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1982 The physical Universe. University Science Books, Mill Valley, CA. 2 Relativity The foundations of modern cosmology were laid during the second and third decade of the 20th century: on the theoretical side by Einstein’s theory of general relativity, which represented a deep revision of current concepts; and on the observational side by Hubble’s discovery of the cosmic expansion, which ruled out a static Universe and set the primary requirement on theory. Space and time are not invariants under Lorentz transformations, their values being diﬀerent to observers in diﬀerent inertial frames.

Another convenient comoving coordinate is χ, deﬁned by integrating over dχ = √ dσ . 35) where Sk (χ) ≡ σ and S1 (χ) = sin χ, S0 (χ) = χ, S−1 (χ) = sinh χ. 35) interchangeably since both oﬀer advantages. During the course of gravitational research many other metrics with diﬀerent mathematical properties have been studied, but none appears to describe the Universe well. Since they are not needed, we shall not discuss them here. Let us brieﬂy digress to deﬁne what is sometimes called cosmic time. In an expanding universe the galaxies are all moving away from each other (let us ignore peculiar velocities) with clocks running at diﬀerent local time, but from our vantage point we would like to have a time value applicable to all of them.

577, 853. [13] Ferreras, I. et al. 2001 Mon. Not. R. Astron. Soc. 327, L47. [14] Shu, F. H. 1982 The physical Universe. University Science Books, Mill Valley, CA. 2 Relativity The foundations of modern cosmology were laid during the second and third decade of the 20th century: on the theoretical side by Einstein’s theory of general relativity, which represented a deep revision of current concepts; and on the observational side by Hubble’s discovery of the cosmic expansion, which ruled out a static Universe and set the primary requirement on theory.