Pioneers Of Science: Einstein's General Relativity

PIONEERS OF SCIENCE: EINSTEIN'S GENERAL RELATIVITY.

In the last essay in this series, we saw that Einstein had found that his Special Theory of Relativity conflicted with Newton, and that this had something to do with the Sun vanishing. So, what would happen if our parent star were to do that? Well, surprisingly, we would go about our lives as if nothing had happened. This is because light travels at finite speed, and at 160 million miles per hour it takes about eight minutes to travel the 149, 597, 870 km to Earth. Think of the oncoming darkness behind the last wave of light as akin to ripples spreading out on a pond after a stone impacts its surface. But, bathing our world in warmth and light is not the only influence the Sun exerts on our planet, because its gravity is what keeps all the planets, moons, asteroids and comets locked in orbit around it. Newton had calculated that the force of gravity was dependent on the mass of an object and the square of the distance between objects. From this assumption, Newton believed that if the Sun were to just vanish gravity would instantly switch off and all the planets would drift away.

(Einstein. Image from wikimedia commons)

It was that assumption that put Newton’s law of gravity in conflict with the Special Theory of Relativity. Although the Earth would not see the Sun vanish until the last light took the eight minutes required to close the distance, it would immediately feel gravity switch off. But this was expressly forbidden by Einstein’s theory. Nothing can travel faster than light, including any kind of information. Therefore, there was no way that news of the Sun’s untimely demise could be transmitted via gravity in an instant. Unfortunately, the Special Theory was not robust enough to deal with problems like this. Remember, that it was special in the sense that it only dealt with constant motion in a straight line. It did not apply to accelerated motion, changes in direction and, most importantly, it ignored gravity.

Although Einstein did not complete his General Theory until November 1915, the thought that set him on the right path had occurred years earlier. In 1907, Einstein was sat in a chair at the patent office in Bern, when his thoughts turned to jumping out of the window. But these were not the imaginings of a suicidal man, it was (according to Einstein) the happiest thought of his life. This was because Einstein realised that if he were to fall from a height, it would seem as if there was no gravitational field. Ignoring the effects of wind resistance, Einstein imagined dropping various objects around him, and saw that they must move relative to him in a state of uniform motion. From this mental picture, Einstein concluded that the force of acceleration must precisely cancel out the force of gravity. This in turn meant that acceleration and gravity must be exactly equal to one another.

Well, that’s nice. But it may still seem an unlikely candidate for the happiest thought of a man’s life. Remember, though, that for three hundred years Newton had shown the world how to calculate the strength of gravity and had shown its hand at work in a falling apple, the rising tides, the curve of a projectile and the orbiting planets. But he had not been able to say what gravity really was. It had a mysterious element to it and that made it a rather daunting subject for study. But now that Einstein had established the equivalence principle he could turn his attention to acceleration and the mathematical insights that he gained from this would apply to gravitation. After all, gravity was acceleration.

THE 'ELEVATOR' THOUGHT EXPERIMENT

So, what can we learn from studying acceleration? A good place to start would be in a lift that is travelling in an upward direction, because this will paint a nice picture of how acceleration can stand in for gravity. Picture yourself in a lift, standing on a set of bathroom scales. The scales are working just fine, but as soon as the lift starts to ascend, the scales go up. But no need to panic, you are not piling on the pounds. What is happening is that as the lift goes up, the floor pushes the scales against your feet, just as a car seat pushes against your back during acceleration. This results in the scales being squeezed, which the mechanism registers as a higher weight.

(A good place to learn about gravity. Image from wikimedia commons)

This fact can form the basis of an interesting thought experiment. Let’s imagine twins (let’s call them Kath and Kim), one standing on a set of scales inside a lift, the other standing in what looks like a lift but is in fact the cleverly disguised interior of a spaceship. The question we put to them is, with no windows to look out of, can they tell who is in the lift and who is in the rocket heading for the depths of Space? Well, at first one might be tempted to say that the set of scales will provide a giveaway clue. Because, as the rocket gets further away from the Earth, gravity will diminish and its occupant will weigh less and less, and this will be reflected in the scales. But remembering the lesson we learned in the lift will reveal the flaw in that argument. As we have seen, upward acceleration can be a stand in for gravity. In principle, it would be possible to increase the acceleration of the upward thrust so that it precisely compensates for the diminishing effects of gravity. From this conclusion, we can add new conditions for observers who claim to be at rest. ’All observers, regardless of their state of motion, can claim to be at rest while everyone else is in motion, so long as they include a suitable gravitational field’.

Ok, now let’s imagine that Kath is walking around inside her lift, eating an apple, when the engines on the spaceship are switched off. As she is far from any planets, she feels the effects of weightlessness. She drops her apple and it stays floating in the air. Surely now it is obvious which twin was in the real lift back home. How could Kim not feel the effects of gravity while stuck inside a lift? Before I said that the scales go up if the lift goes up. So, logically, if the lift descends Kim’s weight would seem to decrease. And if the lift were to plummet in freefall, she would feel all the effects of weightlessness, just as Einstein had imagined would happen, were he to leap out of his office window. As the lift falls down its shaft, Kim could drop her apple and it too would float in the air. In fact, Kim could perform all the neat tricks that astronauts entertain us with in the absence of gravity.

But, let’s suppose that the lift has emergency breaks that bring it to a halt. Now, gravity comes back with a vengeance and the apple drops to the floor, as does Kim. Meanwhile, the engines on Kath’s spaceship are re-ignited and she and the apple seem to fall to the floor as well. Actually, the apple and Kath stayed where they were and the floor rushed up to meet them. As Einstein developed his theory of General Relativity, he saw that these two events would be indistinguishable. Remember the apple that fell in Newton’s day? Newton could either claim that the apple fell to the ground, or that the Earth moved up to meet the apple, and either claim would be just as valid. There was a problem with the latter view, though, because it only worked if the Earth were flat. But the Earth is a globe, and if the Earth moved up to meet an apple in the northern hemisphere, it would move away from objects in the south. So, how can gravity work the same in all four corners of the world?

THE 'MERRY-GO-ROUND' THOUGHT EXPERIMENT

Einstein admitted that this was the hardest problem he had ever tackled and (just as Newton had, three centuries before) he became obsessed with gravity. Then, in 1912 he made an important step when he applied the Special Theory to accelerated motion. Einstein was fond of saying that something should be kept as simple as possible, but no simpler. Adhering to this rule, he imagined a kind of acceleration where only the direction of motion changed while the speed stayed the same. If you have ever ridden on a round-a-bout, you will have experienced this kind of acceleration. Now, let’s imagine that Kath and Kim are each given a ruler, both of which are exactly the same length as the other. Kath is asked to lay the ruler end over end along the outer rim to measure the ride’s circumference. Kim is asked to measure from the centre of the ride to the outer edge and find its radius. When the two measurements are compared, the ratio should be two times pi or roughly 6.28. But Special Relativity shows that this would not be the case if the ride was spinning and Kath moved in the direction of motion. If that were to happen, Kath’s ruler would shrink due to the effects of Lorentz contraction, but because there is no relative motion between her and the ruler, she does not consider it to have changed in any way. As for Kim, Lorentz contraction does not affect her because she is moving at an angle that is ninety degrees to the direction of motion. Because Kath’s ruler has shrunk, her measurement of the circumference will be such that the ratio will be larger than two times pi. They seem to have broken the laws of Euclidian geometry.

(Not just child's amusement, but a handy tool for developing general relativity. Image from wikimedia commons)

Einstein noted that Euclidian geometry (the kind that says parallel lines never meet etc) only applies for flat surfaces. But if you draw a circle on a piece of paper, and then draw an identical circle on a curved surface like a ball or a saddle, the radius will stay the same but the circumference will change. If it was drawn on a surface with positive curvature (like a ball) the circumference is shorter, and if it is drawn on a surface that has negative curvature (like a saddle) its circumference is longer. And that’s what we found when Kath and Kim tried to measure the merrygoround.

Einstein was beginning to solve a mystery that had persisted since the Principia was first published. Newton had shown the world how to use gravity for all kinds of purposes, but had not been able to say what gravity actually was; how it pulled apples to the ground or kept the planets orbiting the Sun. It was Einstein who provided an answer, as he began to see gravity in terms of a curvature of spacetime. As with any scientist, Einstein had the benefit of previous works with which to build his own work on. When he was tempted back to the ETH from the years 1911 to 1914, his old friend Marcel Grossman introduced him to the work of one Georg Friedrich Riessman. Riessman had studied the theory of curved shapes and surfaces, and it was Einstein who saw that it was actually spacetime that was curved. This lead to the partially successful theory by Einstein and Grossman in 1913. The reason that it was only partially successful was because the theory failed to find the equations that related the curvature of spacetime to the mass and energy within it. Interestingly, Einstein had to use a kind of calculus known as tensor theory to find the appropriate equations, just as Newton had required calculus when he developed his theory of gravity. But, unlike Newton, Einstein had no need to invent that particular form of calculus, because tensor theory had been developed in the 19th century by mathematicians like Bruno Christoffel and Gregorio Ricci. It took until November 1915 for Einstein to find the right equations, and he was actually not the first to do so. That person was a mathematician at the University of Gottingham called David Hilbert, whom Einstein had discussions with. He independently discovered the equations in the summer of 1915, but admitted that Einstein deserved all the credit for discovering the curvature of Spacetime.

Because that’s what gravity is: The curvature of Spacetime caused by the presence of mass.

THE 'TRAMPOLINE' THOUGHT EXPERIMENT

To visualise how gravity works, we need a mental picture. The problem with Einstein’s picture of the Universe is that it reveals there are actually four dimensions and not three as we have always thought. But it is impossible to visualise a fourth dimension, so instead we will imagine a two dimensional surface with gravity acting in the third dimension. We need to visualise a trampoline, upon which live creatures who can only perceive the two dimensions of length and breadth. They have no awareness of a third dimension and terms like ’up’ or ’down’ are meaningless to them. Now, imagine that there is a heavy ball sitting on the trampoline. Whenever a creature ventures too near, it finds itself being pulled toward the ball and so it concludes that a force must be coming from it. But we human beings, able to see in three dimensions, can appreciate that the creature is mistaken. What has happened is that the weight of the ball has distorted the fabric of the trampoline. If a creature comes too close, it slides down the warp in the fabric of its ’Universe’, a warp it cannot see because it exists in a higher dimension. But, it can feel this dimension in the form of gravity. It is the same for us. We cannot see the fourth dimension, but feel its effects whenever gravity pulls us Earthward. If you take a ball and drop it, it is not attracted to the Earth by a mysterious force, it simply slides down a depression in the spatial fabric caused by the presence of the Earth’s mass.

(Image from wikimedia commons)

Ok. Now imagine that a smaller ball is bowled along the trampoline. Remembering the 1st law of motion, if friction were absent the ball would travel in a straight line indefinitely (assuming an infinitely long trampoline!). But, as soon as it falls within the warp caused by the larger balls mass, its path will be bent and it will travel round and round the ball. From the perspective of the creatures, the ball will seem to be in orbit. We can again apply this picture to our solar system. An orbit is a straight line that is curved by the warping of Spacetime, a warping caused by the presence of an object with large mass like the Sun. Similarly, the Earth also causes a distortion in the fourth dimension and this curves the path of the Moon.

So, Einstein had finally shown how gravity keeps the Earth locked in orbit and this new model also showed how gravity cannot inform us of the Sun’s disappearance before light can do so. When there is no mass in the Universe, the fabric of spacetime is smooth and flat, but as soon as a large mass is placed in the Universe, the fabric is warped. However, it does not warp instantly, but rather spreads out from the body at finite speed. The same thing occurs when mass is removed. If the Sun were to vanish, the fabric of spacetime would ’unwarp’, again spreading out from the Sun at finite speed. It was Einstein who calculated how fast the disturbances travelled, and he found that they travel at exactly the speed of light. So, in the unhappy event of God deciding to let there be no more light, the Sun would vanish, but we mortals would not know it. Not until eight minutes later would the last lightwave and the distortion of the fabric of space reach our position.

The picture of the bowling ball warping the fabric of the trampoline also highlights the reason behind Newton’s Law of Gravity which states: Every object in the Universe attracts every other object with a force along the line of centre for the two objects that is proportional to the product of their masses and inversely proportional to the square of the separation between the two objects. The more massive an object is, the greater its gravitational ’pull’. The bowling ball analogy shows that the more massive a body is, the greater the area of surrounding space it warps and so the further its influence will extend. One can also see that the warping lessens as one gets further away, and this is why gravity diminishes by the square of the separation between two bodies.

Having sung its praises, though, it is worth pointing out those aspects of the model that are misleading. We have already seen that this is an imperfect representation, since it shows a three dimensional Universe (two of Space and one of Spacetime) whereas the real Universe has four dimensions. Another misleading fact of the trampoline analogy is that when a bowling ball is placed on its surface, it is the gravitational pull of the Earth that pulls it down, causing the fabric to warp. But this does not happen to the Sun, because there cannot be any gravity to pull it ‘down’. Rather, the warping of Spacetime is gravity. There is also a problem with the ball that stands in for an orbiting planet. In the example of the trampoline, a combination of gravity and the contours of the fabric guide it along its path. In reality, a planet like the Earth always moves along the easiest possible path, which happens to be a curve when the fabric of Space is warped. Finally, this model roughly shows how Space is warped, but the General Theory also includes the warping of Time. This aspect is not included, simply because warping Time is difficult to illustrate.

Although it is hard to picture time being warped in the presence of a gravitational field, the way this effect will manifest itself from the perspective of two observers can be explained. Earlier, we saw that accelerating after somebody who has travelled at constant speed away from you breaks the symmetry between the two of you, because you can no longer claim to be at rest, so your clock must be slower. Because acceleration is equivalent to gravity, what applies to one must apply to the other. This leads to the conclusion that time for an observer experiencing a gravitational field must be slower, relative to an observer who does not feel the force of gravity. Imagine our intrepid twins, Kath and Kim, as the former travels toward a suitably massive object, while her sister stays out of its gravitational influence. Their clocks were synchronised before Kath’s journey began but as she heads into the field of gravity, Kath’s clock runs slower. How slow depends upon the strength of gravity (or, how much the surrounding space is warped by the mass) and, as ever, the effects on time are negligible when applied to gravity as we experience it. For instance, if Kath were to be suspended a few miles above the surface of the Sun, her clock would run at 99.9998 per cent the speed of Kim’s. It follows, then, that the Earth must have an even tinier effect on time. People who live at the bottom of a skyscraper experience slower time than those at the top, because gravity lessens as you move away from the centre of the Earth. But this difference is way to slight to be noticed. However, there are certain regions in the Universe where space is warped so severely, the effect this has on Time is dramatic indeed.

EDDINGTON'S OBSERVATION

Now let's turn attention to what turned out to be the key prediction of general relativity; the bending of light. The reason why this was a key prediction was because it established a way of testing Einstein’s theory via experimentation. Einstein once imagined a beam of light crossing a room that was being accelerated. Normally, light travels in a straight line, but if a room is accelerating the opposite wall will speed up and move forward relative to the light in the time it takes to cross the room. From the point of view of an observer, the light would appear bent. Keeping in mind the equivalence principle, Einstein predicted that gravity must also bend light and in November 1915 he calculated the angle through which light passing close to the Sun would be bent. The answer was 1.57 arcseconds or .00049 of a degree. This meant that, if you observed the light from a star as it grazes the Sun on its journey to Earth, and then observe the same star six months later when the Earth is in a suitable position, the star will seem to have moved. Remember, though, that the star will not have moved at all. Rather, the light was bent, causing its apparent position to differ from its actual position.

Actually performing this experiment could only be done during a total eclipse of the Sun, since that is the only time when the Sun is in the sky and its light doesn’t totally swamp the other stars. On May 29th 1919, an expedition lead by the secretary of the Royal Astronomical Society, Sir Arthur Eddington, travelled to the island of Principe, in anticipation of a total eclipse. The results of the experiment had to wait for months of analysis to be carried out. On November the 6th, 1919, the results of the experiment were revealed at a joint meeting of the Royal Society and the Astronomical Society. Addressing the audience who gathered in the main hall of Burlingham House, the Astronomer Royal, Sir Frank Dyson, announced that Einstein’s theory had been exactly confirmed.

(Sir Arthur Eddington. Image from wikimedia commons)

Before this date, Einstein’s theories had already left an indelible mark on the world of physics, but he was far from the household name that he is today. But when the results of the experiment were trumpeted in The Times with a banner reading new theory of the Universe- Newtonian ideas overthrown, Einstein was catapulted into the limelight to become the iconic figure of science. Why his theory should strike such a chord probably had much to do with the political climate of the time. Although the Armistice was signed on 11th November 1918, a state of war still technically existed between Great Britain and Germany until the signing of the Treaty of Versailles on June 28th 1919. The confirmation of Einstein’s theory represented Science’s ability to transcend political differences and unite enemies. After all, here was a theory proposed by a German-born scientist, that was confirmed by an English astronomer.

It should be pointed out that The Times’ headline was a little misleading when it said Newtonian ideas were overthrown. Einstein’s model of gravity does not force us to abandon Newton entirely. Indeed, his theory of gravity is included in the framework of General Relativity. It’s more appropriate to think of Einstein’s theory as a more refined model of gravity, since it can be used in areas where Newtonian mechanics fail. You may remember that the orbit of Mercury deviates from the path predicted by Newton’s laws and this is because such laws break down under the kinds of gravity you feel that close to an object as massive as the Sun. But when its path is plotted using General Relativity, Einstein’s model precisely accounts for the planet’s strange orbit. So, General Relativity improves upon Newton’s theory of gravity because it reveals the reason behind the inverse square law and can be used in the places where Newton’s laws of gravity break down.

In another sense, though, Einstein’s theory really did overturn Newton’s concept of the Universe. To Newton, Space was the static backdrop upon which the history of the Universe unfolded, and Time stretched from the infinite past to the infinite future, beating out the same time for everyone, everywhere. But Einstein showed that Space was far more dynamic in terms of its relationship with its contents. This is summed up by the physicist John Wheeler’s description of gravity: Mass grips Space by telling it how to curve and Space grips mass by telling it how to move.

REFERENCES

The Elegant Universe by Brian Greene

Science: A History by John Gribbin