## Relativity, Cosmology, and Time

### Static Universe and Cosmic Acceleration (2021 Edition)

This is the model of the universe that led Einstein to proclaim "the biggest blunder in my life", when most of the scientific community
had recognized the expansion of the universe by late 1920s. He can be excused for making this mistake because astronomical observation in earlier time indicated a homogeneous distribution of objects in the sky and the view did not seem to change over "long" time. It induced him to assume a constant density in Eq.(18a). Simple mathematics as well as from solution of the equation§ shows that such model describes an expanding universe. In order to avoid this "unwanted" predicament, he introduced a repulsive term = 4G/c2 to make d2R/dt2 = 0. The additional term would make dR/dt = 0 as well if
k = 4GR2/c2 or k = R2. Thus, R = R0 = constant (see mathematical detail). However, this universe is unstable. A small perturbation would induce collapse or expansion forever. With the discovery of

#### Figure 10l1 Einstein and His Cosmic Blunder [view large image]

cosmic acceleration, it is fashionable again to re-introduce the cosmological constant back into the model universe. The crucial difference is that we now know the density is not a constant; it varies with time as the universe expands. The two terms on the right-hand side of Eq.(20f) equal to each other only momentarily about 8 billion years after the Big Bang.
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For demonstration purpose, the mathematical derivation starts from the matter-only Friedmann equation (see the general form of GR):

The above formulation can be carried over to describe cosmic acceleration by dark energy represented by the cosmological constant term,

#### Figure 10l3 Hubble Tension [view large image]

See Figure 10l2 for the Crossover, and Figure 10l3 for Hubble Tension.

Figure 10l1 was taken when Einstein was presenting a lecture at the Mount Wilson Observatory in 1931. The big question mark on the blackboard (after the short form of the equation for the static universe) seems to indicate that he was aware of the blunder already. Actually, the Hubble Law for cosmic expansion had been discovered at the same location in 1929. This picture has become the favourite message board in the internet with various kinds of text scribbled over the original equation.

§N.B. The solution for Eq.(18a) with = constant is:
for k > 0, R = (R0/2)(eHt + e-Ht) = R0cosh(Ht) (R0/2)eHt (as t ), the minimum size Rmin = R0;
for k < 0, R = (R0/2)(eHt - e-Ht) = R0sinh(Ht) (R0/2)eHt (as t ), the minimum size Rmin = 0;
for k = 0, R = R0eHt, there is no minimum size;
where H = (8G/3)1/2, and R0 = (3|k|c2/8G)1/2 (for k 0), otherwise R0 = R(t = 0) (for k = 0).

The formula for k = 0 also describe the cosmic expansion in the steady state universe. The Hubble Law can be readily derived as :
dR/dt = HR,
where H can be considered as the "Hubble constant" for the steady state universe.
The deceleration parameter is defined by:
q = - [R(d2R/dt2)/(dR/dt)2],
which becomes q = -1 for the steady state universe, i.e., it just reiterates the accelerating characteristic of this model universe. The steady state theory states that not only are there no privileged locations in space, there are no privileged moments in time as well. Thus, the global properties of the universe, such as density and Hubble constant remain constant with time. The theory fell out of favor when observational evidence strongly suggested that the global properties do change with time as indicated by the discoveries of the Cosmic Microwave Background and the quasi-stellar objects.

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