## Ds2 nv dx dxv126

where n»v = 0 if » = v and n»» = 1, 1, 1, and -1, for » = 1, 2, 3, and 4, respectively. The principle of equivalence as understood by Einstein in 1913 is equivalent to the condition that at any point in spacetime there is a coordinate transformation that reduces the metric tensor g»v to the Minkowski tensor n»v. Locally, spacetime has the structure of special relativity.

In the Entwurftheory, the motionof aparticle infree fall within a gravitational fieldgivenbyEqn(12.5)isdeterminedfromthevariationalprinciple 5/ ds = 0. In other words, in the absence of any forces, objects move between two points along the shortest path just as they do in special relativity, but now length is being measured by the line element given in Eqn (12.5) and so the resulting trajectories are not straight lines. Gravity is thus no longer a force but, instead, is considered as a deformation of Minkowski's spacetime. A helpful analogy is to think of gravity as a curved surface that can be approximated locally by the tangent plane to the surface. To complete the theory, equations that relate the metric g»v to the distribution of matter are needed - these are the so-called 'field equations of gravitation' - and this is where Einstein and Grossman came up short.

The core of general relativity as we know it today is that all frames of reference should be treated equally, none are preferred.

The general laws of nature are to be expressed by equations which hold good for all systems of coordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant) this requirement... takes away from space and time the last remnant of physical objectivity.

However, the field equations in the Entwurf paper generally are not co-variant, but covariant only with regard to linear transformations, and Einstein

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