3.1.2 Curvature and metric in general relativistic universes

The early relativistic world models are static. Einstein (1917) considered positive space curvature with matter acting as the form-giving factor: the Einstein universe. When he found that his solution was not stable he introduced a compensating factor, the cosmological constant (Sect. 3.2.4).

De Sitter (1917) found all three possible static solutions. For a matter-filled universe it is the solution given by Einstein, the other two solutions pertain to a universe without matter. Assuming = 0, the empty universe is Euclidean and corresponds to an Newtonian universe or to an Einstein universe before the introduction of . A projection of the Einsteinian universe into spaces of different curvature requires that time be always and everywhere the same, i.e. absolute. With 0 an empty universe of positive curvature can be projected into a Euclidean or hyperbolic space under complete invariance of all four variables.

Observational evidence for a hyperbolical universe is discussed by de Sitter (1917) in the following:

``System A: time is the same everywhere and always, the time coordinate is different from the three space coordinates; system B: there is no universal time, no difference between coordinates in four-dimensional space, no physical meaning of the coordinates.
The three-dimensional space of this system of reference is the space with constant negative curvature, or hyperbolical space, or space of Lobatschewski.
In the system B the rays of light are straight lines in hyperbolic space . . . . In space B we have g44 = cos2 . Consequently the frequency of light vibrations diminishes with increasing distance from the origin of co-ordinates. The lines in the spectra of very distant stars or nebulae must therefore be systematically displaced towards the red, giving rise to a spurious positive radial velocity . . . . Of the following three nebulae, the velocities have been determined by more than one observer:

 Andromeda 3 observers - 311 km/sec. N.G.C. 1068 3 " + 925 " N.G.C. 4594 2 " + 1185 "

. . . . If, however, continued observations should confirm the fact that the spiral nebulae have systematically positive radial velocities, this would certainly be an indication to adopt the hypothesis B in preference of A.''

Friedmann (1922) introduced his metric as follows:

. . . . R is proportional to the curvature radius of space, which then may vary with time.
In the expression for the line element ds2 the terms g14, g24, g34 can be made to vanish for an appropriate choice of the time coordinate, or briefly speaking, time is orthogonal to space. For this second assumption, it seems to me, no physical or philosophical reasons can be given; it serves exclusively the purpose of making the computation simpler . . .
With the [above] assumptions ds2 can be brought into the form

whereby R is a function of x4 and M depends in the general case dependent on all four world coordinates.''

Friedmann then shows that both the Einstein universe and the de Sitter universe are special cases of his more general metric. The implicit assumption in Friedmann's metric is that the curvature is positive.

Two years later, Friedmann (1924) extended his models to negative curvature:

``. . . we can say that the stationary world with constant negative curvature of space is only possible with disappearing or negative density of matter....''

but

``. . . the possibility of non-stationary worlds with constant negative curvature of space and positive density [is given].''

Lemaître (1927) also allowed for both positive and negative curvature.

Tolman (1929) develops his metric with the two following conditions:

``As the first condition to be satisfied by the line element, we shall require it to be compatible for a limited region in space and time with the special theory of relativity
. . . As the second condition for the line element we shall require the possibility of writing it on a form which is spherically symmetrical in the spatial variables, symmetrical with respect to the past and future time, and static with respect to the time. These requirements, as is well known, lead necessarily to the form

where and are functions of r alone. The requirement of spherical symmetry is an obvious one to impose, since otherwise the universe regarded on a large scale would have different properties in different directions. The requirement of symmetry with respect to past and future time means that the large-scale behaviour of the universe is reversible, and the static form of the line element means that by and large the universe is in a steady state.''

Robertson (1929) criticized Friedmann and Tolman on grounds of

``untenable assumptions . . . . instead of making full use of the intrinsic uniformity of such a space [homogeneous and isotropic] as we do here.''

His line element is:

where f is an arbitrary function of time.

While positive and negative curvature had been introduced into expanding relativistic universes by 1924, the Euclidean universe - the only one in Newtonian physics - was not considered until the work of Heckmann, published in July 1931:

``The solutions of the Einstein field equations by Friedmann and Lemaître frequently used in recent times, shall in the following be extended. Especially the very simple proof shall be given that besides the assumption of a spherical (or elliptical) closed space, the assumption of a hyperbolical, in a limiting case even Euclidean space, are of entirely equal standing within the framework of the theory of relativity.''

It may be interesting to note, that de Sitter in a Harvard lecture series given in October 1931 and published in 1932, discussed already models of different values for and the three curvature parameters and included a general sketch for three different cases (For a forerunner of this type of diagram see Sect. 3.1.3)

 Figure 23. Four of Heckmann's nine models of the universe (Heckmann 1932). a ``This case is the richest. For M 0 it displays the de Sitter universe . . .'', b . . . . Except for vanishing matter term where the solutions become straight lines intersecting the -axis under 45°, all solutions rise vertically from the -axis to reach inclination 1 asymptotically . . . at infinity.'' c ``. . . With M 0 all solutions rise vertically from the -axis, their inclination approaching 0 monotonously at infinity . . .'' d ``. . . for M 0 the solutions remain always finite and (dz / d)2 has a zero z1, so that z = z1 is envelope to the general solutions . . .''

Heckmann (1932) published the full sequence of possible models with three types of space curvature and three different signs of the cosmological constant . All models constitute solutions of the differential equation

(1)

``(1) has within the framework of the theory of relativity nine different types of solutions, which originate from the combinations of the three cases with the three also possible . These types of solution shall be discussed in the following. The diagrams included for illustration are computed for the special case of a radiation-filled universe (A = 0). Though this case can only command theoretical interest, it is nevertheless typical for the more general cases A 0; all characteristic features appear in this model which, however, has the great advantage that it can be expressed in elementary functions . . .
One can always give (1) the form

where z is proportional to R, proportional to t and assumes the values +1, 0, -1.
M (z) is defined by

where a, b, and 1 are proportional to the corresponding constants A, B, and in (1). C is, as above, +1 for hyperbolic, 0 for Euclidean, -1 for spherical metric.''

For the coordinates z and compare de Sitter (1930), Sect. 3.1.4. Note that the constant C is the curvature parameter k, except that the latter is positive for Euclidean and negative for hyperbolical space.

Following Tolman and others, Walker (1936) published the now generally used metric which includes the curvature parameter k:

Another consequence of Heckmann's paper was an immediately following publication by Einstein and de Sitter (1932) describing the relativistic Euclidean universe, the Einstein-de Sitter universe, which was to become so important within the framework of the inflationary universe (Sect. 4.), opening with the sentence:

``In a recent note in the Göttinger Nachrichten, Dr. O. Heckmann has pointed out that the non-static solutions of the field equations of the general theory of relativity with constant density do not necessarily imply a positive curvature of three-dimensional space, but that this curvature may also be negative or zero.''