**The General Theory of Relativity (extract)**

part of a lecture by Albert Einstein (1921)

(Princeton University, May)

ALL of the previous considerations have been based upon the assumption

that all inertial systems are equivalent for the description of physical phenomena, but that they are preferred, for the formulation of the laws of nature, to spaces of reference in a different state of motion. We can think of no cause for this preference for definite states of motion to all others, according to our previous considerations, either in the perceptible bodies or in the concept of motion; on the contrary, it must be regarded as an independent property of the space-time continuum. The principle of inertia, in particular, seems to compel us to ascribe physically objective properties to the space-time continuum. Just as it was consistent from the Newtonian standpoint to make both the statements, tempus est absolutum, spatium est absolutum, so from the standpoint of the special theory of relativity we must say, continuum spatii et temporis est absolutum. In this latter statement absolutum means not only “physically real”, but also “independent in its physical properties, having a physical effect, but not itself influenced by physical conditions”.

As long as the principle of inertia is regarded as the keystone of

physics, this standpoint is certainly the only one which is justified.

But there are two serious criticisms of the ordinary conception. In

the first place, It is contrary to the mode of thinking in science to

conceive of a thing (the space-time continuum) which acts itself, but

which cannot be acted upon. This is the reason why E. Mach was led to make the attempt to eliminate space as an active cause in the system of mechanics. According to him, a material particle does not move in unaccelerated motion relatively to space, but relatively to the centre of all the other masses in the universe; in this way the series of causes of mechanical phenomena was closed, in contrast to the mechanics of Newton and Galileo. In order to develop this idea within the limits of the modern theory of action through a medium, the properties of the space-time continuum which determine inertia must be regarded as field properties of space, analogous to the

electromagnetic field. The concepts of classical mechanics afford no

way of expressing this. For this reason Mach’s attempt at a solution

failed for the time being. We shall come back to this point of view

later. In the second place, classical mechanics exhibits a deficiency

which directly calls for an extension of the principle of relativity

to spaces of reference which are not in uniform motion relatively to

each other. The ratio of the masses of two bodies is defined in

mechanics in two ways which differ from each other fundamentally; in

the first place, as the reciprocal ratio of the accelerations which

the same motive force imparts to them (inert mass), and in the second place, as the ratio of the forces which act upon them in the same gravitational field (gravitational mass).

The equality of these two masses, so differently defined, is a fact

which is confirmed by experiments of very high accuracy (experiments of Eötvös), and classical mechanics offers no explanation for this equality. It is, however, clear that science is fully justified in assigning such a numerical equality only after this numerical equality is reduced to an equality of the real nature of the two concepts.

That this object may actually be attained by an extension of the

principle of relativity, follows from the following consideration. A

little reflection will show that the law of the equality of the inert

and the gravitational mass is equivalent to the assertion that the

acceleration imparted to a body by a gravitational field is independent of the nature of the body. For Newton’s equation of motion in a gravitational field, written out in full, is (Inert mass) * (Acceleration) = (Intensity of the gravitational field)

* (Gravitational mass)

It is only when there is numerical equality between the inert and

gravitational mass that the acceleration is independent of the nature

of the body. Let now K be an inertial system. Masses which are

sufficiently far from each other and from other bodies are then, with

respect to K, free from acceleration. We shall also refer these masses to a system of co-ordinates K’, uniformly accelerated with respect to K . Relatively to K’ all the masses have equal and parallel

accelerations, with respect to K’ they behave just as if a gravitational field were present and K’ were unaccelerated.

Overlooking for the present the question as to the ’cause’ of such a

gravitational field, which will occupy us later, there is nothing to

prevent our conceiving this gravitational field as real, that is, the

conception that K’ is ‘at rest’ and a gravitational field is present

we may consider as equivalent to the conception that only K is an

‘allowable” system of co-ordinates and no gravitational field is

present. The assumption of the complete physical equivalence of the systems of co-ordinates, K and K’, we call the ‘principle of equivalence’; this principle is evidently intimately connected with the law of the equality between the inert and the gravitational mass, and signifies an extension of the principle of relativity to co-ordinate systems which are in non-uniform motion relatively to each other. In fact, through this conception we arrive at the unity of the nature of

inertia and gravitation. For, according to our way of looking at it

the same masses may appear to be either under the action of inertia

alone (with respect to K ) or under the combined action of inertia and

gravitation (with respect to K’ ).

The possibility of explaining the numerical equality of inertia and

gravitation by the unity of their nature gives to the general theory

of relativity, according to my conviction, such a superiority over the

conceptions of classical mechanics, that all the difficulties encountered in development must be considered as small in comparison with this progress.

What justifies us in dispensing with the preference for inertial

systems over all other co-ordinate systems, a preference that seems so securely established by experience? The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass

moves without acceleration if it is sufficiently far from other

bodies; we know that it is sufficiently far from other bodies only by

the fact that it moves without acceleration. Are there, at all, any

inertial systems for very extended portions of the space-time

continuum, or, indeed, for the whole universe ? We may look upon the principle or inertia as established, to a high degree of approximation, for the space of our planetary system, provided that we neglect the perturbations due to the sun and planets. Stated more exactly, there are finite regions, where, with respect to a suitably chosen space of reference, material particles move freely without acceleration, and in which the laws of the special theory of

relativity, which have been developed above, hold with remarkable

accuracy. Such regions we shall call ‘Galilean regions’. We shall

proceed from the consideration of such regions as a special case of

known properties.

The principle of equivalence demands that in dealing with Galilean

regions we may equally well make use of non-inertial systems, that is,

such co-ordinate systems as, relatively to inertial systems, are not

free from acceleration and rotation. If, further, we are going to do

away completely with the vexing question as to the objective reason

for the preference of certain systems of co-ordinates, then we must

allow the use of arbitrarily moving systems of co-ordinates. As soon

as we make this attempt seriously we come into conflict with that

physical interpretation of space and time to which we were led by the

special theory of relativity. For let K’ be a system of co-ordinates

whose z-axis coincides with the z-axis of K, and which rotates about

the latter axis with constant angular velocity. Are the configurations

of rigid bodies, at rest relatively to K’, in accordance with the laws

of Euclidean geometry?

Since K’ is not an inertial system, we do not know directly the laws

of configuration of rigid bodies with respect to K’ , nor the laws of

nature, in general. But we do know these laws with respect to the

inertial system K, and we can therefore infer their form with respect

to K’. Imagine a circle drawn about the origin in the x’ y’ plane of

K’, and a diameter of this circle. Imagine, further, that we have

given a large number of rigid rods, all equal to each other. We

suppose these laid in series along the periphery and the diameter of

the circle, at rest relatively to K’ . If U is the number of these

rods along the periphery, D the number along the diameter, then, if K’

does not rotate relatively to K, we shall have

U / D = pi

But if K’ rotates we get a different result. Suppose that at a

definite time t , of K we determine the ends of all the rods. With

respect to K all the rods upon the periphery experience the Lorentz

contraction, but the rods upon the diameter do not experience this

contraction (along their lengths!). *note It therefore follows that

U / D > pi

It therefore follows that the laws of configuration of rigid bodies

with respect to K’ do not agree with the laws of configuration of

rigid bodies that are in accordance with Euclidean geometry. If,

further, we place two similar clocks (rotating with K’ ), one upon the

periphery, and the other at the centre of the circle, then, judged

from K, the clock on the periphery will go slower than the clock at

the centre. The same thing must take place, judged from K’ , if we do

not define time with respect to K’ in a wholly unnatural way (that is,

in such a way that the laws with respect to K’ depend explicitly upon

the time). Space and time, therefore, cannot be defined with respect

to K’ as they were in the special theory of relativity with respect to

inertial systems. But, according to the principle of equivalence, K’

may also be considered as a system at rest, with respect to which

there is a gravitational field (field of centrifugal force, and force

of Coriolis). We therefore arrive at the result: the gravitational

field influences and even determines the metrical laws of the

space-time continuum. If the laws of configuration of ideal rigid

bodies are to be expressed geometrically, then in the presence of a

gravitational field the geometry is not Euclidean.

–

* These considerations assume that the behaviour of rods and clocks

depends only upon velocities, and not upon accelerations, or, at

least, that the influence of acceleration does not counteract that of

velocity.