By Harrison D.M.

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At that stage, we shall be able to see much more explicitly how gravity arises from geometry. 1 Spacetime as a Differentiable Manifold Our aim is to construct a mathematical model of space and time that involves as few assumptions as possible, and to be explicitly aware of the assumptions we do make. In particular, we have seen that the theories of relativity call into question the meanings we attach to distances and time intervals, and we need to be clear about these. The mathematical structure that has proved to be a suitable starting point, at least for a non-quantum-mechanical model of space and time, is called a differentiable manifold.

44) The geometrical properties of the metric would be rather peculiar if this were not so, and the existence of the inverse is sometimes included as part of the definition of a metric. So long as the inverse metric does exist, we can say that it raises the index of a one-form to produce a vector: ωµ = g µν ων . 45) In fact, any index of any tensor can be raised or lowered in this way. Since gµν is symmetric, it does not matter which of its indices is contracted. 38 Geometry Now that we have a metric tensor at our disposal, it is clearly possible in practice to regard vectors and one-forms as different versions of the same thing— hence the terms contravariant and covariant vector.

We encounter no difficulty in drawing, say, the street plan of a city on a flat sheet of paper using Cartesian coordinates, but we should obviously be misled if we assumed that this map could be extended straightforwardly to cover the whole globe. By assuming that two-dimensional Euclidean geometry was valid on the surface of the Earth, we should be making a mistake, owing to the curvature of the spherical surface, but the mistake would not become apparent as long as we made measurements only within a region the size of a city.