By Erwin Schrödinger
In line with repeated requests this vintage booklet on space-time constitution through Professor Erwin Schrödinger is now on hand within the Cambridge technology Classics sequence. First released in 1950, and reprinted in 1954 and 1960, this lucid and profound exposition of Einstein's 1915 thought of gravitation nonetheless offers necessary studying for college students and learn staff within the box.
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Example text
Yet they are not sufficient to establish an exhaustive tensor analysis in our continuum. Not even such a simple question as this has any meaning: When is a vector field Ak to be considered as constant throughout a certain region ? For the vanishing of all the derivatives Aki is (as we have seen) not a frame-independent property, because Ak i is not a tensor. The geometrical concept for removing this difficulty will be introduced in Part II. Before doing so, let us dwell more closely on the interesting fact alluded to just above, that the analytical means developed up to here suffice to establish the principal statements of Maxwell's theory, which may duly be called the spiritual ancestor of all field theories that were to follow it.
1) simply state that h has a vanishing invariant derivative. Now let us do the same thing with four linearly independent vectors at P, which we label by a subscript a, thus hva> with a = 1,2,3,4. We might just as well use four different letters, as hvy gv,fv>jv to indicate our four vectors, instead of that subscript, which must not be confused with a tensor-index. The numbering is more convenient, because it is convenient for the purpose of the present investigation to extend the summation convention to this INTEGRABILITY 45 subscript, when it appears twice (though it will always be written as a subscript, being a mere label).
In the most general case (non-symmetric affinity) the 5-tensor has only the one obvious symmetry of being skew in the last two subscripts, A and fi in our present notation. It is easily seen that it then has 4 x 4 x 6 = 96 independent components. 12) that B has in addition the cyclical symmetry To count the independent components in this case, first give v a fixed value. Then take for the subscripts a definite triplet a 4= A 4= /*. Of the three independent components, which the plain skewness in the last couple leaves in this case, one can be expressed by the other two in virtue of the afore-standing relation.