I am very glad to be here in Lindau for the third 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 first put forward by Newton more than 200 years ago.
Newton's theory was a very good theory.
It survived unchallenged for more than 200 years.
And then in the present century a new theory of gravitation was put forward by Einstein.
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 a 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 the interest in the Einstein theory of gravitation rather died away.
People found the equations difficult to work with.
They found other subjects of interest, largely quantum theory.
And for a while one did not hear so much about the Einstein theory in the world of science.
But in recent times, since the war, there has been a revival of interest in the Einstein 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 two 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 international conferences on gravitation, which are held regularly every two years.
The third 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 present reports on their resent researches and discuss the problems, which still remain to be solved.
I think that the people who come to these conferences, the people who work on gravitation
can very clearly be divided into three 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.
(Laughter).
The other two classes are concerned with the actual world.
The physicists are concerned with studying the gravitational field
as a physical field and finding out the physical effects of gravitational forces.
And they hope to be able to detect these effects with their instruments.
The cosmologists 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 one does not need to use the exact equations of the Einstein theory.
One can work with a certain approximation.
The effects of the gravitational field are attributed to a curvature in spacetime.
And for the physicist one 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 theory
applying to the case when the curvature of spacetime is extremely small.
That would not do for the cosmologists, because this approximation would not be a good approximation,
if one were concerned with extremely large distances.
Distances comparable with the distances between disparent nebulae for instance.
But for the distances, which interest the physicist this approximation is certainly a valid one.
Now the exact equations of the Einstein theory is the equation (1) in these notes.
I think a good many of you have the German translation of these notes with you,
so that I need not write down all the equations but can just refer to them.
And I will write down on the blackboard only the more important equations.
If we take the approximate form of the Einstein theory, when it is applied to space,
which is nearly flat, we have this as our basic equation.
We have an equation, which involves a quantity h-mu-nu, which is introduced in this way:
The exact theory of Einstein is based on a certain tensor, which is written like this, G with two suffices mu and nu.
And mu and nu take on the four values 0, 1, 2, 3.
This tensor describes the gravitational field, describes the curvature of spacetime.
And it also fixes the system of coordinates. This tensor describes the gravitational field, describes 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 the values 1 or -1 or 0.
And the differences from flat spacetime we denote by h-mu-nu and we count h-mu-nu as small and neglect quantities,
which are of the second order of smallness.
We then have this as our basic equation of the Einstein theory.
The Laplacian operator, which is denoted by the symbol square applied to h-mu-nu.
First I will write down all the equation and then explain it.
dv-mu by dx-mu plus dv mu by dx mu equal 16 pi gamma rho-mu-nu.
This rho-mu-nu is constructed from the tensor, which describes any matter which is present.
And in particular, this rho vanishes when there is no matter.
Gamma here is the gravitational constant and it counts as very small in the approximation with which we are working.
This v-mu is a certain quantity, which is constructed from the first derivatives of h-mu-nu.
And you will find the expression for v-mu written down in equation (3).
And I need not say more about that.
This is then our fundamental equation.
Now for dealing with this equation people usually choose a system of coordinates, which makes this quantity v-mu vanish.
It's quite a nice condition to impose on the coordinates.
And when people are working with these coordinates, they say that they are working with harmonic coordinates.
This harmonic condition on the coordinates, is one which is used very extensively
and it results in a big simplification in the equation, because with these harmonic coordinates these two terms just vanish.
And we are left with this equation with just those two terms.
Now if we apply that equation to a region of space and time, where there is no matter present,
we have this term also vanishing and we have just this equation left, square h-mu nu equals 0.
And that is just the well-known equation 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 fields,
the theory of Einstein leads to these waves in this quantity h-mu-nu.
In those regions of space and time where there is no matter.
Now an important feature of Einstein's theory is that it is valid for all systems of coordinates.
We are working with a 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 little curvature in our space,
which prevents one from giving a precise meaning to Cartesian coordinates.
But still, we can take coordinates, which are approximately Cartesian
and that is what we are doing when we introduce these quantities h-mu-nu.
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 are really something physical
or whether they are just connected with our system of coordinates.
Now that is really the main difficulty all the time when one is working with the Einstein theory.
It is the difficulty of separating what is real and physical, from what depends simply on our system of coordinates.
And that will be our 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 them.
And we want to discuss exactly what happens.
I shouldn't 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 a solution of this first equation here.
Now solutions to that equation are quite familiar to physicists,
because this equation itself is very similar to the equation, which we have in electrodynamics.
We can look upon this right hand side as generating waves in this quantity h,
in the same way as electric charges and currents generate 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 rho.
There is a solution in terms of retarded potentials.
And this will be the solution, which is the important one - physically - if there are no incoming gravitational waves.
The general solution one gets from this retarded solution by adding on to the retarded solution
certain arbitrary incoming waves.
But it will be mainly the retarded solution that I want to talk to you about.
With this retarded solution we have the h-mu-nu at larger distances proportional to one over r.
r being the distance.
The h-mu-nu are rather like the electromagnetic potentials
and to get something which corresponds to the electromagnetic field,
something which we can count as the gravitational field, we must differentiate this h-mu-nu once.
If we differentiate something, which is of the form of one over r for great distances, then we get two terms appearing:
One term depending on one over r^2 and the other term depending on omega over r.
I should say a term of the order of omega over r, where omega is the frequency of oscillations,
which are occurring in this distribution of matter.
For distances, which are not too large the one over r^2 term is the important term.
That one over r^2 term gives you the Coulomb force in electrodynamics.
And it gives you the Newtonian force in gravitation.
But for much larger distances than that the omega over r term is the dominant term.
And this term corresponds to waves.
So that this will be the important term for our talk today.
This term which dominates the solution at very large distances.
Let us now fix our attention on the waves, which come out in one particular direction.
Let us say the waves, which come out in the direction of the axis x3.
Suppose this is the axis x3.
And then we want to examine the solution of our field equation,
four points out there where x3 has some large and positive value.
And x1 and x2 are small.
In this region out here, we shall 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 d by dx1 and d by dx2 equal to zero.
But we must also put d by dx3 equal to minus d by dx0.
There is an error in the paper, which has been distributed; this minus sign has been omitted so please insert it.
We can no see what is the effect of putting in these conditions into the solution of our field equations.
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, equation (6).
Now as I mentioned before, even with the harmonic conditions there is still some arbitrariness in our system of coordinates.
So we can take this question, let us make a change in our system of coordinates,
a change, 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 (5) in the notes.
Where a-mu is a field function which fixes the change.
And a-mu 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 ten quantities, h-mu-nu.
And these changes are given in the notes.
I don't need to describe them in detail, but I will just say what the important result is:
We find that when we make this change in coordinates six of the h-mu-nu remain invariant
and the other four of these ten quantities can get 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.
So that these four components of h-mu-nu which gets changed arbitrarily, will not have any physical significance.
There are the six invariant ones and of these six invariant ones,
four must be zero because of the harmonic conditions themselves.
And that leaves only two, which are invariant and which are not restricted to be zero.
Those two are the components h12 and h11 minus h22.
These two components we may therefore expect to have a physical meaning.
For waves which are moving in the direction of the axis x3.
They are invariant under any transformations, which we can make,
which preserve the harmonic conditions and they are not restricted to be zero.
Well this means then that we should expect that we have these gravitational waves,
which are physical and that we have these two kinds of polarisation
for gravitational waves moving in the direction of the axis x3.
The question remains: "Should these waves really be counted as something physical?"
And that is rather to the question: "Do these waves carry energy?"
So that brings us to the discussion of the question of the energy of the Einstein field of gravitation.
For this discussion of energy one can set up a certain tensor or tensor density, T-mu-nu,
which has the physical significance that its components are connected with stresses and energy density, momentum density.
And one of its components, the T-0-0 component, can be interpreted as the energy density.
One finds that this T-mu-nu added 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 T-0-0, we get exact conservation of energy.
But there is some trouble with this little t.
I talked about it as a tensor density, but it is not really a tensor density,
it is something, which is called a 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 use this T-0-0 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 shall get a different energy.
Now energy ought to be something, which is physical, we want it to be a really physical thing.
And it should be independent of our system of coordinates.
This T-0-0 is really the best thing, which we can do for discussing energy density.
And we have here a real difficulty.
This difficulty has bothered 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 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 a very logical process, it's not logical at all and it is unreliable.
And on account of that there has been much discussion for very many years
as to whether these gravitational waves really do carry energy or not.
Well with the development of the theory of gravitation,
which has taken place in recent times, this question has been cleared up.
One of the main lines of this recent development
has been the expression of the equations of the Einstein theory in the Hamiltonian form.
Now the Hamiltonian form of writing equations is a form, which has very great mathematical power.
It was discovered more than 100 years ago by Hamilton,
who worked it out simply because of the mathematical beauty connected with it.
And Hamilton himself did not realise the great importance of his form of equations.
But we see now that his form is really of fundamental importance in nature because his form of equations is the form,
which lends itself naturally to a passage to the quantum theory.
Just working from the Newtonian form of equations in motion, one has not got any good way of passing to the quantum theory.
But working from the Hamiltonian form we have well defined rules,
which have been applied successfully in very many cases
for passing from any classical field theory or 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 one must first put the classical theory into Hamiltonian form.
Now, with the Hamiltonian form of the equations one deals with the state at a certain time.
Now the state at a certain time means the state for all values of the coordinates x1, x2, x3,
but for one particular value of the coordinate x0.
Now you see when we discuss the state at a certain time, we are introducing a dissymmetry between the four coordinates.
One of the great features of Einstein's theory, was the fact that we had asymmetry between the four coordinates,
the three coordinates of space and the one time coordinate.
And for a long time people were interested only in developing the Einstein theory in a form, which preserved this symmetry.
It is just within the last few years that people have found,
that they can get a lot of new results by departing from this symmetry
and in particular by working with this concept of the state at a certain time.
Where we go entirely away from this four-dimensional symmetry
and we go back to the old idea of a three-dimensional world changing with a time coordinate.
With the development of the Hamiltonian form, we get this work in which we destroy the four-dimensional symmetry.
And of course in a way it's a pity to destroy the four-dimensional symmetry, everyone would agree with that.
But there are these compensations that one has great mathematical power
and one finds some new features of the equations, which are not so obvious when one keeps to the four-dimensional symmetry.
With the Hamiltonian form one's dynamical variables are all paired off into dynamical coordinates and conjugate momenta.
So far as concerns the gravitational field,
we have the G-mu-nu for all values of x1, x2, x3 appearing as dynamical coordinates.
And we have then momentum variables P-mu-nu appearing as the conjugates of these dynamical coordinates.
Now one of the first things one found when one started to put the theory into Hamiltonian form one got a result,
which was rather unexpected.
Which was of the ten quantities G-mu-nu and their conjugate P-mu-nu,
four of the P-mu-nu and their conjugates drop out from the Hamiltonian equations of motion.
Namely these four: G-mu-0, P-mu-0.
If one of these indices takes on the value 0, either one, either the first or the second because it's symmetrical,
then we get these quantities here.
And these quantities drop out from the Hamiltonian equations
and we are left with Hamiltonian equations involving only the variables G-r-s, P-r-s.
Now these roman letters r 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 0, 1, 2, 3.
We have here just six G-r-s's and six P-r-s's instead of the ten G-mu-nu's and P-mu-nu's.
And that means that with the Hamiltonian formulation,
we start off expecting to have ten degrees of freedom for each point of space.
But four of the degrees of freedom drop out and we're left with just six degrees of freedom for each point of space.
And that is a big simplification and this simplification, which brings out the advantages of the Hamiltonian formalism.
Now this simplification ought not to surprise one too much.
One might have expected it if one just looked into what is really needed for describing the state at a certain time.
The state at a certain time means the state for all regions of space for a certain value of x0.
And that is to be pictured in spacetime as a three-dimensional hypersurface.
The hypersurface x0 equals constant.
Which is to be pictured as existing in four-dimensional spacetime.
Now to describe such a hypersurface, we need only the six G-r-s's.
They are sufficient to describe the geometry of the hypersurface and the coordinate system in the hypersurface.
And these G-mu-0's are needed only to describe the relationship of this hypersurface to a neighbouring hypersurface.
But if you're interested only in describing the state at one particular time, then we only need these six G-r-s's
and we also need their dynamical conjugates.
So that from rather general arguments, arguments of a geometrical and kinematical nature,
one can see that these six degrees of freedom are all that is really necessary.
If we ask this physical question, how should we 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 or the energy density in the region at a certain time
should not depend on these variables G-mu-0, P-mu-0.
Now if you look at the energy density given by the pseudo tensor, this T-0-0, that we had before,
and you work it out, you see that T-0-0 does depend on G-mu-0.
T-0-0 thus involves some quantities which are not really relevant for describing the state at a certain time.
It involves certain things, which are concerned only with the coordinate system.
Now that is quite a bad feature in this energy density, this pseudo energy density.
And we can improve upon this feature by taking a modified expression for the energy density.
We get a modified expression for the energy density by expressing this T-0-0
in a suitable way and substituting for the G-mu-0's which occur in it, their values for flat spacetime.
namely we substitute for G-0-0 the value -1 and for G-1-0, G-2-0, G-3-0, the values 0.
By this procedure we can get an improved expression for the energy density.
An improved expression, which I call W.
Now W still depends on our system of coordinates.
Although not so badly as T-0-0.
It means that we have made some improvement,
with regard to this difficulty of the dependence of the energy on the coordinate system.
But there is still some trouble left.
There is still a dependence of W on the three coordinates x1, x2, x3.
And further, if we want to look at things from the physical point of view,
we should consider W defined on a certain hypersurface in spacetime.
And we may ask ourselves: "What happens if we make just a small deformation in this hypersurface?"
A small deformation of the order of gamma or gravitational constant.
Which is really an extremely small deformation.
But we find that with this extremely small deformation W changes by a quantity of the same order of magnitude as itself.
Well there is still this difficulty but there are some nice features about this expression for the energy density W.
The gravitational part of this energy density can be divided into two terms.
It falls very naturally into two terms.
Which are given by equations (8) and (9).
One of these terms, which I have written W suffice K, can be interpreted as kinetic energy
because it is quadratic in the momentum variables p.
It is quadratic and homogeneous in these momentum variables.
And is just like any ordinary kinetic energy is in physics.
So that can be very naturally interpreted as kinetic energy.
The other term does not involve momentum variables at all.
And we call that the potential energy.
It is quadratic and homogenous in the field quantities, which we get by taking the first derivatives of the h's.
So the gravitational part of the energy density divides into these two terms.
The first of these terms is subject to an uncertainty, when we make a small deformation of the surface.
But is not subject to any uncertainly, when we change the coordinates in the surface.
It is of the correct tensor form with respect to the coordinates in the surface.
The other part, the potential energy, is just the other way around.
That behaves all right when we make a small deformation of the surface.
But that gets disturbed when we change the coordinates in the surface.
Well that is the situation with regard to this improved expression for the energy density.
And that shows that there is still some uncertainly 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 our gravitational waves really carry energy or not.
However there is one example where these difficulties can be eliminated.
And that is the example when we have waves moving only in one direction.
If we apply these expressions for the energy density to the case when there are waves moving in only one direction.
The direction of the axis x3, then we get the expressions written down by equations (12) and (13).
That's what the potential and kinetic energy...(end).