Traversable wormholes necessarily require violations of the averaged null energy condition; this being the definition of "exotic matter". However, the theorems which guarantee the energy condition violation are remarkably silent when it comes to making quantitative statements regarding the "total amount" of energy condition violating matter in the spacetime. We develop a suitable measure for quantifying this notion, and demonstrate the existence of spacetime geometries containing traversable wormholes that are supported by arbitrarily small quantities of "exotic matter".
Mn(I)-PNP pincer catalyzed sequential one-pot homogeneous CO 2 hydrogenation to CH 3 OH by molecular H 2 is demonstrated. The hydrogenation consists of two partsN-formylation of an amine utilizing CO 2 and H 2 , and subsequent formamide reduction to CH 3 OH, regenerating the amine in the process. A reported air-stable and welldefined Mn-PNP pincer complex was found active for the catalysis of both steps. CH 3 OH yields up to 84% and 71% (w.r.t amine) were obtained, when benzylamine and morpholine were used as amines, respectively; and a TON of up to 36 was observed. In our opinion, this study represents an important development in the nascent field of base-metal-catalyzed homogeneous CO 2 hydrogenation to CH 3 OH.
We present a brief review on the Raychaudhuri equations. Beginning with a summary of the essential features of the original article by Raychaudhuri and subsequent work of numerous authors, we move on to a discussion of the equations in the context of alternate non-Riemannian spacetimes as well as other theories of gravity, with a special mention on the equations in spacetimes with torsion (Einstein-Cartan-Sciama-Kibble theory). Finally, we give an overview of some recent applications of these equations in general relativity, quantum field theory, string theory and the theory of relativisitic membranes. We conclude with a summary and provide our own perspectives on directions of future research.
The beginningsAbout half a century ago, general relativity (GR) was young (just forty years old!), and even the understanding of the simplest solution, the Schwarzschild, was incomplete. Cosmology was virtually in its infancy, despite the fact that the FriedmannLemaitre-Robertson-Walker (FLRW) solutions had been around for quite a while. The question about the then-known exact solutions of GR, which worried the serious relativist quite a bit, concerned their singular nature. Both the Schwarzschild and the cosmological solutions were singular. It is well-known that the creator of GR, Einstein himself, was quite worried about the appearance of singularities in his theory. Was there a way out? Was it correct to believe in a theory which had singular solutions? Were singularities inevitable in GR?It was during these days in the early 1950s, Raychaudhuri began examining some of these questions in GR. One of his early works during this era involved the construction of a non-static solution of the Einstein equations for a cluster of radially moving particles in an otherwise empty space [1]. A year before, he had also written an article related to condensations in an expanding Universe [2] 49
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