The circumstance that there is no objective rational division of the four-dimensional continuum into a three-dimensional space and a one-dimensional time continuum indicates that the laws of nature will assume a form which is logically most satisfactory when expressed as laws in the four-dimensional space-time continuum. Upon this depends the great advance in method which the theory of relativity owes to Minkowksi.
—Albert Einstein (The Meaning of Relativity)Light-cone geometry: the key to special relativity
WE HAVE SEEN how an index notation is strikingly helpful in the development of physical formulae for flat three-dimensional space. We found it convenient to work with a fixed Cartesian coordinate system, expressing the components of vectors and tensors with respect to that system. We know, nevertheless, as a matter of principle, that the general conclusions we draw are independent of the particular coordinatization chosen for the underlying space.
We now propose to formulate special relativity in essentially the same spirit. We shall regard space-time as a flat four-dimensional continuum with coordinates xa (a = 0,1,2,3). The points of space-time are called ‘events’, and we are interested in the relations of events to one another. Our purpose here is two-fold: first, to review some aspects of special relativity pertinent to that which follows later; and second, to develop further a number of index-calculus tools which are very useful in general relativity as well as special relativity.
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