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Einstein's Theory of Gravity

Newton's theory of gravity is one of the most significant triumphs of scientific effort. Its predictions are vindicated by observations with extraordinary accuracy. When Adams in England and Le Verrier in France used the motion of Uranus to predict the existence of Neptune they were expressing the trust people have come to place on Newtonian gravitation as a very good approximation to reality. However as with any theory there are regimes where it is no longer applicable. For example the perihelion of Mercury is observed to precess at the rate of 43 arc-seconds every century. This ``fast'' precession cannot be explained by Newtonian gravitation. Another issue that came back to haunt Newton's theory of gravity was its ``action at a distance'' nature. Following the success of Maxwell's field theory of electromagnetic forces it was becoming increasingly necessary to find a similar field theory for gravitation. It was to meet this challenge that Einstein's theory of gravitation rose. It is an even better approximation to physical reality than Newton's theory, and reduces to Newton's theory when applied to the regime where Newton's theory is expected to work. Perhaps even more importantly it provides a geometric understanding of gravitation, which makes it more satisfying than the action at a distance nature of Newtonian gravitation.

Special Theory of Relativity

Before we enter Einstein's theory of gravitation we should take a detour into another one of his very famous works. One of the problems that faced physics at the turn of this century was one of relativity. In Newtonian physics there were a set of observers, called the inertial observers, to whom the laws of dynamics were the same. They all observed that the Newton's laws were satisfied. These observers were all un-accelerated, i.e., at rest or moving at uniform speed. The laws of dynamics were relative to the inertial observers. This is referred to as Galilean relativity. This was the situation until Maxwell's theory of electro-magnetism. Inspite of its stunning successes, it had one huge problem. Not all inertial observers saw the same Maxwell's laws. Physics was no longer relative. Two inertial observers at uniform motion relative to each other would measure different speeds of light. Maxwell himself was aware of this and attempted to solve the problem by defining his theory to be applicable only to observers stationary relative to a frame defined by a substance called Ether that filled all space. This however was not just an unsatisfying solution, it was also untenable from an observational point of view. No amount of observations led to any detection of earth's motion through this Ether that filled space. Einstein decided that the problem was not with Maxwell's laws but with Galilean relativity. He restored relativity to all of physics by defining a new set of inertial observers who all agreed on the laws of physics. This meant of course Newton's laws couldn't be correct. In fact it turned out that they were only true for inertial observers at rest or moving very slowly (compared to the speed of light). The laws of dynamics had to be changed for inertial observers moving very fast. One of the most important changes was that nothing could move faster than the speed of light. This was a consequence of the requirement that all inertial observers (regardless of how fast they moved) had to measure the speed of light in vacuum to be the same number c. Newton's second law was still true in the sense that the rate of change of momentum of a body was proportional to the force applied, but the meaning of momentum was now different. It became a function of the speed of the body such that it is increasingly more difficult to accelerate a body as it gets faster. For a body of finite mass it takes an infinite amount of energy to accelerate it to the speed of light. Only particles with zero mass (like the light particle, photons) can travel at the speed of light.

Gravitation

One consequence of the new theory of relativity meant of course that Newton's theory of gravitation was no longer relative to inertial observers moving relative to each other either. This meant that a new theory of gravitation had to be found which would be relative according to the special theory of relativity. Now a basic tenet of Newtonian relativity is the concept of absolute space. According to Newton there was an absolute space that did not change or alter and did not care about the state of the observer. The experiment that led him to this conclusion was one he performed himself. He strung a bucket full of water with a rope. He rotated the bucket so the rope was twisted and then let it go. The bucket started spinning around. The water level of course is flat when the bucket starts to spin. Then gradually the water starts to pick up the rotation from the bucket. Eventually the water is rotating at the same speed as the bucket. At this point the surface of the water is curved into a parabola. When the bucket was moving relative to the water the level was flat. It was only when water wasn't moving relative to the bucket that the surface became curved. So Newton concluded it wasn't the motion of the bucket that changed the surface of the water, because by the time the surface of water was affected the water wasn't moving relative to the bucket anymore, but that it was the motion of the water itself which was significant. Somehow the water was aware of the fact that it was in rotation. And so Newton concluded that there was an absolute space that decided what did and didn't have a force acting on it. And only observers at rest or in uniform motion relative to that absolute space could be inertial observers.

Ernst Mach, an Austrian physicist who worked in the end of the last century disagreed with Newton's interpretation of the bucket experiment. He held that all knowledge was derived from sensations. So he refused to admit any statement in natural science that wasn't empirically verifiable. This led him to dismiss Newton's absolute space. He argued instead that the water was responding the mass around it, like the Earth, that it was rotating relative to. As he pointed out, the bucket was of small mass, but if it was made ``several leagues thick'' no one was competent to say how the water would react. His theory of relativity thus did not allow any absolute space. Inertial observers in his theory were at rest or in uniform motion relative to some space that was defined by all the material in the Universe.

This new statement of relativity strongly influenced Einstein while he was developing his theory of gravity. He postulated the strong principle of equivalence. According to this principle, in the presence of a gravitational field (say on Earth or around Sun), at each point it is possible over a small volume to define an observer for whom the laws of physics are identical to that of an unaccelerated observer (i.e., an observer who has no forces acting on him/her). Thus in some ways Einstein's theory of gravity falls in between Machian and Newtonian relativity. The material around our small volume of interest does define the frame identical to the unaccelerated frame (as far as physical laws go), but once that frame is defined the details of the mass distribution no longer matter for the entire small volume.

Stating the equivalence principle in this way brings it very close to a different field that had been worked on before at some length. Imagine a curved line. If we magnify a small portion of it, it looks less curved. We can continue our focusing into smaller and smaller sections of the curved line. Eventually we'll be looking at such a small portion of the curved line that it will look straight to us, much like the way the Earth seems flat at close range even though we know that it is spherical. Note how close this is to the idea that in a small enough volume about a point in a gravitational field we can define a frame where we can forget that there is a gravitational field. This analogy led Einstein to conclude that the gravitational field was infact a statement about the geometry of space-time itself. The presence of massive objects like the Sun causes space time to curve like the surface of the Earth is curved. The curvature of space time is much harder to visualize than the curved surface of a balloon or earth of course. The Earth is a two dimensional curved surface occupying a three dimensional volume. Its far harder to imaging a curved three or four dimensional volume. Regardless the mathematics is very similar and it is possible to carry over the same mathematical arguments from curved two dimensional surfaces to higher dimensions. In the absence of mass the geometry of space time is flat, in the way the table top is a flat two dimensional surface. When there is mass present the space time curves to act like the surface of Earth or the surface of water in the rotating bucket. Except that the curved ``surface'' is in four dimensions (three space dimensions and one time) rather than two. As is to be expected, indeed is required, the Einstein's theory of gravity reduces to Newtonian gravity when the gravitational field involved is not very intense, like in the solar system. Which is necessary given the wonderful successes of Newtonian gravity in explaining the motions of planets. Newtonian gravity fails at Mercury, closest to Sun, where the gravitational field is the most intense. And there the corrections made by Einstein's gravity match with observations perfectly. The unexplained fast precession of Mercury is exactly what is to be expected once corrections due to Einstein's theory of gravity is taken into account.

Some consequences of Einstein's theory of gravity fall straight out of the basic propositions. Because the presence of mass curves space time, ``straight lines'' are no longer straight in the way we think of them in flat space time. Imagine two people back to back on the North Pole starting to walk out in what they think are straight lines. If the Earth were flat they would never meet. Of course we know that that won't be the case and they will meet face to face at the South Pole. This is drastically different from what we learnt in Euclidean geometry where parallel lines never meet and a straight line extends out in two directions infinitely. Whereas in curved space they turn around and meet. Similarly in curved space time ``straight lines'' behave strangely. Two rays of light for example that are parallel in a flat space time would carry on parallel to each other never reducing or increasing the distance to each other. In curved space time they would change the distance between each other. This was observed by Sir Arthur Eddington when he looked for stars near the Sun during a complete solar eclipse and found that they were in different positions to where they were when the Sun was in a different part of the sky. This meant that the light rays reaching us from the stars had been bent by the curved space time due to the Sun's mass.

Another consequence that follows from the equivalence principle is the dilation of time. Imagine two points in a gravitational field, say the Earth and the Pluto in the Sun's gravitational field. According to the strong equivalence principle around each point there is a small volume where we can define an observer (inertial observer) where the physical laws are the same as an un-accelerated observer. But the volume must be small. So the observer who is inertial in the small volume around Pluto won't appear inertial on Earth at all. Imagine a clock in the hand of this observer on Pluto. It will work according to the time scale the inertial observer at Pluto thinks is appropriate. This won't be appropriate at all to the inertial observer at the Earth. So the identical clock on Earth will run differently from the clock on Pluto. In fact as we get closer to Sun, i.e., as the gravitational field intensifies the clock ticks slower compared to what it would in the absence of the Sun. Of course sitting on Earth we can't tell that it is ticking slower, because our standards of time (like how fast we grow old, etc.) are also slowed (remember all physical laws are affected). But if we were to compare clocks in Pluto and Earth we would be able to tell the difference. In fact this is what happens when we look at an atom radiating light. This is a natural clock and consequently will run slower on Earth than on Pluto. As the photon is radiated on Earth it has the color appropriate to the difference of energies of the stationary states the atom is making the transition between. But because of the difference in ticks of the clocks on Pluto the color will appear inappropriate for the same transition. It will appear redder on Pluto where gravity due to Sun is less intense than on Earth. This is called gravitational redshift and has been observed for light coming from white dwarves which have an intense gravitational field.

There are other theories of gravity as well, some not as elegant as Einstien's theory, others of comparable elegance. However the final word in science comes not from prejudices about heuristics but empirical evidence. And from that point of view, Einstein's theory is far ahead of its nearest competitors. But it continues to be challenged by observational tests. One of the most interesting ones being done today involves a yet unobserved prediction of the theory. Just as Maxwell's theory of Electromagnetic fields predicted the presence of waves, later identified as light, Einstein's theory for gravitational fields predict the presence of gravitational waves. However these are extremely difficult to observe and require sensitivity that is yet to be achieved with current technology. However there already exists indirect evidence through pulsars that have been observed to be slowing down because of energy radiated out in gravitational waves. These observations led to Nobel prizes for Joseph Taylor and Russell Hulse. However efforts continue to directly detect these waves in the geometry of the very space time we exist in.


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