A pedagogical cosmology illustrates general relativity concepts, without requiring general or special relativity. Topics examined are the existence of a global time scale, proper vs coordinate variables, the variation of light speed in an expanding universe, the look-back paradox, the horizon, the red shift, the age of the universe, and the dynamics of the universe. An Appendix is devoted to space and time in general relativity, but can be skipped by readers unfamiliar with general relativity.

## REFERENCES

1.

The three spatial dimensional model for constant expansion is called the Milne model. See

Abraham N.

Silverman

, “Resolution of a cosmological paradox using concepts from general relativity

,” Am. J. Phys.

54

, 1092

–1096

(1986

), especially p. 1095.2.

See

W. M.

Stuckey

, “Kinematics between comoving photon exchangers in a closed matter-dominated universe

,” Am. J. Phys.

60

, 554

–560

(1992

), where Stuckey emphasizes and clearly illustrates the concept of proper distance.3.

One arrives at Eq. (2.1) by noting that, for a dot galaxy traveling at constant speed $V,$ $x=Vt$ for any time

*t*and, at time $t0$ in particular, $x0=Vt0.$ Eliminating $V$ between these two distance equations gives Eq. (2.1).4.

The one spacial dimension FRW metric is given by $(ds)2=\u2212c2(dt)2+[a(t)/a0]2(dxc)2,$ where $a(t)/a0=t/t0$ is used so far in this article, although $a(t)/a0$ will shortly be replaced by the actual FRW $(t/t0)2/3.$ Hence light travel is given by $dxc/dt=\xb1c/(a(t)/a0).$ Substituting $xc=x/(a(t)/a0)$ from Eq. (2.1) gives $dx/dt=\xb1c+x/t$ for light, which is Eq. (2.3), arrived at in this article in a different, non-GR way.

5.

Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler,

*Gravitation*(W. H. Freeman, New York, 1973), pp. 721, 722.6.

Steven Weinberg,

*Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity*(Wiley, New York, 1972), p. 483, Eq. (15.3.14);Joseph Silk,

*The Big Bang*(W. H. Freeman, San Francisco, 1980), p. 335.7.

The grid speed that is added to

*c*in Eq. (2.5) is found by differentiating the logarithm of Eq. (2.4) to give $(dx/dt)proper,\u200agrid\u200arelative\u200ato\u200aruler=2x/3t.$8.

For emphasis, the comoving time is not the time on clocks on the rubber band, but merely a useful mathematical time to accompany the comoving distance.

9.

See

Robert C.

Fletcher

, “Light exchange in an expanding universe in fixed coordinates

,” Am. J. Phys.

62

, 648

–656

(1994

). Fletcher presents an elegant and detailed calculation for this paradox for a matter-dominated universe of three spatial dimensions.10.

Jayant V.

Narlikar

, “Spectral shifts in general relativity

,” Am. J. Phys.

62

, 903

–907

(1994

), especially Eq. (1).11.

Robert C.

Fletcher

, “Light exchange in an expanding universe in fixed coordinates

,” Am. J. Phys.

62

, 648

–656

(1994

).12.

The decrease in the overall rate of grid expansion as time proceeds hastens the slowdown of light effect.

13.

Steven Weinberg,

*Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity*(Wiley, New York, 1972), p. 489.14.

Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler,

*Gravitation*(W. H. Freeman, New York, 1973), p. 709, Eq. (10).15.

Steven Weinberg,

*Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity*(Wiley, New York, 1972), p. 483, Eq. 15.3.15; Silk,*ibid.*, p. 335.16.

G. Börner,

*The Early Universe*(Springer-Verlag, Berlin, 1988), p. 7.17.

In GR, Eqs. (4.134.144.15) arise from a solution of the GR tensor field equations, where the only mass-energy for the mass-energy tensor is that of the mass of the galaxies. Gravitational potential energy is not a part of this tensor setup; it is incorporated in the metric one uses for GR. The limitations of Eqs. (4.124.134.144.15) occur in GR as the representation of a flat RW universe, often chosen by assigning a value of $k=0$ to the more general structure of the RW interval, for all universe densities, as we have done in this paper. See Stephen Weinberg,

$(ds)2=\u2212c2(dt)2+[a(t)/a(t0)]2\u22c5{(dr)2/1\u2212k\u22c5r2+r2[(d\theta )2+sin2\u200a\theta (d\phi )2]}$

*Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity*(Wiley, New York, 1972), p. 412, Eq. (14.2.1).18.

The RW universe with $k=\u22121$ corresponds to our model universe with positive total mechanical energy, while the RW universe with $k=+1$ corresponds to our model universe of negative total mechanical energy.

19.

Bernard F. Schultz,

*A First Course in General Relativity*(Cambridge University Press, Cambridge, 1985), p. 22, Fig. 1.14.20.

See Ronald J. Adler, Maurice J. Bazin, and Menahem Schiffer,

*Introduction to General Relativity*, 2nd ed. (McGraw-Hill, New York, 1975), pp. 122–124.21.

Constant

*t*must be used in the integration because $dt=0,$ according to the definition of proper distance.22.

Robert C.

Fletcher

, “Light exchange in an expanding universe in fixed coordinates

,” Am. J. Phys.

62

, 648

–656

(1994

).23.

Ronald

Gautreau

, “Curvature coordinates in cosmology

,” Phys. Rev. D

29

(2

), 186

–197

(1984

);Ronald

Gautreau

, “Newton’s absolute time and space in general relativity

,” Am. J. Phys.

68

, 350

–366

(2000

).
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2001

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