by John Gribbin
Einstein’s excitement that day was slightly dampened by the arrival of a letter from Hilbert containing a copy of his own work, which, in spite of the disclaimer in the previous letter, turned out to be along very similar lines to Einstein’s work. He replied (before giving his lecture!) with a carefully worded letter, clearly intended to establish his priority and worth quoting at length:
The system you furnish agrees – as far as I can see – exactly with what I found in the last few weeks and have presented to the Academy … Three years ago with my friend Grossmann I had already taken into consideration the only covariant equations, which have now been shown to be the correct ones. We had distanced ourselves from it, reluctantly, because it seemed to me that the physical discussion yielded an incongruity with Newton’s law. Today I am presenting to the Academy a paper in which I derive quantitatively out of general relativity, without any guiding hypothesis, the perihelion motion of Mercury. No gravitational theory has achieved this until now.
Hilbert replied by return, offering his congratulations. But he also, on 20 November, sent his version to a science journal based in Göttingen for publication. Just how much – or how little – Einstein was influenced by Hilbert’s work we will never know for sure. But on 25 November 1915, in his fourth lecture to the Prussian Academy, under the title ‘The Field Equations of Gravitation’, Einstein presented the covariant field equations that are the basis of the General Theory – the gμ equations that mean as much to the General Theory as E = mc2 does to the Special Theory. The General Theory of Relativity was essentially complete.e
Was it all Einstein’s own work? The evidence suggests it was, although Hilbert was within a hair’s breadth of getting there first. Although Hilbert’s published paper was dated 20 November, five days before Einstein’s fourth lecture, a set of page proofs survive which shows that he made some corrections to the paper before finally returning it to the publisher on 16 December. Those corrections were necessary because the original equations were not fully covariant – and they seem to have been made in the light of what Einstein said on 25 November. In other words, Hilbert corrected his work to match Einstein, not the other way around. As Hilbert himself put it when presenting the equations, they were ‘first introduced by Einstein’. And it was, of course, Einstein’s theory that Hilbert was working on, as Hilbert always acknowledged. Einstein mailed printed copies of his four lectures to his colleagues (full publication took place in 1916), describing them as containing ‘the most valuable discovery of my life’. He was right. At the age of 36 he had changed our view of the Universe more profoundly than anyone except Newton. To explain why, I shall describe the legacy of the General Theory, before returning to Einstein’s life and later work.
Footnotes
a In German, die glücklische Gedanke, which is traditionally quoted as ‘happiest’ but might be better interpreted as ‘luckiest’.
b These are, of course, hypothetical silent rockets that make no vibration!
c This number was later revised to 0.85 seconds of arc.
d Doubly wrong, as we shall see.
e The presentations to the Academy were published bit by bit as the talks went along, but the definitive formal paper putting it all together, ‘The Foundation of the General Theory of Relativity’, appeared in 1916 in the Annalen der Physik, volume 49, p. 79.
4
Legacy
Black holes and timewarps; Beyond reasonable doubt; Making waves; The Universe at large
The first person to sit up and take notice of Einstein’s General Theory – apart from the people he had been discussing it with – was Karl Schwarzschild, the former director of the Potsdam Observatory, who was now serving with the German army on the Eastern Front, calculating the trajectory of artillery shells. By early 1916 he had not only received copies of Einstein’s presentations to the Prussian Academy, but had found an exact solution to the equations, which he mailed to Einstein, who in turn presented it to the next meeting of the Academy on 16 January.
Schwarzschild’s breakthrough was more of an achievement than it might seem to a non-mathematician. Einstein had found a description of gravity in terms of a set of ten coupled nonlinear partial differential equations (the full version of the gμ), which on the face of things were very difficult to solve fully. It was one thing to apply them to a specific problem, such as the calculation of the orbit of Mercury, but quite another to get an exact solution which described the complete behaviour of spacetime near a mass. But that is what Schwarzschild had done. First, he solved the equations to provide a description of the behaviour of space everywhere outside a spherical, non-rotating mass, such as a star; then, in a second communication to Einstein, he provided a description of how things are inside such an object. Both results implied the possibility of what are now known as black holes.
Black holes and timewarps
Schwarzschild’s solution to Einstein’s equations (known today simply as the Schwarzschild solution) implied that if any mass were squeezed into a small enough volume, two things would happen. The space around the object would be bent round upon itself so that it would be sealed off from the outside world and nothing could escape (not even light; hence the later introduction of the term ‘black hole’); and the matter inside this ‘Schwarzschild radius’ would collapse all the way down to a mathematical point, a singularity. The appearance of a singularity in a theory is usually a sign that something is wrong – that the equations that are being applied do not work under such extreme conditions. Einstein himself never believed that such Schwarzschild singularities could exist in the real world, or even that real objects could shrink within the appropriate Schwarzschild radius. But what the equations implied was that the Sun would become a black hole if it were squeezed within a radius of 2.9 kilometres, and that the Earth would do so if squeezed to the size of a large bean, with a radius of 0.88 centimetres. The bigger the mass involved, the bigger the Schwarzschild radius and the easier it would be to make a black hole.
It took decades for these ideas to even begin to be taken seriously as applying to real objects in the Universe, and Schwarzschild, who contracted a rare skin disease and died on 11 May 1916, never knew what he had started. The first hint that the Schwarzschild radius might at least be approached by collapsing objects in the real world came when Subrahmanyan Chandrasekhar, an Indian physicist based in England, applied the then-new understanding of quantum mechanics to calculate the fate of a star that had run out of nuclear fuel and could no longer generate heat in its interior to provide the pressure to hold itself up against the pull of gravity. It was thought that a star at the end point of its life would become a solid object with the atoms jammed tightly together, something about the size of the Earth but containing about as much mass as the Sun. But at the beginning of 1930 Chandrasekhar showed that if such a white dwarf star had more than about 1.5 times the mass of the Sun then the atoms themselves would be crushed and the star would collapse even further. His ideas were not widely accepted at the time, but in Russia Lev Landau suggested that under such extreme conditions electrons and protons would be fused to make neutrons, and the star (or at least, its core) would become a ball of neutrons like a huge atomic nucleus. The idea was born of a neutron star, containing about as much mass as the Sun contained in a volume about the same as that of Mount Everest. But it turned out that even this was not the last word. In 1938, Robert Oppenheimer and George Volkoff, in America, found that even a neutron star could not hold itself up against the pull of gravity if its mass exceeded a certain amount, now known as the ‘Oppenheimer-Volkoff limit’. This is about three times the mass of the Sun. For greater masses, they wrote in a paper published in 1939, ‘the star will continue to contract indefinitely, never achieving equilibrium’.
Oppenheimer discussed the implications in another paper, published later in 1939:
When all the thermonuclear sources of energy are exhausted a sufficiently heavy star will collapse. Unless fission due to rotation, the radiati
on of mass, or the blowing off of mass by radiation, reduce the star’s mass to the order of that of the sun, this contraction will continue indefinitely … the radius of the star approaches asymptotically its gravitational radius … The total time of collapse for an observer co-moving with the stellar matter is … of the order of a day.1
Or in everyday language, the star collapses into a black hole in a matter of a few hours.
But hardly anyone believed that neutron stars, let alone black holes, really existed. Surely there must be some law of nature to prevent such an absurdity? There things stood until 1967, when the discovery of pulsars forced a rethink. The only workable explanation for these rapidly flickering radio sources turned out to be that they were neutron stars, spinning quickly and flicking beams of radio noise around like the beams of a lighthouse. If neutron stars were real, astronomers realised, so might black holes be. It is no coincidence that the revival of interest in black holes, and John Wheeler’s coining of the name in December 1967, followed hot on the heels of the discovery of pulsars. Since then, it has become abundantly clear, especially from observations using X-ray telescopes orbiting the Earth, that black holes are not only real but far from rare. ‘Stellar mass’ black holes have been discovered orbiting ordinary stars and revealing their presence by the energy radiated by matter falling into the holes; and ‘supermassive’ black holes, millions of times as massive as our Sun and as big as the Solar System, have been identified by their similar (but bigger) energetic outbursts at the hearts of galaxies, including our Milky Way. Perhaps Einstein would have been astonished by all this. Or perhaps not, in view of some of his own ideas about what happens to space and time when they are distorted by matter.
What happens inside a black hole? The short answer is, nobody knows. But the equations give us some clues. A key feature of those equations is that it is possible to define a kind of interval in spacetime that is the same for all observers, however they are moving. In physical terms, relative motion may make time stretch and space shrink, so that a kind of average of the two stays the same. This is called a ‘metric’. Different metrics apply in different kinds of curved spacetime, and the description of a black hole found by Schwarzschild can be referred to as the Schwarzschild metric. As early as 1916, the Austrian Ludwig Flamm noticed that the Schwarzschild solution to Einstein’s equations actually describes not one black hole (using modern parlance) but two, connected by some kind of tunnel, which became known as a ‘wormhole’. This led to some intriguing speculation by physicists. If an electron, or some other particle, existed literally at a point in space, then the spacetime around it would be described by the Schwarzschild metric. Could it be that particles such as electrons actually existed in pairs, perhaps widely separated in space, but connected by wormholes? The appeal of this idea was that it raised the possibility that particles – matter – might be described entirely in terms of curved spacetime. This idea turned out to be a blind alley.a But it did lead Einstein, working with Nathan Rosen at Princeton in the mid-1930s, to investigate the mathematical description of larger versions of such wormholes, which became known to relativists as ‘Einstein-Rosen bridges’.
An Einstein-Rosen bridge is like a tunnel through spacetime, linking two flat regions of space which may be far apart – even on opposite sides of the Universe. If such an object existed – and it is a big if – and if the entrances were big enough, you could jump in one end of the wormhole and come out of the other. This is the basis of many a science fiction story, so Star Trek is part of Einstein’s legacy. The big if, though, is related to the fact that even if such an object formed, it would collapse much more quickly than the time it would take to traverse it. But the idea was revived, 50 years later, when theorists realised what should have been obvious from the start – a tunnel through spacetime can take you to a different place or to a different time (or both!). The equations were telling us that if an Einstein-Rosen bridge could be kept open, it would operate as a time machine. There are immense practical difficulties (see my and Mary’s book Time Travel for Beginners). But the bottom line is that according to Einstein’s equations – the best description of the Universe at large that we have – time travel is not impossible, just very, very difficult. Could Einstein be wrong? Well, so far the general theory has passed every test, the latest as recently as 2013, with flying colours.
Beyond reasonable doubt
The most famous test of the General Theory came in 1919, when the light bending effect was observed during an eclipse of the Sun. But that story belongs in the next chapter, as a key event in Einstein’s life. The explanation of the shift in the perihelion of Mercury was another contemporary proof that Einstein was right. But the most impressive proofs of the accuracy of his theory are part of his legacy, having been obtained after his death.
Two related kinds of test of the General Theory depend on a phenomenon known as the ‘gravitational redshift’. The simplest way to understand how this arises is to go back to the image of a freely falling lift derived from Einstein’s vision that ‘if a person falls freely he will not feel his own weight’. If gravity and acceleration are equivalent, everything going on inside the lift must look the same to observers inside and outside the lift. Inside the lift, a pulse of light is emitted from a laser at the bottom, and proceeds vertically towards the roof. While it is on its way, an observer inside the lift measures its wavelength. The lift and everything it contains is in free fall, so the light wave travels at the speed of light and with unchanging wavelength. But if the pulse of light is observed through a window by someone outside the lift, sitting still on the surface of the Earth, because of the motion of the lift the light will be squashed to shorter wavelengths – a ‘blueshift’ (such an observer would have to have supersensitive instruments, or use a high-speed movie camera and play back the scene in slow motion). The only physical difference is that the outside observer is sitting still in a gravitational field. So the only way the two observers can measure exactly the same wavelength for the light pulse is if gravity causes a redshift, stretching the light, which exactly cancels out the squashing, producing the blueshift. This, of course, can all be described mathematically by Einstein’s equations. The prediction is that for light near the surface of the Earth, the measured frequency of a laser beam should be shifted by one trillionth of 1 per cent for a difference in height of 100 metres. This is the gravitational redshift. Amazingly, in the 1960s physicists were able to carry out appropriate experiments to this precision.
The experiments were carried out by Robert Pound and colleagues at Harvard University. The Jefferson Tower of the physics building at Harvard has a lift shaft 22.5 metres tall, and Pound’s team measured the change in wavelength (actually a blueshift, because the light is going down, not up) of radiation emitted at the top of the tower and recorded at the bottom of the tower to a precision of two parts in a thousand trillion. This required an emitter that produced a very precise wavelength (or frequency) of light, and a detector that could measure the wavelength with exquisite precision. The source was a sample of radioactive material, iron-57, which emits gamma rays with a very precise frequency, corresponding to a wavelength of 0.86 Ångstrom, or 0.086 nanometres. The same material will absorb gamma rays, but only if they have exactly the right frequency. So a detector incorporating iron-57 was placed at the foot of the shaft on a hydraulic lift which moved slowly downwards to produce a Doppler redshift, cancelling out the gravitational blueshift, and the speed of the lift at the point where so-called ‘resonance’ occurred and the gamma rays were absorbed was measured. The numbers are mind-boggling, but some idea of the precision of the experiment is shown by the fact that resonance occurred only when the hydraulic lift was moving downwards at two millimetres per hour. This was, of course, exactly in line with the predictions of the General Theory.
This gravitational influence affects not only measurements of length such as the wavelength of light, but also time, so that clocks in a stronger gravitational field should run m
ore slowly than identical clocks in a weaker gravitational field. This is actually the same thing as the gravitational redshift, looked at from a different perspective. In effect, we measure the wavelength of light by counting how many wave peaks go past a fixed point in one tick of the clock. If the wave is redshifted, fewer peaks go past in the same time. But we would get exactly the same effect if the ticks of the clock were closer together, so there was less time for the waves to get past. The reverse is true for a blueshift. In our falling lift experiment, the observer on the ground says the light is redshifted, but the observer in the lift says the clock on the ground is running slow.
The experiments that tested this variation on the theme were more complicated than the Harvard lift experiment, since they involved moving clocks (on aeroplanes: the only way to get them to a high enough altitude to measure the effect), which meant that the effects described by both the Special Theory and the General Theory had to be taken into account. So it wasn’t until 1971 that direct proof of gravitational time dilation was obtained.
The experimenters who took on this daunting task were Joseph Hafele, of Washington University, St Louis, and Richard Keating, of the US Naval Observatory in Washington, DC. If they had been able to charter a private jet to fly their ultra-precise atomic clocks in, life would have been reasonably simple. But government financial rules restricted their budget, so they had no choice but to fly their clocks around the world on commercial aircraft on scheduled flights. They couldn’t even travel first class, but in economy, with the clocks strapped securely to the front wall of the passenger cabin and connected to the aircraft’s power supply. The experiment involved flying a pair of clocks eastward, right around the world, between 4 and 7 October 1971, and westward around the world between 13 and 17 October. The two flights in opposite directions were necessary in order to make allowance for the rotation of the Earth. And, being commercial flights, the circumnavigations involved many stopovers, changes in speed, altitude and direction, all of which had to be carefully logged and fed into the calculations when the times recorded on the traveling clocks were compared with the times recorded on counterparts which had sat quietly at the US Naval Observatory while all this was going on.