## Riemanns Theory of Gravity

Riemann attempted to extend his three-dimensional geometry of curved spaces into a unified theory of an ether of gravity, electricity and magnetism. As part of this effort, he developed a specific Ansatz of a gravitational ether, which was later partly realized in Einstein's general relativity. Unlike Einstein, however, Riemann intended to found a theory where, in domains with nonvanishing matter density, there is an absorption of the flux of ether by matter causing a gravitational interaction of masses filling such regions. In our paper, Riemann's Ansatz of a gravitational theory is reconstructed, and it is shown that, in a four-dimensional version, it can be formulated as islands of non-Riemannian geometry lying in the sea of Riemann-Einstein geometry.

Riemann considered his formulation of differential geometry (Riemann [8]) as a first step towards a unified geometrical theory of an "ether of gravity, electricity and magnetism" [6]. He identified the ether with the physical properties of the structured three-dimensional spatial manifold. One finds the same idea of a gravitational ether later in Einstein's 1920 talk in Leiden (cf. also Einstein [3]), now of course referred to as a four-dimensional curved space-time. This idea was subsequently taken over from Riemann and Einstein by Weyl [11].

As to the gravitational interaction, Riemann formulated an explicit Ansatz, however, in which he expresses his belief that this ether would prove to be the unified ether of gravity, electricity, magnetism and light. He started from the observation that, in contrast to electricity and magnetism, the gravitational flow <x -3kO has sinks but no sources. Indeed, due to the fact that there are only positive gravitational charges, i.e., positive masses, one always has AO> 0, such that the flow is negative. Anticipating the modern conception, according to which there exists a connection between space-time mirror symmetry and charge conjugation Q ^ -Q, he concluded that, due to the missing negative masses and thus the missing charge conjugation, the gravitational law cannot be invariant under time inversion. Therefore, he assumed the gravitational interaction to be a dissipative ether flow. In regions with ponderable matter, the differential equation for the gravitational potential O should show a time-dependence of the ether describing this dissipation. From this point of view, it was logical to postulate in matter-dominated regions a differential equation containing a term with the first time derivative d/dt in order to exclude the invariance of this equation under time inversion. Accordingly, in regions filled with ponderable matter, the ether and thus the three-dimensional metric show a secular variation.

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In more detail, Riemann [7] assumed the ether to be a perfect fluid satisfying the equation of continuity where the ether density is given by the square-root of the determinant g of the metric gk, and the velocity field u corresponds with the potential flow x -dk O of the ether,

Here K is a constant which will be determined below (cf. (9a)).

In empty space domains, Riemann's ether is stationary and, as a consequence of the definition of its density, incompressible, i .e.,

In matter-dominated regions with a = -\[ga0 > 0 the stationarity of the ether flow, and thus its incompressibility, is destroyed because the matter represents sinks of the ether flow, d (pu') = kpca0 = 2c/dgl2a0 = 2kc^a , (3)

Here the light velocity c is assumed to be the dissipation velocity, i.e., the velocity of the ether absorption; accordingly the absorption coefficient k has the dimension g1 cm2. Due to the validity of the continuity equation demanded by Riemann, d_

one then obtains in matter regions, where a >0, for the density p = 2/dyfg the relation dp = 2^ g12 =-2^kcgl'2aa. (5)

dt dt

The solution of this equation, p = 2^exp(-kca0t), g x exp(-kc<70t), (6)

shows that the ether density continuously decreases.

As a consequence of the identification of the velocity of the ether flow with the gradient of the gravitational potential assumed in (1), one obtains in vacuum regions from (2) the Laplace equation

and in matter regions the Poisson equation

(here G0 denotes Newton's gravitational constant and G the effective gravitational coupling constant).

The comparison of (8) and (3) shows that and thus

so that the Poisson equation (8) can be written as

4nG0 0 dt 2 dt

Thus, in Riemann's theory, one finds an absorption in matter regions that, as equations (6) and (8) show, can also be interpreted as a variation of the effective gravitational constant. In cosmology, this means that for a universe with a finite average mass density a0 > 0 this provides a secular decrease of the gravitational interaction as postulated by Dirac [5]. (It will be shown below that the absorption coefficient is that of Bottlinger and Majorana (see the paper by v. Borzeszkowski and Treder in this volume)).

In Riemann's theory, the gravitational attraction of N masses is a "hydro-dynamic action-at-a-distance force" which was introduced by C. A. Bjerknes [1] and V. Bjerknes [2] (where C. AA. Bjerknes [1] proposed however another theoretical Ansatz than Riemann's). According to V. Bjerknes, it is due to a "kinetic buoyancy" what can be seen by discussing Euler's equation which, beside the continuity equation, forms the basis of Riemann's theory.

Euler's equation written in a curved three-dimensional space reads du' =d u' + u'kuk = 0, (11)

dt dt where the semicolon denotes the covariant derivation with respect to the Christoffel symbols of this space. Together with the continuity equation (6), this leads to Bernoulli's equation which provides a condition for the continuity of the energy flow (1/2) p uu = j yfg uu of the ether. With Riemann's Ansatz (5), one then has jd, (gi'2u v ) = jg

0 dt

With u « c and du2 j dt « dc2/ dt , (12) defines a dissipation of "the kinetic energy of the ether" caused by completely inelastic impacts of the (continuously distributed) infinitesimal ether particles on the ponderable matter (u « c is the velocity of the ether particles, p = 2j the mass density of the ether flow,

C. A. Bjerknes was a disciple of Riemann's predecessor , P. L. Dirichlet, in Goettingen. Dirichlet influenced Riemann's and Bjorknes' investigations on the motion of bodies in ideal fluids (cf. Riemann

kM the cross section of the mass of the heavy ponderable bodies

Using V. Bjerknes' method of kinetic buoyancy, for (G0 / r )\ad3 x << c and r = 0, one can derive from the above relations the following hydrody-namic action-at-a-distance force acting between two masses Mj and M2,

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