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Ladies and gentlemen, Einstein is by all criteria the most distinguished physicist of this century.
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No physicist in this century has been accorded a greater acclaim.
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But it is an ironic comment that even though most histories of 20th century physics
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starts with the proforma statement that this century began with two great revolutions of thought –
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the general theory of relativity and the quantum theory.
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The general theory of relativity has not been a stable part of the education of a physicist,
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certainly not to the extent quantum theory has been.
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Perhaps on this account a great deal of my ethology has created an own Einstein’s name
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and the theory of relativity which he founded seventy years ago.
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Even great physicists are not exempt from making statements which, if not downright wrong, are at least misleading.
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Let me quote for example a statement by Dirac made in 1979 and on occasion celebrating Einstein’s 100th birthday.
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This is what he said:
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“When Einstein was working on building up this theory of gravitation,
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he was not claimed to account for some results of observation, far from it.
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His entire procedure was to search for a beautiful theory, a theory of a type that nature will choose.
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He was guided only by considerations of the beauty of his equations.”
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Now this contradicts statements made by Einstein himself on more than one occasion.
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Let me read what he said in 1922 in a lecture he gave titled “How I Came to Discover the General Theory of Relativity”.
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I read Einstein’s statement:
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“I came to realise that all the natural laws, except the law of gravity,
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could be discussed within the framework of the special theory of relativity.
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I wanted to find out the reason for this but I could not attain this goal easily.
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The most unsatisfactory point was the following:
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although the relationship between inertia and energy was explicitly given by the special theory of relativity,
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the relationship between inertia and weight or the energy of the gravitational field was not clearly elucidated.
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I felt that this problem could not be resolved within the framework of the special theory of relativity.
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The breakthrough came suddenly, one day I was sitting on a chair in a patent office in Bern, suddenly a thought struck me.
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If a man falls freely, he would not feel his weight.
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I was taken aback, this simple thought experiment made a deep impression on me.
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This led me to the theory of gravity.
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I continued my thought:
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A falling man is accelerated, then what he feels and judges is happening in the accelerated frame of reference.
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I decided to extend the theory of relativity to the reference frame with acceleration.
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I felt that in doing so I could solve the problem of gravity at the same time.
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A falling man does not feel his weight because in his reference frame there is a new gravitational field
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which cancels the gravitational field due to the earth.
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In the accelerated frame of reference we need a new gravitational field.
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Perhaps it is not quite clear from what I have read precisely what he had in mind.
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But two things are clear.
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First, he was guided principally by the equality of the inertial and gravitational mass,
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an empirical fact which has been really accurately determined and in fact probably the most well established experimental fact.
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The second point is that this equality of the inertial and the gravitational mass
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led him to formulate a principle which he states very briefly and which has now come to be called ‘the principle of equivalence’.
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Let me try to explain more clearly what is involved in these statements I have just made.
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The first point in order to do that I have to go back in time,
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in fact I have to go back 300 years to the time when Newton wrote the Principia.
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The fact that the publication of the Principia is 300 years old was celebrated last year in many places.
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Now,Newton notices already within the first few pages of the Principia
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that the notion of mass and weight are two distinct concepts based upon two different notions.
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The notion of mass follows from his second law of motion which states that if the subject is a body till it falls,
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then it experiences an acceleration in such a way that the quantity, which we call the mass of the body,
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times the acceleration is equal to the force.
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Precisely, if you apply a force to one cubic centimetre of water and measure the acceleration which it experiences
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and then you find that another piece of water, when subjected to the same force, experiences ten times the acceleration,
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then you can conclude that the mass of the liquid you have used is one tenth cubic centimetre.
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In other words then, the notion of mass is a consequence of his law of motion,
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it is a consequence of proportionality in the relation that the force is equal to the mass times the acceleration.
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But the notion of weight comes in a different way.
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If you take a piece of matter and it is subject to the gravitational field, say of the earth,
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then you find that the attraction which it experiences in a given gravitational field is proportional to what one calls the weight.
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For example, if you take a piece of liquid, say water, and you find that the earth attracts it by force,
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which you measure and you find that another piece of the same matter experiences gravitational attraction
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which is, say ten times more, then you say the rate is ten times greater.
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In other words the notion of weight and the notion of mass are derived from two entirely different sets of ideas.
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And Newton goes on to say that the two are the same and in fact, as he says,
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as I have found experiments with pendula made accurately.
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The way he determined the quality of the inertial and the gravitational mass was simply to show that a period of a pendulum,
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a certain pendulum, depends only on its length and not upon the weight or the mass or the body or the constitution of it.
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And he established the equality of the inertial and gravitational mass to a few parts in a thousand.
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Essentially later Wentzel improved the accuracy to a few parts in several tens of thousands.
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Earlier in this century yet was shown the equality to one part at ten to the eleven.
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And more recently the experiments of Dicke and Braginskii have shown that they are equal to one part at ten to the minus thirteen.
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Now this is a very remarkable fact, the notion of mass and weight are fundamental in physics.
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And when one equates them by the fact of experience and this of course is basic to the Newtonian theory.
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Herman Wilde called it an element of magic in the Newtonian theory.
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And one of the objects of Einstein’s theory is to illuminate this magic.
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But the question of course is, you want to illuminate the magic, but how?
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For this Einstein developed what one might call today the principle of equivalence.
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Now let me illustrate his ideas here.
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Here is the famous experiment with an elevator or a lift, which Einstein contemplated.
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Now the experiment is following, here is a lift with a rocket booster.
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Let us imagine that this lift is taken to a region of space which is far from any other external body.
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If this elevator is accelerated by a value equal to the acceleration of gravity,
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then the observer will find that if he drops a piece of apple or a ball,
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it will fall down towards the bottom with a certain acceleration.
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On the other hand, if the rockets are shut off and the rocket simply cools, then if he leaves the same body,
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then it remains where it was.
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Now you perform the experiment now on the same elevator shaft on the earth and you find that if he leaves the body,
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then it falls down to the ground in the same way as it did when this was accelerated and not subject to gravity.
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Now suppose this elevator is put in a shaft and falls freely towards the centre of the earth.
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Then,when you leave the body there, it remains exactly as it was.
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In other words, in this case the action of gravity and the action of acceleration are the same.
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On the other hand you cannot conclude from this that action of gravity and the action of uniform acceleration are the same.
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Let us now perform the same experiment in which the observer has two pieces of bodies instead of just one.
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Then if the rocket is accelerated, then he will find that both of them fall along parallel lines.
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And again, if the acceleration is stopped, then the two bodies will remain at the same point.
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Now if you go to the earth and similarly you have these two things,
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then the two will fall but not exactly in parallel lines, if the curvature of the earth is taken into account.
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The two lines in which they will drop will intersect at the centre of the earth.
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Now if the same experiment is performed with a lift which is falling freely,
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then as the lift approaches the centre of the earth, the two objects will come close together.
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And this is how Einstein showed the equivalence locally of a gravitational field, of a uniform gravitational field,
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with a uniform acceleration, but showed nevertheless that if the gravitational field is not uniform,
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then you can no longer make that equivalence.
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Now, in order to show how from this point Einstein derived his principle of equivalence in a form
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in which he could use it to find gravity, I should make a little calculation.
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Now, everyone knows that if you describe the equations of motion in, say Cartesian coordinates,
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then the inertial mass times the acceleration
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is given by the gravitational mass times the gradient of the potential, gravitational potential.
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There are similar equations for X and Y.
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Suppose we want to realign this equation in a coordinate system which is not XYZ, but a general curvilinear coordinates.
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That is, instead of XYZ you change to coordinates Q1, Q2, Q3.
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And you can associate with the general curvilinear system in metric in the following way:
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The distance between two neighbouring points in Cartesian framework is the DX2 and DZ2.
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On the other hand, if you find the corresponding distance for general curvilinear coordinates,
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it will be a certain quantity each alpha beta with the two index quantity
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which will be functions of the coordinates times DQ Alpha, DQ Beta.
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For example in spherical polar coordinates it will be DR2 + R2 D Theta2 + R2Psi2Theta D Phi2.
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But more generally that will be the kind of equation you will have.
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Now let us suppose you write this equation down and ask what the gravitational equations become,
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then you find that M inertial times this quantity Q.Beta that is the Q..Beta,
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the acceleration in the coordinate beta times this quantity contracted
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is equal to the minus the inertial mass times a certain quantity Gamma, Alpha, Beta, Gamma, called the Christoffel symbols,
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but it doesn’t matter what they are, they are functions of the coordinate, functions of the geometry, times Q.Beta, Q.Gamma,
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and then minus the same gravitational mass times the gradient of the potential.
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You see, the main point of this equation is to show that the acceleration in the coordinate, which is corresponding to that,
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when you write it down in general curvilinear coordinates, the acceleration consists of two terms.
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A term which is geometrical in origin which is a co-efficient the inertial mass in a term from the gravitational field.
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And if you accept the equality between the inertial mass and the gravitational mass,
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then the geometrical part of the acceleration and the gravitational part are the same.
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And this is Einstein’s remark, he said
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“why make this distinction, why not simply say that all acceleration is metrical in origin?”
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And that is the starting point of his work.
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He wanted to abolish the distinction between the geometrical part of the acceleration and the gravitational part
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by saying that all acceleration is metrical in origin.
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Einstein’s conclusion that in the context of gravity all accelerations are metrical in origin
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is as staggering in its own way as rather for its conclusion
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when Geiger and Marsden first showed him the result of the experiments on the large angle scattering of alpha-rays,
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another remark was it was as though you had fired a fifteen inch shell at a piece of tissue paper
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and it had bounced back and hit you.
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In the case of Rutherford, he was able to derive his scattering over night but it took Einstein many years,
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in fact ten years almost, to obtain his final field equations.
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The transition from the statement that all acceleration is metrical in origin to the equations of the field
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in terms of the Riemann tensor is a giant leap.
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And the fact that it took Einstein three or four years to make the transition is understandable,
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indeed it is astonishing that he made the transition at all.
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Of course, one can claim that mathematical insight was needed to go from his statement
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about the metrical origin of gravitational forces to formulating those ideas in terms of Riemannian geometry.
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But Einstein was not particularly well disposed to mathematical treatments
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and particularly geometrical way of thinking in his earlier years.
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For example, when Minkowski wrote, a few years after Einstein had formulated his special theory of relativity
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by describing the special relativity in terms of what we now call Minkowski geometry,
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in which we associate geometric in spacetime, which is DT2– DX2- DY 2- DC2.
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And he showed that rotations in a spacetime with this metric is equivalent to a special relativity.
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Einstein’s remark on Minkowski’s paper was first that, well, we physicists show how to formulate the laws of physics,
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and mathematicians will come along and say how much better they can do it.
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And indeed he made the remark that Minkowski’s work was “überflüssige Gelehrsamkeit” -“unnecessary learnedness”.
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But it was only in 1911 or 1912 that he realised the importance of this geometrical way of thinking
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and particularly with the aide and assistance of his friend Marcel Grossmann,
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he learned sufficient differential geometry to come to his triumph and conclusion with regard to his field equations in 1915.
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But even at that time Einstein’s familiarity with Riemannian geometry was not sufficiently adequate.
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He did not realise that the general co-variants of his theory required
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that the field equations must leave four arbitrary functions free.
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Because of his misunderstanding here, he first formulated his field equations
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by equating the rigid tensor with the energy-momentum tensor.
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But then he realised that the energy-momentum tensor must have its co-variant divergence zero,
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but the covariant divergence of the digitants is not zero and he had to modify it to introduce what is the Einstein tensor.
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Now I do not wish to go into the details more, but only to emphasise that the principle motive of the theory
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was a physical insight and it was the strength of this physical insight
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that led him to the beauty of the formulation of the field equations in terms of Riemannian geometry.
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Now I want to turn around and say that why is it that we believe in the general theory of relativity.
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Of course there has been a great deal of effort during the past two decades to confirm the predictions of general relativity.
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But these predictions relate to very, very small departures from the predictions of the Newtonian theory.
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And in no case more than a few parts in a millionth, the confirmation comes from the reflection of light,
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as light traverses a gravitational field and the consequent time delay.
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The procession of the perihelion of Mercury and the changing period of double stars, the close double stars as pulsars,
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due to the emission of gravitational radiation.
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But in no instance is the effect predicted more than a few parts in a million departures from Newtonian theory.
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And in all instances it is no more than verifying the values of one or two or three parameters
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in expression of the equations of general relativity, in what one calls the post-Newtonian approximation.
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But one does not believe in a theory in which only the approximations have been confirmed.
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For example, if you take the Dirac theory of the electron and the only confirmation you had
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was the fine structure of ionised helium in partial experiments, a conviction would not have been as great.
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And suppose there had been no possibility in the laboratory of obtaining energies of a million electron volts,
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then the real experiment, the real verification of Dirac’s ideas, prediction of anti-matter,
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the creation of electron-positron-paths would not have been possible.
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And of a conviction in the theory would not have been as great.
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But it must be stated that in the realm of general relativity no phenomenon
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which requires the full non-linear aspects of general relativity have been confirmed, why then do we believe in it?
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I think of a belief in general relativity comes far more from its internal consistency and from the fact that,
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whenever general relativity has an interface with other parts of physics it does not contradict any of them.
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Let me illustrate these two things in the following way.
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We all know that the equations of physics must be causal.
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Essentially what it means is that if you make a disturbance at one point,
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the disturbance cannot be followed on another point for a time light will take it from one point to another.
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Technically one says that the equations of physics must allow an initial value formulation.
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That is to say you give the initial data on a space like surface and you show that the only part of the spacetime
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in which the future can be predicted is that which is determined by sending out light rays
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from the boundary of the spacetime region to the point.
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In other words, if for example, suppose you have a space like slice,
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then you send a light ray here and you light a region here, it is in that region that the future is defined.
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Now, when Einstein formulated the general theory of relativity,
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he does not seem to have been concerned whether his equations allowed an initial value formulation.
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And in fact to prove, in spite of the non-numerity of the equations,
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the initial value formulation is possible in general relativity, was proved only in the early ‘40s by Lichnerowicz in France.
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So that even though, when formulating the general theory of relativity,
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the requirement that satisfied the laws of causality was not included.
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In fact it was consistent with it.
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Or let us take the notion of energy.
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In physics, the notion of energy is of course central, we define it locally and it is globally concerned.
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In general relativity for a variety of reasons I cannot go into you cannot define a local energy.
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On the other hand you should expect on physical grounds that you have an isolated matter and even if it really emits energy,
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then globally you ought to be able to define a quantity which you could call the energy of the system.
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And that, if the energy varies it can only be because gravitational waves cross the boundary at a sufficiently large distance.
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There’s a second point, of course the energy of a gravitating system must include the potential energy of the field itself,
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but the potential energy in the Newtonian theory has no lower bound.
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By bringing two points sufficiently close together you can have an infinite negative energy.
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But in general relativity you must expect that there is a lower bar to the energy of any gravitating system.
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And if you take a reference with this lower bar as the origin of measuring the energy, then the energy must always be positive.
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In other words, if general relativity is to be consistent with other laws of physics,
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you ought to be able to define for an isolated system,
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yet global meaning for its energy and you must also be able to show that the energy is positive.
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But actually this has been the so-called positive energy conjecture for more than sixty years.
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And only a few years ago it was proved rigorously by Ed Witten and Yau.
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Now, in other words then that, even though Einstein formulated the theory from very simple considerations,
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like all accelerations must be metrical in origin, and putting it in the mathematical framework of Riemannian geometry,
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it nevertheless is consistent in a way in which its originator could not have contemplated.
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But what is even more remarkable is that general relativity does have interfaces with other branches of physics.
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I cannot go into the details but one can show that if you take a black hole
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and have the Dirac waves reflected and scattered by a black hole,
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then there are some requirements of the nature of scattering which the quantum theory requires.
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But even though in formulating this problem in general relativity no aspect of quantum theory is included,
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the results one gets are entirely consistent with the requirements of the quantum theory.
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In exactly the same way general relativity has interfaces with thermodynamics
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and it is possible to introduce the notion of entropy, for example in the context of what one generally calls Hawking radiation.
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Now, certainly thermodynamics must not be incorporated in founding general relativity,
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but one finds that when you find the need to include concepts from other branches of physics
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in consequences of general relativity, then all these consequences do not contradict branches of other parts of physics.
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And it is this consistency with physical requirements, this lack of contradiction with other branches of physics,
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which was not contemplated in its founding, and it´s these which gives one confidence in the theory.
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Now, I'm afraid I do not have too much time to go into the other aspect of my talk
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namely why is the general theory an excellent theory.
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Well, let me just make one comment.
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If you take a new physical theory, then it is characteristic of a good physical theory,
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that it isolates a physical problem which incorporates the essential features of that theory
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and for which the theory gives an exact solution.
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For the Newtonian theory of gravitation you have the solution to the Kepler problem.
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For quantum mechanics, relativistic or non-relativistic, you have the predictions of the energy of the hydrogen atom.
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And in the case of the Dirac theory, I suppose the creation of formula and the pair production.
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Now, in the case of the general theory of relativity you get asked,
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is there a problem which incorporates the basic concepts of general relativity in its purest form.
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In its purest form, the general theory of relativity is a theory of space and time.
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Now, a black hole is one whose construction is based only on the notion of space and time.
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The black hole is an object which divides the three dimensional space into two parts,
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an interior part and an exterior part, bound by a certain surface which one calls a horizon.
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And the reason for calling it that “the horizon” is that no person,
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no observer in the interior of the horizon can communicate with the space outside.
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So your black hole is defined as a solution of Einstein’s vacuum equations which has a horizon, which his convex
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and which is asymptotically flat, in the sense that this spacetime is minkowskian at sufficiently large distances.
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It is a remarkable fact that these two simple requirements provide, in the basis of general relativity,
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a unique solution to the problem.
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A solution which has just two parameters, the mass and the angular momentum.
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This is a solution discovered in 1962.
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The point is that if you ask what a black hole solution consistent to general relativity is,
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you find that there’s only one simple solution, its two parameters and all black holes which occur in nature must belong to it.
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One can say the following:
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If you see macroscopic objects, then you see microscopic objects all around us,
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if you want to understand them it depends upon a variety of physical theories,
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a variety of approximations and you understand it approximately.
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There is no example in macroscopic physics of an object which is described exactly and with only two parameters.
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In other words one could say that almost by definition the black holes are the most perfect objects in the universe,
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because their construction requires only the notions of space and time.
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It is not vulgarised by any other part of physics with which we are mostly dealing with.
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And one can go on and point out the exceptional mathematical perfectness of the theory of black holes.
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Einstein, when he wrote his last paper, his first paper announcing his field equations stated,
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that anyone, scarcely anyone who understands my theory can escape its magic.
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For one practitioner at least,
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the magic of the general theory of relativity is in its harmonious mathematical character and the harmonious structure of its consequences.
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Thank you.
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(Applause).