Relativity: The Special and General Theory quotes

In your schooldays most of you who read this book made acquaintance with the noble building of Euclid’s geometry, and you remember—perhaps with more respect than love—the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers.

By reason of our past experience, you would certainly regard everyone with disdain who should pronounce even the most out-of-the-way proposition of this science to be untrue. But perhaps this feeling of proud certainty would leave you immediately if some one were to ask you:

Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning.

We cannot ask whether it is true that only one straight line goes through two points.

We can only say that Euclidean geometry deals with things called “straight lines,” to each of which is ascribed the property of being uniquely determined by two points situated on it.

[2] Here we have assumed that there is nothing left over i.e. that the measurement gives a whole number. This difficulty is got over by the use of divided measuring-rods, the introduction of which does not demand any fundamentally new method.

Every description of the scene of an event or of the position of an object in space is based on the specification of the point on a rigid body (body of reference) with which that event or object coincides.

This applies not only to scientific description, but also to everyday life. If I analyse the place specification “Trafalgar Square, London”[3] I arrive at the following result. The earth is the rigid body to which the specification of place refers;

We thus obtain the following result:

Every description of events in space involves the use of a rigid body to which such events have to be referred.

The resulting relationship takes for granted that the laws of Euclidean geometry hold for “distances;” the “distance” being represented physically by means of the convention of two marks on a rigid body.

I stand at the window of a railway carriage which is travelling uniformly, and drop a stone on the embankment, without throwing it. Then, disregarding the influence of the air resistance, I see the stone descend in a straight line.

A pedestrian who observes the misdeed from the footpath notices that the stone falls to earth in a parabolic curve.

I now ask: Do the “positions” traversed by the stone lie “in reality” on a straight line or on a parabola? Moreover, what is meant here by motion “in space”? From the considerations of the previous section the answer is self-evident.

I now ask: Do the “positions” traversed by the stone lie “in reality” on a straight line or on a parabola? Moreover, what is meant here by motion “in space”? From the considerations of the previous section the answer is self-evident.

In the first place we entirely shun the vague word “space,” of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by “motion relative to a practically rigid body of reference.”

The positions relative to the body of reference (railway carriage or embankment) have already been defined in detail in the preceding section.

It is not clear what is to be understood here by “position” and “space.”

I stand at the window of a railway carriage which is travelling uniformly, and drop a stone on the embankment, without throwing it.

Then, disregarding the influence of the air resistance, I see the stone descend in a straight line. A pedestrian who observes the misdeed from the footpath notices that the stone falls to earth in a parabolic curve. I now ask:

There is hardly a simpler law in physics than that according to which light is propagated in empty space.

Every child at school knows, or believes he knows, that this propagation takes place in straight lines with a velocity c = 300,000 km./sec.

At all events we know with great exactness that this velocity is the same for all colours, because if this were not the case, the minimum of emission would not be observed simultaneously for different colours during the eclipse of a fixed star by its …

Lightning has struck the rails on our railway embankment at two places A and B far distant from each other.

I make the additional assertion that these two lightning flashes occurred simultaneously. If I ask you whether there is sense in this statement, you will answer my question with a decided “Yes.”

Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity).

Every reference-body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.

Now before the advent of the theory of relativity it had always tacitly been assumed in physics that the statement of time had an absolute significance, i.e. that it is independent of the state of motion of the body of reference.

of a metre.

The rigid rod is thus shorter when in motion than when at rest, and the more quickly it is moving, the shorter is the rod.

For the velocity v = c we should have

seconds, i.e. a somewhat larger time.

As a consequence of its motion the clock goes more slowly than when at rest.

Here also the velocity c plays the part of an unattainable limiting velocity.

The non-mathematician is seized by a mysterious shuddering when he hears of “four-dimensional” things, by a feeling not unlike that awakened by thoughts of the occult.

And yet there is no more common-place statement than that the world in which we live is a four-dimensional space-time continuum.

The non-mathematician is seized by a mysterious shuddering when he hears of “four-dimensional” things, by a feeling not unlike that awakened by thoughts of the occult.

And yet there is no more common-place statement than that the world in which we live is a four-dimensional space-time continuum.

Space is a three-dimensional continuum.

“If we pick up a stone and then let it go, why does it fall to the ground?”

The usual answer to this question is: “Because it is attracted by the earth.” Modern physics formulates the answer rather differently for the following reason.

As reference-body let us imagine a spacious chest resembling a room with an observer inside who is equipped with apparatus. Gravitation naturally does not exist for this observer.

He must fasten himself with strings to the floor, otherwise the slightest impact against the floor will cause him to rise slowly towards the ceiling of the room.

To the middle of the lid of the chest is fixed externally a hook with rope attached, and now a “being” (what kind of a being is immaterial to us) begins pulling at this with a constant force.

Not in the least. Electrostatics is contained in electrodynamics as a limiting case; the laws of the latter lead directly to those of the former for the case in which the fields are invariable with regard to time.

No fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case.

In the example of the transmission of light just dealt with, we have seen that the general theory of relativity enables us to derive theoretically the influence of a gravitational field on the course of natural processes, the laws of which are already known when …

… have a kind of centre in which the density of the stars is a maximum, and that as we proceed outwards from this centre the group-density of the stars should diminish, until finally, at great distances, it is succeeded by an infinite region of emptiness.

The stellar universe ought to be a finite island in the infinite ocean of space.[24]

[24] Proof—According to the theory of Newton, the number of “lines of force” which come from infinity and terminate in a mass m is proportional to the mass m.

… less satisfactory because it leads to the result that the light emitted by the stars and also individual stars of the stellar system are perpetually passing out into infinite space, never to return, and without ever again coming into interaction with other objects of nature.

Such a finite material universe would be destined to become gradually but systematically impoverished.

In order to escape this dilemma, Seeliger suggested a modification of Newton’s law, in which he assumes that for great distances the force of attraction between two masses diminishes more rapidly than would result from the inverse square law.

… conceivable that our universe differs only slightly from a Euclidean one, and this notion seems all the more probable, since calculations show that the metrics of surrounding space is influenced only to an exceedingly small extent by masses even of the magnitude of our sun.

We might imagine that, as regards geometry, our universe behaves analogously to a surface which is irregularly curved in its individual parts, but which nowhere departs appreciably from a plane: something like the rippled surface of a lake.

Such a universe might fittingly be called a quasi-Euclidean universe. As regards its space it would be infinite. But calculation shows that in a quasi-Euclidean universe the average density of matter would necessarily be nil.

Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms. We call such a system of thought a theory.

The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the “truth” of the theory lies.

Corresponding to the same complex of empirical data, there may be several theories, which differ from one another to a considerable extent.