occur in different temporal sequences for others. For ordinary velocities, the differences are so small that they cannot be detected, but when the velocities approach the speed of light, they give rise to measurable effects. In high energy physics, where the events are interactions between particles moving almost at the speed of light, the relativity of time is well established and has been confirmed by countless experiments.**
The relativity of time also forces us to abandon the Newtonian concept of an absolute space. Such a space was seen as containing a definite configuration of matter at every instant; but now that simultaneity is seen to be a relative concept, depending on the state of motion of the observer, it is no longer possible to define such a definite instant for the whole universe. A distant event which takes place at some particular instant for one observer may happen earlier or later for another observer. It is therefore not possible to speak about ‘the universe at a given instant’ in an absolute way; there is no absolute space independent of the observer.
Relativity theory has thus shown that all measurements involving space and time lose their absolute significance and has forced us to abandon the classical concepts of an absolute space and an absolute time. The fundamental importance of this development has been clearly expressed by Mendel Sachs in the following words:
The real revolution that came with Einstein’s theory … was the abandonment of the idea that the space-time coordinate system has objective significance as a separate physical entity. Instead of this idea, relativity theory implies that the space and time coordinates are only the elements of a language that is used by an observer to describe his environment.5
This statement from a contemporary physicist shows the close affinity between the notions of space and time in modern physics and those held by the Eastern mystics who say, as quoted before, that space and time ‘are nothing but names, forms of thought, words of common usage’.
Since space and time are now reduced to the subjective role of the elements of the language a particular observer uses for his or her description of natural phenomena, each observer will describe the phenomena in a different way. To abstract some universal natural laws from their descriptions, they have to formulate these laws in such a way that they have the same form in all coordinate systems, i.e. for all observers in arbitrary positions and relative motion. This requirement is known as the principle of relativity and was, in fact, the starting point of relativity theory. It is interesting that the germ of the theory of relativity was contained in a paradox which occurred to Einstein when he was only sixteen. He tried to imagine how a beam of light would look to an observer who travelled along with it at the speed of light, and he concluded that such an observer would see the beam of light as an electromagnetic field oscillating back and forth without moving on, i.e. without forming a wave. Such a phenomenon, however, is unknown in physics. It seemed thus to the young Einstein that something which was observed by one observer to be a well-known electromagnetic phenomenon, namely a light wave, would appear as a phenomenon contradicting the laws of physics to another observer, and this he could not accept. In later years, Einstein realized that the principle of relativity can be satisfied in the description of electromagnetic phenomena only if all spatial and temporal specifications are relative. The laws of mechanics, which govern the phenomena associated with moving bodies, and the laws of electrodynamics, the theory of electricity and magnetism, can then be formulated in a common ‘relativistic’ framework which incorporates time with the three space coordinates as a fourth coordinate to be specified relative to the observer.
In order to check whether the principle of relativity is satisfied, that is, whether the equations of one’s theory look the same in all coordinate systems, one must of course be able to translate the space and time specifications from one coordinate system, or ‘frame of reference’, to the other. Such translations, or ‘transformations’ as they are called, were already well known and widely used in classical physics. The transformation between the two reference frames pictured on p. 181, for example, expresses each of the two coordinates of observer A (one horizontal and one vertical, as indicated by the arrow-headed cross in the drawing) as a combination of the coordinates of observer B, and vice versa. The exact expressions can be easily obtained with the help of elementary geometry.
In relativistic physics, a new situation arises because time is added to the three space coordinates as a fourth dimension. Since the transformations between different frames of reference express each coordinate of one frame as a combination of the coordinates of the other frame, a space coordinate in one frame will in general appear as a mixture of space and time coordinates in another frame. This is indeed an entirely new situation. Every change of coordinate systems mixes space and time in a mathematically well-defined way. The two can therefore no longer be separated, because what is space to one observer will be a mixture of space and time to another. Relativity theory has shown that space is not three-dimensional and time is not a separate entity. Both are intimately and inseparably connected and form a four-dimensional continuum which is called ‘space-time’. This concept of space-time was introduced by Hermann Minkowski in a famous lecture in 1908 with the following words:
The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.6
The concepts of space and time are so basic for the description of natural phenomena that their modification entails an alteration of the whole framework we use in physics to describe nature. In the new framework, space and time are treated on an equal footing and are connected inseparably. In relativistic physics, we can never talk about space without talking about time, and vice versa. This new framework has to be used whenever phenomena involving high velocities are described.
The intimate link between space and time was well known in astronomy, in a different context, long before relativity theory. Astronomers and astrophysicists deal with extremely large distances, and here again the fact that light needs some time to travel from the observed object to the observer is important. Because of the finite velocity of light, the astronomer never looks at the universe in its present state, but always looks back into the past. It takes light eight minutes to travel from the Sun to the Earth, and hence we see the Sun, at any moment, as it existed eight minutes ago. Similarly, we see the nearest star as it existed four years ago, and with our powerful telescopes we can see galaxies as they existed millions of years ago.
The finite velocity of light is by no means a handicap for astronomers but is a great advantage. It allows them to observe the evolution of stars, star clusters or galaxies at all stages just by looking out into space and back into time. All types of phenomena that happened during the past millions of years can actually be observed somewhere in the skies. Astronomers are thus used to the importance of the link between space and time. What relativity theory tells us is that this link is important not only when we deal with large distances, but also when we deal with high velocities. Even here on Earth, the measurement of any distance is not independent of time, because it involves the specification of the observer’s state of motion and thus a reference to time.
The unification of space and time entails—as mentioned in the previous chapter—a unification of other basic concepts, and this unifying aspect is the most characteristic feature of the relativistic framework. Concepts which seemed totally unrelated in nonrelativistic physics are now seen to be but different aspects of one and the same concept. This feature gives the relativistic framework great mathematical elegance and beauty. Many years of work with relativity theory have made us appreciate this elegance and become thoroughly familiar with the mathematical formalism. However, this has not helped our intuition very much. We have no direct sensory experience of the four-dimensional space-time, nor of the other relativistic concepts. Whenever we study natural phenomena involving high velocities, we find it very hard to deal with these concepts both at the level of intuition and ordinary language.
For example, in classical physics it was always assumed that rods in motion and at rest have the same length. Relativity theory has shown that this is not true. The length of an object depends on its motion relative to the observer and it changes with the velocity of that motion. The change is such that the object contracts in the direction of its motion. A rod has its maximum length in a frame of reference where it is at rest, and it becomes shorter with increasing velocity relative to the observer. In the ‘scattering’ experiments of high-energy physics, where particles collide with extremely high velocities, the relativistic contraction is so extreme that spherical particles are reduced to ‘pancake’ shapes.
It is important to realize that it makes no sense to ask which is the ‘real’ length of an object, just as it makes no sense in our everyday life to ask for the real length of somebody’s shadow. The shadow is a projection of points in three-dimensional space on to a two-dimensional plane, and its length will be different for different angles of projection. Similarly, the length of a moving object is the projection of points in four-dimensional space-time on to three-dimensional space, and its length is different in different frames of reference.
What is true for lengths is also true for time intervals. They, too, depend on the frame of reference, but contrary to spatial distances they become longer as the velocity relative to the observer increases. This means that clocks in motion run slower; time slows down. These clocks can be of varying types: mechanical clocks, atomic clocks, or even a human heartbeat. If one of two twins went on a fast round-trip into outer space, he would be younger than his brother when he came back home, because all his ‘clocks’—his heartbeat, bloodflow, brainwaves, etc.—would slow down during the journey, from the point of view of the man on the ground. The traveller himself, of course, would not notice anything unusual, but on his return he would suddenly realize that his twin brother was now much older. This ‘twin paradox’ is perhaps the most famous paradox of modern physics. It has provoked heated discussions in scientific journals, some of which are still going on; an eloquent proof of the fact that the reality described by relativity theory cannot easily be grasped by our ordinary understanding.
The slowing down of clocks in motion, unbelievable as it sounds, is well tested in particle physics. Most of the subatomic particles are unstable, i.e. they disintegrate after a certain time into other particles. Numerous experiments have confirmed the fact that the lifetime* of such an unstable particle depends on its state of motion. It increases with the speed of the particle. Particles moving with 80 per cent of the speed of light live about 1.7 times as long as their slow ‘twin brothers’, and at 99 per cent of the speed of light they live about 7 times as long. This, again, does not mean that the intrinsic lifetime of the particle changes. From the particle’s point of view, its lifetime is always the same, but from the point of view of the laboratory observer the particle’s ‘internal clock’ has slowed down, and therefore it lives longer.
All these relativistic effects only seem strange because we cannot experience the four-dimensional space-time world with our senses, but can only observe its three-dimensional ‘images’. These images have different aspects in different frames of reference; moving objects look different from objects at rest, and moving clocks run at a different rate. These effects will seem paradoxical if we do not realize that they are only the projections of four-dimensional phenomena, just as shadows are projections of three-dimensional objects. If we could visualize the four-dimensional space-time reality, there would be nothing paradoxical at all.
The Eastern mystics, as mentioned above, seem to be able to attain non-ordinary states of consciousness in which they transcend the three-dimensional world of everyday life to experience a higher, multidimensional reality. Thus Aurobindo speaks about ‘a subtle change which makes the sight see in a sort of fourth dimension’.7 The dimensions of these states of consciousness may not be the same as the ones we are dealing with in relativistic physics, but it is striking that they have led the mystics towards notions of space and time which are very similar to those implied by relativity theory.
Throughout Eastern mysticism, there seems to be a strong intuition for the ‘space-time’ character of reality. The fact that space and time are inseparably linked, which is so characteristic of relativistic physics, is stressed again and again. This intuitive notion of space and time has, perhaps, found its clearest expression and its most far-reaching elaboration in Buddhism, and in particular in the Avatamsaka school of Mahayana Buddhism. The Avatamsaka Sutra, on which this school is based,* gives a vivid description of how the world is experienced in the state of enlightenment. The awareness of an ‘interpenetration of space and time’—a perfect expression to describe space-time—is repeatedly emphasized in the sutra and is seen as an essential characteristic of the enlightened state of mind. In the words of D. T. Suzuki,
The significance of the Avatamsaka and its philosophy is unintelligible unless we once experience … a state of complete dissolution where there is no more distinction between mind and body, subject and object … We look around and perceive that … every object is related to every other object … not only spatially, but temporally. … As a fact of pure experience, there is no space without time, no time without space; they are interpenetrating.8
One could hardly find a better way of describing the relativistic concept of space-time. In comparing Suzuki’s statement to the one, quoted before, by Minkowski, it is also interesting to note that both the physicist and the Buddhist emphasize the fact that their notions of space-time are based on experience; on scientific experiments in one case, and on mystical experience in the other.
In my opinion, the time-minded intuition of Eastern mysticism is one of the main reasons why its views of nature seem to correspond, in general, much better to modern scientific views than do those of most Greek philosophers. Greek natural philosophy was, on the whole, essentially static and largely based on geometrical considerations. It was, one could say, extremely ‘non-relativistic’, and its strong influence on Western thought may well be one of the reasons why we have such great conceptual difficulties with relativistic models in modern physics. The Eastern philosophies, on the other hand, are ‘space-time’ philosophies, and thus their intuition often comes very close to the views of nature implied by our modern relativistic theories.
Because of the awareness that space and time are intimately connected and interpenetrating, the world views of modern physics and of Eastern mysticism are both intrinsically dynamic views which contain time and change as essential elements. This point will be discussed in detail in the following chapter, and constitutes the second main theme recurring throughout this comparison of physics and Eastern mysticism, the first being the unity of all things and events. As we study the relativistic models and theories of modern physics, we shall see that all of them are impressive illustrations of the two basic elements of the Eastern world view—the basic oneness of the universe and its intrinsically dynamic character.
The theory of relativity discussed so far is known as the ‘special theory of relativity’. It provides a common framework for the description of the phenomena associated with moving bodies and with electricity and magnetism, the basic features of this framework being the relativity of space and time and their unification into four-dimensional space-time.
In the ‘general theory of relativity’, the framework of the special theory is extended to include gravity. The effect of gravity, according to general relativity, is to make space-time curved. This, again, is extremely hard to imagine. We can easily imagine a two-dimensional curved surface, such as the surface of an egg, because we can see such curved surfaces lying in three-dimensional space. The meaning of the word curvature for two-dimensional curved surfaces is thus quite clear; but when it comes to three-dimensional space—let alone four-dimensional space-time—our imagination abandons us. Since we cannot look at three-dimensional space ‘from outside’, we cannot imagine how it can be ‘bent in some direction’.
To understand the meaning of curved space-time, we have to use curved two-dimensional surfaces as analogies. Imagine, for example, the surface of a sphere. The crucial fact which makes the analogy to space-time possible is that the curvature is an intrinsic property of that surface and can be measured without going into three-dimensional space. A two-dimensional insect confined to the surface of the sphere and unable to experience three-dimensional space could nevertheless find out that the surface on which he is living is curved, provided that he can make geometrical measurements.
To see how this works, we have to compare the geometry of our bug on the sphere with that of a similar insect living on a flat surface.* Suppose the two bugs begin their study of geometry by drawing a straight line, defined as the shortest connection between two points. The result is shown overleaf. We see that the bug on the flat surface drew a very nice straight line; but what did the bug on the sphere do? For him, the line he drew is the shortest connection between the two points A and B, since any other line he may draw will be longer; but from our point of view we recognize it as a curve (the arc of a great circle, to be precise). Now suppose that the two bugs study triangles. The bug on the plane will find that the three angles of any triangle add up to two right angles, i.e. to 180°; but the bug on the sphere will discover that the sum of the angles in his triangles is always greater than 180°. For small triangles, the excess is small, but it increases as the triangles become larger; and as an extreme case, our bug on the sphere will even be able to draw triangles with three right angles. Finally, let the two bugs draw circles and measure their circumference. The bug on the plane will find that the circumference is always equal to 2Π times the radius, independent of the size of the circle. The bug on the sphere, on the other hand, will notice that the circumference is always less than 2Π times the radius. As can be seen in the figure below, our three-dimensional point of view allows us to see that what the bug calls the radius of his circle is in fact a curve which is always longer than the true radius of the circle.
As the two insects continue to study geometry, the one on the plane should discover the axioms and laws of Euclidean geometry, but his colleague on the sphere will discover different laws. The difference will be small for small geometrical figures, but will increase as the figures become larger. The example of the two bugs shows that we can always determine whether a surface is curved or not, just by making geometrical measurements on the surface, and by comparing the results with those predicted by Euclidean geometry. If there is a discrepancy, the surface is curved; and the larger the discrepancy is—for a given size of figures—the stronger the curvature.
In the same way, we can define a curved three-dimensional space to be one in which Euclidean geometry is no longer valid. The laws of geometry in such a space will be of a different, ‘non-Euclidean’ type. Such a non-Euclidean geometry was introduced as a purely abstract mathematical idea in the nineteenth century by the mathematician Georg Riemann, and it was not considered to be more than that, until Einstein made the revolutionary suggestion that the three-dimensional space in which we live is actually curved. According to Einstein’s theory, the curvature of space is caused by the gravitational fields of massive bodies. Wherever there is a massive object, the space around it is curved, and the degree of curvature, that is, the degree to which the geometry deviates from that of Euclid, depends on the mass of the object.
The equations relating the curvature of space to the distribution of matter in that space are called Einstein’s field equations. They can be applied not only to determine the local variations of curvature in the neighbourhood of stars and planets, but also to find out whether there is an overall curvature of space on a large scale. In other words, Einstein’s equations can be used to determine the structure of the universe as a whole. Unfortunately, they do not give a unique answer. Several mathematical solutions of the equations are possible, and these solutions constitute the various models of the universe studied in cosmology, some of which will be discussed in the following chapter. To determine which of them corresponds to the actual structure of our universe is the main task of present-day cosmology.
Since space can never be separated from time in relativity theory, the curvature caused by gravity cannot be limited to three-dimensional space, but must extend to four-dimensional space-time and this is, indeed, what the general theory of relativity predicts. In a curved space-time, the distortions caused by the curvature affect not only the spatial relationships described by geometry but also the lengths of time intervals. Times does not flow at the same rate as in ‘flat space-time’, and as the curvature varies from place to place, according to the distribution of massive bodies, so does the flow of time. It is important to realize, however, that this variation of the flow of time can only be seen by an observer who remains in a different place from the clocks used to measure the variation. If the observer, for example, went to a place where time flows slower, all her clocks would slow down too and she would have no means of measuring the effect.
In our terrestrial environment, the effects of gravity on space and time are so small that they are insignificant, but in astrophysics, which deals with extremely massive bodies, like planets, stars and galaxies, the curvature of space-time is an important phenomenon. All observations have so far confirmed Einstein’s theory and thus force us to believe that space-time is indeed curved. The most extreme effects of the curvature of space-time become apparent during the gravitational collapse of a massive star. According to current ideas in astrophysics, every star reaches a stage in its evolution where it collapses due to the mutual gravitational attraction of its particles. Since this attraction increases rapidly as the distance between the particles decreases, the collapse accelerates and if the star is massive enough, that is, if it is more than twice as massive as the Sun, no known process can prevent the collapse from going on indefinitely.
As the star collapses and becomes more and more dense, the force of gravity on its surface becomes stronger and stronger, and consequently the space-time around it becomes more and more curved. Because of the increasing force of gravity on the star’s surface, it becomes more and more difficult to get away from it, and eventually the star reaches a stage where nothing—not even light—can escape from its surface. At that stage, we say that an ‘event horizon’ forms around the star, because no signal can get away from it to communicate any event to the outside world. The space around the star is then so strongly curved that all the light is trapped in it and cannot escape. We are not able to see such a star, because its light can never reach us and for this reason it is called a black hole. The existence of black holes was predicted on the grounds of relativity theory as early as 1916 and they have lately received a great deal of attention because some recently discovered stellar phenomena might indicate the existence of a heavy star moving around some unseen partner which could be a black hole.
Black holes are among the most mysterious and most fascinating objects investigated by modern astrophysics and illustrate the effects of relativity theory in a most spectacular way. The strong curvature of space-time around them prevents not only all their light from reaching us, but has an equally striking effect on time. If a clock, flashing its signals to us, were attached to the surface of the collapsing star, we would observe these signals to slow down as the star approached the event horizon, and once the star had become a black hole, no clock signals would reach us any more. To an outside observer, the flow of time on the star’s surface slows down as the star collapses and it stops altogether at the event horizon. Therefore, the complete collapse of the star takes an infinite time. The star itself, however, experiences nothing peculiar when it collapses beyond the event horizon. Time continues to flow normally and the collapse is completed after a finite period of time, when the star has contracted to a point of infinite density. So how long does the collapse really take, a finite time or an infinite time? In the world of relativity theory, such a question does not make sense. The lifetime of a collapsing star, like all other time spans, is relative and depends on the frame of reference of the observer.
In the general theory of relativity, the classical concepts of space and time as absolute and independent entities are completely abolished. Not only are all measurements involving space and time relative, depending on the state of motion of the observer, but the whole structure of space-time is inextricably linked to the distribution of matter. Space is curved to different degrees and time flows at different rates in different parts of the universe. We have thus come to apprehend that our notions of a three-dimensional Euclidean space and of linear flowing time are limited to our ordinary experience of the physical world and have to be completely abandoned when we extend this experience.
The Eastern sages, too, talk about an extension of their experience of the world in higher states of consciousness, and they affirm that these states involve a radically different experience of space and time. They emphasize not only that they go beyond ordinary three-dimensional space in meditation, but also—and even more forcefully—that the ordinary awareness of time is transcended. Instead of a linear succession of instants, they experience—so they say—an infinite, timeless, and yet dynamic present. In the following passages, three Eastern mystics speak about the experience of this ‘eternal now’; Chuang Tzu, the Taoist sage; Hui-neng, the Sixth Zen Patriarch; and D. T. Suzuki, the contemporary Buddhist scholar.
Let us forget the lapse of time; let us forget the conflict of opinions. Let us make our appeal to the infinite, and take up our positions there.9
Chuang Tzu
The absolute tranquillity is the present moment. Though it is at this moment, there is no limit to this moment, and herein is eternal delight.10
Hui-neng
In this spiritual world there are no time divisions such as the past, present and future; for they have contracted themselves into a single moment of the present where life quivers in its true sense … The past and the future are both rolled up in this present moment of illumination, and this present moment is not something standing still with all its contents, for it ceaselessly moves on.11
D. T. Suzuki
Talking about an experience of timeless present is almost impossible, because all words like ‘timeless’, ‘present’, ‘past’, ‘moment’, etc., refer to the conventional notions of time. It is thus extremely difficult to understand what the mystics mean in passages like those quoted; but here again, modern physics may facilitate the understanding, as it can be used to illustrate graphically how its theories transcend ordinary notions of time.
In relativistic physics, the history of an object, say a particle, can be represented in a so-called ‘space-time diagram’ (see figure opposite). In these diagrams, the horizontal direction represents space,* and the vertical direction, time. The path of the particle through space-time is called its ‘world line’. If the particle is at rest, it nevertheless moves through time, and its world line is, in that case, a straight vertical line. If the particle moves in space, its world line will be inclined; the greater the inclination of the world line, the faster the particle moves. Note that the particles can only move upwards in time, but can move backwards or forwards in space. Their world lines can be inclined towards the horizontal to various degrees, but can never become completely horizontal, since this would mean that a particle travels from one place to the other in no time at all.
Space-time diagrams are used in relativistic physics to picture the interactions between various particles. For each process, we can draw a diagram and associate a definite mathematical expression with it which gives us the probability for that process to occur. The collision, or ‘scattering’, process between an electron and a photon, for example, may be represented by a diagram like the one overleaf. This diagram is read in the following way (from the bottom to the top, according to the direction of time): an electron (denoted by e- because of its negative charge) collides with a photon (denoted by γ— ‘gamma’); the photon is absorbed by the electron which continues its path with a different velocity (different inclination of the world line); after a while, the electron emits the photon again and reverses its direction of motion.
The theory which constitutes the proper framework for these space-time diagrams, and for the mathematical expressions associated with them, is called ‘quantum field theory’. It is one of the main relativistic theories of modern physics whose basic concepts will be discussed later on. For our discussion of space-time diagrams, it will be sufficient to know two characteristic features of the theory. The first is the fact that all interactions involve the creation and destruction of particles, like the absorption and emission of the photon in our diagram; and the second feature is a basic symmetry between particles and antiparticles. For every particle, there exists an antiparticle with equal mass and opposite charge. The antiparticle of the electron, for example, is called the positron and is usually denoted by e+. The photon, having no charge, is its own antiparticle. Pairs of electrons and positrons can be created spontaneously by photons, and can be made to turn into photons in the reverse process of annihilation.
The space-time diagrams, now, are greatly simplified if the following trick is adopted. The arrowhead on a world line is no longer used to indicate the direction of motion of the particle (which is unnecessary, anyway, since all particles move forwards in time, i.e. upwards in the diagram). Instead, the arrowhead is used to distinguish between particles and antiparticles: if it points upwards, it indicates a particle (e.g. an electron), if it points downwards an antiparticle (e.g. a positron). The photon, being its own antiparticle, is represented by a world line without any arrowhead. With this modification, we can now omit all the labels in our diagram without causing any confusion: the lines with arrowheads represent electrons, those without arrowheads, photons. To make the diagram even simpler, we can also leave out the space axis and the time axis, remembering that the direction of time is from the bottom to the top, and that the forward direction in space is from left to right. The resulting space-time diagram for the electron-photon scattering process looks then as follows.
If we want to picture the scattering process between a photon and a positron, we can draw the same diagram and just reverse the direction of the arrowheads:
So far, there has been nothing unusual in our discussion of space-time diagrams. We have read them from the bottom to the top, according to our conventional notion of a linear flow of time. The unusual aspect is connected with diagrams containing positron lines, like the one picturing the positron-photon scattering. The mathematical formalism of field theory suggests that these lines can be interpreted in two ways; either as positrons moving forwards in time, or as electrons moving backwards in time! The interpretations are mathematically identical; the same expression describes an anti-particle moving from the past to the future, or a particle moving from the future to the past. Our two diagrams can thus be seen as picturing the same process evolving in different directions in time. Both of them can be interpreted as the scattering of electrons and photons, but in one process the particles move forwards in time, in the other they move backwards.* The relativistic theory of particle interactions shows thus a complete symmetry with regard to the direction of time. All space-time diagrams may be read in either direction. For every process, there is an equivalent process with the direction of time reversed and particles replaced by antiparticles.**
To see how this surprising feature of the world of subatomic particles affects our views of space and time, consider the process shown in the diagram overleaf. Reading the diagram In the conventional way, from the bottom to the top, we will interpret it as follows: an electron (represented by a solid line) and a photon (represented by a dashed line) approach each other; the photon creates an electron-positron pair at point A, the electron flying off to the right, the positron to the left; the positron then collides with the initial electron at point B and they annihilate each other, creating a photon in the process which flies off to the left. Alternatively, we may also interpret the process as the interaction of the two photons with a single electron travelling first forwards in time, then backwards, and then forwards again. For this interpretation, we just follow the arrows on the electron line all the way through; the electron travels to point B where it emits a photon and reverses its direction to travel backwards through time to point A; there it absorbs the initial photon, reverses its direction again and flies off travelling forwards through time. In a way, the second interpretation is much simpler because we just follow the world line of one particle. On the other hand, we notice immediately that in doing so we run into serious difficulties of language. The electron travels ‘first’ to point B, and ‘then’ to A; yet the absorption of the photon at A happens before the emission of the other photon at B.
The best way to avoid these difficulties is to see space-time diagrams like the one above not as chronological records of the paths of particles through time, but rather as four-dimensional patterns in space-time representing a network of interrelated events which does not have any definite direction of time attached to it. Since all particles can move forwards and backwards in time, just as they can move left and right in space, it does not make sense to impose a one-way flow of time on the diagrams. They are simply four-dimensional maps traced out in space-time in such a way that we cannot speak of any temporal sequence. In the words of Louis De Broglie:
In space-time, everything which for each of us constitutes the past, the present, and the future is given en bloc … Each observer, as his time passes, discovers, so to speak, new slices of space-time which appear to him as successive aspects of the material world, though in reality the ensemble of events constituting space-time exist prior to his knowledge of them.12
This, then, is the full meaning of space-time in relativistic physics. Space and time are fully equivalent; they are unified into a four-dimensional continuum in which the particle interactions can stretch in any direction. If we want to picture these interactions, we have to picture them in one ‘four-dimensional snap shot’ covering the whole span of time as well as the whole region of space. To get the right feeling for the relativistic world of particles, we must forget the lapse of time’, as Chuang Tzu says, and this is why the space-time diagrams of field theory can be a useful analogy to the space-time experience of the Eastern mystic. The relevance of the analogy is made evident by the following remarks by Lama Govinda concerning Buddhist meditation:
If we speak of the space-experience in meditation, we are dealing with an entirely different dimension … In this space-experience the temporal sequence is converted into a simultaneous co-existence, the side by side existence of things … and this again does not remain static but becomes a living continuum in which time and space are integrated.13
Although the physicists use their mathematical formalism and their diagrams to picture interactions ‘en bloc’ in four-dimensional space-time, they say that in the actual world each observer can only experience the phenomena in a succession of space-time sections, that is, in a temporal sequence. The mystics, on the other hand, maintain that they can actually experience the full span of space-time where time does not flow any longer. Thus the Zen master Dogen:
It is believed by most that time passes; in actual fact, it stays where it is. This idea of passing may be called time, but it is an incorrect idea, for since one sees it only as passing, one cannot understand that it stays just where it is.14
Many of the Eastern teachers emphasize that thought must take place in time, but that vision can transcend it. ‘Vision’, says Govinda, ‘is bound up with a space of a higher dimension, and therefore timeless.’15 The space-time of relativistic physics is a similar timeless space of a higher dimension. All events in it are interconnected, but the connections are not causal. Particle interactions can be interpreted in terms of cause and effect only when the space-time diagrams are read in a definite direction, e.g. from the bottom to the top. When they are taken as four-dimensional patterns without any definite direction of time attached to them, there is no ‘before’ and no ‘after’, and thus no causation.
Similarly, the Eastern mystics assert that in transcending time, they also transcend the world of cause and effect. Like our ordinary notions of space and time, causation is an idea which is limited to a certain experience of the world and has to be abandoned when this experience is extended. In the words of Swami Vivekananda,
Time, space, and causation are like the glass through which the Absolute is seen … In the Absolute there is neither time, space, nor causation.16
The Eastern spiritual traditions show their followers various ways of going beyond the ordinary experience of time and of freeing themselves from the chain of cause and effect—from the bondage of karma, as the Hindus and Buddhists say. It has therefore been said that Eastern mysticism is a liberation from time. In a way, the same may be said of relativistic physics.
* To derive this result it is essential to take into account the fact that the speed of light is the same for all observers.
** Note that in this case, the observer is at rest in his laboratory, but the events he observes are caused by particles moving at different velocities. The effect is the same. What counts is the relative motion of observer and observed events. Which of the two moves with respect to the laboratory is irrelevant.
* A small technical point should perhaps be mentioned. When we speak about the lifetime of a certain kind of unstable particle, we always mean the average lifetime. Due to the statistical character of subatomic physics, we cannot make any statement about individual particles.
* See p. 111.
* The following examples are taken from R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1966), vol. II, Ch. 42.
* Space, in these diagrams, has only one dimension; the other two dimensions have to be suppressed to make a plane diagram possible.
* The dashed lines are always interpreted as photons, whether they move forwards or backwards in time, because the antiparticle of a photon is again a photon.
** Recent experimental evidence suggests that this might not be true for a particular process involving a ‘super-weak interaction’. Apart from this process, for which the role of time-reversal symmetry is not yet clear, all particle interactions seem to show a basic symmetry with regard to the direction of time.