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Printed Pamphlet Accompanying The Rhodes Lectures, Oxford, 1931, by Albert Einstein

By Professor Albert Einstein


The special theory of relativity is formally characterized by a pseudo-Euclidean metric in a four-dimensional space. Physical basis: Generally, Maxwell's electromagnetic equations and more especially the aberration of light and the experiments of Fizeau and Michelson. Time on the one hand and three-dimensional space on the other lose their absolute character; the four-dimensional 'Space' alone retains it.
The general theory of relativity is formally characterized by a Riemann metric in four dimensions. Physical basis: The equality of gravitational and inertial mass; the physical relationships (equivalence) between an acceelerated co-ordinate system and a special type of gravitational field. The Riemann 's metric describes, on the one hand, the metrical properties of 'Space', i.e. the abstractions which can be made from measurements with clocks and scales, on the other hand, the gravitational field, i.e. it determines the paths along which electrically neutral particles move.
The major problem of the general theory of

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relativity: Search for the field equations of gravitation. These are completely determined by the requirements of co-variance if one assumes that they are linear and of the second order in the metrical co-efficients. Criticism of the general theory of relativity as based upon the Riemann metric: Whilst providing a satisfactory theory of gravitation it does not provide naturally for the electromagnetic field and matter.

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If one describes a three-dimensional sphere large enough to contain many stars about any point in the universe and divides the enclosed mass by the volume of the sphere one obtains a mean density [Greek lower-case rho] which may be assumed to reach a limiting value as the radius increases. It is to be assumed that this value is not zero, since otherwise the universe would be virtually empty.
It is natural to assume that this density [rho] is the same everywhere in the universe. The question arises: Are the field equations of the general theory of relativity compatible with this assumption?
For a preliminary investigation one can imagine the local irregularities smoothed out and replace [rho] by a density constant throughout the universe. In this way space with irregular local curvatures is replaced by one of constant curvature.
On first approaching this problem I made the further assumption that the density thus defined (and consequently the constitution of space on a large scale) did not change with time.
Calculations show that the field equations of the general theory of relativity cannot be solved with these assumptions throughout a space of constant

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curvature. But it appeared that a solution was possible if an arbitrary term (the [Greek lower-case lambda] term) was added to the field equations, a procedure compatible with the general relativity postulate. A spherical space resulted whose radius [Greek upper-case RHO] was defined uniquely by the density [RHO alpha (kappa rho) to the power of minus a half], where [kappa] denotes the gravitational constant of the general theory of relativity.
In 1922 A. Friedmann discovered that these equations (with [lambda] term) also have solutions in which [RHO] and [rho] are functions of the time. De Sitter, Lemaitre, and Tolman also investigated these problems mathematically. The question has entered a new phase, since Hubble has shown in the course of the last few years by spectral investigation of the extra galactic nebulae that these show a Doppler effect (displacement towards the red of spectral lines), increasing linearly with the distance, and that these systems (each analogous to the Milky Way) are distributed approximately uniformly in space. If these red displacements are to be interpreted as radial motions, an interpretation we have at present no right to call into question, then the assumption that [rho] does not vary with the time is considerably weakened.
Basing ourselves on Friedmann's formulae it may be shown that the equations of the general theory of relativity can be satisfied by an expanding space and, what is more, without introducing the [lambda] term, which was theoretically unsatisfactory and empirically uncalled for. According to this the order of magnitude of the varying world radius depends not only on the mean density [rho] but also upon Hubble's constant for

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the red displacement (D = Doppler effect [over] distance) according to the equations
[RHO alpha (kappa rho) to the power of minus a half],
[RHO alpha 1 over D].
Difficulties: Small value of the world radius (10[to the power of 8] light years); comparatively short period during which the space structure can have been expanding (10[to the power of 10] years); difficulty of extrapolating in time farther back into the past.

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An attempt to complete the theory of relativity by superposing a directional structure on the simple metrical one. If the electromagnetic field is to be inherent in the theory, then this must be represented, together with the gravitational metric, in a single unitary structure of space. Since a physical basis for this is lacking we can only be guided by considerations of mathematical simplicity.
Fundamental assumptions. The relative orientation of the local systems of Cartesian co-ordinates embodying the Riemann metric has an objective (physical) meaning, i.e. it is possible to state the law for the infinitesimal displacement of a vector, which leaves its magnitude (determined by the Riemann metric) unchanged and which leads to a vector independent of the path after any (finite) displacement. (Integrability of the law of displacement.)
This spatial structure is described in four dimensions by sixteen functions. Besides the metric these determine a structure for the direction relationship which may be regarded as providing an interpretation of the electromagnetic field. It may further be hoped that the phenomena which have already been provisionally co-ordinated in the atomic and quantum theories may follow from the field law by postulating freedom from singularities.
Description of the field. The local co-ordinate

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system consists of four orthogonal contravariant vectors (s = 1, 2, 3, 4) whose components referred to a system of Gaussian co-ordinates are h[superscript v & subscript s]. The fundamental problem consists in seeking the differential equations of the field for the sixteen vector components h, which we assume once again to be of the second order and linear in the second order terms. The gravitational equations of the original theory suggested the line along which the solution of the problem might be found. The ten field equations (tensor equations) in the original theory contain as we know four identities. This is necessary, otherwise the number of equations would be too great; for owing to the condition of general co-variance our choice of four of the field variables must be completely free; hence four of the equations cannot be independent of the others. The field equations of the new theory must have the same characteristics.
An investigation carried out by myself and my colleague, W. Mayer, has established that this condition determines in the main the field equations, but not so unambiguously as in the simple metrical theory. Actually there are four types of field equations, each of which still contain one or two universal constants not determined by the theory: of these types two contain the old gravitational equations as special cases.
Whether the fundamental assumption which we have made brings us any nearer to the structure of real space can only be decided by the (very difficult) integration of the equations.

[Transcription of a printed pamphlet issued in conjunction with Einstein's Rhodes Lectures, 1931, summarising each lecture; the Museum's blackboard was written for or during the second lecture and refers to the questions of the size, age, and density of the universe dealt with at the end of the lecture; note that the Museum does not hold the original pamphlet, it has been copied from one in a private collection]

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