Einstein 1938: The Time-Space Continuum

Reference: Evolution of Physics

This paper presents Chapter III, section 9 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.

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The Time-Space Continuum

“The French revolution began in Paris on the 14th of July 1789.” In this sentence the place and time of an event are stated. Hearing this statement for the first time, one who does not know what “Paris” means could be taught: it is a city on our earth situated in long. 2° East and lat. 49° North. The two numbers would then characterize the place, and “14th of July 1789” the time, at which the event took place. In physics, much more than in history, the exact characterization of when and where an event takes place is very important, because these data form the basis for a quantitative description.

For the sake of simplicity, we considered previously only motion along a straight line. A rigid rod with an origin but no end-point was our c.s. Let us keep this restriction. Take different points on the rod; their positions can be characterized by one number only, by the co-ordinate of the point. To say the co-ordinate of a point is 7.586 feet means that its distance is 7.586 feet from the origin of the rod. If, on the contrary, someone gives me any number and a unit, I can always find a point on the rod corresponding to this number. We can state: a definite point on the rod corresponds to every number, and a definite number corresponds to every point. This fact is expressed by mathematicians in the following sentence: all points on the rod form a one-dimensional continuum. There exists a point arbitrarily near every point on the rod. We can connect two distant points on the rod by steps as small as we wish. Thus the arbitrary smallness of the steps connecting distant points is characteristic of the continuum.

Arbitrarily small units of distance are possible. Thus we have a one-dimensional continuum.

Now another example. We have a plane, or, if you prefer something more concrete, the surface of a rectangular table. The position of a point on this table can be characterized by two numbers and not, as before, by one. The two numbers are the distances from two perpendicular edges of the table. Not one number, but a pair of numbers corresponds to every point on the plane; a definite point corresponds to every pair of numbers. In other words: the plane is a two-dimensional continuum. There exist points arbitrarily near every point on the plane. Two distant points can be connected by a curve divided into steps as small as we wish. Thus the arbitrary smallness of the steps connecting two distant points, each of which can be represented by two numbers, is again characteristic of a two-dimensional continuum.

We also have a two-dimensional continuum in terms of the same arbitrarily small unit.

One more example. Imagine that you wish to regard your room as your c.s. This means that you want to describe all positions with respect to the rigid walls of the room. The position of the end-point of the lamp, if the lamp is at rest, can be described by three numbers: two of them determine the distance from two perpendicular walls, and the third that from the floor or ceiling. Three definite numbers correspond to every point of the space; a definite point in space corresponds to every three numbers. This is expressed by the sentence: Our space is a three-dimensional continuum. There exist points very near every point of the space. Again the arbitrary smallness of the steps connecting the distant points, each of them represented by three numbers, is characteristic of a three-dimensional continuum.

Our space is a three-dimensional continuum described by arbitrarily small units.

But all this is scarcely physics. To return to physics, the motion of material particles must be considered. To observe and predict events in nature we must consider not only the place but also the time of physical happenings. Let us again take a very simple example.

A small stone, which can be regarded as a particle, is dropped from a tower. Imagine the tower 256 feet high. Since Galileo’s time we have been able to predict the co-ordinate of the stone at any arbitrary instant after it was dropped. Here is a “timetable” describing the positions of the stone after 0, 1,2, 3, and 4 seconds. Five events are registered in our “timetable”, each represented by two numbers, the time and space coordinates of each event. The first event is the dropping of the stone from 256 feet above the ground at the zero second. The second event is the coincidence of the stone with our rigid rod (the tower) at 240 feet above the ground. This happens after the first second. The last event is the coincidence of the stone with the earth.

We could represent the knowledge gained from our “timetable” in a different way. We could represent the five pairs of numbers in the “timetable” as five points on a surface. Let us first establish a scale. One segment will correspond to a foot and another to a second. For example:

We then draw two perpendicular lines, calling the horizontal one, say, the time axis and the vertical one the space axis. We see immediately that our “timetable” can be represented by five points in our time-space plane.

The distances of the points from the space axis represent the time co-ordinates as registered in the first column of our “timetable”, and the distances from the time axis their space co-ordinates.

Exactly the same thing is expressed in two different ways: by the “timetable” and by the points on the plane. Each can be constructed from the other. The choice between these two representations is merely a matter of taste, for they are, in fact, equivalent.

Let us now go one step farther. Imagine a better “timetable” giving the positions not for every second, but for, say, every hundredth or thousandth of a second. We shall then have very many points on our time-space plane. Finally, if the position is given for every instant or, as the mathematicians say, if the space co-ordinate is given as a function of time, then our set of points becomes a continuous line. Our next drawing therefore represents not just a fragment as before, but a complete knowledge of the motion.

We may associate space with time in arbitrarily small units of both space and time.

The motion along the rigid rod (the tower), the motion in a one-dimensional space, is here represented as a curve in a two-dimensional time-space continuum. To every point in our time-space continuum there corresponds a pair of numbers, one of which denotes the time, and the other the space, co-ordinate. Conversely: a definite point in our time-space plane corresponds to every pair of numbers characterizing an event. Two adjacent points represent two events, two happenings, at slightly different places and at slightly different instants.

The time-space coordinate describes an event.

You could argue against our representation thus: there is little sense in representing a unit of time by a segment, in combining it mechanically with the space, forming the two-dimensional continuum from the two one-dimensional continua. But you would then have to protest just as strongly against all the graphs representing, for example, the change of temperature in New York City during last summer, or against those representing the changes in the cost of living during the last few years, since the very same method is used in each of these cases. In the temperature graphs the one-dimensional temperature continuum is combined with the one-dimensional time continuum into the two-dimensional temperature-time continuum.

Just like there is a relationship among the units of the three dimensions of space, there may be a relationship between natural units of space and time. There could be a conversion factor between space and time.

Let us return to the particle dropped from a 256-foot tower. Our graphic picture of motion is a useful convention since it characterizes the position of the particle at an arbitrary instant. Knowing how the particle moves, we should like to picture its motion once more. We can do this in two different ways.

We remember the picture of the particle changing its position with time in the one-dimensional space. We picture the motion as a sequence of events in the one-dimensional space continuum. We do not mix time and space, using a dynamic picture in which positions change with time.

But we can picture the same motion in a different way. We can form a static picture, considering the curve in the two-dimensional time-space continuum. Now the motion is represented as something which is, which exists in the two-dimensional time-space continuum, and not as something which changes in the one-dimensional space continuum.

Both these pictures are exactly equivalent, and preferring one to the other is merely a matter of convention and taste.

Nothing that has been said here about the two pictures of the motion has anything whatever to do with the relativity theory. Both representations can be used with equal right, though classical physics favoured rather the dynamic picture describing motion as happenings in space and not as existing in time-space. But the relativity theory changes this view. It was distinctly in favour of the static picture and found in this representation of motion as something existing in time-space a more convenient and more objective picture of reality. We still have to answer the question: why are these two pictures, equivalent from the point of view of classical physics, not equivalent from the point of view of the relativity theory?

A static time-space picture is more useful.

The answer will be understood if two c.s. moving uniformly, relative to each other, are again taken into account.

According to classical physics, observers in two c.s. moving uniformly, relative to each other, will assign different space co-ordinates, but the same time coordinate, to a certain event. Thus in our example, the coincidence of the particle with the earth is characterized in our chosen c.s. by the time co-ordinate “4” and by the space co-ordinate “0”. According to classical mechanics, the stone will still reach the earth after four seconds for an observer moving uniformly, relative to the chosen c.s. But this observer will refer the distance to his c.s. and will, in general, connect different space co-ordinates with the event of collision, although the time co-ordinate will be the same for him and for all other observers moving uniformly, relative to each other. Classical physics knows only an “absolute” time flow for all observers. For every c.s. the two-dimensional continuum can be split into two one-dimensional continua: time and space. Because of the “absolute” character of time, the transition from the “static” to the “dynamic” picture of motion has an objective meaning in classical physics.

With change in inertia, both space and time units change.

But we have already allowed ourselves to be convinced that the classical transformation must not be used in physics generally. From a practical point of view it is still good for small velocities, but not for settling fundamental physical questions.

According to the relativity theory the time of the collision of the stone with the earth will not be the same for all observers. The time co-ordinate and the space co-ordinate will be different in two c.s., and the change in the time co-ordinate will be quite distinct if the relative velocity is close to that of light. The two-dimensional continuum cannot be split into two one-dimensional continua as in classical physics. We must not consider space and time separately in determining the time-space co-ordinates in another c.s. The splitting of the two-dimensional continuum into two one-dimensional ones seems, from the point of view of the relativity theory, to be an arbitrary procedure without objective meaning.

We must not consider space and time separately in determining the time-space co-ordinates in another c.s. because time and space are closely related as duration and extents of the substance through its inertia.

It will be simple to generalize all that we have just said for the case of motion not restricted to a straight line. Indeed, not two, but four, numbers must be used to describe events in nature. Our physical space as conceived through objects and their motion has three dimensions, and positions are characterized by three numbers. The instant of an event is the fourth number. Four definite numbers correspond to every event; a definite event corresponds to any four numbers. Therefore: The world of events forms a four-dimensional continuum. There is nothing mysterious about this, and the last sentence is equally true for classical physics and the relativity theory. Again, a difference is revealed when two c.s. moving relatively to each other are considered. The room is moving, and the observers inside and outside determine the time-space co-ordinates of the same events. Again, the classical physicist splits the four-dimensional continua into the three-dimensional spaces and the one-dimensional time-continuum. The old physicist bothers only about space transformation, as time is absolute for him. He finds the splitting of the four-dimensional world-continua into space and time natural and convenient. But from the point of view of the relativity theory, time as well as space is changed by passing from one c.s. to another, and the Lorentz transformation considers the transformation properties of the four-dimensional time-space continuum of our four-dimensional world of events.

Inertia is considered constant in classical mechanics, but in long range as covered by the theory of relativity it is a variable. Change in inertia means change in the substance, its extent (space) and its duration (time). Duration (time) cannot be taken as constant.

The world of events can be described dynamically by a picture changing in time and thrown on to the background of the three-dimensional space. But it can also be described by a static picture thrown on to the background of a four-dimensional time-space continuum. From the point of view of classical physics the two pictures, the dynamic and the static, are equivalent. But from the point of view of the relativity theory the static picture is the more convenient and the more objective.

Even in the relativity theory we can still use the dynamic picture if we prefer it. But we must remember that this division into time and space has no objective meaning since time is no longer “absolute”. We shall still use the “dynamic” and not the “static” language in the following pages, bearing in mind its limitations.

Time makes the picture dynamic, but it can be portrayed in static terms in relationship to space.

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FINAL COMMENTS

Inertia is considered constant in classical mechanics, but in the range covered by the theory of relativity inertia becomes a variable. Change in inertia means change in the characteristics of substance. The key characteristics of substance are its extents and duration. Extent (space) and duration (time) cannot be assumed to have fixed characteristics.

Arbitrarily small units of both distance and time are possible. We, therefore, have a continuum of three space dimensions and one time dimension. These dimensions represent the extents and duration of a substance respectively. Since it is the same substance, its extents and duration are not independent of each other. There seems to be an exact relationship among space, time and inertia. There could even be a conversion factor between space and time.

Time makes the picture dynamic, but it can be portrayed in static terms in relationship to space. A static time-space picture is more useful as used in the theory of relativity. The time-space coordinate describes an event.

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