EXAMPLES

CHAPTER4

IV. EXAMPLES

As we have noted before, when all parameters dk are zero, the general solution

(3.18H3.22) reduces to the solution (2.4) of Sec. II that was obtained from the inte-

rior Schwarzschild metric. From now on, we will denote by the symbol ( dk ) the unique colliding plane wave solution (3.18H3.22) which corresponds to a given choice of the parameters dk . When all dk are zero except for the parameter d _{0 ,} the solution (dk }=(d₀,0,0,0, ... ) is again equal to (2.4), except in this case the mass M =-,Jab /2 (and the boost-invariant product -{(if; of the characteristic wavelengths) is rescaled by a factor e -do corresponding to a monopolar distortion of the solution (2.4 ).

To illustrate the evaluation of the function

1\u

,v) by Eqs. (3.20H3.22), we write down below the functions

v

2(u ,v) and

y

2(u ,v) corresponding to the solution {0,0,1,0,0, ... }, where all dk are zero except for d₂=1. The metric in the interaction region of this solution is locally isometric to an interior Weyl metric (3.13); this Weyl solution can be interpreted as the interior metric of a Schwarzschild black hole dis- torted by a static, quadrupolar matter distribution outside the horizon:

A 1 [ ⁰

2[

^u

v]

⁰

2[

^u

v] 2[

^u

v]

⁰

2[

^u

v]]

'1'2(u ,v

)="2

^{3 Sill}

-;;+[;

Sill -;;-[; +cos -;;-[; -sill

-;;+[; ,

^(4ol)

(4o2)

where

..f;ii; ^{0 [} u

v]

^{0 [ u}

v]

z

=--

₂-sm

-;;+[;

sill -;;-[; , ^(4.3)

q

f sds

I ₁(q;a,b,c,d)=. ~ ,

o ...fs+c (s+a+ s²+bs+d)

q ds

I2(q;a,b,c,d)=f ~

o...fs+c (s+a+ s²+bs+d)

We now tum to our second example of a colliding plane-wave solution in the family (3ol8H3o22): a solution which describes colliding purely anastigmatic plane

waves.⁷- 9 According to Eqs. (3.18)-(3.22), the metric gn on the region II (Fig. 3) of a solution { dk } is given by

where

+e21jr¹¹

(u)[

^{l+sin(u Ia )] dx2}

1-sin(u/a) (4.4)

(4.5a)

(4.5b) The metric (4.4)-(4.5) describes the geometry of one of the two incoming colliding plane waves in the solution {dk}. As before, the metric gm in region ill (Fig. 3) (which describes the remaining incoming wave) is obtained by replacing u Ia by v /b in the above equations.

Unfortunately, the polynomials (2t-1)k¹²Pk(ttht-1) are not orthogonal polyno- mials with respect to any weight function, since they fail the Darboux-Christoffel test (Ref. 14, Sec. 8.90). However, we shall construct one particular infinite sequence (dk) -

[Eqs. (4.11) below], such that for the corresponding solution {dk} the function "'n(u) [Eqs. (4.5)] has the right asymptotic behavior as u ^--Hta /2 to make the incoming plane wave in region IT [Eq. (4.4)] purely anastigmatic.⁷-9 Clearly, because of the u ~v symmetry of our solutions, with this choice of the parameters dk the other incoming

plane wave (which is represented by the metric gm on region III) will also be purely anastigmatic. Also note that Eqs. (4.11) represent only one particular solution in our family of solutions for which the incoming plane waves are purely anastigmatic; the details of the construction below will make it clear that infinitely many different solu- tions {dk} with the same property can be found in the fanilly (3.18)-(3.22).

To proceed, consider the following function

f

(t ), defined on the interval (-1,1) :

f

(t )=ln(1-t)

f

^{(t )=ln(1} +t)

Since

f

(t) is even, there is an expansion

00

! <t

⁾⁼

'L

dk P 2k <t) .

k=O

for t~,

for t<O. (4.6a)

(4.6b)

Since

f

(t) is square integrable on (-1,1), this expansion converges absolutely every- where on ( -1,1) with the exception of the point t =0 at which

f

(t) is not C ¹. In fact,

1

A 4k+1

J

dk = - -

f

(t )P 2k (t )dt 2 -1

1

=( 4k + 1) Jln(l-t )P _2k (t )dt .

0

Now consider the solution {dk}, - where dk are defined by

fork even, fork odd.

(4.7)

(4.8)

Then, the function Vn(u) [Eqs. (4.5)] for this solution

tc4}

is given by

(4.9)

where t::sin²(u/a ). However, fortE (0,1) and for all k ~1 we have¹⁴

I (2t-1)kP₂k[

~

_2t-1 ^{t ] I< 1}

⁼

^{sup IP2k(t)l .}

IE(<f,l)

Therefore, the series (4.9) converges absolutely to a continuous function on the inter- val (0,1). We can write

+

i:

^{8k ( 1-t )k '}

k=l

where the second series is convergent being the difference of two convergent series on (0,1). Hence, by Eq. (4.9) and Eqs. (4.6)

where

lim S (u )=0.

U-Hta/2

We thus obtain the following for the asymptotic behavior of Vn(u) as u ---t1ta /2

2Wu(u) [1 · "] 2

[1 · "] 2

e - -sm- +sm- ,

u-tlta/2 a a ^(4.10a)

-2~u(u)

^{[ 1 . U l-2 [ 1+ . U l-2}

e - -sm- sm-

u-Hta/2 a a ^(4.10b)

The asymptotic behavior of the function "'m(v) [which is the counterpart of "'n(u) in region III of the solution { dk } ] will have the analogous form near the focal plane ( v=nb/2}. From Eqs. (4.10) we conclude, by inspecting the metric 8n (and 8m) given by Eq. (4.4), that for our solution {dk} both incoming plane waves are purely anastig- matic, i.e., for both incoming plane waves the metric coefficients 8xx and 8yy vanish simultaneously on the respective focal planes { u ^=1ta /2} and { v =nb /2}.

- -

TI1e coefficients dk for the solution { dk } can be calculated explicitly using Eqs.

(4.7) and (4.8). This gives¹⁴

- " 4k+l k (-1)' (4k-2/)!

d

²

k=dk=-~;~J

(2k-2l+l)l !(2k-!)!(2k-2!)!

X(\j/(2k-2/ +2)-\j/(1)) , (4.11a)

where

\jl(x )= dx d [lnr(x)]

is Euler's psi function,¹⁴ and

(4.11b)

for any k~O.

V. CONCLUSIONS

The infinite-parameter family of colliding plane-wave solutions we have con- structed in Sec. ill have the following new features.

(i) The interaction regions of our solutions are locally isometric to interior Weyl black-hole solutions. An observer who enters the interaction region will not be able to distinguish, by local measurements that she performs completely within the interac- tion region, the geometry of the surrounding spacetime from the geometry in the inte- rior of a black hole.

(ii) The metric functions of our solutions have oscillatory forms in a suitable coordinate system [Eqs. (3.18H3.22)].

(iii) By constructing an infinite series expansion for the function "'(u',v) [Eq.

(3.19)], we have built a colliding plane-wave solution in our family for which both of the two incoming colliding plane waves are purely anastigmatic, i.e., for which both incoming waves have equal focal lengths (Sec. IV).

On the other hand, our solutions suffer from the following drawbacks, some of which are common to all presently known exact solutions for colliding plane waves.

(i) As with the fatuous Khan-Penrose solution,⁵ so also here, there are <>-function contributions to the Riemann curvature (gravitational shock waves) on the boundaries between the adjacent regions (Fig. 3 ); i.e., the metric is not C ¹. The reason for this discontinuous behavior is the particular prescription that we use to extend the metric beyond the interaction region. It is clear, from the form of our metric as described by

Eqs. (3.18H3.22), that no finite sum (3.19) will eliminate the <>-function shocks

across the boundaries as long as we use the Penrose prescription for extending the metric beyond the interaction region. Since infinite sums of the form we have dis- cussed in Sec. IV will in general converge only in the mean (i.e., in the L ² sense), we

cannot reliably employ infinite series expansions of the form (3.19) to construct smoother wave forms.

(ii) Except for the characteristic wavelengths a and b which can be freely adjusted by scale transformations, the two incoming colliding plane waves in our solu- tions have exactly the same functional form. The reasons for this are the u ~v sym- metry of the metric (3.18) in the interaction region, and the u ~v symmetry of the Penrose prescription for extending the metric beyond the interaction region.

(iii) The incoming colliding plane waves in our solutions are not of sandwich type in general; i.e., the spacetime regions II and ill (Fig. 3) representing the incoming waves are not flat near the respective focal planes of these waves. As we have dis- cussed in detail in Sec. IT, this property is responsible for the curvature singularities that are present at the focal planes of our solutions. The technique of "cutting off' the incoming waves just before their focal planes, which we have discussed in Sec. II, successfully avoids this difficulty from a physical viewpoint; however, it appears exceedingly difficult to determine whether the resulting solution (e.g., the solution depicted in Fig. 4) can be expressed in closed form as a member of the family of solu- tions (3.18H3.22) that we have constructed.

We conclude by listing some open questions which suggest directions for further research on some of the issues that we have raised in this paper.

(i) Are there different prescriptions for extending the metric (3.18) beyond the interaction region which could resolve some of the drawbacks in our solutions listed above?

(ii) Can the technique of using static axisymmetric black-hole metrics to generate parallel-polarized colliding plane-wave spacetimes be generalized to stationary axisymmetric solutions? Such a generalization presumably would yield an infinite-

parameter family of solutions representing colliding plane waves with nonparallel polarizations.

ACKNOWLEDGMENTS

The author wishes to thank Kip S. Thome for a critical reading of the first draft of this paper, and Sandip Chakrabarti for stimulating discussions which initiated this research. This work was supported by the National Science Foundation under Grant No. AST 85-14911.

REFERENCES FOR CHAPTER 4

1 B.K. Harrison, Phys. Rev. Lett. 41, 1197 (1978); G. Neugebauer, J. Phys. A 12, L67 (1979); C. Hoenselaers, W. Kinnersley, and B.C. Xanthopoulos, Phys. Rev. Lett. 42, 481 (1979); J. Math. Phys. 20, 2530 (1979); D. Kramer and G. Neugebauer, Phys.

Lett. A 75,259 (1980); W. Dietz and C. Hoenselaers, Proc. R. Soc. London A382, 221 (1982).

2 S. Chandrasekhar and V. Ferrari, Proc. R. Soc. London A396, 55 (1984); S. Chan-

drasekhar and B.C. Xanthopoulos, ibid. A398, 223 (1985); A402, 37 (1985); A402, 205 (1985); A403, 189 (1986).

3 P. Szekeres, J. Math. Phys. 13, 286 (1972).

4 U. Yurtsever, Phys. Rev. D 38, 1706 (1988) (Chapter 5).

5 K. Khan and R. Penrose, Nature (London) 229, 185 (1971).

6 S. Chandrasekhar and B.C. Xanthopoulos, Proc. R. Soc. London A (to be published).

7 U. Yurtsever, Phys. Rev. D 36, 1662 (1987).

8 R. Penrose, Rev. Mod. Phys. 37, 215 (1965).

9 U. Yurtsever, Phys. Rev. D 37, 2803 (1988) (Chapter 3).

1

°

F.J. Tipler, Phys. Rev. D 22, 2929 (1980); KipS. Thorne, in Nonlinear Phenomena in Physics, proceedings of the 1984 Latin American School of Physics, edited by F.

Claro (Springer, Berlin, 1985).

11 R. A. Matzner and F.J. Tipler, Phys. Rev. D 29, 1575 (1984).

12 H. Weyl, Ann. Phys. 54, 117 (1917); L. Mysak and P. Szekeres, Can. J. Phys. 44, 617 (1966).

13 S. Chakrabarti, Ph. D. thesis, University of Chicago, 1985.

14 I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

FIGURE CAPTIONS FOR CHAPTER 4

FIG. 1. The two-dimensional geometry of the characteristic initial-value problem for colliding plane waves. The null surfaces N₁={u=O) and N₂={v=O) are the past wave fronts of the incoming plane waves 1 and 2. Initial data corresponding to waves 1 and 2 are posed, respectively, on the upper portions of the surfaces N₂ and N₁ that are adjacent to the interaction region I. The geometry in region IV is flat, and the geometry in regions II and III is given by the metric describing the incoming waves 1 and 2, respectively. The geometry of the interaction region I is uniquely determined by the solution of the above initial-value problem.

FIG. 2. The region J in Schwarzschild spacetime to which the interaction region of the colliding plane-wave solution (2.4) is locally isometric. This region J is shown shaded in this figure which is drawn in a { t=const), { <j>=O, 1t) plane. As explained in the text, the geometry in region J is extended nonanalytically beyond the null surfaces r=M(1+cos9) and r=M(l-cos9), which correspond to the wave fronts {u=O) and { v =0}, respectively. Consequently, all coordinate singularities are avoided and the Schwarzschild metric on the shaded region J is lifted from S²xR² to R⁴, on which the final metric (2.4) is defined.

FIG. 3. A two-dimensional projection of the geometry of the colliding plane-wave solution (2.4). The metric is analytic throughout each of the regions I, IT, ill and IV, but it suffers discontinuities in its derivatives across the boundaries between the adja- cent regions. The interaction region I is locally isometric to region J (Fig. 2) of the interior Schwarzschild solution (2.1). The curvature singularity at (u Ia )+(v lb )=1t/2 corresponds, under this isometry, to the Schwarzschild singularity at r =0.

FIG. 4. Geometry of the colliding plane-wave solution that results from "cutting off' the incoming, colliding plane waves described by the solution (2.4 ). As explained in the text, the introduction of the secondary shocks along the surfaces { u =uc ) and { v =v c } removes the curvature singularities on the focal planes { u =u1 ) and { v =v 1 } . However, the geometry in the interaction region of this new solution is not every- where described by the metric (2.4); the regions Ia and Ib are described by a different metric.

~ II

. . _..

. •

. .

./

/ N

i f

i / i ·~'

.l i

.? /

,

l

.;t .;

/ I

j

I I

.j

,

/

I

!

r=M(1-cos 8) r=M(1 +COS 8)

8=0 (coordinate singularity) r=M(1 +COS 8) r=2M, the horizon

I

-

VI

(coordinate singularity)

~

r=M(1-cos 8) 8=1t (coordinate singularity)

>-..

-

^•-4 _~

-

^{'00 ::J}

•-4 ~

...._,

II\

•

...

..Q

""' ^;::.

...,

... +

~ =

...,

~

>

~ ~

01 01

~

~

>

~

~

_4-1

\..

Ll