The authors show that the recently proposed pure-connection action for general relativity with a cosmological constant by Capovilla, Dell and Jacobson (1989) correctly yields the usual constraints of general relativity as given by Ashtekar (1986). They also point out alternative possibilities for the Capovilla-Dell-Jacobson action.

It is well known that vacuum equation of arbitrary Lovelock order for static spacetime ultimately reduces to a single algebraic equation, we report the results in 1 showing that the same continues to hold true for pure Lovelock gravity of arbitrary order N for topology S (n) × S (n) . Thermodynamical stability of black hole discerns between odd and even N , and consequently between negative and positive Λ and it favors the former while rejecting the latter.

The application of the uncertainty relation to position and velocity of a source point represented by a general asymptotically flat static metric leads to fluctuations of the metric and the light cone structure. The fluctations in the coordinate photon velocity are given by the formula [Formula: see text] where k is the surface gravity and m is the mass of the source point.

We argue that like velocity of light, Λ is a constant of spacetime, and these are the only two most fundamental constants of spacetime structure. Like gravitational field energy, vacuum energy must gravitate but not through a stress tensor instead more subtly in line with the former by enlarging spacetime framework. Unfortunately this enlargement would not become visible until there comes about a quantum theory of spacetime or gravity. The possible ways could be that it would be automatically taken care of when we quantize geometry, the left hand side of Einstein equation, as it happened for self interaction in GR or it gravitates through higher dimension. The important message that is that Λ has therefore nothing to do with vacuum energy and hence is free to have any value as determined by accelerating expansion of the Universe. The incorrigibly embarrassing number 10 120 is therefore nothing but the statement that in terms of the Planck area, the Universe measures as much! First and formost we wish to argue that Λ is a true constant of spacetime structure 1 on the same footing as velocity of light. Note that it is homogeneity of spacetime characterizing absence of all forces and dynamics (i.e. maximal symmetry) that demands both c and Λ as constants of spacetime structure. They are the only two constants that enjoy this unique distinction and hence are the two most fundamental constants of Nature. No othe constant can claim this degree of fundamentalness.It is however true that stress tensor generated by quantum fluctuations, so called vacuum energy, relative to flat spacetime has the same form as Λ in Einstein's gravitational equation. It then gets slated against the Planck length giving rise to the number 10 120 indicating monumental mismatch between the two. We would like to point out that it all hinges on the question, how does vacuum energy gravitate through a stress tensor as is generally done or by some other subtle way? We should however note an important point of difference between vacuum energy and other matter fields that it is created by the latter. Hence it does not its own independent existence. It is therefore a scondary source which is produced by the primary source, matter. Would it therefore be in principle appropriate to make it sit on the same footing alongside matter on the right of gravitational equation?Let us begin at the very beginning. In the classical mechanics a force-free state of space and time is characterized by space being homogeneous and isotropic and time being homogeneous. As space is homogeneous, we can freely interchange x and y. Since time is also homogeneous which means both space and time are homogeneous, it should equally well permit interchange of x and t. But their dimensions don't match, one is measured in cm while the other in sec. Homogeneity is a general property which should always be respected and hence we need to match dimension which could only be done by demanding existence of an invariant velocity, c, then

The authors establish two general results. Firstly, every solution with the cosmological constant and self-dual Weyl tensor of Einstein's equation comes from the Samuel (1988) or Ashtekar-Renteln (1987) ansatz; and secondly, self-duality for spherically symmetric spacetimes implies conformal flatness.