The Cosmological Constant

Lemaître favours a “cosmological constant” in the equations of gravitation.

I think that such a constant is not convincing because it makes my general relativity logically complicated.

Hubble discovered the “expansion” of the stellar system.

Friedmann discovered that the unsupplemented equations involve the possibility of an average positive density of matter in an expanding universe. This makes such a constant unjustified.

This is complicated by the entire duration of space-expansion to the present is shorter than the known age of terrestrial minerals.

A “cosmological constant” does not help us escape the difficulty caused by Hubble’s expansion-constant and the age-measurement of minerals, completely independent of any cosmological theory, provided that one interprets the Hubble-effect as Doppler effect.

Everything finally depends on the question: Can a spectral line be considered as a measure of a “proper time”?*

A yes made the invention of General Relativity psychologically possible.

However, this supposition is logically not necessary. Relativity needs the following:

1. Physical things are described by continuous functions, field-variables of 4 coordinates. As long as the topological connection is preserved, these latter can be freely chosen

2. The field-variables are tensor-components; among the tensors is a symmetrical tensor gik for the description of the gravitational field.

3. There are physical objects, which (in the macroscopic field) measure the invariant ds.

If (1) and (2) are accepted, (3) is plausible, but not necessary. The construction of mathematical theory rests exclusively on (1) and (2).

A complete theory of physics as a totality, in accordance with (1) and (2) does not yet exist.

If it did exist, there would be no room for the supposition (3). For the objects used as tools for measurement do not lead an independent existence alongside of the objects implicated by the field-equations.

It is not necessary that one should permit one’s cosmological considerations to be restrained by such a sceptical attitude; but neither should one close one’s mind towards them from the very beginning.

These reflections bring me to Karl Menger’s essay.

For the quantum-facts suggest the suspicion that doubt may also be raised concerning the ultimate usefulness of the program characterised in (1) and (2).

There exists the possibility of doubting only (2) and, in doing so, to question the possibility of being able adequately to formulate the laws by means of differential equations, without dropping (1).

The more radical effort of surrendering (1) with (2) appears to me — and I believe to Dr. Menger also — to lie more closely at hand. So long as no one has new concepts, which appear to have sufficient constructive power, mere doubt remains.

This is, unfortunately, my own situation.

Adhering to the continuum originates with me not in a prejudice, but arises out of the fact that I have been unable to think up anything organic to take its place. How is one to conserve four-dimensionality in essence (or in near approximation) and [at the same time] surrender the continuum?

L. Infeld’s essay is an independently understandable, excellent introduction into the so-called “cosmological problem” of the theory of relativity, which critically examines all essential points.

Max von Laue: An historical investigation of the development of the conservation postulates, which, in my opinion, is of lasting value. I think it would be worth while to make this essay easily accessible to students by way of independent publication.

In spite of serious efforts I have not succeeded in quite understanding H. Dingle’s essay, not even as concerns its aim.

Is the idea of the special theory of relativity to be expanded in the sense that new group-characteristics, which are not implied by the Lorentz-invariance, are to be postulated?

Are these postulates empirically founded or only by way of a trial “posited”? Upon what does the confidence in the existence of such group-characteristics rest?

I think that Kurt Gödel’s essay is an important contribution to General Relativity, especially to the analysis of the concept of time.

The problem here involved disturbed me already at the time of the building up of General Relativity, without my having succeeded in clarifying it.

Entirely aside from the relation of the theory of relativity to idealistic philosophy or to any philosophical formulation of questions, the problem presents itself as follows:

If P is a world-point, a “light-cone” (ds2= 0) belongs to it.

We draw a “time-like” world-line through P and on this line observe the close world-points B and A, separated by P.

Does it make any sense to provide the world-line with an arrow, and to assert that B is before P, A after P?

Is what remains of temporal connection between world-points in the theory of relativity an asymmetrical relation, or would one be just as much justified, from the physical point of view, to indicate the arrow in the opposite direction and to assert that A is before P, B after P?

In the first instance, the alternative is decided in the negative, if we are justified in saying: If it is possible to send (to telegraph) a signal (also passing by in the close proximity of P) from B to A, but not from A to B, then the one-sided (asymmetrical) character of time is secured, i.e., there exists no free choice for the direction of the arrow.

What is essential is that the sending of a signal is, in the sense of thermodynamics, an irreversible process. It is connected with the growth of entropy. According to our present knowledge, all elementary processes are reversible.

If, therefore, B and A are two, sufficiently neighbouring, world-points, which can be connected by a time-like line, then the assertion: “B is before A,” makes physical sense.

But does it still make sense, if the points, which are connectable by the time-like line, are arbitrarily far separated from each other?

Certainly not, if there exist point-series connectable by time-like lines in such a way that each point precedes temporally the preceding one, and if the series is closed in itself. In that case the distinction “earlier-later” is abandoned for world-points which lie far apart in a cosmological sense, and those paradoxes, regarding the direction of the causal connection, arise, of which Mr. Gödel has spoken.

Such cosmological solutions of the gravitation-equations (with not vanishing A-constant) have been found by Mr. Gödel. It will be interesting to weigh whether these are not to be excluded on physical grounds.

P.S. The preceding remarks refer to essays which were in my hands at the end of January 1949. Inasmuch as the volume was to have appeared in March, it was high time to write down these reflections.