Bending space–time: a commentary on Dyson, rsta.royalsocietypublishing.org Eddington and Davidson (1920) ‘A determination of the Review deflection of light by the Sun’s Cite this article: LongairM.2015Bending gravitational field’ space–time: a commentary on Dyson, Eddington and Davidson (1920) Malcolm Longair ‘A determination of the deflection of light by the Sun’s gravitational field’. Phil.Trans.R. Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK Soc. A 373: 20140287. http://dx.doi.org/10.1098/rsta.2014.0287 The famous eclipse expedition of 1919 to Sobral, Brazil, and the island of Principe, in the Gulf of Guinea, led by Dyson, Eddington and Davidson was One contribution of 17 to a theme issue a turning point in the history of relativity, not only ‘Celebrating 350 years of Philosophical because of its importance as a test of Einstein’s Transactions: physical sciences papers’. General Theory of Relativity, but also because of the intense public interest which was aroused by the success of the expedition. The dramatic sequence of events which occurred is reviewed, as well as the long- Subject Areas: term impact of its success. The gravitational bending relativity, astrophysics of electromagnetic waves by massive bodies is a subject of the greatest importance for contemporary Keywords: and future astronomy, astrophysics and cosmology. general relativity, light bending by the Sun, Examples of the potential impact of this key tool eclipse expedition 1919, gravitational lensing, of modern observational astronomy are presented. This commentary was written to celebrate the 350th Eddington anniversary of the journal Philosophical Transactions of the Royal Society. Author for correspondence: Malcolm Longair e-mail: [email protected] 1. Einstein and bent space–time The famous expedition to measure the deflection of the positions of stars caused by the curvature of space–time in the gravitational field of the Sun had a profound Thefeaturedarticlecanbeviewedat http://dx.doi.org/10.1098/rsta.1920.0009. 2015 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited. impact upon the acceptance of the General Theory of Relativity and in arousing public interest in 2 all things relativistic. Although it was not expressed in this way at the time, the concept that the rsta.royalsocietypublishing.org fabric of the four-dimensional space–time we live in is determined by the distribution of mass– ......................................................... energy in the Universe lies at the heart of relativistic theories of gravity. The story began in the late eighteenth century when non-Euclidean geometries started to be taken seriously by mathematicians who realized that Eudlid’s fifth postulate—that parallel lines meet only at infinity—might not be essential for the construction of a self-consistent geometry [1,2].1 The fathers of non-Euclidean geometry were Nikolai Ivanovich Lobachevsky in Kazan in Russia and János Bolyai in Transylvania, then part of Hungary. In the 1820s, they independently solved the problem of the existence of non-Euclidean geometries and showed that Euclid’s fifth postulate could not be deduced from the other postulates. Non-Euclidean geometries were placed on a firm theoretical basis by the German Phil.Trans.R.Soc.A mathematician Bernhard Riemann, and the English-speaking world was introduced to these ideas through the works of British mathematicians William Kingdon Clifford and Arthur Cayley. Until Einstein’s discovery of the General Theory of Relativity, considerations of the geometry of space and the role of gravity in defining the structure of the Universe were separate questions. After 1915, they were inextricably linked. The history of the discovery of General Relativity is admirably told by Abraham Pais 373 in his scientific biography of Albert Einstein, Subtle is the Lord: the Science and the Life of :20140287 Albert Einstein [3], in which many of the technical details of the papers published in the period 1907–1915 are discussed. Equally recommendable is the survey by John Stachel of the history of the discovery of both theories of relativity [4]. In seeking a fully self-consistent relativistic theory of gravity, Einstein was entering uncharted territory and for many years he ploughed a lone furrow, making the ultimate spectacular success of the theory in 1915 all the more remarkable. It is simplest to quote Einstein’s words from his Kyoto address [5] of December 1922: In 1907, while I was writing a review of the consequences of Special Relativity,...Irealised that all the natural phenomena could be discussed in terms of Special Relativity except for thelawofgravitation.Ifeltadeepdesiretounderstandthereasonbehindthis...Itwasmost unsatisfactory to me that, although the relation between inertia and energy is so beautifully derived [in Special Relativity], there is no relation between inertia and weight. I suspected that this relationship was inexplicable by means of Special Relativity. He goes on: I was sitting in a chair in the patent office in Bern when all of a sudden a thought occurred to me: ‘If a person falls freely he will not feel his own weight’. I was startled. This simple thought made a deep impression upon me. It impelled me towards a theory of gravitation. In his comprehensive review of the Special Theory of Relativity published in 1907 [6], Einstein devoted the whole of the last section, Section V, to the Principle of Relativity and Gravitation. In the very first paragraph, he raised the question, Is it conceivable that the principle of relativity also applies to systems that are accelerated relative to one another? He had no doubt about the answer and stated the Principle of Equivalence explicitly for the first time: 1I have given a more detailed account of the origin of non-Euclidean geometries and the origins of General Relativity in my books [1,2]. ...in the discussion that follows, we shall therefore assume the complete physical 3 equivalence of a gravitational field and a corresponding acceleration of the reference rsta.royalsocietypublishing.org system. ......................................................... From this postulate, he derived the time-dilation formula in a gravitational field, dt = dτ(1 + φ/c2), where φ is the gravitational potential, recalling that φ is always negative, τ is proper time and t is the time measured at zero potential. Then, applying Maxwell’s equations to the propagation of light in a gravitational potential, he found that the equations are form- invariant, provided the speed of light varies in the gravitational potential as c(r) = c[1 + φ(r)/c2], according to an observer at zero potential. Einstein realized that, as a result of Huyghens’ principle, or equivalently Fermat’s principle of least time, light rays are bent in a non-uniform gravitational field. He was disappointed to find that the effect was too small to be detected in any Phil.Trans.R.Soc.A terrestrial experiment. Einstein published nothing further on gravity and relativity until 1911, although he was undoubtedly wrestling with these problems through the intervening period. In his paper of that year [7], he reviewed his earlier ideas, noting that the gravitational dependence of the speed of light would result in the deflection of the light of background stars by the Sun. Applying Huyghens’ principle to the propagation of light rays with a variable speed of light, he found the 373 standard ‘Newtonian’ result that the angular deflection of light by a mass M would amount to :20140287 θ = 2GM/pc2,wherep is the collision parameter of the light ray. In the case of grazing incidence p = R and the deflection amounted to 0.87 arcsec. Einstein urged astronomers to attempt to measure this deflection. Intriguingly, Einstein’s prediction had been derived by Johann Georg von Soldner in 1801 on the basis of the Newtonian corpuscular theory of light [8].2 Following the famous first Solvay Conference of 1911 in Brussels, Belgium, Einstein returned to the problem of incorporating gravity into the Theory of Relativity and, from 1912 to 1915, his efforts were principally devoted to formulating the Relativistic Theory of Gravity. It was to prove to be a titanic struggle. In summary, his thinking was guided by four ideas: — the influence of gravity on light, — the principle of equivalence, — Riemannian space–time, and — the principle of covariance. During 1912, he realized that he needed more general space–time transformations than those of Special Relativity. Two quotations illustrate the evolution of his thought. The simple physical interpretation of the space-time coordinates will have to be forfeited, and it cannot yet be grasped what form the general space-time transformations could have [9]. If all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them. [5] Towards the end of 1912, he realized that what was needed was non-Euclidean geometry. Einstein consulted his old school friend, the mathematician Marcel Grossmann, about the most general forms of transformation between frames of reference for metrics of the form ds2 = μ ν gμν dx dx . Although outside Grossmann’s field of expertise, he soon came back with the 2In fact, Henry Cavendish appears to have come to essentially the same result as von Soldner in an unpublished manuscript of about 1784, inspired by John Michell’s paper of the previous year on the escape of light from a massive body. Clifford Will [8] provides an intriguing comparison of how Cavendish and von Soldner may have each derived their result. Soldner’s result θ/ = πε / 2 2 can be found most easily from the deviation of particles under Rutherford scattering, for which cot 2 (4 0m zZe )pv0, 2 by replacing the electrostatic zZe /4πε0 by the gravitational GMm and setting v0 = c and p = R .