Finally T represents the matter, the energy-momentum, existing in space-time. Note that M by itself does not have such geometric structure there are no distances between points in M alone, no straight lines, and so forth. The metric tensor defines the metric and geometric structure of the space-time: distances between points A and B, whether points A, B and C are collinear, whether line L is a straight line (geodesic) or curved, and so on. For example some models of GTR have M structurally identical to □ 4, which means that space-time can be coordinatized (all the points labeled) with four-dimensional Cartesian coordinates. The manifold is a collection of points with a local and global topology built-in. ) Notice that each of these objects is four -dimensional, representing not just how things are at a specific time but rather how things are over the entire history of the (model-) universe. (In the latter case we say the space-time is "empty," but it may still have an interesting structure as encoded in g. Like g, T is defined everywhere in the space-time, but unlike g, T may be exactly equal to zero at some or even all points of space-time. Einstein's field equations describe these interactions, and delimit the set of models, or physically possible worlds, corresponding to the theory.Ī model of GTR is usually presented as a triple consisting of a four-dimensional, continuously differentiable manifold M, a metric-field tensor g (representing the geometry of space-time) defined everywhere on the manifold, and a stress-energy tensor T representing the material substances in space-time. GTR describes the dynamical interaction of material substances in space-time with other material substances, as well as their interactions with the variably-curved structure of space-time itself. Subsequently philosophers have explored the status of general covariance, and therefore of the hole argument, in the domain of quantum gravity theories. Regardless of which viewpoint is better supported, it is indisputable that Earman and Norton's hole argument led to a huge resurgence of interest in the interpretation of space-time in GTR, and lies at the core of much of the philosophy of space-time theories published since 1987. Instead most philosophers came to think that the hole argument's indeterminism is merely an artifact of a particular interpretation of the mathematical structure of GTR that we are not logically compelled to accept. But within a few years this view of the argument's significance was widely rejected. Earman and Norton argued that the problem is reason enough to justify rejecting a substantival view of space-time in GTR. A close cousin of Einstein's hole argument was put forth by John Earman and John Norton (1987) as an argument claiming to show that, if one embraces a substantival view of space-time, then in a generally covariant theory such as the GTR, one is committed to an unpleasant form of indeterminism. Seven decades later, after the rediscovery of Einstein's argument by John Stachel and John Norton, history repeated itself. The indeterminism allegedly shown by the hole argument is spurious, and the argument cuts no ice in favor of any particular theory or interpretation of the nature of space-time. From his second point of view the argument rests on a mistaken interpretation of the mathematics of general covariance. The second use of the hole argument came in 1915 when Einstein came to see the argument, taken in its first form, as a mistake. Einstein was not fully satisfied with that theory, in part because he believed that general covariance was necessary if a theory were to capture a fully general relativity of motion, and so the hole argument served to help Einstein reconcile himself (temporarily and only partially) to the Entwurf theory. First before the discovery of his final field equations for the General Theory of Relativity (GTR), the argument was put forward as a justification for accepting non -generally covariant field equations, namely those of the 1913 Einstein-Grossman Entwurf theory. Einstein put the argument to two different uses. The point of the argument may be put as follows: If a physical theory's equations are generally covariant (that is, invariant under a wide group of continuous coordinate transformations) then the theory is in a certain specific sense indeterministic. The original "hole argument" ( lochbetrachtung ) was created by Albert Einstein.
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