9 Gravitational Waves

9.1 Static Cylinder

The general external solution may be reduced to

This is equivalent to Wilson’s result (1920), who finds by considerations of geodesic motion.

Consider solutions inside the cylinder (radius ) corresponding to metrics of the type (not the most general)

of which (9.1) is a particular case.

Suppose we take , , where and are constants determined by boundary conditions. Field equations give

is small ( , say) since . Pressures are therefore small compared with density which is positive and approximately constant. is larger than other pressures and balances gravitational attraction. Integrating gives where depends on distribution. Considerations of periodicity of show . Other solutions for , give alternative but less interesting solutions. Geodesic motion shows that free particles outside the cylinder have, in general, an acceleration in the -direction.

9.2 Periodic Waves

Take metric of form

with , , functions , (not to be confused with previous , ). Rosen solves the case [field equations , , )] in free space, and takes as periodic wave solution (denoted by , 8 on detailed paper).

and a corresponding which has a factor and contains an aperiodic term, . We have superimposed the static exterior solution (transformed) on to Rosen’s solution and extended to within the cylinder obtaining a system which is initially physical, for of order . The aperiodic term makes the system unphysical by changing the sign of the density at . This is the first unphysical behavior of the system. The mass of the cylinder/unit length is initially , showing that vanishes at . The density changes sign at at .

9.3 Pulse Waves

A solution of field equations relating to (9.3) is

where represents the source of the pulse.

• If: (1) is of finite duration

• or: (2) is of infinite duration with a finite number of sign changes,

• or: (3) is of infinite duration with an infinite number of sign changes,

then , , , , for fixed , showing, by boundary conditions, etc., that when the pulse has travelled to infinity the mass of the cylinder reverts to its value prior to the pulse. Considerations of a particular form of pulse show that the mass does change during early motion of the pulse. It is clear that and derivatives are not of such a form as to allow the mass to return to its original value whatever the form of , as such functions as do not correspond to true pulses, but to travelling waves such as those considered above, which clearly radiate mass.

***

DE WITT asked, “If the mass eventually returns to its original state, are there any waves left out in space?”

BONDI replied that the waves travel out toward infinity. They carry no energy with them. If one integrates the energy momentum tensor through the interior during the time the motion is going on, then while the wave is being sent out the mass is decreasing, but as the wave dies down the mass returns to its original value.

In answer to a question as to why it was necessary for the process of emission of gravitational waves to be so unpredictable, BONDI replied that then and only then is information being transmitted.

DE WITT added, “In other words, if I know at an initial time that I am going to give you a yes answer, then the field already contains it.”

BELINFANTE asked if there were any incoming waves. BONDI replied that there were no incoming waves in this example at all. He then continued with the analogy between electromagnetic and gravitational field: “To my mind the electromagnetic field is like money spent. I do not get it back unless someone is very charitable. The gravitational field is more like my breathlessness when I do my exercise. When I stop, I regain my breath. If I do not stop (as in the periodic case) I will collapse. In the finite case, which to my mind is the more physical one, no irreversible change has taken place.”

BELINFANTE asked if one would find, although one starts with no fields, as in this case, that at a certain time the energy ceases to go outward and starts to go inward. Bondi replied that he has not so far examined this, so he does not have the answer to the questions.

WHEELER remarked, “How one could think that a cylindrically symmetric system could radiate is a surprise to me. There seems to be a far-reaching analogy between this case and the problem of emission of electromagnetic radiation from a zero-zero transmission in an atom or nucleus. The charge can oscillate spherically symmetrically, but the system doesn’t radiate. However, if we have an electron in the neighborhood, internal conversion can take place, with still no electromagnetic radiation emitted. This would correspond to the uptake of energy of the gravitational disturbance created by the ‘cylindrical symmetric’ exercise of yours.”

BONDI replied that he has had suspicions on that side also. To put it crudely, what stops the emission of electromagnetic radiation in the atom is the law of conservation of charge, and what stops gravitational radiation from taking place is the conservation of mass and of momentum. But he does not think there is necessarily anything against radiation of cylindrical symmetry. However, he hopes to be able to demonstrate some day that if one has a cylinder surrounded by a shell of matter one can transfer energy from the cylinder to the shell by means of cylindrically symmetric motions. This, he agrees, is required to complete the problem.

BONDI then reported on some of his own research. He takes a finite three-dimensional region in which there is mass, and considers the gravitational field at large distances from the mass. In this region the metric has the approximate form

The coefficients are functions of the variables in the brackets. He investigates what one gets if one proceeds to the approximation which includes the term. His strong impression is that this coefficient is independent of time. If we again consider the transmitter, only this time a much more general one (the system is three-dimensional) - but suppose the system to be spherically symmetrical both before and after emission - the “ ” of the two Schwarzschild solutions is then the same: hence no energy will have been lost.

The next speaker was WEBER, who made some remarks on cylindrical waves, using the pseudo-tensor formalism. He considered the Einstein-Rosen metric, and its relevant solution. Because of the linear nature of the equation one can construct a pulse which, for example, can implode from large distances. This cylindrical pulse will have a metric which is well behaved at large distances and this implies immediately that the energy per unit length of this wave will be zero. This “horse sense” argument is bolstered by an exact calculation of the pseudo-tensor. If one calculates and , he finds that they vanish everywhere. This has the consequence that energy cannot be transferred around as long as one has this type of symmetry and the above metric.

BONDI: “Where did you feed in the condition that the solution is well behaved at infinity?”

WEBER replied that he did not make use of this at all, but if it is well behaved at infinity, then nothing can happen. He continued: If one has a wave with energy per unit length which is zero, a particle put in the system will move in some fashion. Since this seems nonsensical, one thinks of some alternative. A possible one is that the total energy of the wave is not zero. He has shown that the energy of the entire disturbance is zero. He has also obtained an approximate solution of the Hamilton-Jacobi equation of a particle which interacts with this wave. The approximate solution says that if the particle is initially at rest, it will be at rest also after the disturbance has passed over it. The wave interacts with the particle in this respect like a conservative field - it takes no energy from the wave. In connection with the question of whether or not these solutions are trivial, he has calculated all of the components of the Riemann tensor, and has found that not all of them vanish.

BONDI remarked that it is vital, in this confusing subject, to make sure that one can physically detect what one is talking about. Also a single particle is a very poor absorber of any sort of energy.

WEBER replied that he thinks one could carry this sort of calculation out for a couple of particles.

BERGMANN remarked that if we try to bring in decent conditions at infinity, we are licked before we start. Also if we have a spatially limited area of disturbance, we cannot assume too low forms of symmetry because we are limited by four conservation laws, which have rigorous significance.

The next speaker was PIRANI. An attempt was made to formulate a definition of gravitational radiation in an invariant way. The definition was arrived at by making two assumptions: (1) gravitational radiation is characterized by the Riemann tensor, and (2) radiation must be propagated along the null cone. From this point of view, on a wave front one could expect to find a discontinuity in the Riemann tensor. One takes a space-time in which Lichnerowicz’s conditions hold, and calculates the permitted discontinuity across the wave front in the Riemann tensor. To make a physical interpretation one introduces a vierbein at any space-time event; its interpretation is that the time-like vector is the observer’s four-velocity, and the three space-like vectors are the Cartesian coordinate axes which he happens to be using at that time. In order to write down the permissible discontinuities, it is convenient to introduce the six-dimensional formalism. One re-labels the physical components of a skew tensor, regarding it as a six dimensional vector. The curvature tensor can be similarly cast in the six dimensional formalism, appearing as a symmetric 6-tensor. If one imposes the empty space-time field equations one finds that its form must be

where and are symmetric arrays. The spurs of both and are zero. Lichnerowicz’s continuity conditions permit the following array of discontinuities (taking the discontinuity across the surface ):

and are independent numbers. This form is obtained by taking a particular choice of and axes. Here can be made to vanish by a suitable choice of and axes. The above array, using the preceding definition, characterizes a gravitational wave front (or shock wave). The remarks from here on refer only to the pure gravitational field, which is analogous to the pure electromagnetic field.

In the electromagnetic case, one can define an invariant Poynting vector

If the field is not of the self conjugate type, then an observer following the field can make the Poynting vector vanish by acquiring a suitable 4-velocity. If it is self-conjugate, then the field contains pure radiation, and the Poynting vector vanishes only if the observer acquires the velocity of propagation of the wave.

Similarly, in the gravitational case one can define a certain timelike eigenvector, in terms of the Riemann tensor. This vector is interpreted as the 4-velocity of an observer following the field, and if for some field this vector collapses onto the null cone, one has radiation. The eigenvectors of the gravitational field are defined with the aid of Petrov’s classification of empty space-time Riemann tensors into three canonical types. These are:

The ’s and ’s are independent scalar invariants of the Riemann tensor. For example, in the Schwarzschild solution, which is of the first type,

The Einstein-Rosen metric gives a Riemann tensor of the second type. Using these forms one may work out the eigen 6-vectors of the Riemann tensor, and if one takes the intersections of the corresponding 2-planes in pairs, one obtains 4-vectors, the Riemann principal vectors. The definition of gravitational radiation is that if the Riemann tensor is of the second or third type, one has radiation, and if it is of the first type, one does not have radiation. It can be shown that the difference between the non-radiation type and one of the radiation types can be made to correspond to the discontinuity in the Riemann tensor across a wave front permitted by Lichnerowicz’s conditions. (Details of this work are to be published in the Physical Review, February 1957.)

Next, SCHILD presented some remarks on an apparent possibility to get indirect information on gravitational radiation by looking for gravitational radiation reaction force. The idea is built on an analogy to electromagnetic radiation reaction force. In the electromagnetic case, the equations of motion of a particle are

One proceeds by analogy and attempts to write the geodesic equations in the gravitational case as

where, in analogy with the electromagnetic case

A procedure like this does not work in principle, and this for the following reasons: For an equation like this to make sense, one must go to a limiting background field, letting the mass tend to zero. That is, what one really is considering is the limit of the field containing mass as the mass goes to zero:

This means that we are considering different Riemannian 4-spaces, with no natural one-to-one correspondence between the points of any two spaces. Thus we must demand invariance under independent transformations in different spaces:

Since the equations are tensorial under

it remains to demand (to order in considered) invariance under

However, the equations are not invariant under such a transformation. It appears from an investigation still in progress that any right hand side can be wiped out by a suitable choice of such a transformation.

Geometrically one may picture this as follows:

If one changes coordinate systems in (or equivalently, a one-to-one correspondence ) one can move into coincidence with .

PIRANI reported the following communication from ROSEN.