A Brief Guide to the Great Equations
Page 19
Einstein wrote nothing about gravitation from 1907 to 1911. For one thing, there were personal disruptions; the birth of another son in 1910, a move to Prague with his wife Mileva and two children in March of 1911. His new jobs – associate professorships in Zürich at the Eidgenössiche Technische Hochschule (ETH), then in Prague in 1911 – brought more pressures. And he became absorbed in the problems of quantum theory, in the course of which he showed the world of physics how to banish what Kelvin had called ‘Cloud No. 2’: the difficulty of applying the Maxwell-Boltzmann theory to certain experimental results, which Einstein showed to be due to effects relating to the quantum idea.
By the time Einstein returned to gravitation, a promising new door had opened in the maze, though he passed it by for the moment. It had been opened by a former ETH professor of his, Hermann Minkowski (1864–1909). Minkowski had not been especially enamored of his student, whom he had once called a ‘lazy dog’; and on seeing Einstein’s 1905 paper remarked, ‘Imagine that! I would have never expected such a smart thing from that fellow.’9 But Einstein turned out to be Minkowski’s dream student, one who borrowed, assimilated, and transformed what he had learned, instructing his teacher.
The door Minkowski opened, inspired by Einstein’s theory of special relativity, was a mathematical approach to space and time that put them on equal footing. Minkowski described objects as having not only an x, y, and z of position, but also a t corresponding to their position on a fourth axis, time. This new approach, that is, conceived of objects as moving along a time ‘line’ in just the way they moved along spatial lines. An object that stayed in the same place would be represented by a straight line, with only its position on the t-axis changing. An object moving uniformly along the x-axis would be represented by a curved line, for it had not only moved along the x-axis but also uniformly along the t-axis. More complexly moving objects followed more complex routes. The distance between an object in one position and another had four terms rather than three; it was like adding yet another ‘side’ to the Pythagorean theorem corresponding to the time difference between the two events. While the Pythagorean theorem for three-dimensional Euclidean space is s2 = x2 + y2 + z2, where s2 is the length that is invariant for all observers, the extension of the theorem to ‘space-time’ now meant that it was s2 = x2 + y2 + z2 − (ct)2.
Minkowski’s formulas extended covariance further, by including in the conditions for objective existence an object’s behaviour in time. A description of something that gives only the x, y, and z of position without a position in time is like a snapshot of a person compared to a video – only a partial view. In space-time, a full description of an object is more like a video, for it charts the object’s spatial location and time at once. To generate the formulas, Minkowski made use of mathematical tools, now called ‘tensors’, that translate sets of quantities from one coordinate system to another. Tensors are mathematical objects that are ‘ranked’ based on their indices, which is a measure of how complex they are. A tensor of rank 0 is a constant, of rank 1 is a vector, and of rank 2 is a set of complex matrices with multiple components that can be used to translate one set of coordinates into another. Rank 2 tensors allowed Minkowski to bring space and time together, in the process redefining them as thoroughly as Einstein had simultaneity. In 1908, Minkowski gave a talk in Cologne that began:
Gentlemen, the views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.10
But for the moment Einstein passed by this door in the maze, dismissing his teacher’s work as ‘superfluous learnedness.’11 He would be forced to return.
In 1911, now in Prague with Mileva and his children, Einstein published what amounted to a sequel to his 1907 paper, entitled ‘On the Influence of Gravitation on the Propagation of Light.’12 In an earlier paper, he began, I addressed the question of whether gravitation affects light. But ‘my former treatment of the subject does not satisfy me…because I have now come to realize that one of the most important consequences of that analysis is accessible to experimental test.’ Two tests, in fact. According to the equivalence principle, ‘one can no more speak of the absolute acceleration of the reference system than one can speak of a system’s absolute velocity in the ordinary theory of relativity.’ This had implications for how light behaves in a gravitational field. One, which he discussed in section 3, was the existence of a gravitational red shift, that light emitted by sources ‘further down’ in such a field – at the sun’s surface, for instance – would be shifted slightly toward the red when compared to similar sources ‘higher up’, such as on the earth. Measuring the difference, though difficult, would be one important test of general relativity.
A second test, discussed in section 4, was the bending of starlight. For ‘rays of light passing near the sun experience a deflection by its gravitational field, so that a fixed star appearing near the sun displays an apparent increase of its angular distance from the latter, which amounts to almost one second of arc.’ Suppose astronomers took photographs of the stars in the background during a total solar eclipse, and compared those with photographs of the same stars without the sun. If, as Einstein suspected, the starlight bent as it brushed past the sun, the two photographs would be different, and the stars would appear farther from the sun in the first picture. He now predicted how much.
A ray of light going past the sun would accordingly undergo deflection to the amount of 4.10−6 = .83 seconds of arc… As the fixed stars in the parts of the sky near the sun are visible during total eclipses of the sun, this consequence of the theory may be compared with experience… It would be a most desirable thing if astronomers would take up the question here raised. For apart from any theory there is the question whether it is possible with the equipment at present available to detect an influence of gravitational fields on the propagation of light.
This value, shortly corrected to .87 seconds of arc, simply reflects the fact that light, because it has mass according to E = mc2, is as affected by gravitation as stones and apples, and is based on Newtonian principles; in fact, it is known as the ‘Newtonian value.’
In this 1911 paper, Einstein still could say nothing about the still-mysterious precession of Mercury’s perihelion, which would become the third key prediction of general relativity. But the first two predictions – especially the bending of starlight – set in motion events that would culminate 8 years later in the famous meeting, and make headlines all over the world. For Einstein began to agitate for astronomers to look into these predictions.
In August 1911, he sent his ‘Influence of Starlight’ paper to an astronomer at the University of Utrecht who had authored a paper about the solar redshift; Einstein wrote that he had arrived at the ‘somewhat daring’ conclusion that ‘the gravitational potential difference’ might be the cause. ‘A bending of light rays by gravitational fields also follows from these arguments.’ If something else is responsible, Einstein added, ‘then my darling theory must go in the wastebasket.’13 That month, too, Einstein sent his paper to Erwin Freundlich, a young Berlin astronomer who would become relativity’s advocate among astronomers. Freundlich offered to look for the influence of Jupiter on starlight, and to investigate photographs taken during eclipses. Jupiter proved not massive enough to bend starlight to any detectable degree. ‘If only we had an orderly planet larger than Jupiter!’ Einstein lamented. ‘But Nature did not deem it her business to make the discovery of her laws easy for us.’14 Freundlich also set out to see if stellar deflections could be detected from photographs taken of past eclipses, but this proved impossible. And he initiated an expedition to photograph an eclipse in Brazil in October 1912. But the expedition was rained out, one organizer wryly commenting that ‘we…suffered a total eclipse instead of observing one.’15
Einstein tried to help where he could, and endured the setbacks with some impatience. But the collapse of the attempts proved beneficial to him in the long run. For by the time of the Russian expedition Einstein had entered a different part of the maze, causing him to revise his predictions.
Second Step: Geometry of Space-Time
In the meantime, thanks in part to his struggles with the bending of light in a gravitational field, Einstein had realized an even deeper implication of the principle of equivalence – that he would have to link his equations with different geometries of space. For a space to have a geometry does not mean that the space is literally curved – something can only be curved with respect to something else taken to be straight – but rather has to do with the way that the measurements of a path of something going through that space (a beam of light, say) add up. If light were bent by a gravitational field, the measurements of its trajectory add up in a way that corresponds to the mathematics of a certain geometry. This involved treating gravitation not as a force, that is, as something that exerts a tug, but as a property of space itself, as having a structure or architecture that must be followed by things passing through it. Writing equations without such architecture – without the geometry of space – Einstein wrote, would be like ‘describing thoughts without words.’17 Moreover, he also now realized the virtues of his old (and now deceased) teacher Minkowski’s work, and how the use of tensors would vastly simplify the task he was undertaking. Accordingly, he retraced his steps in the maze and entered the door Minkowski had opened. But there he found himself overwhelmed by the task of finding tensors to work for curved, non-Euclidean, geometries.
Fortunately, he knew where to turn. In summer 1912 Einstein had moved from Prague to Zürich, recruited by his former companion and classmate Marcel Grossmann. In school, Einstein used to borrow Grossmann’s mathematics lecture notes so he could concentrate on physics; now, Grossmann was dean of the ETH’s mathematics-physics section. ‘Grossmann’, Einstein wrote in desperation, ‘you must help me or else I’ll go crazy!’18 Grossmann then told him about non-Euclidean geometries that mathematicians had produced – by Bernhard Riemann (who had a twenty-component curvature tensor), Gregorio Ricci-Curbastro (who had developed a contracted version with ten components), and Tullio Levi-Cività – and explained to him about tensor calculus. With Grossmann’s help, Einstein set out to develop a generalized, Minkowski-like tensor for a four-dimensional surface to represent the gravitational field. In August he excitedly wrote to one friend, ‘The work on gravitation is going splendidly. Unless I am completely wrong, I have now found the most general equations.’19 In October he wrote to another:
I am now working exclusively on the gravitation problem and believe that I can overcome all difficulties with the help of a mathematician friend of mine here. But one thing is certain: never before in my life have I troubled myself over anything so much, and I have gained enormous respect for mathematics, whose more subtle parts I considered until now, in my ignorance, as pure luxury! Compared with this problem, the original theory of relativity is child’s play.20
The next year, 1913, Einstein and Grossmann published an article, ‘Outline of a General Theory of Relativity and a Theory of Gravitation’, that comes within a ‘hair’s breadth’ of the final equations.21 It is in two parts, part I on physics by Einstein, and part II on mathematics by Grossmann. In writing it, they encountered hints that general covariance was impossible. This finding greatly disturbed Einstein, who called it an ‘ugly dark spot’ in the theory.22 He abandoned his hopes for general covariance of the field equation, and embarked on what would turn out to be a dead end from which it would take him 2 years to extricate himself.
Early in 1914, Einstein left Zürich for a new position in Berlin, leaving behind not only Grossmann but also Mileva, their marriage broken. At a farewell dinner thrown by an old ETH friend, Einstein expressed some apprehension, and complained of being treated as a ‘prize hen.’ As he told the person who walked him home, ‘I don’t even know whether I’m going to lay another egg.’23
Third Step: Covariant Equations
Once in Berlin, Einstein threw himself into work on general relativity, ignoring everything else; entering Einstein’s study one day, Freundlich saw a meat hook hanging from the ceiling, from which the scientist impaled letters that he had no time to read.24 Between October and November 1915, Einstein finally realized his mistake, and turned back to seeking a formulation that was generally covariant. That time, he wrote a friend, would be ‘one of the most stimulating, exhausting times of my life.’25 On November 4, Einstein told the Prussian Academy that he had ‘lost trust in the field equations’ that he had earlier reported. He explained that he had returned to the ‘demand of general covariance’ for the field equations, ‘a demand from which I parted, though with a heavy heart, three years ago’, and had produced a truly covariant theory. ‘Nobody who really grasped it can escape from its charm.’
Sometime during the next 2 weeks, he made a discovery that, one biographer wrote, was ‘by far the strongest emotional experience in Einstein’s scientific life, perhaps in all his life.’26 He noticed that his new theory perfectly accounted for the precession of Mercury’s perihelion. The prediction he had mentioned in 1907, and so diligently worked on, now popped right out of his new theory without any extra hypotheses or assumptions. ‘I was beside myself with joy and excitement for days’, he wrote to Lorentz.27 He told another friend he had heart palpitations, and yet another that something ‘snapped’ inside him.28
On November 18, Einstein announced to the members of the Prussian Academy that his new, covariant gravitational field equations account for Mercury’s orbit, and also – something he realized for the first time – that the theory of curved space-time also implies a prediction of the deflection of starlight around the sun of twice the amount he had predicted earlier. ‘This theory, however, produces an influence of the gravitational field on a light ray somewhat different from that given in my earlier work… A light ray grazing the surface of the sun should experience a deflection of 1.7 sec of arc instead of 0.85 sec. of arc.’ The old Newtonian value was .85 seconds of arc; the new ‘Einsteinian value’ was 1.7 seconds of arc.29
And, on November 25, in a talk called ‘The Field Equations of Gravitation’, he wrote it down as follows, in the familiar way:30
Gim = −κ(Tim − ½gimT )
though it is sometimes written with Rs instead of Ts, and with additional terms.31
The equation comes in two parts. The left-hand side refers to a set of terms that characterize the geometry of space. The right-hand side refers to a set of terms that describe the distribution of energy and momentum. The left is the geometry side, the right is the matter side. As physicist John Wheeler liked to describe it, reading from left to right is space-time telling mass how to
move; reading from right to left is mass telling space-time how to curve. Though the equation predicts only the slightest of deviations experimentally from the world as we know it – the positions of a few stars out of place – it amounts to a conceptual revolution from the world of Newton. In this new world, there is no absolute time nor space, and gravitation is not a force – not a tug between one object and another – but a property of space and time.
Einstein was utterly confident that it was right. He sent physicist Arnold Sommerfeld a postcard: ‘You will become convinced of the General Theory of Relativity as soon as you have studied it. Therefore I will not utter a word in its defence.’32 But while the core of general relativity was now clear to Einstein, it was still a maze to nearly all others. ‘[T]he basic formulas are good, but the derivations abominable’, Einstein wrote to Lorentz.33 Early in 1916, therefore, Einstein sat down to compose a logical route that others could follow. The result was a fifty-page paper published in a March issue of the Annalen der Physik, ‘The Foundation of the General Theory of Relativity.’ The paper was such a stunning success that it was reprinted in booklet form, ran through several printings, and was translated into English. The final section states the three experimental predictions made by the theory:34
the spectral lines of light reaching us from the surface of large stars must appear displaced towards the red end of the spectrum.
a ray of light going past the Sun undergoes a deflection of 1.7 [seconds of arc]; and a ray going past the planet Jupiter a deflection of about .02 [seconds of arc].
The orbital ellipse of a planet undergoes a slow rotation… Calculation gives for the planet Mercury a rotation of the orbit of 43 [seconds of arc] per century, corresponding exactly to astronomical observation (Leverrier).