Einstein's Masterwork

by John Gribbin

  The results were not as precise as the Harvard lift shaft experiment, but matched the predictions of the General Theory within the limits of experimental error. For the westward flight, the clock gained 273 nanoseconds (billionths of a second), compared with a prediction of 275 nanoseconds, two-thirds of this attributable to the gravitational blueshift resulting from being at altitude. The situation was more complicated for the eastward flight, where the time dilation effect of the Special Theory was expected to cause the clock to lose more than it would gain from the gravitational blueshift. The observed loss was 59 nanoseconds, but with a range of errors caused by inaccuracies in the flight data allowing for the possibility of anything from 39 to 79 nanoseconds; the prediction was 40 nanoseconds. Suggestive, but not on its own compelling.

  But if anyone doubted the reality of the gravitational redshift, those doubts were laid to rest in June 1976, when a rocket-borne experiment, devised by Robert Vessot and Martin Levine and known as Gravity Probe A, soared into space to an altitude of 10,000 kilometres on a sub-orbital flight.b During the flight, the ‘ticking’ of an atomic clock in the payload was compared over a radio link with an identical clock on the ground, monitoring the way the changing Doppler shift resulting from the motion of the rocket relative to the ground interacted with the gravitational blueshift at altitude to produce an overall difference between the two clocks. The results matched the predictions of relativity theory to a precision of 70 parts per million, 7 thousandths of 1 per cent.

  Another test of Einstein’s theory, known as Gravity Probe B, took place at lower altitudes, but in orbit rather than on a sub-orbital flight. This was a long time in the preparation, and although it produced results towards the end of the first decade of the 21st century, it is close to my heart since it was the subject of one of the first articles I ever wrote for the magazine New Scientist, 40 years earlier, back in 1968 when I was a PhD student.

  Gravity Probe B was also known as ‘the Stanford weightless gyro experiment’. This neatly sums up what it was all about. The lead scientists on the project came from Stanford University, under the leadership of Francis Everitt, and it involved a set of gyroscopes floating weight-lessly in orbit around the Earth in an unmanned satellite. The aim of the experiment was to measure the effect on spacetime of the rotation of the Earth, which has been likened to the distortion in a bowl of maple syrup produced by twisting a spoon round in the middle of the syrup. It is known as the ‘Lense-Thirring effect’, because it was suggested by Austrian physicists Josef Lense and Hans Thirring two years after Einstein published his General Theory of Relativity. The effect is also known as ‘frame dragging’, and it is tiny, which is why it took so long to develop instruments sensitive enough to measure it and get them into space. Happily, all we have to worry about here are the results.

  There were four gyros on board the satellite, which orbited the Earth at an altitude of 650 kilometres passing over both poles. It was launched on 20 April 2004, and the mission lasted for sixteen months. The launch had to be timed to a precision of one second to get the payload into the precise orbit needed for the experiment. But this was trivial compared with the precision of the experiment itself. The gyros, about the size of table-tennis balls, were the most perfectly round objects ever made. They were spherical to within 40 atoms and if one of them could have been expanded to the size of the earth, the biggest irregularity on the surface would be only 2.4 metres high. The spheres themselves were made of fused quartz, coated with an extremely thin layer of the metal niobium. They could never touch the walls of their container, so they were suspended with electric fields, and made to spin with a squirt of helium gas; the directions in which they were pointing (their spin axes) were monitored using magnetic fields produced by the niobium layer. Full results from the experiment appeared in the journal Physical Review Letters in 2011. They revealed that for all four gyroscopes the ‘drift’ in the direction the gyros were pointing matched the predictions of the General Theory. The measured frame-dragging drift rate was 37.2 milliarcseconds per year, with possible errors of ±7.2 mas/yr; the prediction was 39.2 mas/yr.

  But matter doesn’t only drag spacetime round with it. It can also make ripples in spacetime, and in 2013 one consequence of such gravitational radiation provided the most precise confirmation yet of the accuracy of Einstein’s theory.

  Making waves

  Gravity Probe B confirmed that massive objects, such as the Earth or a star, drag spacetime around with them as they rotate. If they move back and forth, they can also generate waves in the fabric of spacetime, known as ‘gravitational waves’, or ‘gravitational radiation’, like the ripples you can make in a bowl of water by wiggling your finger about in it. The resulting ripples in the fabric of space are very weak, unless a very large mass is involved in fairly rapid motion. But the waves were predicted by Einstein in a paper published in 1916, where he showed that they should move at the speed of light. Physicists have been trying for decades (as yet unsuccessfully) to detect gravitational radiation using very sensitive detectors here on Earth, and plan to put even more sensitive detectors into space. But meanwhile absolute proof of the accuracy of his prediction has come from observations of compact objects far away in space – the latest, and most precise, of these observations being reported in 2013.2 The objects involved are compact binary stars, systems in which one star orbits closely around another – or rather, where both stars orbit around their common centre of mass, like a whirling dumbbell or the twirling mace of a drum majorette. The first of these systems extreme enough to test Einstein’s prediction was a ‘binary pulsar’, studied in the mid-1970s.

  A binary pulsar exists when two neutron stars, one of which is a pulsar, are in orbit around one another, forming a binary star system. The term is also used to refer to a pulsar in orbit about any other star, for example, a white dwarf. More than twenty binary pulsars are now known, but astronomers reserve the term ‘the binary pulsar’ for the first one to be discovered, which is also known by its catalog number, as PSR 1913+16.

  The binary pulsar was discovered in 1974 by Russell Hulse and Joseph Taylor, of the University of Massachusetts, working with the Arecibo radio telescope in Puerto Rico. This pulsar was at the time the most accurate clock yet discovered. What they found that summer was so important that in 1993 the pair received the Nobel Prize for their work on the binary pulsar.

  The first hint of the existence of the binary pulsar came on 2 July, when the instruments recorded a very weak signal. Had it been just 4 per cent weaker still, it would have been below the automatic cutoff level built into the computer program running the search, and would not have been recorded. The source was especially interesting because it had a very short period, only 0.059 seconds, making it the second fastest pulsar known at the time. But it wasn’t until 25 August that Hulse was able to use the Arecibo telescope to take a more detailed look at the object.

  Over several days following 25 August, Hulse made a series of observations of the pulsar and found that it varied in a peculiar way. Most pulsars are superbly accurate clocks, beating time with a precise period measured to six or seven decimal places; but this one seemed to have an erratic period which changed by as much as 30 microseconds (a huge ‘error’ for a pulsar) from one day to the next. Early in September 1974, Hulse realised that these variations themselves followed a periodic pattern, and could be explained by the Doppler Effect caused by the motion of the pulsar in a tight orbit around a companion star. Taylor flew down to Arecibo to join the investigation, and together he and Hulse found that the orbital period of the pulsar around its companion (its ‘year’) is seven hours and 45 minutes, with the pulsar moving at a maximum speed (revealed by the Doppler effect) of 300 kilometers per second – one tenth of the speed of light – and an average speed of about 200 kilometers per second, as it zipped around its companion. The size of the orbit traced out at this astonishing speed in just under eight hours is about 6 million kilometres, roughly the circumference of the Sun. In
other words, the average separation between the pulsar and its companion is about the radius of the Sun, and the entire binary pulsar system would neatly fit inside the Sun.

  All pulsars are neutron stars; the orbital parameters showed that in this case the companion star must also be a neutron star. The system was immediately recognised as an almost perfect test bed for the General Theory – and, indeed, for the Special Theory, as well. As I have explained, one of the key tests of the General Theory is the advance of the perihelion of Mercury. The equivalent effect in the binary pulsar (the shift in the ‘periastron’) would be about 100 times stronger than for Mercury, and whereas Mercury only orbits the Sun four times a year, the binary pulsar orbits its companion 1,000 times a year, giving much more opportunity to study the effect. It was duly measured and found to conform exactly to the predictions of Einstein’s theory – the first direct test of the General Theory made using an object outside the Solar System. By feeding back the measurements of the shift into the orbital data for the system, the total mass of the two stars in the system put together was eventually determined to unprecedented accuracy, as 2.8275 times the mass of our Sun.

  But this was only the beginning of the use of the binary pulsar as a gravitational laboratory in which to test and use Einstein’s theory. Extended observations over many months showed that, once allowances were made for the regular changes caused by its orbital motion, the pulsar actually kept time very precisely. Its period of 0.05903 seconds increased by only a quarter of a nanosecond (a quarter of a billionth of a second) in a year, equivalent to a clock that lost time at a rate of only 4 per cent in a million years.

  The numbers became more precise as the observations mounted up. For 1 September, 1974, the data were: period, 0.059029995271 sec; rate of increase, 0.253 nanoseconds per year; orbital period, 27906.98163 seconds; rate of change of periastron, 4.2263 degrees of arc per year.

  The accuracy of the observations soon made it possible to carry out more tests and applications of the theory of relativity. One involves the time dilation predicted by the Special Theory of Relativity. Because the speed of the pulsar around its companion is a sizeable fraction of the speed of light, the pulsar ‘clock’ is slowed down, according to our observations, by an amount which depends on its speed. Since the speed varies over the course of one orbit (from a maximum of 300 km/sec down to ‘only’ 75 km/sec), this will show up as a regular variation of the pulsar’s period over each orbit. And because the pulsar is moving in an elliptical orbit around its companion, its distance from the second neutron star varies. This means that it moves from regions of relatively high gravitational field to regions of relatively low gravitational field, and that its timekeeping mechanism should be subject to a regularly varying gravitational redshift.

  The combination of these two effects produces a maximum measured variation in the pulsar period of 58 nanoseconds over one orbit, and this information can be fed back in to the orbital calculations to determine the ratio of the masses of the two stars. Since the periastron shift tells us that the combined mass is 2.8275 solar masses, the addition of these data reveals that the pulsar itself has 1.42 times the mass of our Sun, while its companion has 1.40 solar masses. These were the first precise measurements of the masses of neutron stars.

  But the greatest triumph of the investigation of the binary pulsar was still to come. Almost as soon as the discovery of the system had been announced, several relativists pointed out that in theory the binary pulsar should be losing energy as a result of gravitational radiation, generating ripples in the fabric of spacetime that would carry energy away and make the orbital period speed up as the binary pulsar and its companion spiraled closer together as a result.

  Even in a system as extreme as the binary pulsar, the effect is very small. It would cause the orbital period (about 27,000 seconds) to increase by only a few tens of a millionth of a second (about 0.0000003 per cent) per year. The theory was straightforward, but the observations would require unprecedented accuracy. In December 1978, after four years of work, Taylor announced that the effect had been measured, and that it exactly matched the predictions of Einstein’s theory. The precise prediction of that theory was that the orbital period should decrease by 75 millionths of a second per year. By 1983, nine years after the discovery of the binary pulsar, Taylor and his colleagues had measured the change to a precision of 2 millionths of a second per year, quoting the observed value as 76+2 millionths of a second per year. Since then, the observations have been improved further and show an agreement with Einstein’s theory that has an error less than 1 per cent. This was a spectacular and comprehensive test of the General Theory, and effectively ruled out any other theory as a good description of the way the Universe works.

  But astronomers were not prepared to rest on their laurels, and kept searching for other objects which might be used to test the General Theory. Their latest success involves a neutron star and a white dwarf star orbiting around each other some 7,000 light years from Earth. The neutron star – another pulsar, dubbed PSR J0348+0432 – was discovered by radio astronomers using the Green Bank Telescope, and its companion was soon detected in optical light, with the system being studied using both optical and radio telescopes around the world from late 2011. The two stars orbit around each other once every 2.46 hours, with the pulsar spinning on its axis once every 39 milliseconds – that is, roughly 25 times per second. The same kind of analysis as that used for the binary pulsar reveals that in this case the neutron star has a mass just over twice that of the Sun, with a diameter of about twenty kilometres, while the white dwarf has a mass a bit less than 20 per cent of the mass of the Sun. The distance between the two stars is about 1.2 times the radius of the Sun, just over half the Sun’s diameter; so once again the whole system would fit inside the Sun. With the measured orbital properties, this implies that gravitational radiation should make the orbit ‘decay’ at a rate of 2.6 × 10–13 seconds per second; the measured rate is 2.7 × 10–13 seconds per second, with an uncertainty of + 0.5.c Over a whole year, this amounts to just 8 millionths of a second. This is an even better test of the General Theory, partly because of the larger mass of the pulsar (the most massive neutron star yet discovered) compared with the neutron stars in the original binary pulsar system.

  Over the years ahead, continuing observations will provide even more precise tests of the General Theory. But the accuracy of the test is already so precise, and the agreement with the predictions of Einstein’s theory is so good, that the General Theory of Relativity can now be regarded as one of the two most securely founded theories in the whole of science, alongside quantum electrodynamics. So what can this spectacularly successful theory tell us about the Universe at large?

  The Universe at large

  The name ‘General Theory of Relativity’ actually has a double meaning. It is general because it applies to accelerated motion and gravity, not just to objects moving in straight lines at constant speed. This is the sense in which Einstein originally used the term. But it is also general in the sense that it applies to everything – the entire Universe, all of space and time and all it contains. Indeed, strictly speaking the General Theory only applies to an entire universe. When we use it to describe the orbit of Mercury or the behaviour of a neutron star, what we are really doing is using an approximation in which the local metric is stitched onto the metric of surrounding flat spacetime, assumed to extend off forever. The influences of other parts of the Universe are ignored, because they are so small. But Einstein, as I describe in the next chapter when I get back to his life story, realised at once that what he had discovered was a description of the whole Universe, and in 1917 he published solutions of his equation, the first of the ‘cosmological models’, as these descriptions of spacetime are called, derived from the General Theory. Note the plural. There is no unique cosmological solution to the equations, but several possibilities, describing different kinds of universe. The lower case (universe) is used when talking about these models;
the capital (Universe) is reserved for when we are discussing the Universe we live in.

  This came as a surprise to Einstein. In 1917, he was looking for a unique solution to his equations, a description of the Universe. At that time, what we now know to be our home galaxy, the Milky Way, was thought to be the entire Universe, and the Milky Way seemed to be an essentially static collection of stars, eternal and unchanging on the largest scale, apart from random motions. The solution Einstein found nearly matched this expectation, with one irritating defect. His model would not stay still. Depending on how you looked at it, it would either expand forever or contract. He solved the problem by adding another term to the equations, what became known as the ‘cosmological constant’, which had the sole purpose of holding the model still, to match the way he thought the Universe was. According to the physicist George Gamow, Einstein later described this as the ‘biggest blunder’ of his career, for reasons that will become clear.d