The Einstein tensorGij is symmetric, divergence free, and a concomitant of the metric tensorgab together with its first two derivatives. In this paper all tensors of valency two with these properties are displayed explicitly. The number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors.

The Einstein Tensor and Its Generalizations

The Classical Field Theories

Classical mechanics was first envisaged by Newton, formed into a powerful tool by Euler, and brought to perfection by Lagrange and Laplace. It has served as the paradigm of science ever since. Even the great revolutions of 19th century phys icsnamely, the FaradayMaxwell electro-magnetic theory and the kinetic t h e o r y w e r e viewed as further support for the complete adequacy of the mechanistic world view. The physicist at the end of the 19th century had a coherent conceptual scheme which, in principle at least, answered all his questions about the world. The only work left to be done was the computing of the next decimal. This consensus began to unravel at the beginning of the 20th century. The work of Planck, Einstein, and Bohr simply could not be made to fit. The series of ad hoc moves by Bohr, Eherenfest, et al., now called the old quantum theory, was viewed by all as, at best, a stopgap. In the period 1925-27 a new synthesis was formed by Heisenberg, Schr6dinger, Dirac and others. This new synthesis was so successful that even today, fifty years later, physicists still teach quantum mechanics as it was formulated by these men. Nevertheless, two foundational tasks remained: that of providing a rigorous mathematical formulation of the theory, and that of providing a systematic comparison with classical mechanics so that the full ramifications of the quantum revolution could be clearly revealed. These tasks are, of course, related, and a possible fringe benefit of the second task might be the pointing of the way 'beyond quantum theory'. These tasks were taken up by von Neumann as a consequence of a seminar on the foundations of quantum mechanics conducted by Hilbert in the fall of 1926. In papers published in 1927 and in his book, The Mathemat ical Foundations of Quantum Mechanics, von Neumann provided the first completely rigorous

The mathematical foundations of quantum mechanics

It is frequently explicitly or implicitly assumed that if a variational principle is invariant under some symmetry groupG, then to test whether a symmetric field configuration ϕ is an extremal, it suffices to check the vanishing of the first variation of the action corresponding to variations ϕ + δϕ that are also symmetric. We show by example that this is not valid in complete generality (and in certain cases its meaning may not even be clear), and on the other hand prove some theorems which validate its use under fairly general circumstances (in particular ifG is a group of Riemannian isometries, or if it is compact, or with some restrictions if it is semi-simple).

/pdf/the-principle-of-symmetric-criticality-ph4ixz8kp6.pdf

The principle of symmetric criticality

All tensors of contravariant valency two, which are divergence free on one index and which are concomitants of the metric tensor, together with its first two derivatives, are constructed in the four‐dimensional case. The Einstein and metric tensors are the only possibilities.

The four-dimensionality of space and the einstein tensor

Space, time, matter /

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