Astron. Nachr. 312 (1991) 5, 307-308

Foppl’s repulsive cosmos and nyperbolic F’riedmannian matter cosmos

H.- J . TREDER, Potsdam-Babelsberg

Einstein-Laboratorium

Received 1991 May 7; accepted 1991 May 26

Fijppl’s repulsive cosmos and the hyperbolic Friedmannian matter cosmos are compared. Both worlds have positive energy integrals. In the repulsive cosmos no cosmic singularity exists.

Key words: cosmology - Fijppl cosmos

A A A subject classification: 161

In a repulsive cosmos with Foppl’s symmetry (Foppl 1897) between positive and negative gravitational masses the factor in eq.(4) of Treder (1991) is k2 = 1 and the equation of expansion becomes

+ = v o ( l - - q - ) 2 fM .

T is the mean distance between the cosmical particles (Treder 1975). i is the velocity of expansion and 2ro its asymptotic value for T -+ 00 ( v g is the maximum off ) ; M means the sum of inertial masses, i.e., M is the absolute value of the gravitational mass in the Foppl cosmos (Foppl 1897, Treder 1991). The iiegative sign of f M means cosmic repulsion. The solution of en.( 1) is:

= vo(t - t ’ ) 2fM 4fM r (1 - F) + T a r c t a n

The ”world radius” r becomes real for r 2 y. For t = t‘ the minimal radius

TI = - 2fy and +(t’) = 0 00

results - in the repulsive cosmos no ”cosmic singularity” exists.

metrics (Einstein 1955, Jordan 1955): The repulsive cosmos is analogous to the hyperbolic Friedmannian matter cosmos with the Robertson-Walker

ds2 = c2dt2 - R2(t) (1 - ;2,4) (dx2 + dY2 + d t 2 )

with r2 = x 2 + y2 + 2’. The Friedmann equation gives

with the mass constant

4* 3

M = -pR3 ( p = matter density).

The positive sign of f M means gravitational attraction. The velocity of light c is the asymptotic value of the expansion velocity R and its minimum. The solution of eq. (3) is (Jordan 1955):

308

= c( t - t‘) . 1 2 f M R (1 + x) ‘I2 - ( 2 f M c - ~ + R)’f2 + R1t2 ( 2 f Mc-2 + R)’f2 - R’t2 log [ -

Astron. Nachr. 312 (1991) 5

(4)

In this model a cosmic singularity exists: for t = t’ it becomes

R(t’) = 0 and d(t’) -+ 0.

The hyperbolic Friedmann cosmos and the Foppl cosmos have positive energy integrals (Treder 1975). In the Foppl cosmos it is

and in the Friedmann cosmos it is

R2 f M c2 _ _ - - - 2 R - 2 ’

Both integrals have for vo = c the same value. The expansion velocity 1: of the repulsive cosmos increases from 1: = c for t = 2’ to 1: = c for t -+ 00. In the attractive Friedmann cosmos R decreases from R -f 00 for t = t’ to R = 0 for t -, 00.

We notice the analogies between the repulsive cosmos with Foppl’s symmetry between positive and negative gravitational masses and Alfvkn’s repulsive cosmos with symmetry between matter and antimatter elaborated by 0. Klein (Alfvkn 1969). In both models a minimal radius of the system exists and no cosmological singularity occurs.

References

AlfvBn, H.: Einstein, A.: Foppl, H.: Jordan, P.: Treder, H.- J.: Treder, H.-J.:

1969, Kosmologie und Antimaterie, Xed. Frankfurt. 1955, The Meaning of Relativity, 5.ed. Princeton.

1955, Schwerkraft und Weltall, 2.ed. Braunschweig. 1897, obe r eine Erweiterung des Gravitationsgesetzes. Miinchen.

1975, Elementare Kosmologie. Berlin. 1991, Astron. Nachr. 312, 229.

Address of the author:

H.- J. Treder Einstein-Laboratorium Rosa-Luxemburg-Str. 17a D-0-1590 Potsdam-Babelsberg Federal Republic of Germany