## 9.1 Static Cylinder

The general external solution may be reduced to

9.1 |

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)

9.2 |

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

9.3 |

with
,
,
functions
,
(not to be confused with previous
,
). Rosen

and a corresponding
which has a factor
and contains an aperiodic term,
. We have superimposed the static exterior solution (transformed) on to Rosen’s

Fig. 9.1

## 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

In answer to a question as to why it was necessary for the process of emission of gravitational waves

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

BELINFANTE

WHEELER

Fig. 9.2

BONDI

BONDI

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

The next speaker was WEBER,

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

WEBER

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

BERGMANN

The next speaker was PIRANI

where
and
are symmetric
arrays. The spurs of both
and
are zero. Lichnerowicz’s

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

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

The
’s and
’s are independent scalar invariants

The Einstein-Rosen *Physical Review*, February 1957.)

Next, SCHILD

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:

Fig. 9.3

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

PIRANI