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Gravitational Waves

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Gravitational Waves
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Buick Century Year Particle physics Heisenberg, Werner
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Video Graviton Electric power distribution Nanotechnology Particle physics Material Fire apparatus Heisenberg, Werner
I am very glad to be here in Lindau for the 3rd time and to have this opportunity of talking to you about problems that I have
been working on recently this time I would like to talk to you about the theory of gravitation rather different from my previous topics you all know that a theory of gravitation was 1st put forward by Newton more than 200 years ago Newton's theory was a very good theory it survived an unchallenged for more than 200 years and then in the present century a new theory of
gravitation was put forward by I'm stand Einstein's theory was connected with his principle of relativity and he showed how gravitation could be explained as an effect arising from the curvature of space and time Einstein's theory was very beautiful theory mathematically and also it was found to be in good agreement with observation and so it became generally accepted in the world of science Einstein's theory aroused an enormous scientific interest very many people worked very intensively on this theory for a good many years and then at the interest in the Einstein's theory of gravitation rather died away people found the Gration's difficult to work with they found other subjects of interest they largely quantum theory and for
a while when did not hear so much about the Einstein's theory in the world of science but in recent times
since the war there has been a revival of interest in the Einstein's theory of gravitation and at present there are more and more people working on it this revival of interest can be explained I think or accounted for by 2 reasons partly there have been new mathematical methods developed for dealing with it and partly people have been continually getting new observations about the very
distant parts of our universe they have been getting these observations with the help of the very big telescopes which are now available and also with the help of the new technique of radio astronomy so there is this revival of interest in Einstein's theory of gravitation and as a consequence there are now at international conferences on gravitation which are held regularly every 2 years the 3rd of these international conferences was held only last week near Paris at these conferences people who are working on the theory of gravitation from all countries come and meet together and presented reports on their recent researches and discuss the problems which still remain to be solved I think that the people who come to these conferences people who work on
gravitation can very clearly be divided into 3 classes there are the mathematicians the physicists and the cosmologists the mathematicians are concerned with getting exact solutions of Einstein's equations they are interested in all kinds of exact
solutions independently of whether these solutions have anything to do with our actual world or not the other 2 classes are concerned with the actual world the physicists are concerned with studying the gravitational field
as the physical field and finding out the physical effects of gravitational forces and they hope to be able to detect these effects with their instruments they cause edges are concerned with the universe as a whole they are dealing with what the universe is like at extremely large distances and their main problem is whether the universe is closed up or whether it is an open Universe I want to talk to you today only about the point of view of the physicist for the physicist there is a fortunate circumstance in that 1 does not need to use the exact equations of the Einstein's theory 1 can work with a certain approximation the effects of the gravitational field are attributed to a curvature in spacetime and for the physicist 1 can count this curvature as extremely
small in the space of the physicist this curvature is certainly extremely small and it is sufficient to work with the approximations of the
Einstein's theory applied to the case when the curvature of space time is extremely small that would not do for the cause mom because this approximation would not
be a good approximation if 1 were concerned with extremely large distances distance is compatible with the distances between the spiral nebulae for instance but for the distances which interest the physicist this approximation is certainly a valid 1 now the exact equations of the Einstein's theory is the equation 1 in these
notes I think a good many of you have a German translation of the notes with you show that I need not write down all the creations but can just refer to them and will write down on the blackboard of the more important aggressions if we take the approximate form of the Einstein's theory when it is applied to space which is nearly flat we have this as our basic equations we have an equation and which involves a quantity H new new which is introduced in this way the exact theory of Einstein it is based on the circumstances it is recognized this gene with 2 separate new and new and you and you take on the form values the old 1 2 3 this Spencer describes the gravitational field describe the curvature of spacetime and it also fixes the system of coordinates now for space
which is nearly flat this tensor differs only by a small quantity from its value for flat
spacetime for flat spacetime these different elements all have values 1 or minus 1 or not and the differences from flat space-time we denote by H you and we count h you knew a small and neglect quantities which are of the 2nd order of smallness we then have this as our basic equation of the Einstein's theory the Laplacian operator which is denoted
by the symbols square applied to a you knew plus all right down on immigration and then explain it you think about it you that's in the United States I was sitting in a room you you this will you new is constructed from the tensor each describes any matter which is present and in particular in this rural vanishes when there is no matter gamma here is the gravitational constant and it counts is very small in the approximation in which we are working this the new is a certain quantity which is constructed from the 1st derivatives of new new and you will find expression for being you written down in equation 3 and then the note say more about that this is then are fundamental
equation now for dealing with this equation people usually choose a system of coordinates which makes this quantity the new
vanish it is quite a nice condition to impose on the coordinates and that when people are working with these coordinates
they say that they're working with harmonic coordinates this harmonic condition on the coordinates is 1 which is used very extensively and it results in a big
simplification in the equation because with these harmonic coordinates the script and just that and we're left with this equation we just those 2 terms
now if we apply that equation into a region of space and time where there is no matter present we have this term also vanishing and we have just this integration there squared feets the new peak was not that is just the well-known equations for wave propagation the equation which we have for all kinds of fields when there are waves which propagate with the speed of light so that we can say that in this approximation of weak field is the theory of leads to these waves in this quantity H Union in those regions of space and time where there is no matter an important feature of Einstein's theory is that it is valid for all systems of coordinates we are working with the
case when the field is weak and the natural thing to do under those conditions is to work with a system of coordinates which is
approximately Cartesian we cannot say that it is exactly Cartesian because there is still a bit of curvature in our space which prevents 1 from giving a precise meaning to Cartesian coordinates but still we can take coordinates which are approximately Cartesian and that is what we're doing when we introduce these quantities H you knew but even with these coordinates which are approximately Cartesian and even with the harmonic condition there is still
some arbitrariness left in our system of coordinates and because of this arbitrariness which is still left in our system of coordinates we cannot be very sure about the meaning of these waves whose existence is shown by this equation we cannot be sure whether these waves really something physical or whether they are I just connected with our system of coordinates now that is really the main difficulty all the time when 1 is working with the Einstein's theory it is the difficulty all separating what is real and physical from what it depends simply on our system of coordinates and that they are the main problem of discussion today now in order to fix our ideas rather more precisely let us suppose that we have some actual physical problem we have
some masses coming together perhaps even with high speeds interacting with each other in some way and we have gravitational forces
between and we want to discuss exactly what happens I should say exactly what happens I should say we want to discuss what happens in this approximation of weak gravitational fields we then have to look for the solution of this 1st equation here no solutions of that equation are quite familiar to
physicists because this equation itself is very similar to the equation which we have an elected dynamics we can look upon this right hand
side as generating waves in this quantity H in the same way as the electric charges and currents generated electromagnetic waves and from our familiarity with a solution of electromagnetic equations we can immediately get information about the solution of this equation here we know that there is a solution when we are given the value of role there is a solution in terms of retarded potentials and this would be the solution which is the important 1 physically if there are no incoming gravitational waves the general solution 1 gets from this regarded solution by adding on to the retarded solution certain arbitrary incoming waves but it will be mainly the retarded solutions that I want to talk to you about with this retarded solution we
had the H you knew at large distances proportional to 1 over R R being the distance the H U Nu are rather than
like the that community potentials and to get something which corresponds to the electromagnetic field of something that we can count as the gravitational field we must differentiate is H new new ones if we differentiate something which is of the form of 1 over R for great distances then we get 2 terms appearing when German depending on whether where our squares and the other terms depending on only
good over our I should say in terms of the order of only go over R where omega is the frequency of oscillations which are occurring in this distribution of matter 4 distances which are not too large the 1 over r squared term is the important terms that 1 over r squared terms gives you the cool force in the victory dynamics and it gives you the Newtonian force in gravitation but for a much larger distances the net we only go over R term is the dominant term and this term corresponds to so that this would be the important term for our talk today this term which dominates the solution and very large distances let us now fix tension on the waves which come out in 1 particular direction let us say the
waves which come out in the direction of the axis it's suppose this is the axis X 3 and then we want
to examine the solution of our field equation 4 points out here where x 3 year has some large positive
value and X 1 and X 2 are small in this region out here we should have waves moving radially outward and those will be the dominant part of our solution to examine the solution in that region of space we must put the bad
X 1 and the guy next to the 0 we must also put imply that it's really people to bias the value of X naught there is an error in the paper which has been distributed this minus has been omitted so please inserted we can now see what is the effect of putting in these conditions into the solution of our field equations the if we examine the harmonic conditions in that region of space and time we get a set of equations which is written down in the notes he Gration's 6 now as I mentioned before even with a harmonic conditions there is still some arbitrariness in our system of coordinates so we can take this question that is to make a change in our system of coordinates the changes which preserves the
harmonic conditions I don't want to make a general change in the coordinates which is going to disturb the harmonic conditions but I'm going to make a change which preserves the harmonic
conditions and such a change is described by the equation find in the notes where a new a field
function which fixes the changes and the same you must satisfy the wave equation which is written immediately after equation 5 well the effect of making this change is to bring in certain changes in all the 10 quantities each you knew and
these changes are given in the notes I don't need to describe them in detail but I would just say what the important result is we find that when we make this change in coordinates 6 of the H new new get altered no sex of the age you new remain invariant and the other for these 10 quantities can they changed and they can be changed arbitrarily things which can be changed arbitrarily when we make a change in our system of coordinates cannot have any physical meaning show that these 4 components of each new new which gets changed arbitrarily I will not have any physical significance there are other 6 invariant ones and all these 6 invariant ones FIL for must be 0 because of the harmonic conditions themselves and that leaves only 2 which are
invariant and which are not restricted to be 0 those 2 highly the this they want and take what was the last thing to do the 2 components we may therefore expect to have a physical meaning the formant wave which are moving in the direction of the X 6 3 they are invariant under the transformations which we can
make it with because the harmonic conditions and they are not used to be 0 well what this means then that we should expect that we have these gravitational
waves which are physical that we have these 2 kinds of polarization for gravitational waves moving in the direction of the axis takes the the question remains should these waves really be counted as something physical and that is rather equivalent to the question to these waves of high energy so that brings us to the discussion of the question of the energy of the Einstein's theory of gravitation for these discussions of energy 1 can set up a certain dancer or tensor density the new which has the physical significance that its components are connected with stresses and and from the momentum density and that 1 of its components the team's not not component can be interpreted as the energy density 1 finds that this team you knew I had it on to a suitable
tensor describing the matter satisfies the conservation law which is written down in the notes there so that if we define the energy
density as Denault Nault we get the exact conservation of energy but there is some trouble with this the the I talked about it as a tensor density but it is not really a dense density it is something which is called pseudo tensor density
because when we make a change in coordinates it does not transform correctly to be a tensor density and that means that
if we used this gene not naught as the energy density and we work out the energy in a certain region then if we make a change in our system of coordinates we should get a different energy that energy ought to be something which is physical we want to be a really physical thing and it should be independent of our system of coordinates this gene or not is really the best thing we can do for discussing energy density and we have here is a real difficulty this difficulty has a ball of people for very many years and it has led to a procedure in practice when people want to discuss energy in connection with the Einstein's theory they adopt some nice system of coordinates and they assume that if the energy is calculated with this nice system of coordinates the result will have some physical meaning but that
of course is not very logical processes not logical at all and it is undeniable and on account of that there has been much discussion
for their many years as to whether these gravitational waves really do carry energy or not but with the development of the theory of gravitation which has taken place in recent times this question has been cleared up 1 of the main lines of this recent development has been the expression of the equations of the Einstein's theory in the Hamiltonian form now the Hamiltonian form of writing the
creations is of the form which has a very great mathematical power it was discovered more than a hundred years ago by Hamilton who worked it out simply because of the mathematical beauty connected with it and Hamilton himself did not realize the great importance of his form of equations but we see now that is form is really of fundamental importance in nature because his form of equations he is the former which lends itself naturally to the facet to the quantum theory just working from the Newtonian form of equations of motion 1 has not got any good way of passing to the quantum theory but working from the Hamiltonian form we have well defined rules which had been applied successfully in many cases for passing from any classical field theory for the classical theory of particles to the corresponding quantum theory a good deal of the recent interest in the theory of gravitation has been
concerned with obtaining a quantum theory of gravitation and for that purpose 1 must 1st put the classical theory
into Hamiltonian fall now with the Hamiltonian form of the equations 1 deals with the state at a certain time now the state at a certain time means the state for all values of the coordinates X 1 X 2 X 3 but for 1 particular value of the coordinate x
not now you see when we discuss the state at a certain time we are introducing they d symmetry between the 4 coordinates 1 of the great
features of Einstein's theory was the fact that we had a symmetry between the 4 coordinates this recordings of space and the 1 time coordinate and for a long time people were interested only in developing the Einstein's theory in a form which preserve this symmetry it is just within the last few years that he could have found found that they can get a lot of new results by departing from the symmetry and in particular by working with this concept of all the state at a certain time where we go entirely away from these 4 domains the symmetry and we go back to the old idea of a 3 dimensional world changing with a time coordinate with the development of the Hamiltonian former we get this work in which we destroy the 4 symmetry and of course in a way it is a pity to destroy the 4 dimensional symmetry everyone would agree with that but there are these
compensations that 1 has greater mathematical power and when find some new features of the equations which are not so obvious when what keeps to the four-dimensional symmetry
with the Hamiltonian form once dynamical variables are all paired off into dynamical coordinates and conjugate momenta so far as concerns the gravitational field we have the gene new new for all values of X 1 X 2 X 3 appearing as dynamical coordinates and we have then momentum there was beam you knew appearing as the conjugates of the dynamical coordinates
now 1 of the 1st things 1 found when once started to put the theory into Hamiltonian form 1 got a result which was rather unexpected which was on the 10 quantities did you know and their content being you knew further of the gene you knew and their conjugates drop out from the Hamiltonian equations of motion namely these for the know all the you know what if 1 of these indices takes on the value and all either 1 either the 1st or the 2nd because a significant then we get these quantities here and quantities drop out from the Hamiltonian equations and we are left with Hamiltonian equations involving only the variables G R S being caller it's now
he's roman letters and s take on the values 1 2 and 3 and they are to be sharply distinguished from the Greek letters which take on the values not
1 2 3 we have here just 6 G R a season 60 or instead of the 10 and G you news and be new news and that means that with the Hamiltonian formulation we start off expecting to have 10 degrees of freedom for each point of space but for of the degrees of freedom got out and we're left with just 6 degrees of freedom for each point is space that is a big simplification and this is this
simplification which brings out the advantages of the Hamiltonian formalism now this simplification ought not to surprise when too much 1 might have expected it if 1 just looked into what is really needed for describing the state at a certain time this think at a certain time means the state for all regions of space for a certain value of X naught and that is to be pictured in space time as the three-dimensional hypersurfaces the hypersurface fixed not was constant which is to be pictured as existing in four-dimensional spacetime not to describe such a hypersurface we need only the 6 is they are sufficient to describe the geometry of the hypersurface and the coordinate system and the hypersurface and these gene you also I needed only to describe the
relationship of this hypersurface to enable hypersurfaces but if you are interested only in describing the state at 1 particular time then we only
need these 6 OSes and we also need a dynamic the continents so that from religion of arguments arguments of the geometrical and kinematic in nature 1 can see that these 6 degrees of freedom all of that is really necessary if we ask this
physical question how should be set up the energy at a certain time then it seems clear that
this energy should not depend on any variables which are not needed for describing the state at that time so the energy at a certain time of the energy density in the region has a certain time should not depend on these various you not being you know what now if you look at the energy density given by the pseudo tensor is the that we had before and you work it out you see that t not what does depend on you know T. Naur note thus involves some quantities which are not really relevant for describing the state at a certain time it involves certain things which we are concerned only with the coordinate system now that is quite a bad features energy density this due energy density and we can improve upon this features by taking a modified expression for the energy density we get a
modified expression for the energy density by expressing this gene or not in a suitable way and substituting for
the gene you knots which occur in it their values for fastest time namely we substitute for gene or not the value minus 1 and 4 G 1 naughty to free not the values not by this procedure we can get an improved expression for the energy density and include expression which I call W now w still depends on our system of coordinates although not so badly as the north north it
means that we have made some improvement with regard to this difficulty of the dependence of the energy on the court in a system but to there is still some trouble there is the dependence of W on the 3 coordinates X 1 X 2 X 3 and further if we want to look at things from the physical point of view but we should consider W defined on a certain hypersurface in spacetime and that we may ask ourselves what happens if we make just a small the formation in this hypersurface the small deformation of the order of gamma are gravitational constant which is really an extremely small the formation but we find that with this extremely small the formation w changes by quantity of the same order of magnitude as a itself there is town this difficulty but there
are some nice features about this expression for the energy density governance the gravitational part of this energy density can be divided into
2 terms it follows very naturally into 2 terms which are given by equations 8 and 9 1 of these
terms which are written w suffix K can be interpreted as kinetic energy because it is quadratic in the momentum there a be it is quadratic and homogeneous these momentum variables and it just like any ordinary kinetic energy is in physics so that can be very natural interpreters as
kinetic energy the other term does not involve momentum there was a tall and we call that the potential energy it is quadratic and homogeneous in the few quantities which we get by taking the 1st derivatives of the features so the gravitational part of the energy density divides into these 2 terms the 1st of these terms is subject to an uncertainty when we make a small the formation of the surface but is not subject to any uncertainty when we changed the chord in this in the surface it is of the correct tensor form with respect to the codons in the surface the other part of the potential energy is just the other way round that behaves alright when we make a small the formation of the surface but that gets disturbed when we changed the coordinates in the surface well that is the the situation with regard to
this improved expression for the energy density and that shows that there is still some uncertainty in the improved expression for the energy density depending on our system of coordinates so that we are still in difficulties with regard to the question of whether gravitational waves really k energy or not however there is
1 example where these difficulties can be eliminated and that is the example when we
have way this moving only in 1 direction if we apply these expressions for the the energy density to the case when there are
waves of moving in only 1 direction the direction of the axis x 3 you then we get the expressions written down by immigration is 12 and 13 that's what the potential and kinetic energy becomes in that case let us now make a change in our system of coordinates which preserves the condition that we have waves moving in only 1 direction we can calculate how these expressions 12 and 13 change under those conditions now if we have ways moving only in the direction x 3 and we make a disturbance and we make a change in our system of coordinates change the tire explore close to it thinks of this size be part of the size is infinite digital 1 of these things we
are functions of X 1 X 2 X 3 which fixed the change in our system of coordinates and if we make such a change in such a way as to preserve condition that will have
waves moving only in the direction it's 3 then this quantity here which is the change in our system of coordinates we must also involve only waves is moving in the direction and that means that the derivatives of B R will vanish the derivatives in all directions except the direction x 3 when we take those conditions into account we find that there is no change in the potential energy when we make this change in the system of
coordinates there is no change in the potential energy when applied to this example with waves moving in only 1 direction so we have quite a well-defined potential energy for this case we can do the corresponding thing for the time attic energy making a small the formation in the service we make a small the formation of the surface by shifting each point of the surface through a normal distance given by a function and silence any any being a function of X 1 X 2 X 3 if we do that currency how our potential energy changes under those conditions we get this result it changes by who think that the the 2 say that the the we get something which does not immediately seem to vanish but we have some which we can make use of haven't talked about yet some equations which
connect our Hamiltonian variables at 1 instant of time equations which I call the supplementary conditions some of
these equations are applied to the case when there is no matter present just leads to the vanishing of b 3 3 year in the example when they're waves moving only in the direction of the axis it's the so with the help of these supplementary creations we find that the potential energy I'm sorry this is the kinetic energy is also invariant under changes of our system of coordinates preserve the conditions that we have
waves moving in only 1 direction well we now have a situation which we can understand physically very well we have this situation that there is an uncertainty in the energy density even using our improved expression there is still an uncertainty in our energy density depending in our system of coordinates but there is no uncertainty when we have ways of moving in only 1 direction if we superposed waves moving in 1 direction with waves moving in a different direction and you know that the total energy density is not just the sum of the energy density of these waves and the energy density of these waves there is a 3rd term coming in what we might call the interference energy density which is linear in the amplitude of these waves and in the amplitude of these waves now it is this interference energy
density in which we have the uncertainty connected with the system of coordinates the uncertainty arising from the system of coordinates
involves only the interference energy density of waves moving in different directions and there is no uncertainty when we have waves moving in only 1 direction a further results that 1 finds is but if we take these expressions which I have for energy density kinetic and potential energy densities for ways moving in 1 direction and if we make a certain because transformations with the help of a supplementary conditions we get this expression for the energy density as a
positive definite form a sum of squares and that makes it quite definite that's when we have waves moving in 1 direction only these waves carried a positive energy ways of so this answers quite definitely the question of whether the waves in this theory of gravitation carry energy in this clear case of ways moving only 1 direction they certainly do carry energy now this 1 clear case is just what we want for discussing the retarded solutions which I talked about earlier the solution in terms of retarded potentials we have some accelerating masses here and we work out the retarded solution at great distances for this retarded solutions we have waves moving in only 1 direction that is just a condition which we have now been examining and where where we have seen that there is no uncertainty in the energy
density show that for this retarded solution we can certainly say that there is a positive amount of energy being radiated out words know what perhaps to
mention that the energy of the gravitational field is not always positive there's a difference there between the gravitational field and the electromagnetic field because of that part of the gravitational field which is concerned with Newtonian forces which corresponds to the common on field of electrodynamics this part has a negative definite energy the Newtonian field around a particle
as a native definite energy while the gravitational waves have positive definite energy and of course in general gravitational field where we have the 2 combined the energy may be either positive or negative this that difference from the that education where the field is such that the energy is always positive well the answer is our main questions about the reality of the gravitational waves they are to be understood as something which is physically real and which carries energy the other waves in physics we can apply quantum theory to these waves we have already seen what the polarization of the waves is and this polarization gives a after quantization spin with the values plus or minus 2 in the direction of motion we have a quantum of energy in these gravitational waves corresponding to the
photons of electrodynamics these quantum gravitational energy are called gravitons the have have a spin of plus or minus 2 in the direction of
motion they cannot have less of that the spin that corresponds to the fact that the photon cannot have spin 0 there are just these 2 values as spin corresponding to the 2 states all polarization well that is the situation with regard to the theory which can be deduced Einstein's equations the question arises of what is the importance of these results the experimental
physicist where gravitational forces really extremely weak you might know things they're week when you fall down but that is because 1 of the boldest taking part in this process is a very large body the but when you compare the gravitational forces in a fair way it with the other forces of physics you find that the gravitational forces are extremely weak there are 4 forces known to physicists nowadays the strongest of all is the force which holds a protons and neutrons together in nuclear forces which are associated with the pi-mesons then the 2nd strongest force this is the electromagnetic force which is associated with that photons the third-strongest force is the force which arises in the weak nuclear reactions and which is associated with that
neutrinos and then the 4th force the weakest of all is the gravitational force the other 3 forces have been observed on observed very extensively by physicists
that we just of those 3 the neutrino force has only been recently observed directly with the absorption of neutrinos the question of observing gravitons will be still more difficult the question of how observing neutrinos but in this not to have hopelessly difficult and physicists we are now actively concerned with the question of
trying to devise a means of observing gravity gravitons and in particular those Professor Weber with the University of Maryland who just gave a report on this subject at the conference last week near Paris and he thinks that the situation is not entirely hopeless for observing gravitons 1 could observe them by having some bits so electric crystal and the gravitational waves falling on this crystal which set up oscillations which would be transferred to the kinetic oscillations to electric oscillations and that it might just be possible to observe these under suitable conditions there are really 2 questions of interest 1st to observe whether there are gravitons falling on the earth from outer space there might be some coming from an unknown source the known sources would be pretty weak but there might be some unknown sources just like there are unknown
sources for the cosmic rays and just because the known sources are weak 1 should not be too discouraged and it would be worthwhile
to try to construct apparatus for observing whether there are gravitons coming in from outer space and this secondly the question now actually producing gravitons 5 laboratory means and then observing them and that is a quite a difficult question but they're still not altogether beyond the bounds of possibility and the
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