A general line element and a general metric tensor are defined as functions of two parameters $\alpha$ and $\alpha'$. The related Einstein's field equations of a gravitational potential field in a vacuum, including parameter $\Lambda$, have been derived. The parameters $\alpha$ and $\alpha'$ are identified in a gravitational field by the solution of the Einstein's field equations. Parallel with this, it has been find out that the so‐called cosmological constant $\Lambda$, is not really constant, but a function of gravitational radius, $\Lambda = f(r)$. This discovery is very important, among the others, for cosmology. One of the consequences is the new form of the acceleration equation of the universe motion that can be attractive (negative) or repulsive (positive). According to the observations, the repulsive acceleration gives rise to accelerating expansion of the universe at the present time. The obtained solution of the diagonal line element can be applied in a very strong gravitational field. Besides, this solution gives the Ricci scalar equal to zero, $R=0$. This is in an agreement with the current observation that our universe is flat.

from Novakovic B.M., Novakovic D.B. & Novakovic A.B. (2004). The Cosmological Constant $\Lambda$ is not Really Constant but the Function of a Gravitational Radius, AIP Conference Proceedings, 718 133. DOI: 10.1063/1.1787318

The magnitude-redshift relation is one of the tools for a direct observational approach to cosmology. The discovery of high redshift Type Ia supernovae (SNIa) and their use as "standard candles" has resurrected interest in this approach. Recently collected data have been used to address the problem of measuring the cosmological parameters of the universe. Analysed in the framework of homogeneous models, they have yielded, as a primary result, a strictly positive cosmological constant. However, a straight reading of the published measurements, conducted with no a priori idea of which model would best describe our universe at least up to redshifts $z \sim 1$, does not exclude the possibility of ruling out the Cosmological Principle - and cosmological constant - hypotheses. It is therefore shown here how the large scale homogeneity of this part of the universe can be tested on our past light cone, using the magnitude-redshift relation, provided sufficiently accurate data from sources at redshifts approaching $z=1$ would be available. An exemple of an inhomogeneous model with zero cosmological constant reproducing the current observations is given. The presently published SNIa data can thus be interpreted as implying either a strictly positive cosmological constant in a homogeneous universe or large scale inhomogeneity with no constraint on $\Lambda$. An increase in the number and measurement accuracy of the candidate "standard candles" at very high redshift is therefore urgently needed, for progress in both fundamental issues of the Cosmological Principle and cosmological constant.

from Célérier M.N. (2000). Do we really see a cosmological constant in the supernovae data?, Astron. Astrophys., 353 63-71. arXiv: astro-ph/9907206v4

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