In Pursuit of the Unknown (45 page)

The usual way to picture this is to forget time, drop the dimensions of space down to two, and get something like
Figure 51
(
left
). The flat plane of Minkowski space(-time) is distorted, shown here by an actual bend, creating a depression. Far from the star, matter or light travels in a straight line (dotted). But the curvature causes the path to bend. In fact, it looks superficially as though some force coming from the star attracts the matter towards it. But there is no force, just warped space-time. However, this image of curvature deforms space along an extra dimension, which is not required mathematically. An alternative image is to draw a grid of geodesics, shortest paths, equally spaced according to the curved metric. They bunch together where the curvature is greater,
Figure 51
(
right
).

Fig 51
Left
: Warped space near a star, and how it bends the paths of passing matter or light.
right
: alternative image using a grid of geodesics, which bunch together in regions of higher curvature.

If the curvature of space-time is small, that is, if what (in the old picture) we think of as gravitational forces are not too large, then this formulation leads to Newton's law of gravity. Comparing the two theories, Einstein's constant
κ
turns out to be 8π
G/c
⁴
, where
G
is Newton's gravitational constant. This links the new theory to the old one, and proves that in most cases the new one will agree with the old. The interesting new physics occurs when this is no longer true: when gravity is large. When Einstein came up with his theory, any test of relativity had to
take place outside the laboratory, on a grand scale. Which meant astronomy.

Einstein therefore went looking for unexplained peculiarities in the motion of the planets, effects that didn't square with Newton. He found one that might be suitable: an obscure feature of the orbit of Mercury, the planet closest to the Sun, subjected to the greatest gravitational forces – and so, if Einstein was right, inside a region of high curvature.

Like all planets, Mercury follows a path that is very close to an ellipse, so some points in its orbit are closer to the Sun than others. The closest of all is called its
perihelion
(‘near Sun' in Greek). The exact location of this perihelion had been observed for many years, and there was something funny about it. The perihelion slowly rotated about the Sun, an effect called precession; in effect, the long axis of the orbital ellipse was slowly changing direction. That was all right; Newton's laws predicted it, because Mercury is not the only planet in the Solar System and other planets were slowly changing its orbit. The problem was that Newtonian calculations gave the wrong rate of precession. The axis was rotating too quickly.

That had been known since 1840 when François Arago, director of the Paris Observatory, asked Urbain Le Verrier to calculate the orbit of Mercury using Newton's laws of motion and gravitation. But when the results were tested by observing the exact timing of a transit of Mercury – a passage across the face of the Sun, as viewed from Earth – they were wrong. Le Verrier decided to try again, eliminating potential sources of error, and in 1859 he published his new results. On the Newtonian model, the rate of precession was accurate to about 0.7%. The difference compared with observations was tiny: 38 seconds of arc every century (later revised to 43 arc-seconds). That's not much, less than one ten thousandth of a degree per year, but it was enough to interest Le Verrier. In 1846 he had made his reputation by analysing irregularities in the orbit of Uranus and predicting the existence, and location, of a then undiscovered planet: Neptune. Now he was hoping to repeat the feat. He interpreted the unexpected perihelion movement as evidence that some unknown world was perturbing Mercury's orbit. He did the sums and predicted the existence of a small planet with an orbit closer to the Sun than that of Mercury. He even had a name for it: Vulcan, the Roman god of fire.

Observing Vulcan, if it existed, would be difficult. The glare of the Sun was an obstacle, so the best bet was to catch Vulcan in transit, where it would be a tiny dark dot against the bright disc of the Sun. Shortly after
Le Verrier's prediction, an amateur astronomer named Edmond Lescarbault informed the distinguished astronomer that he had seen just that. He had initially assumed that the dot must be a sunspot, but it moved at the wrong speed. In 1860 Le Verrier announced the discovery of Vulcan to the Paris Academy of Science, and the government awarded Lescarbault the prestigious Légion d'Honneur.

Amid the clamour, some astronomers remained unimpressed. One was Emmanuel Liais, who had been studying the Sun with much better equipment than Lescarbault. His reputation was on the line: he had been observing the Sun for the Brazilian government, and it would have been disgraceful to have missed something of such importance. He flatly denied that a transit had taken place. For a time, everything got very confused. Amateurs repeatedly claimed they had seen Vulcan, sometimes years before Le Verrier announced his prediction. In 1878 James Watson, a professional, and Lewis Swift, an amateur, said they had seen a planet like Vulcan during a solar eclipse. Le Verrier had died a year earlier, still convinced he had discovered a new planet near the Sun, but without his enthusiastic new calculations of orbits and predictions of transits – none of which happened – interest in Vulcan quickly died away. Astronomers became skeptical.

In 1915, Einstein administered the
coup de grâce
. He reanalysed the motion using general relativity, without assuming any new planet, and a simple and transparent calculation led him to a value of 43 seconds of arc for the precession – the exact figure obtained by updating Le Verrier's original calculations. A modern Newtonian calculation predicts a precession of 5560 arc seconds per century, but observations give 5600. The difference is 40 seconds of arc, so about 3 arc-seconds per century remains unaccounted for. Einstein's announcement did two things: it was seen as a vindication of relativity, and as far as most astronomers were concerned, it relegated Vulcan to the scrapheap.
⁵

Another famous astronomical verification of general relativity is Einstein's prediction that the Sun bends light. Newtonian gravitation also predicts this, but general relativity predicts an amount of bending that is twice as large. The total solar eclipse of 1919 provided an opportunity to distinguish the two, and Sir Arthur Eddington mounted an expedition, eventually announcing that Einstein prevailed. This was accepted with enthusiasm at the time, but later it became clear that the data were poor, and the result was questioned. Further independent observations from 1922 seemed to agree with the relativistic prediction, as did a later reanalysis of Eddington's data. By the 1960s it became possible to make the
observations for radio-frequency radiation, and only then was it certain that the data did indeed show a deviation twice that predicted by Newton and equal to that predicted by Einstein.

The most dramatic predictions from general relativity arise on a much grander scale: black holes, which are born when a massive star collapses under its own gravitation, and the expanding universe, currently explained by the Big Bang.

Solutions to Einstein's equations are space-time geometries. These might represent the universe as a whole, or some part of it, assumed to be gravitationally isolated so that the rest of the universe has no important effect. This is analogous to early Newtonian assumptions that only two bodies are interacting, for example. Since Einstein's field equations involve ten variables, explicit solutions in terms of mathematical formulas are rare. Today we can solve the equations numerically, but that was a pipedream before the 1960s because computers either didn't exist or were too limited to be useful. The standard way to simplify equations is to invoke symmetry. Suppose that the initial conditions for a space-time are spherically symmetric, that is, all physical quantities depend only on the distance from the centre. Then the number of variables in any model is greatly reduced. In 1916 the German astrophysicist Karl Schwarzschild made this assumption for Einstein's equations, and managed to solve the resulting equations with an exact formula, known as the Schwarzschild metric. His formula had a curious feature: a singularity. The solution became infinite at a particular distance from the centre, called the Schwarzschild radius. At first it was assumed that this singularity was some kind of mathematical artefact, and its physical meaning was the subject of considerable dispute. We now interpret it as the event horizon of a black hole.

Imagine a star so massive that its radiation cannot counter its gravitational field. The star will begin to contract, sucked together by its own mass. The denser it gets, the stronger this effect becomes, so the contraction happens ever faster. The star's escape velocity, the speed with which an object must move to escape the gravitational field, also increases. The Schwarzschild metric tells us that at some stage, the escape velocity becomes equal to that of light. Now nothing can escape, because nothing can travel faster than light. The star has become a black hole, and the Schwarzschild radius tells us the region from which nothing can escape, bounded by the black hole's event horizon.

Black hole physics is complex, and there isn't space to do it justice here.
Suffice it to say that most cosmologists are now satisfied that the prediction is valid, that the universe contains innumerable black holes, and indeed that at least one lurks at the heart of our Galaxy. Indeed, of most galaxies. In 1917 Einstein applied his equations to the entire universe, assuming another kind of symmetry: homogeneity. The universe should look the same (on large enough scale) at all points in space and time. By then he had modified the equations to include a ‘cosmological constant' Λ, and sorted out the meaning of the constant
κ
. The equations were now written like this:

The solutions had a surprising implication: the universe should shrink as time passes. This forced Einstein to add the term involving the cosmological constant: he was seeking an unchanging, stable universe, and by adjusting the constant to the right value he could stop his model universe contracting to a point. In 1922 Alexander Friedmann found another equation, which predicted the universe should expand and did not require the cosmological constant. It also predicted the rate of expansion. Einstein still wasn't happy: he wanted the universe to be stable and unchanging.

For once Einstein's imagination failed him. In 1929 American astronomers Edwin Hubble and Milton Humason found evidence that the universe
is
expanding. Distant galaxies are moving away from us, as shown by shifts in the frequency of the light they emit – the famous Doppler effect, in which the sound of a speeding ambulance drops as it passes by, because the sound waves are affected by the relative speed of emitter and receiver. Now the waves are electromagnetic and the physics is relativistic, but there is still a Doppler effect. Not only do distant galaxies move away from us: the more distant they are, the faster they recede.

Running the expansion backwards in time, it turns out that at some point in the past, the entire universe was essentially just a point. Before that, it didn't exist at all. At that primeval point, space and time both came into existence in the famous Big Bang, a theory proposed by French mathematician Georges Lemaître in 1927, and almost universally ignored. When radio telescopes observed the cosmological microwave background radiation in 1964, at a temperature that fitted the Big Bang model, cosmologists decided Lemaître had been right after all. Again, the topic deserves a book of is own, and many have been written. Suffice it to say
that our current most widely accepted theory of cosmology is an elaboration of the Big Bang scenario.