The Founding of General Relativity and Its Excellence
This is a modal window.
Das Video konnte nicht geladen werden, da entweder ein Server- oder Netzwerkfehler auftrat oder das Format nicht unterstützt wird.
Formale Metadaten
| Titel |
| |
| Serientitel | ||
| Anzahl der Teile | 340 | |
| Autor | ||
| Lizenz | CC-Namensnennung - keine kommerzielle Nutzung - keine Bearbeitung 4.0 International: Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nicht-kommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. | |
| Identifikatoren | 10.5446/55120 (DOI) | |
| Herausgeber | ||
| Erscheinungsjahr | ||
| Sprache |
Inhaltliche Metadaten
| Fachgebiet | ||
| Genre | ||
| Abstract |
|
00:00
Chandrasekhar-GrenzeChandrasekhar, SubrahmanyanFACTS-AnlageNewtonsche AxiomeKalenderjahrSpantBuick CenturyDruckkraftBeschleunigungTagUmlaufbahnFernordnungPagerMasse <Physik>FeldeffekttransistorGewichtsstückFeldquantSatz <Drucktechnik>TrägheitComte AC-4 GentlemanWasserdampfFlüssigkeitErsatzteilKompendium <Photographie>
09:14
BootChandrasekhar, SubrahmanyanElementarteilchenphysikChandrasekhar-GrenzeGruppenlaufzeitKreuzfahrtschiffGewichtsstückMasse <Physik>UmlaufzeitH-alpha-LinieBeschleunigungProfilwalzenTagesschachtBetazerfallRaketeBiegenFeldeffekttransistorErderHochbahnGammaquantBuick CenturyErsatzteilTrenntechnikBahnelementMagic <Funkaufklärung>FACTS-AnlageFernordnungBecherwerkVideotechnikPendelAbstandsmessung
18:27
BrennofenBootChandrasekhar, SubrahmanyanElementarteilchenphysikChandrasekhar-GrenzeGruppenlaufzeitKreuzfahrtschiffStoffvereinigenAhle <Werkzeug>UmlaufzeitFACTS-AnlagePräzessionWarmumformenKalenderjahrErsatzteilKlangeffektEmissionsvermögenNewtonsche FlüssigkeitKristallgitterBeschleunigungElektronRutherford-StreuungLuftionisationLichtstreuungPaarerzeugungAtomhülleWalken <Textilveredelung>AlphazerfallLichtPerihelPatrone <Munition>TheodolitTissueOberflächeRotationszustandStrahldivergenzBlatt <Papier>FeldstärkeTransporttechnik
27:41
BootChandrasekhar-GrenzeKreuzfahrtschiffChandrasekhar, SubrahmanyanSchwarzes LochWeltraumErsatzteilOberflächeFeldquantSichtverbindungBeschleunigungPatrone <Munition>Magic <Funkaufklärung>Blatt <Papier>AbstandsmessungSchwächungPaarerzeugungNegativ <Photographie>KristallgitterWasserstoffatomKalenderjahrNewtonsche FlüssigkeitFACTS-AnlageStrahlungHauptsatz der Thermodynamik 2Kompendium <Photographie>Masse <Physik>LunkerBasis <Elektrotechnik>Vorlesung/Konferenz
Transkript: Englisch(automatisch erzeugt)
00:12
Ladies and gentlemen, Einstein is by all criteria the most distinguished physicist
00:23
of the century. No physicist in this century has been accorded a greater acclaim. But it is an ironic comment that even though most histories of 20th century physics
00:40
starts with the pro forma statement that this century began with two great revolutions of thought, the general theory of relativity and the quantum theory. The general theory of relativity has not been a staple part of the education of a
01:03
physicist, certainly not to the extent quantum theory has been. Perhaps on this account, a great deal of mythology has accreted around Einstein's name and the theory of relativity which he founded 70 years ago.
01:25
And even great physicists are not exempt from making statements which if not downright wrong or at least misleading.
01:41
Let me quote, for example, a statement by Dirac made in 1979 on an occasion celebrating Einstein's 100th birthday. This is what he said, when Einstein was working on building up this theory of gravitation,
02:05
he was not trying to account for some results of observation. Far from it. His entire procedure was to search for a beautiful theory, a theory of a type that
02:22
nature will choose. He was guided only by considerations of the beauty of his equations. Now, this contradicts statements made by Einstein himself on more than one occasion.
02:42
Let me read what he said in 1922 in a lecture he gave titled, how I came to discover general theory of relativity. And I read Einstein's statement. I came to realize that all the natural laws except the law of gravity could be discussed
03:06
within the framework of the special theory of relativity. I wanted to find out the reason for this, but I could not attain this goal easily. The most unsatisfactory point was the following.
03:25
Although the relationship between inertia and energy was explicitly given by the special theory of relativity, the relationship between inertia and weight or the energy of the
03:41
gravitational field was not clearly elucidated. I felt that this problem could not be resolved within the framework of the special theory of relativity. The breakthrough came suddenly one day.
04:00
I was sitting on a chair in my patent office in Bern. Suddenly, a thought struck me, if a man falls freely, he would not feel his weight. I was taken aback. The simple thought experiment made a deep impression on me.
04:23
This led me to the theory of gravity. I continued my thought, a falling man is accelerated. That then what he feels and judges is happening in the accelerated frame of reference.
04:41
I decided to extend the theory of relativity to the reference frame with acceleration. I felt that in doing so, I could solve the problem of gravity at the same time. A falling man does not feel his weight because in his reference frame, there is
05:02
a new gravitational field which cancels the gravitational field due to the earth. In the accelerated frame of reference, we need a new gravitational field. Perhaps that is not quite clear from what I have read, precisely what he had in mind.
05:24
But two things are clear. First, he was guided principally by the equality of the inertial and the gravitational mass in empirical fact, which has been very accurately determined and in fact probably the most
05:46
well-established experimental fact. And the second point is that this equality of the inertial and the gravitational mass led him to formulate a principle which he states there very briefly and which has
06:03
now come to be called the principle of equivalence. Let me try to explain more clearly what is involved in these statements I have just made. The first point, in order to do that, I have to go back in time.
06:24
In fact, I have to go back 300 years to the time when Newton wrote the Principia. The fact that the publication of the Principia is 300 years old was celebrated last year in many places. Now, Newton notices already within the first few pages of the Principia that the notion
06:48
of mass and weight are two distinct concepts based upon two different notions. The notion of mass follows from his second law of motion which states that if you subject
07:05
a piece of body to a force, then it experiences an acceleration in such a way that the quantity which we call the mass of the body times the acceleration is equal to the force.
07:22
Precisely, if you apply a force to one cubic centimeter of water and measure the acceleration which it experiences, and then you find that another piece of water, when subjected to the same force, experiences ten times the acceleration, then you conclude
07:46
that the mass of the liquid you have used is one-tenth cubic centimeter. In other words, then, the notion of mass is a consequence of his law of motion. It is a constant of proportionality in the relation that the force is equal to the
08:05
mass times the acceleration, but the notion of weight comes in a different way. If you take a piece of matter and it is subject to the gravitational field, say of the Earth, then you find that the attraction which it experiences in a given gravitational
08:28
field is proportional to what one calls the weight. For example, if you take a piece of liquid, say water, and you find that the Earth attracts
08:43
it by a force which you measure, and you find that another piece of the same matter experiences a gravitational attraction which is, say, ten times more, then you say the weight is ten times greater.
09:01
In other words, the notion of weight and the notion of mass are derived from two entirely different sets of ideas, and Newton goes on to say that the two are the same, and in fact, as he says, as they have found by experiments with pendula made accurately.
09:22
The way he determined the equality of the inertial and the gravitational mass was simply to show that the period of a pendulum, a simple pendulum, depends only on its length and not upon the weight or the mass of the body or the constitution of it, and he established
09:43
the equality of the inertial and gravitational mass to a few parts in a thousand. A century later, Bessel improved the accuracy to a few parts in several tens of thousands. Earlier this century, it was shown the equality to one part in ten to the eleventh,
10:04
and more recently the experiments of Dicke and Brzezinski have shown that they are equal to one part in ten to the minus thirteen. Now this is a very remarkable fact. The notion of mass and weight are fundamental in physics, and one equates them by a fact
10:28
of experience, and this of course is basic to the Newtonian theory. Hermann Weyl called it an element of magic in the Newtonian theory, and one of the objects
10:42
of Einstein's theory is to illuminate this magic, but the question of course is you want to illuminate the magic, but how? And this Einstein, for this Einstein developed what one might call, is what one calls today
11:04
the principle of equivalence, and let me illustrate his ideas here. Here is the famous experiment with an elevator or a lift which Einstein contemplated.
11:21
Now the experiment is following. Now here is a lift with a rocket booster, and let us imagine that this lift is taken to a region of space which is far from any other external body, and if this elevator
11:42
is accelerated by a value equal to the acceleration of gravity, then the observer will find that if he drops a piece of apple or a ball, it will fall down towards the bottom with a sudden acceleration.
12:00
On the other hand, if the rockets are shut off, and the rocket simply cools, then if he leaves the same body, then it remains where it was. Now you perform the experiment now on the same elevator shaft on the earth, and you find that if he leaves the body, then it falls down to the ground in the same way as it
12:26
did when this was accelerated and not subject to gravity. Now suppose this elevator is put in a shaft and falls freely towards the center of the earth, then when you leave the body there, it remains exactly as it was.
12:46
In other words, in this case, the action of gravity and the action of acceleration are the same. On the other hand, you cannot conclude from this that action of gravity and the action of uniform acceleration are the same.
13:03
Let us now perform the same experiment in which the observer has two pieces of bodies instead of just one. Then if the rocket is accelerated, then you will find that both of them fall towards
13:21
the same along parallel lines. And again, if the acceleration is stopped, then the two bodies will remain at the same point. Now if you go to the earth, and similarly you have these two things, then the two will fall, but not exactly in parallel lines if the curvature of the earth is taken into
13:44
account. The two lines in which they will drop will intersect at the center of the earth. Now if the same experiment is performed with a lift which is falling freely, then as the
14:02
lift approaches the center of the earth, the two objects will come close together. And this is how Einstein showed the equivalence locally of a gravitational field, of a uniform gravitational field with a uniform acceleration, but showed nevertheless that if the gravitational
14:28
field is non-uniform, then you can no longer make that equivalence. Now in order to show how from this point, Einstein derived his principle of equivalence
14:45
in a form in which he could use it to find gravity. I should make a little calculation. Now everyone knows that if you describe the equations of motion in, say, Cartesian
15:03
coordinates, then the inertial mass times the acceleration is given by the gravitational mass times the gradient of the potential, gravitational potential. There are similar equations for x and y.
15:22
Suppose we want to rewrite this equation in a coordinate system which is not x, y and z, but a general curvilinear coordinates, that is, instead of x, y and z, you change two coordinates, q1, q2, q3, and you can associate with the general curvilinear system
15:47
a metric in the following way. The distance between two neighboring points in Cartesian framework is dx squared plus dz squared. On the other hand, if you find the corresponding distance for general curvilinear coordinates,
16:07
it will be a certain quantity, 8 alpha beta, with a two-index quantity, which will be functions of the coordinates times dq alpha dq beta.
16:22
For example, in spherical polar coordinates, it will be dr squared plus r squared d theta squared plus r squared sine squared theta d phi squared. But more generally, that will be the kind of equation you will have.
16:41
Now, let us suppose you write this equation down and ask what the gravitational equations become. Then you find that m inertial times this quantity q dot beta, that is the q double dot beta, the acceleration in the coordinate beta times this quantity contracted is equal
17:07
to the minus the inertial mass times a certain quantity gamma alpha beta gamma. We call the Christoffel symbols, but it doesn't matter what they are. They are functions of the coordinate, functions of the geometry times q dot beta q dot gamma
17:28
and then minus the same gravitational mass times the gradient of the potential. You see, the main point of this equation is to show that the acceleration in the coordinate,
17:45
which is corresponding to that, when you write it down in general curvilinear coordinates, the acceleration consists of two terms, a term which is geometrical in origin, which is a coefficient, the inertial mass, and a term from the gravitational field.
18:04
And if you accept the equality between the inertial mass and the gravitational mass, then the geometrical part of the acceleration and the gravitational part are the same. And this is Einstein's remark. He said that why make this distinction? Why not simply
18:27
say that all acceleration is metrical in origin? And that is the starting point of his work. He wanted to abolish the distinction between the geometrical part of the acceleration
18:45
and the gravitational part by saying that all acceleration is metrical in origin. Einstein's conclusion that in the context of gravity, all accelerations are metrical
19:01
in origin is as staggering in its own way as Rutherford's conclusion when Geiger and Marsden first showed him the results of the experiments on the large-angle scattering of alpha rays. And Rutherford's remark was it was as though you had fired a 15-inch
19:22
shell at a piece of tissue paper, and it had bounced back and hit you. In the case of Rutherford, he was able to derive his law of scattering overnight. But it took Einstein many years, in fact 10 years almost, to obtain his final field equations.
19:46
The transition from the statement that all acceleration is metrical in origin to the equations of the field in terms of the Riemann tensor is a giant leap. And the fact that it took Einstein three or four years to make the transition is understandable.
20:06
Indeed, it is astonishing that he made the transition at all. Of course, one can claim that mathematical insight was needed to go from his statement
20:21
about the metrical origin of gravitational forces to formulating those ideas in terms of Riemannian geometry. But Einstein was not particularly well-disposed to mathematical treatments and particularly geometrical way of thinking in his earlier years.
20:45
For example, when Minkowski wrote a few years after Einstein had formulated his special theory of relativity by describing the special relativity in terms of what we now call
21:00
Minkowski geometry, in which we associate a metric in space-time, which is dt squared minus dx squared minus dy squared minus dc squared. And he showed that rotations in a space-time with this metric is equivalent to a special relativity.
21:23
Einstein's remark on Minkowski's paper was first that, well, we physicists show how to formulate the laws of physics and mathematicians will come along and say how much better they can do it. And indeed, he made the remark that Minkowski's work
21:42
was überfelissiger, überfelissiger Giletsamtkeit, unnecessary learnedness. But it is only in 1911 or 1912 that he realized the importance of this geometrical way of thinking and particularly with the aid and assistance of his friend Marcel Grossman,
22:05
he learned sufficient differential geometry to come to his triumphant conclusion with regard to his field equations in 1915. But even at that time, Einstein's familiarity with Riemannian geometry was not sufficiently adequate. He did not realize
22:28
that the general covariance of his theory required that the field equations must leave four arbitrary functions free. Because of his misunderstanding here, he first
22:42
formulated his field equations by equating the Ritchie tensor with the energy-momentum tensor. But then he realized that the energy-momentum tensor must have its covariant divergence zero, but the covariant divergence of the Ritchie tensor is not zero and he
23:02
had to modify it to introduce what is the Einstein tensor. Now, I do not wish to go into the details more, but only to emphasize that the principal motive of the theory was his physical insight and it was the strength of his physical
23:23
insight that led him to the beauty of the formulation of the field equations in terms of Riemannian geometry. Now, I want to turn around and say that why is it that we believe in the general theory of relativity? Of course, there has been a great deal of
23:45
effort during the past two decades to confirm the predictions of general relativity, but these predictions relate to very, very small departures from the predictions of the Newtonian theory, and in no case more than a few parts in a million. The confirmation comes
24:06
from the deflection of light as light traverses a gravitational field and the consequent time delay, the precession of the perihelion of mercury, and the change in period of double stars, close double stars as pulsars, due to the emission of gravitational radiation.
24:26
But in no instance is the effect predicted more than a few parts in a million departures from Newtonian theory, and in all instances it is no more than verifying the values of
24:41
one or two or three parameters in expansion of the equations of general relativity in what one calls the post-Newtonian approximation. But one does not believe in a theory of which only the approximations have been confirmed. For example, if you take the Dirac
25:01
theory of the electron, and the only confirmation you had was the fine structure of ionized helium and partial experiments of a conviction would not have been as great. And suppose there had been no possibility in the laboratory of obtaining energies of
25:21
a million electron volts, then the real experiment, the real verification of Dirac's structure, the creation of electron-positron pairs, would not have been possible, and our conviction in the theory would not have been as great. But it must be stated
25:41
that in the realm of general relativity, no phenomenon which requires the full non-linear aspect of general relativity have been confirmed. Why then do we believe in it? I think our belief in general relativity comes far more from its internal consistency
26:04
and from the fact that whenever general relativity has an interface with other parts of physics, it does not contradict any of them. Let me illustrate these two things in the following way. We all know that the equations of physics must be causal.
26:26
Essentially, what it means is that if you make a disturbance at one point, the disturbance cannot be felt at another point for a time which light will take to go from one point to another. Technically, one says that the equations of physics must allow an initial
26:45
value formulation. That is to say, you give the initial data on a space-like surface and you show that the only part of the space-time in which the future can be predicted is that which is determined by sending out light rays from the boundary of the space-time region
27:07
to the point. In other words, if for example, you have a space-like slice, then
27:22
you send a light ray here and you light a light ray here and it is in that region that the future is defined. Now, when Einstein formulated the general theory of relativity, he does not seem to have been concerned whether his equations allowed an initial
27:41
value formulation. In fact, to prove, in spite of the nonlinearity of the equations, the initial value formulation is possible in general relativity was proved only in the early 40s by Lichner or in France. So that even though when in formulating
28:01
the general theory of relativity, the requirement that it satisfy the laws of causality was not included. In fact, it was consistent with it. Or let us take the notion of energy. In physics, the notion of energy is, of course, central. We define it locally and it is globally
28:22
concerned. In general relativity, for a variety of reasons I cannot go into, you cannot define a local energy. On the other hand, you should expect on physical grounds that if you have an isolated matter and even if it radiates energy, then globally you ought
28:42
to be able to define a quantity which you could call the energy of the system. And that, if the energy varies, it can only be because gravitational waves cross the boundary at a sufficiently large distance. And that is the second point. Of course, the energy
29:04
of a gravitating system must include the potential energy of the field itself. But the potential energy in the Newtonian theory has no lower bound. By bringing two points sufficiently close together, you can have an infinite negative energy. But in general
29:24
relativity, you must expect that there is a lower bound to the energy of any gravitating system. And if you take a reference with this lower bound as the origin of measuring the energy, then the energy must always be positive. In other words, if general relativity
29:45
is to be consistent with other laws of physics, you ought to be able to define for an isolated system a global meaning for its energy and you must also be able to show that the energy is positive. But actually, this has been a so-called positive energy
30:05
conjecture for more than 60 years. And only a few years ago, it was proved rigorously by Ed Witten and Yohol. Now, in other words then, that even though Einstein formulated
30:20
the theory from very simple considerations, like all accelerations must be metrically in origin, and putting it in the mathematical framework of Riemannian geometry, it must nevertheless be consistent in a way in which its originator could not have contemplated.
30:43
But what is even more remarkable is that general relativity does have interfaces with other branches of physics. I cannot go into the details, but one can show that if you take a black hole and have the Dirac waves reflected and scattered, it is possible that
31:05
by a black hole, then there are certain requirements of the nature of scattering which the quantum theory requires. But even though in formulating this problem in general relativity, no aspect of quantum theory is included. The results one gets are entirely consistent with the
31:26
requirements of the quantum theory. In exactly the same way, general relativity has interfaces with thermodynamics, and it is possible to introduce the notion of entropy, for example in the context of what one generally calls Hawking radiation. Now, certainly thermodynamics
31:46
must not be incorporated in founding general relativity, but one finds that when you find the need to include concepts from other branches of physics in consequences of general relativity, then all these consequences do not contradict branches of other parts of
32:06
physics. And it is this consistency with physical requirements, this lack of contradiction with other branches of physics, which was not contemplated in its founding, and it
32:22
is these which gives one confidence in the theory. Well, I'm afraid I do not have too much time to go into the other aspect of my talk, namely, why is the general theory of an excellent theory? Well, let me just make one comment. If you take any new physical
32:42
theory, then it is characteristic of a good physical theory that it isolates a physical problem which incorporates the essential features of that theory and for which the theory gives an exact solution. For the Newtonian theory of gravitation, you have the solution to the
33:03
Kepler problem. For quantum mechanics, relativistic or non-relativistic, you have the predictions of the energy of the hydrogen atom. And in the case of the Dirac theory, I suppose the Klein-Nishina formula and the pair production. Now, in the case of the
33:20
general theory of relativity, you can ask, is there a problem which incorporates the basic concepts of general relativity in its purest form? In its purest form, the general theory of relativity is a theory of space and time. Now, a black hole is one whose
33:40
construction is based only on the notions of space and time. The black hole is an object which divides the three-dimensional space into two parts, an interior part and an exterior part, bound by a certain surface which one calls the horizon. And the reason
34:03
for calling that the horizon is that no person, no observer in the interior of the horizon can communicate with the space outside. So, a black hole is defined as a solution of Einstein's vacuum equations which has a horizon which is convex and
34:25
which is asymptotically flat in the sense that the space-time is Minkowski at sufficiently large distances. It is a remarkable fact that these two simple requirements provide in the basis of general relativity a unique solution to the problem, a solution which
34:45
has just two parameters, the mass and the angular momentum. This is a solution of Kerr discovered in 1962. The point is that if you ask what a black hole solution
35:02
consisted of general relativity is, you find that there is only one simple solution with two parameters and all black holes which occur in nature must belong to it. And the one can say the following. If you see macroscopic objects, then you see macroscopic objects
35:23
all around us. And if you want to understand them, it depends upon a variety of physical theories with a variety of approximations and you understand it approximately. There is no example in macroscopic physics of an object which is described exactly and with
35:45
only two parameters. In other words, one could say that almost by definition, the black holes are the most perfect objects in the universe because their construction requires only the notions of space and time. It is not vulgarized by any other
36:07
part of physics with which we are mostly dealing with. And one can go on and point out the exceptional mathematical perfectness of the theory of black holes. Einstein, when
36:22
he wrote his last paper, his first paper announcing his field equations stated that anyone, scarcely anyone who understands my theory can escape its magic. For one practitioner at least, the magic of the general theory of relativity is in its harmonious mathematical
36:46
character and the harmonious structure of its consequences. Thank you.