Caption: A sphere with great circle paths on the spherical surface between points and antipodal points (i.e., points half a great circle away).

Einstein universe (1917) discovered by Albert Einstein (1879--1955) in 1917 (but only later called the Einstein universe) is the 3-dimensional analogue of 2-dimensional spherical surface. It is the 3-dimensional surface of a 4-dimensional hypersphere: i.e., it is finite, but unbounded, hyperspherical space (Bo-98; No-520; CL-28--29,159; O'Raifeartaigh 2019, p. 16). The Einstein universe has a very simple cosmic curvature.

General relativity (1915) gives NO meaning to off the surface of the the 3-dimensional surface.

Features of the Einstein universe:

1. The Einstein universe in original conception was a stars, NOT galaxies.

2. The Einstein universe is a static cosmological model. The stars have small peculiar velocities, but there is NO overall scaling up or down of distances.

3. If you move along geodesic (the shortest distance between any 2 close points and the analogue to great circle paths) you eventually reach an antipodal point and if you continue, you return to where you started.

   Light rays always follow geodesics.

   If light from the back of your head is unimpeded and has enough time, you will see the back of your head.

   Yours truly CANNOT find any statement from Einstein discussing seeing the back of his head.

4. If you move sufficiently small distances in the Einstein universe, the curved space is approximately a 3-dimensional Euclidean space (i.e., flat space).

5. Einstein universe obtained the static condition for the Einstein universe by introducing the cosmological constant (AKA Lambda, Λ) into the Einstein field equations. The cosmological constant was the simplest of all modifications to the original Einstein field equations and it only has a significant effect on cosmological scales. It is a sort of anti-gravity that opposes the gravity that arises from mass-energy in general relativity. But we do NOT call the cosmological constant anti-gravity since it is NOT what people ordinarly mean by anti-gravity.

   The cosmological constant has to be adjusted to give the static condition and Einstein could only give a very approximate value given the data available in in 1917 or for decades thereafter.

6. In the development of the Einstein universe, Einstein was aiming at a static cosmological model, but, in fact, he could find NO cosmological model static or otherwise without the cosmological constant though there are non-static cosmological models to be found.

   It has to be understood that Einstein's development was a pioneering effort and he did NOT have access to techniques that were developed later.

   In fact, he failed to find the Friedmann equations which Alexander Friedmann (1888--1925) would derive from general relativity in the early 1920s.

   The Friedmann equation allows a simple derivation of all simple cosmological models allowed by general relativity including the Einstein universe. We call such cosmological models the Friedmann equation (FE) models.

7. The Friedmann equation shows that the Einstein universe is obviously unstable. General global perturbations in density will start it expanding or contracting (see Cormac O'Raifeartaigh et al., Einstein's 1917 Static Model of the Universe: A Centennial Review, 2017, p. 40--41).

   Now, in fact, global perturbations in density are NOT realistic. However, it is intuitively obvious that local density perturbations will cause regions of expansion and contraction.

   Thus, the Einstein universe could only be approximately static for some period of time.

   Without the Friedmann equation, the instability of the Einstein universe was NOT obvious and Einstein did NOT learn of it until well into the 1920s.