The year was 1915. Albert Einstein was struggling to discover the fundamental equation of his relativistic theory of gravity. His struggles would take a few more months before his revolutionary conception would be completed – on November 25, matter and geometry would be revealed to have been equal partners in the scheme of things, and general relativity would be born. But we’re getting ahead of ourselves, let’s step back a bit.
It was Newton’s insight that there is a common law governing the fall of an apple to the ground and the ‘fall’ of the moon around Earth, which led him to the law of gravity. It states that between two bodies of masses M1 and M2, there is a force of attraction which is proportional to the product of their masses (M1 times M2) and inversely proportional to the square of the distance between them. This was the discovery of a new law of nature, distinct from Newton’s second law of motion.
There are two distinct ‘kinds’ of mass involved here. The gravitational mass, Mg – which ’causes’ the gravitational attraction by producing a gravitational field – and the inertial mass, Mi, which appears in Newton’s second law of motion as a measure of a body’s reluctance to respond to any applied force. It is a curious fact that these two masses happen to be equal (Mg = Mi), for no apparent reason.
In 1905, when developing his special theory of relativity, Einstein had already merged the once-distinct space and time of Newtonian physics into a unified entity called space-time. This was a consequence of the speed of light in vacuum being a constant – about 300,000 kms/second – independent of the motion of its source. It meant that both the where and when of an event, described by three location labels and a time stamp, assigned by two different observers (named Alice and Bob) are necessarily different when the observers are moving relative to each other. The fixed speed of light also prescribed how the labels of two uniformly moving observers are to be related through a mathematical dictionary called the Lorentz transformations. The transformations implied that no object can move faster than at the speed of light.
Special relativity also stipulated a rule to obtain physical distances and elapsed times between pairs of events given their location labels and time stamps. Such rules define a metric geometry. In that vein, the fixed, unchanging geometry that describes space-time according to special relativity is known as Minkowski geometry.
From special to general relativity
For Einstein, this was not satisfactory and comprehensive enough. Special relativistic rules could be used only by observers in uniform relative motion. Its geometry remained fixed, irrespective of the presence of any material bodies. Finally, the highly successful Newton’s law of gravity contradicted special relativity by violating light’s speed limit.
Resolving this contradiction occupied Einstein for eight years. The search began with his “happiest thought” – the realisation that gravity is relative. A person falling freely under gravity, will not detect any force of gravity in the vicinity! This was an implication of the curious equality of gravitational and inertial mass. The equality was actually a clue about the nature of gravity. Say Alice and her entire lab are under free fall. In her freely falling lab, Alice will feel no gravity and is free to use special relativity. But Bob is standing on the ground and sees Alice and her lab accelerating as they plummet down. Einstein demanded that the descriptions of both Alice and Bob must be equivalent, and called that the principle of equivalence.
This principle offered an opportunity to apply special relativity even in the presence of a gravitational field by simply referring to a freely falling observer. An arbitrary gravitational field now means arbitrary relative motion. And Einstein, using the principle of equivalence, applied the phenomenon of special relativistic length contraction – where objects moving faster appear to be shorter to a parallelly moving observer – deduced that the geometry of space is changed by the presence of a gravitational field. But gravitational field is determined by the distribution of matter. So, geometry must also be determined by matter distribution.
This was a highly satisfactory synthesis: the limitation on observers is removed, space-time geometry need not be fixed and can depend on the presence of material bodies, and it’s the gravitational force that makes all this possible. The precise articulation of these qualitative considerations is called general relativity. In technical terms, it uses the framework of Riemannian geometry, with Minkowski space-time as a special case of a flat geometry (Riemannian geometry describes a world where the angles of a triangle don’t add up to 180º and the ratio of a circle’s circumference to its diameter isn’t π). The precise law itself, that relates space-time geometry with matter-energy distribution, is given by the Einstein field equation. This is a much richer theory and has diverse implications.
Clock rates and GPS
The geometrical structure of Riemannian space-time introduces a new source for affecting the rates at which clocks tick. Special relativity already predicted that a moving clock ticks at a slower rate. General relativity predicts that clocks at sea level run faster than those at mountain tops. The atomic clocks on-board the satellites of the global positioning system (GPS) are affected by both these effects. These rate changes must be factored in correctly for the ubiquitous GPS system to work.
Unlike Newtonian gravity, general relativity predicts that a light beam passing near a massive object like the Sun bends. And like a mirage, someone looking at the beam is misled into believing the beam originated from a different location in the universe. This phenomenon, called gravitational lensing, was famously verified by the Eddington expedition of 1919. This phenomenon has led to the discovery of many gravitational lenses in the cosmos, and has become an important astronomical tool to infer the presence of faint matter distributions on galactic scales.
Stars to black holes
The stars we see dotting the night sky, as well as our Sun, are understood to be straddling a fine balance – between the inward pull of their own gravity and the outward forces of the nuclear reactions in their interiors. When the two balance each other, we have an astrophysical body. But while a star abiding by Newton’s description of gravity remains stable, a ‘relativistic’ star can become unstable.
Consider: A massive star needs more outward pressure to withstand its own implosive gravity, which in turn increases the gravitational pull, and so forth, until an uncontrollable positive-feedback mechanism becomes possible. Einstein’s work predicts that for a sufficiently massive star, a runaway process ensues, leading to an unstoppable and complete collapse of the star, and a black hole is born (or perhaps the more bizarre naked singularity). This instability is in fact the basis for the prediction of astrophysical black holes.
A non-omnipresent universe
Climbing up to cosmological scales, general relativity naturally accommodates a dynamical universe. Together with Edwin Hubble’s observations that our universe is expanding, general relativity implies that as long as the universe has a normal kind of matter, it must have evolved from a point-like size a finite time ago! This is the prediction of the Big Bang. And the expanding space-time geometry is a crucial ingredient in developing the detailed picture of how our universe evolved to its present composition.
Waves in the geometry
We are familiar with water waves, sound waves, electromagnetic waves, etc. Can there be waves of gravity? Yes, says general relativity. And like the other waves, these too can carry away energy. This provides a new mechanism for a pair of orbiting stars to lose energy, come closer and circle faster. The discovery of just such a system received the Nobel Prize for physics in 1993. In the system, the stars’ orbits were shrinking at a rate predicted by general relativity. However, a direct detection of gravitational waves is still awaited.
In fact, there’s a proposal to set-up a gravitational wave observatory in India.
Impressive as it is on the observational front, general relativity has its shortcomings, too, while dropping hints about how they can be rectified. For example, in the two extreme situations of complete gravitational collapse and the Big Bang creation of the universe, the theory predicts ‘singularities’ – points in space-time where the theory breaks down. For another: the theory says black holes should have zero temperature and be absolutely cold, yet the same theory also indicates that they behave as ‘hot bodies’.
The British physicist Stephen Hawking, renowned for his work on theories describing black holes, recognised that the inaccuracies of general relativity could be offset by resorting to quantum mechanical fluctuations near the horizons of black holes. Quantum mechanics allows new types of processes beyond what a classical framework allows. And using them, he was able to explain how black holes could behave as ‘hot bodies’. The implication was that classical space-time may possibly have a granular microstructure much like the atomic structure of matter. Taking such an insight forward to its logical conclusion has become the daunting responsibility of a quantum theory of gravity.
And so, the conceptual revolution unleashed a hundred years ago is already anticipating a further revision.
Ghanashyam Date is a professor of physics at the Institute of Mathematical Sciences, Chennai. He can be reached at [email protected]