SCID007 - Relativity Made Easy on its Centenary Year Nov 2015

Prof S. M. Deen,

University of Keele

Email: s.m.deen@keele.ac.uk

 

For a long time I was trying to see if I can explain in the simplest possible way the essence (in my opinion) of the Theory of Relativity in this Centenary of Einstein’s General Theory of Relativity. This is what I have come up with. Have a drink and a pause.

Below you have: Special Relativity, some easily-readable descriptive para on General Relativity, and an interesting true story on Relativity.

Special Theory of Relativity, which says

1.       The universe is four-dimensional, with one dimension of time and three of space, together constituting what is called the space-time.

2.       Every object in the universe travels at the speed of light in space-time

3.       Nothing can travel faster than the speed of light, which is constant in vacuum at 186,000 miles (300, 000 Km) per second, commonly denoted by the letter c, which stands for the constancy of the speed of light.  If someone is travelling at the speed T in the time direction and S in the space direction, then T² + S² = c². This formula is crucial and used below.

Have another pause. I regret that even my attempted simplest explanation needs diagrams. Consider you are standing at the corner (which we shall call position O) of a football ground, in front of you (i.e. facing you) is one boundary line, which we shall call the X-axis, and on your left-hand direction is another boundary (at right angle to the X-direction) the Y-axis (see Fig 1). Assume you are going to travel for one min(ute).  If you travel in the X–direction, you would travel zero distance in the Y-direction. Equally if you travel on the Y-direction, you would travel zero distance in the X-direction (Fig 1). The chord in the diagram shows the one-min distance from your starting position O.  If you travel in between two directions, say in the direction OR (Fig 2), you would travel the distance ORx on the X-direction and ORy on the Y-direction of the total distance of one min (Fig 2).   I trust all these are crystal clear – if not, please read again.

 

Now I bring in the time direction. Let us suppose the Y-direction is the time direction and the X-axis represents a direction in space. Since the time dimension is orthogonal to any space direction, X-axis can represent any space direction you wish to choose. Assume two friends A and B sitting at the position O. As they are sitting at O, both of them are moving in the time direction at the speed of light c. Now say B has decided to move away from A very fast. Any direction B chooses to move, it would be a space direction, represented here by the X-direction. Below by X-direction I shall mean any space direction chosen by B.

We first examine the equation T² + S² =  c² of Relativity (given earlier), where S is the speed of B in the space direction X, and T is B’s speed in the time direction. If B’s speed S = 0, (that is, B is sitting tight at point O), then T =  c for B, that is, B will be travelling wholly in the time direction Y at the speed of light. On the other hand if S = c for B, then B’s speed in the time direction will be zero (since S = c, T has to be zero for that equation). Therefore for any speed S of B, above zero and below c, there will be a component (i.e. a contribution) to the time direction. If you have understood this equation as explained here, then you have understood the pivotal concept I am using here. The rest is straightforward. So please re-read the explanation until it is crystal clear to you.  By the way, if you are familiar with our normal 3D spatial geometry, then you would know that if you were travelling in the X-direction, there will be no component to the orthogonal Y-direction. But this is not true when the Y-direction is the time-direction in a 4D Universe of Einstein.

If B could move away from A in any space direction X at the speed of light c (the maximum speed possible), then  B will make zero contribution to the Y-direction, as explained above. In that scenario, B would never age, and would be immortal so far as A sitting at O is concerned (in the jargon, it is the A’s Frame of Reference). If B moves away from A at a speed less than c, then it would have a component (i.e. a contribution) in the time direction Y, the actual size of that component would depend on B’s speed.  Say, B moves along OR. After one min, A would reach the point Q on the Y-axis and B on the point R. At that instant, B meets A on A’s position (i.e. A’s Frame of Reference). A’s position would be the point Q, when A’s watch would show that just one min has gone. But the B’s watch which B carried with him in the journey, would show less than one min. This is the famous issue of time dilation in the Theory of Relativity, when someone travelling faster than you would age slower than you.

How slow? For this I offer a small example. If the speed S = 0.6c, that is, 60% of the speed of light c, then from the eqn  T2 + S2 = c2 will become:  T2 + (0.6c)2 = c2, or T2  = (1 – 0.36) c2,  or T2  = 0.64c2, and hence T = 0.8c. Thus our A sitting tight at O travels in the time direction at the speed of light c, while B travelling in the space direction at the speed S = 0.6c will travel at the speed 0.8c in the time direction, thus he aging only 80% of the age of A.  If A has aged 1 min, B will age 0.8 min. 

Why so? Let us revisit the crime-scene as it were.  As we know A sat at O and hence passed the whole of its one min travelling in the Y-direction (the time axis), and hence his watch correctly records that one min has passed. But from A’s perspective, B sped away, and hence B’s one-min journey to R has two components (each a contribution), a Y-component (the time axis) and X-component (the space X-axis), each component being a fraction of a minute. The Y-component is ORy and the X-component is ORx (Fig 2) where:   

              (ORy)²+(ORx)² = (OR)² = (1 min)² = 1 min,

which demonstrates that time and space in space-time are interchangeable. How?  B has travelled only the fraction ORy on the time dimension (Y-axis), which is less than one min as shown by B’s watch, but A perceived a full one min on B’s journey, the difference being made up of the space-component ORx.  Thus space become time – an unheard of situation, i.e. space and time are not independent of each other. This is Relativity for you.

I restate that B had travelled one min only from A’s perspective, and only a fraction ORy from B’s perspective, the difference being made up by the space component ORx. Two important consequences are:

1.       There is no absolute time, all times are relative.  What is one min for someone could be less than one min for someone else  – it all depends on relative speed. And hence the name Theory of Relativity.

2.       The assertion that B had travelled one min to reach the point R is true from A’s perspective, but for B it had travelled only for the fraction ORy.

In my humble opinion, the concept of Time Dilation is the most difficult one to understand conceptually, but found to be true by all experiments. The other formulae, such as, E= mc² can be derived and understood mathematically.

Now the General Theory of Relativity (TGR). For the adherents it is the most beautiful theory. When someone (of course a newspaper man) asked Einstein: “What would happen if your theory is proved wrong?”  He replied “Oh no – in that case a most beautiful theory will be wasted”. Obviously beauty lies in the eyes of the beholder. However TGR is very complex which defies most minds, certainly mine. So I shall state only the substance of it here. In TGR there is no force of gravitation between masses (objects).  Instead of attracting masses, gravity curves the space-time.  The more massive an object is, the higher is its gravity and therefore more it (the gravity) curves the space-time towards the object.  Not only all physical objects, but also light (I repeat light) follow the curves around other objects, giving the impression of being pulled by gravity.  Under Newton’s Gravitation, light is not affected by gravity, but nevertheless his theory works as an approximation of TGR on small scale. Observe that under General Relativity, the time dilation is affected by both speed in the space direction and gravity, and it is slightly less pronounced than in Special Relativity. Finally every prediction of each aspect of Relativity, Special or General Relativity, has been found to be correct by all experiments carried out in the last 100 years.

 Newton and Einstein: Robert Hooke, the first Professional Scientist in the early Royal Society,  produced with some friends the idea of an attractive force between physical bodies, and he approached Newton, the mathematician, for a mathematical formulation of that Idea.  Newton not only did not acknowledge Hooke’s contribution, but also when he became the powerful President of the Royal Society, behaved abominably with Hooke, eventually driving him out of the Royal Society, with vengeance. For Relativity, it was wholly the idea of a single human being, Einstein, who worked alone with paper and pencil and from his pure thought, he produced the remarkable edifice,  the Special and General Theory of Relativity. 

 TGR explains the behaviour, not only of a falling apple on someone’s head, but also of all physical  objects (that have mass) from planets, stars, galaxies in the Universe and indeed the Universe itself. It is more than astonishing that one man thinking alone could produce such a fantastic theory from his thoughts alone – a theory every prediction of which, including that of blackholes, has been found to be correct. Hundreds of physicists would like to prove this theory wrong, but in the last hundred years no such fault has been found. Einstein was not a genius, he was more than that.

Need for a Unified Theory:  Quantum Mechanics (QM) applies to small objects, such as atoms and its constituents where the force of gravity is unimportant.  In contrast TGR applies to larger objects where gravity is significant but Quantum effect is unimportant. But there is a need for a Unified Theory where both Quantum effect and gravity are significant, as in a blackhole.

A blackhole is formed when a star several times more massive than our sun dies (i.e. it stops emitting light, and hence the name blackhole). It then collapses on itself by its huge gravity, crashing itself to a physical size, some claim, of an atom. Some blackholes have the mass of apparently millions or even billions of suns. A blackhole gobbles up other stars close by. Many blackholes have been identified from effect of their huge gravity around their neighbouring stars.  Both QM and TGR apply to blackholes, QM because of the small physical size of a blackhole, and TGR because of its large gravity.  Some physicists working in the area say that TGR is so beautifully integrated with all its parts in a compact way, that it is not feasible to take out one part, modify it and insert it back – apparently one needs to unpick the whole TGR, and nobody has a clue of how to do it. Similar problems also exist in adding gravity into QM.  Some top physicists say that we perhaps need a new Einstein who perhaps woking alone somewhere on this earth will one day produce a Unified Theory of QM and TGR, which is sorely needed.

Einstein and Britain (in fact England)

I shall now end this hard stuff, and tell the true story behind the Theory of General Relativity with a little bit of salt and pepper for a flavour in a Christmas/NewYear spirit. Einstein came up with this beauty in (Nov?) 1915 in the middle of the War when the Brits were not particularly warm towards the Germans. Einstein, a great admirer of Newton, said (I am using a bit of imagination for that tricky situation): “Oh King of Science, your idea of a force of attraction is plain bonkers – leave attraction to young men and women in love. The apple might have hurt your head, but it was not attracted to your august scull. Forgive me for being blunt, but please don’t send me to the Tower. I hear beautiful Mary the Scot, had rather a dreadful time there”.

 But the Brits particularly the English, were not amused, they did want to send him to the Tower, or at least to put him down completely.  On the top of the War, comes this intolerable onslaught from a German upstart, insulting our Newton the greatest scientist, naturally an Englishman, ever lived.  The English were livid. An Englishman with the name Arthur Eddington declared: “I know how to finish that upstart. I shall prove his TGR wrong. Think of the newspaper headline: An Englishman proved the German wrong?” He immediately became the darling of the English Establishment who poured in money for Arthur. Now who was this Arthur? Subrahmanyan Chandrasekhar from India came to study a Physics PhD under our Arthur, but being bored in the long boat trip from Bombay to England, he used the time on trivial pursuits which turned out to be the basis of a theory on the evolution and death of stars.  On seeing this work, Arthur advised Subra: “Forget this nonsense and do some real physics with me”. Years later Subra got a Nobel Prize for that nonsense work. But here our confident Arthur raced ahead to prove TGR wrong.

With “Mad dog and Englishman”, you never know where they would end up. As he looked at TGR, Arthur fell in love with this “beauty”. When a newspaperman asked him: Is it is true that only three persons in the world understand TGR? Arthur paused for a moment and then responded: “I wonder who could be the third?” This is called exclusive love where no rivals are admitted.

 In the 1919 solar eclipse, using some unorthodox procedure, Arthur demonstrated that the light from a planet passing by the side of the sun was curved towards the sun as predicted by TGR. And thus he showed TGR is right.  We English are never daunted by such mishaps – we always rise to the occasion. Came a newspaper headline in England: An Englishman saved the German. Einstein did not get a Nobel Prize for Relativity, but Arthur got a Knighthood.

Finally: to the eternal shame of the Nobel Committee, Einstein was not given Nobel Prize on Relativity, one of the greatest contributions of all time to Physics. He later got a Nobel Prize for his work on an area related to the Quantum theory.

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