A Brief Guide to the Great Equations (28 page)

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
=
mc
²
, 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

Undaunted, Freundlich set out himself on an expedition to Russia – one of several – for an eclipse that would take place in August 21, 1914. Einstein excitedly wrote Ernst Mach in June 1913, ‘Next year, during the solar eclipse, we shall learn whether light rays are deflected by the sun, or in other words, whether the underlying fundamental assumption of the equivalence of the acceleration of the reference system, on the one hand, and the gravitational field, on the other hand, is really correct.’
¹⁶
That fall, he wrote astronomer George Hale asking if a powerful enough telescope could detect stars near the sun during the day, so that he would not have to wait for a solar eclipse; but the eminent astronomer dashed his hopes, and Einstein resigned himself to waiting for the August 1914 eclipse. But that expedition, too, met disaster, for political as well as meteorological reasons. The assassination of Archduke Francis Ferdinand on June 28 was followed by Austria-Hungry’s invasion of Serbia, and Germany’s declaration of war on Russia on August 1. Freundlich’s expedition was an early war casualty. He was arrested – though soon ransomed and able to return to Berlin – and his equipment impounded. Other expeditions were stymied by bad weather.

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.

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

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