## Ds2 gV dx dxV125

where gßV (which depends on the coordinates xis a symmetric tensor of rank 2.62 Here we have replacedx, y, z, and ct,by x1, x2, x3, andx4, respectively,

60 For a thorough discussion of Einstein's interpretation of the principle of equivalence, see Norton (1985).

See Whitrow and Morduch (1965), an article that tabulates the predictions of numerous relativistic theories of gravitation for redshift, light deflection, and perihelion advance. A similar comparison is given in Harvey (1965). Nordstrom's second theory (which is equivalent to the theory proposed by Littlewood (1953a)) actually leads to a regression with one-sixth the general relativistic value (Pirani (1955)). It is clear that the redshift is the least restrictive of the three tests, as the same value is predicted by almost all theories. Much more on Nordstrom's theory of gravity and its influence on Einstein can be found in Norton (1992, 1993). The theory of tensors is described in most textbooks on general relativity (see, for example, Weinberg (1972), pp. 93-100). Tensors are collections of quantities (components) that transform in a certain way and are the natural objects with which to express the mathematical invariance of physical laws. A subscript indicates what is known as a 'covariant transformation rule', whereas a superscript is used for 'contravariant transformations'. This distinction is superfluous for three-dimensional Cartesian tensors, as is illustrated nicely in the treatment given by Foster and Nightingale (1995).

and used a notational convenience that Einstein himself introduced in 1916, and which is now referred to as the 'Einstein summation convention'. If the index in an expression occurs twice - once as a subscript and once as a superscript (1 and v in the above equation)-then a summation is implied. Hence, Eqn(12.5) is shorthand for ds2 = 1 = 4 = i S1v dx 1 dxv. The difficulties Einstein faced when developing general relativity were not helped by the fact that initially his notation was much more cumbersome.

This would be a radical departure from any previous type of theory, in which the underlying structure of space and time was given to begin with. In his new approach, the geometry of spacetime was one of the things gravity had to explain. Also, it made everything much more complicated, since the single variable, c, had been replaced by the ten unknowns . Clearly, Einstein found the work incredibly difficult:

At present I occupy myself exclusively with the problem of gravitation and now believe that I shall master all difficulties with the help of a friendly mathematician here. But one thing is certain, in all my life I have labored not nearly as hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now. Compared with this problem, the original relativity is child's play.

The collaboration with Grossman led to the publication in 1913 of an Outline of a Generalized Theory of Relativity and of a Theory of Gravitation, often referred to as the 'Entwurf theory'. This was Einstein's first attempt at a tensor theory, and came remarkably close to the final theory that would emerge just over 2 years later.

It was Minkowski who had first formulated the equations of special relativity in modern tensor form, but Einstein initially was unimpressed by this formal simplification to the theory (he described it as 'superfluous learnedness'). The

63 It seems likely that one of the inspirations behind Einstein's new approach was Born's work on the definition of a rigid body in special relativity (see Stachel (1980), Pais (1982), pp. 214-16, and Maltese and Orlando (1995)). The latter paper quotes Einstein, from a lecture given in 1921, as saying 'In a system of reference rotating relatively to an inertial system, the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction; thus if we admit non-inertial systems on an equal footing, we must abandon Euclidean geometry.'

From a letter to Sommerfeld in October 1912. Quoted from Pais (1982), p. 216. Entwurf einer verallgemeinerten Relativitatshtheorie und einer Theorie der Gravitation, published in Z. Math. undPhys. 62. The first part of the paper was physics, written by Einstein; the second part was mathematics, written by Grossman. Some details of the content are given in Norton (1984).

Die Grundgleichenungen für die elektromagnetischen Vorgange in bewegten Körpen, published in Göttinger Nachr. (1908). An Appendix to this paper contained a treatment of gravitation that influenced strongly the later theory of Nordström (see Pyenson (1977)).

Minkowski metric can be written

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