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Uber den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes, published in Ann. Phys. 35. English translation in Lorentz, et al. (1923).

In an experiment in 1889, Eotvos determined that the difference in the ratio of inertial to gravitational mass for wood and platinum was less than 10-9 (see Weinberg (1972), pp. 11-13), but Einstein did not learn of this result until 1912 (Pais (1982), p. 216). In 1678, Huygens published Le traite de la lumiere containing his wave theory of light in which the wavefront at any instant is the envelope of spherical wavelets emanating from every point on the wavefront at the previous instant (with the understanding that the wavelets have the same speed as the overall wave). This principle enabled Huygens to establish the laws of reflection and refraction which were subsequently derived analytically by Gustav Kirchhoff.

57 Unless specified otherwise, c will always refer to the special relativistic value.

Einstein (just like the rest of the physics community) was entirely unaware that almost exactly the same prediction had been made over 100 years before him by combining Newtonian optics and mechanics! If we envisage light as being made up of material corpuscles, then these will be attracted by gravity just like any other matter. It is straightforward to compute the deflection of a particle in a hyperbolic orbit passing close to a point mass, M, given that its speed at infinity is c, and this appears to have been done first in 1784 by Cavendish, who obtained the value 2 sin-:(e/(1 + e)), where e = GM/c2A. This agrees with Einstein's formula, since e is small. A very similar calculation was carried out in 1801 by Johann von Soldner (the only difference being that in von Soldner's calculation the speed of light was taken as c at the point of closest approach), with the same conclusion. Cavendish never published his result, but von Soldner's calculation appeared in a major astronomical journal -and was largely ignored. In 1921, Philip Lenard tried to discredit Einstein by resurrecting von Soldner's work, publishing part of the original paper in the Annalen der Physik.

In his 1911 paper, Einstein also quantified the gravitational redshift, showing that, again to the first approximation, the spectral lines when measured on the Earth should be shifted toward the red by a frequency of @/c2, @ being the difference in potential between the surface of the Sun and the Earth. Einstein calculated this figure as 2 x 10—6, and noted that such an effect had been observed but attributed to physical phenomena on the surface of the Sun. The measurement of the gravitational red shift is, in fact, extremely difficult, as it is masked by convection currents in the solar atmosphere, which lead to Doppler shifts of the same order of magnitude.

At this point, Einstein had gone about as far as possible toward a new theory of gravity within the kinematical framework of special relativity. He realized that the Lorentz transformations were not by themselves sufficient to crack the problem, but quite what new idea was required he did not know.

Einstein was not the only person working on gravitation at this time. In 1912, Max Abraham, a staunch opponent of relativity,59 published a theory in which the direction of the force of gravity is given by Vc, with the speed of light given as a function of the gravitational potential by c = c0(1 + 2$ /c^)1/2, whichagreeswithEinstein's(Eqn(12.4))tofirstorderin1/c^. SinceE = mc2

58 Cavendish's calculation was only found among his papers early in the twentieth century (see Will (1988)). The story surrounding von Soldner's work, including a translation of his essay, is told in Jaki (1978).

Einstein and Abraham became embroiled in a very public (and at times unpleasant) debate over the merits of relativity and their respective theories of gravity, and Abraham took great pleasure in the fact that Einstein was now advocating a variable c, in direct conflict with special relativity (see Pais (1982), pp. 231-2).

and the total energy of a body in a gravitational field is a function of the potential it follows that either m or c (or both) must depend on Both Einstein and Abraham had c = c($), but theories in which m = m($) and c retained its constant special relativity value were put forward by Gunnar Nordstrom and Gustav Mie. As far as Einstein was concerned, only the second of two theories put forward by Nordstrom needed to be taken seriously.

In all these attempts to extend relativity to include gravitation, the field was determined from a single scalar function (as it is in Newtonian mechanics) and Einstein developed his own scalar theory of a static gravitational field during the same period. Although the theory had problems, one positive outcome was the unpleasant realization that the equations of gravitation had to be nonlinear, since the gravitational field possesses energy that acts as its own source. This then led Einstein to the reluctant conclusion that the principle of equivalence could only hold locally. In other words, it is not possible, in general, to find a global coordinate transformation that removes the effect of gravity completely, but that this is possible at each point in spacetime. Though he did not work it out at the time, Einstein's scalar theory of gravity leads to two-thirds of the general relativistic advance for the perihelion of Mercury.

It was in 1912 that Einstein was struck by the similarities between the problems he was facing and the theory of surfaces due to Gauss. With the help of Grossman (Einstein was now back in Zürich) he learned about the 'absolute differential calculus' developed through the work of Riemann, Christoffel, Ricci, and Levi-Civita, and began to investigate whether gravitation could be described through a curved space defined via the metric

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