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We are presenting a new class of well-behaved solutions in embedding class-I. We proceed our calculations by assuming a new type of g r r metric potential and solved for the other metric g t t using Karmarkar condition. The necessary condition that any solutions of Einstein’s field equations to be class-I is to satisfy Karmarkar condition and sufficient condition is to satisfy Pandey–Sharma condition i.e. R 2323 ≠ 0 . The solution also satisfies strong energy condition, null energy condition, dominant energy condition and weak energy condition. The obtained compactness parameter is within Buchdahl limit i.e. 2 M / R ≤ 8 / 9 . The solution also satisfies the causality condition and can represent stable stellar fluid system as the adiabatic index Γ > 4 / 3 and the stability factor holds − 1 ≤ v t 2 − v r 2 ≤ 0 good. Finally, we have tuned our solution for two compact stars PSR J1614-2230 and 4U1608-52 which are well-behaved in all respects.

The semiclassical approximation to the functional integral for four-dimensional Euclidean gauge theories is discussed in detail for general stationary points of the action. It is shown how to take the limit from a compact space to flat space, and the zero modes corresponding to global gauge transformations are carefully discussed. The results are specialised to general self-dual multi-instanton gauge fields given by the general construction of Atiyah et al. It is shown how the normalization matrix of the zero modes can be determined and the complete expression for the functional measure is given for the two instanton case. This is shown to factorise for well-separated instantons. Some technical matters are discussed in an appendix and a resume of results for multi-instanton functional determinants is included.