The de Sitter Space-Time and the Riemannian Space H^4

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Two Four-Dimensional Surfaces of Hyperboloids Embedded within the Five-Dimensional Minkowski Space-Time

The de Sitter space-time is of far higher than just historical interest. In fact, the inflationary models of the universe, the sole energy content of which is located in the vacuum, are based on de Sitter’s form of the Robertson-Walker metric. Another example is Hoyle’s steady-state universe.

A very interesting feature of the de Sitter space-time is that it can be mathematically described as a four-dimensional surface embedded within the five-dimensional Minkowski space-time. Owing to its high symmetry—it is, in fact, the equivalent to a sphere in Euclidean space—there are a plethora of possible ways of defining four independent co-ordinates in which the metrical tensor of the de Sitter space-time can be expressed. The freedom of choice reaches so far that the de Sitter metric can be represented by such metrical coefficients as are independent of time and so make the universe appear static, as well as by such as are of the Robertson-Walker form, the scale factor of which exhibits an exponential dependence on time, so that the universe seems to expand inflationarily.

All of these, contradictory as they may seem to be at first sight, are, in fact, results of possible parametrisations of the de Sitter space-time, and there is no physical parameter that requires the choice of one particular co-ordinate system among the mighty manifold of others. This is the most astounding feature of this space-time: the aspect that the universe reveals to a fictitious observer, static or inflationarily expanding, with or without a Big Bang, solely depends on the way in which he set his clocks and adjusted his yardsticks.

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