Unified Bertotti-Robinson and Melvin Spacetimes

Unified Bertotti-Robinson and Melvin spacetimes

Department of Physics, Eastern Mediterranean University, G. Magusa, North Cyprus, Mersin 10, Turkey (Received 28 May 2013; published 10 September 2013)

We present a solution for the Einstein-Maxwell equations which unifies both the magnetic Bertotti-Robinson and Melvin solutions as a single metric in the axially symmetric coordinates ft;
; z; ’g. Depending on the strength of magnetic field the spacetime manifold, unlike the cases of separate Bertotti-Robinson and Melvin spacetime, develops singularity on the symmetry axis (
¼ 0). Our analysis shows, beside other things, that there are regions inaccessible to all null geodesics.

DOI:10.1103/PhysRevD.88.064021 PACS numbers: 04.20.Jb, 04.20.
q, 04.40.Nr

I. INTRODUCTION

The Bertotti-Robinson (BR) [1] and Melvin (ML) [2] solutions of Einstein-Maxwell (EM) theory have been well known for a long time; these had significant impacts on different aspects of general relativity. For decades they remained in fashion and found applications in connection with stellar objects, cosmology, string theory, etc. A recent study discusses the similarities/differences between these spacetimes [3]. It is shown, among other things in [3] for instance, that the only geodesically complete static EM spacetimes are the BR and ML solutions. Since they share more common properties than contrasts, the natural ques-tion arises whether it is possible to describe both soluques-tions in a common metric. This is precisely what we show in the axially symmetric (i.e., t,
, z, ’ coordinates) geometry in this paper. It should be added that large classes of Einstein-Maxwell-Kundt solutions (for the Kundt solution see [4]) found a long time ago by Plebanski and Demianski (PD) [5,6] both admitted separate BR and ML limits in different coordinates through specific limits. We work out our solu-tion entirely in the axially symmetric ft;
; z; ’g coordi-nates and express our metric in those coordicoordi-nates. Our solution admits the BR limit but not the separate ML limit. In other words the BR universe forms the background of our spacetime on which ML is added. In obtaining the solution we choose the magnetic phase of the BR solution so that the total magnetic potentialcð
; zÞ is expressed as a superposition,cð
; zÞ ¼c_BRð
; zÞ þc_MLð
; zÞ. The EM solution constructed from cð
; zÞ is what we dub as the ‘‘unified BR and ML spacetime.’’ The solution involves two parameters, ₀ (for BR charge) and B₀ (for ML charge). The ranges of parameters are 0 < j0j < 1 and
1 < B₀< 1, so that our solution does not admit the ML limit.

BR spacetime is conformally flat whereas ML is cylindrically symmetric which becomes flat near the axis
! 0. For a finite
and jzj ! 1 ML is not flat. Both are singularity free; a feature that makes them

attractive in cosmology and string theory. We remark also that the BR solution can be obtained by a coordi-nate transformation [7] from a spacetime of colliding electromagnetic waves known as the Bell-Szekeres solution [8]. Using the Ernst formalism we showed long ago that within this formalism Bell-Szekeres and Khan-Penrose [9] solutions can be combined through a suitable seed function [10]. Also Schwarzschild and BR spacetimes were interpolated by the electromagnetic parameter in the oblate spheroidal coordinates [11]. Within similar context superposition of spinning sphe-roids [12] from harmonic seed functions in the Zipoy-Voorhees metric [13] were obtained. It is remarkable that interpolation of BR and ML solutions takes place in the static axial coordinates ð
; zÞ instead of oblate/ prolate coordinates. The latter coordinate systems are known to admit separability in the Laplace equation and had much impact in the development of solution gen-eration techniques. One of the important conclusions to be drawn in this study is that two electromagnetic fields, which separately yield regular spacetimes, namely the BR and ML, may yield a singular spacetime upon their combination. Physical interpretation suggests that the mutual magnetic fields focus each other strong enough to result in a singularity. The singularity at
¼ z ¼ 0 (for B0

0< 0 and B0

0> 1) does not exhibit directional properties [14], that is, the Kretschmann scalar diverges irrespective of the way of approach and (
¼ 0, z > 0) is the only singularity in our solution for arbitrary parame-ters. An exact solution of null geodesics reveals that we have a null-geodesically incomplete manifold. Beside null geodesics we study the radial motion for massless/ massive particles and also the circular motion in the z ¼ 0 plane. From the analysis of the potential the circular motion admits stable orbits.

Organization of the paper is as follows. In Sec. II we introduce magnetic fields in axial symmetry, solve the equations, and derive the metric of unified BR and ML spacetimes. The geodesic equation and its solutions are investigated in Sec.III. The paper ends with a conclusion in Sec.IV.

*habib.mazhari@emu.edu.tr

II. MAGNETIC FIELDS IN STATIC AXIAL SYMMETRY

To review the basics of an axially symmetric spacetime we start with the line element

ds2_¼
e2U_dt2_{þ e}
2U_½e2K_ðd
2_{þ dz}2_{Þ þ }2_d’2_{ (1)} in which U, K and  are functions of
and z alone. The EM field equations can be derived from a variational principle of the action

I ¼Z Ld
dz; (2)

where

L ¼ K

þ Kzz
½U
2þ Uz2
e
2Uðc2
þc2zÞ: (3) Here f
=fzdenotes partial derivative of a function fð
; zÞ with respect to
=z and c is a magnetic potential. Upon variation the metric function  is fixed as  ¼
, while the two basic equations take the forms

ð
U
Þ
þ
Uzz

e
2Uðc2
þc2zÞ ¼ 0; (4) ð
e
2U_c

Þ
þ
ðe
2UczÞz¼ 0: (5) The K function is determined more appropriately by the set

K
¼
ðU2

U2zÞ þ
e
2Uðc2z
c2
Þ; (6) Kz¼ 2
U
Uz
2
e
2Uc
cz (7) whose integrability condition is satisfied by virtue of the field equations. The magnetic vector potential is chosen simply by

A¼  ’


(8)

for a function
ð
; zÞ which is related toc above through

¼
e
2Ucz; (9)

z¼

e
2Uc
: (10)

The dual of the field tensor?_F

ti¼ciimplies the absence of any electric components which is our choice here. In [3] BR and ML solutions are summarized in detail so that we can only record them in what follows.

A. BR and ML solutions 1. The BR solution U ¼ U_BR¼ ln ₀þ 1 2ln ð
2_{þ z}2_{Þ; (11)} c ¼_cBR¼ ₀ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2þ z2 q ; (12) K ¼ K_BR¼ const; ð₀ ¼ constantÞ: (13)

Note that the more familiar AdS2 S2 version of BR spacetime is given upon the transformation

¼sin  r ; (14) z ¼cos  r ; (15) by 2 0ds2¼ 1_r2ð
dt2þ dr2Þ þ d2þ sin2d’2 ðt ¼ 2 0tÞ: (16) 2. The ML solution U ¼ U_ML ¼ ln  1 þB 2 0 4
2  ; (17) c ¼_cML ¼ B₀z; (18) K ¼ K_ML¼ 2 ln  1 þB 2 0 4
2  ðB₀ ¼ constantÞ: (19)

B. A combined BR and ML solution

We proceed now to combine the foregoing solutions. For this purpose we take the magnetic potential as the superposition of the two foregoing, namely,

c ¼cBRþcML¼ 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2_{þ z}2 q

þ B₀z; (20) where B₀and ₀are the constants of ML and BR solutions which are restricted by 0 < j0j < 1 and
1 < B0< 1. To get an idea about this superposition we resort to the axial gauge A¼ ð0; 0; 0; A’Þ in flat space,

ds2_¼
dt2_{þ d}
2_{þ dz}2_þ
2_d’2_{: (21)} Let Að1Þ’ ¼ 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2þ z2 p

and Að2Þ’ ¼ B0z be two magnetic potentials where both solve the Maxwell equations @FðiÞ¼ 0, (i ¼ 1, 2) with FðiÞ ¼ @AðiÞ
@AðiÞ. It can be checked easily that their superposition A ¼ ð0; 0; 0; Að1Þ’ þ Að2Þ’ Þ solves the superposed Maxwell equa-tion @_½
ðF1_{þ F}2_{Þ ¼ 0. Upon this observation we} seek an analogous behavior in the curved spacetime and we find out that indeed it works with some difference. Integration of the field equations from Eq. (4) to Eq. (7) yields the following results:

eU_{¼ F; (22)} eK _¼ F2
2_{þ z}2 0 @
1þ B0 20 z þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2þ z2 1 A 2B0 0 ; (23)

F ¼ ₀  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2_{þ z}2 q cosh _B 0 ₀ ln

z sinhB0 ₀ ln
 : (24) It is observed easily that setting B₀¼ 0 recovers the BR solution with a charge ₀. However, the limit ₀ ¼ 0 does not exist, which means that although in flat spacetime our electromagnetic field is a superposition of BR and ML potentials, in curved spacetime the solution gives only the BR limit correctly. This is in contrast with the 7-parametric PD class of EM solutions [5,6] which admits electromagnetic fields even in the flat space limit.

In our case existence of the BR is essential while the ML limit cannot be interpolated. Let us add also that the metric functions of PD are expressed in its most general-ity in quartic polynomial forms whereas our solution involves decimal powers as well. These distinctive prop-erties suggest that our solution is not included in the general class of PD. The two are expressed in different coordinates/symmetries so that transition between the two for arbitrary cases cannot be expressed in closed forms. More specifically, the type-D metric of the PD class that yields separately the ML and BR limits is as follows: (i) ds2 ML ¼ p2 
QðqÞdt2_þ dq2 QðqÞ  þPðpÞ p2 d  2_þ p2 PðpÞdp 2_{: (25)} Letting QðqÞ¼ 1, PðpÞ¼_B42 0ðp
1Þ, (B0 ¼ constant), p ¼ 1 þ B2 0 4
2, q ¼ B2 0

2 z,  ¼ ’, and an overall scaling gives the ML metric. (ii) ds2 BR¼ b2 
QðqÞdt2_þ dq2 QðqÞ  þ 2 _PðpÞ p2 d2þ p2 PðpÞdp 2  : (26) Letting b ¼  ¼ 1, QðqÞ ¼ q2 _¼
2_{þ z}2_{, PðpÞ ¼ 1
p}2 _¼
2

2_þz2,  ¼ ’, gives the BR metric in axial symme-try with a unit charge. It remains to be seen, however, that (25) and (26) follow from the PD class of solutions in the same coordinate patch, i.e., without further transformations in the ðp; qÞ coordinates.

Furthermore, it is worthwhile to look at the form of invariants of the spacetime. The complete form of the Kretschmann scalar is complicated enough that we only give it in a series form around z ¼ 0 i.e.,

K¼

4 2_{þ4 ½A}

1þA2
2 þA3
4 þA4
6 þA5
8  4 0ð1þ
2 Þ8 þ

4 2_þ4
1 ½B₁þB₂
2 _þB 3
4 þB4
6 þB5
8 þB6
10 þB7
12 þB8
14  4₀ð1þ
2 Þ11 z þ

4 2_þ4
2 ½C₁þC₂
2 _þC 3
4 þC4
6 þC5
8 þC6
10 þC7
12 þC8
14 þC9
16 þC10
18 þC11
20  4 0ð1þ
2 Þ14 z2_þOðz3_{Þ; (27)} in which ¼B0

0
0 and Ai, Bi and Ci are all some polynomial functions of only. Having up to second order explicitly is enough to conclude that the solution is singular at
¼ 0 and z
0 for all values of . This is due to the terms C_1
4
4 2þ4
2

0ð1þ
2 Þ14

z2 and B_1
4
4 2þ4
1 0ð1þ
2 Þ11

z which for z
0 diverge for all . The coefficients C₁ and B₁ are given explicitly by B₁¼ 2048 
1 2   6
₃ 5_{þ 31} 4 4
₂₁ 2 3_{þ 16} 2
₄₅ 4 þ 9 2  (28) and C₁¼ 256ð
692 5_{þ 380} 6_{þ 855} 2_{þ 63
128} 7 þ 1100 4
_351
1196 3_{þ 32} 8_{Þ (29)} which cannot be both zero. At z ¼ 0

K¼

4 2_þ4 4

0ð1þ
2 Þ8

½A₁þA₂
2 _þA

3
4 þA4
6 þA5
8  (30) which for regularity at
¼ 0 we must have

4 2_{þ 4  0: (31)}

A₁¼ 768
2304 þ3328 2
₂₃₀₄ 3_þ1792 5_þ256 6 (32)

which has no real roots. The condition (31) implies that for 0   1 the origin is a regular point. Having clarified the role of the BR parameter ₀, i.e., that ₀
0, so that in the rest of our analysis we may set ₀ ¼ 1 without loss of generality. In brief for z ¼ 0 and
¼ 0 the solution is regular if 0 B0

₀ 1 and singular for other values of B0 ₀. Once more we recall that ¼B0

₀ ¼ 0 corresponds to the BR limit whose Kretschmann scalar is_84

0

and the solution is regular everywhere. The Maxwell 2-form of our solution is expressed by

F ¼ ð

d
þ
zdzÞ ^ d’; (33) where

and
z are defined by (9) and (10). As a result we obtain for the Maxwell invariants

I₁ ¼ 1 2FF _{¼ }2 0e
2K  1 þB 2 0 2 0 þ 2B0z ₀pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2_{þ z}2  ; (34) I₂ ¼ 1 2F ?_F_{¼ 0 (35)}

in which K was found in (23).

Nevertheless the following transformations,

¼
þ iz; du ¼ dt

e
2Ud’;

dv ¼
2e2U_dt; (36)

cast (1) into the Kundt form [4]

ds2¼ duðdv þ HduÞ þ P
2dd  (37) in which H ¼ e2U_{and P ¼ e}ðK
UÞ_.

III. GEODESIC MOTION IN CYLINDRICAL COORDINATES

The geodesic equations for the metric given in (1) are (without loss of generality we choose ₀ ¼ 1)

d ds _@L @_x 
@L @x ¼ 0 (38) in which 2L ¼
e2U_{_t}2_{þ e}
2U_½e2K_{ð _}
2_{þ _z}2_{Þ þ}
2 _{_’}2_{ (39)} and a dot means_dsd. From the t and ’ equations one finds

_t ¼ Ee
2U; _’ ¼‘ 2

2e2U: (40) Using L ¼ " ¼
1, þ1, 0 for timelike, spacelike, and null geodesics the other two equations are

d dsðe
2U_e2K _{_
Þ ¼
U}
E2e
2Uþ ðK

U
Þ " þ E2_e
2U
‘4
2e2U 
‘4
2U
e2U þ‘4
3e2U (41) and d dsðe
2U_e2K_{_zÞ ¼
U} zE2e
2Uþ ð
Uzþ KzÞ " þ E2e
2U
‘ 4
2e 2U 
‘4
2Uze 2U_: (42)

We parametrize now
with z so that
0¼d
_dz and express geodesics in a single equation,

 E2_e
2U_{þ  þ}‘4
2e2U  ðU


0_U zÞ
ðK

Kz
0Þ
‘4
3e2Uþ  1 þ
02
00_{¼ 0; (43)} where  ¼ " þ E2_e
2U
‘4

2e2U. Let us consider the null (" ¼ 0) geodesics in a plane of ’ ¼ ’0 which implies ‘ ¼ 0 and therefore (43) becomes (with E2 ¼ 1)

ð2U

K
Þ

0ð2Uz
KzÞ þ
00

1 þ
02¼ 0: (44) The explicit form of the latter equation reads as

d2
dz2 ¼ 2  1 þ _d
dz ₂  z
2þ z2 _d
dz

z  þ _{ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi}B0
2_{þ z}2 p _d
dzþ z
 þB20 2
 (45)

which is still complicated enough for an exact solution. Luckily we obtain an exact solution valid for jB₀j > 2 given by
¼ 8 > < > :  B0 2 ₂
11=2 z; B₀<
2; z > 0
B₀ 2 ₂
11=2 z; B₀> 2; z < 0: (46)

d
d¼

const

8_ð3
8_{þ 1Þ}2; (47) where  is the affine parameter for null geodesics. A similar equation follows also for_ddz. Equation (47) yields a highly localized solution for
ðÞ [and zðÞ] justifying the expected incompleteness. Another interesting solution for (45) can be found exactly when B₀ ¼ 1. The solution in this case is a circle of arbitrary radius a in the plane of ð
; zÞ with equation
2_{þ z}2_{¼ a}2_{. Figure 1 displays the} numerical plot from Eq. (45) for specific initial conditions.

A. Geodesic motion in the z ¼ 0 plane for B0 ¼
0 As we have shown above, the plane z ¼ 0 has no singu-larity if 0 B0

0  1. This makes it distinguished from the other planes z ¼ z₀
0. Setting B0

0¼ 1 is also the only value in this interval which makes the power of
integer. Therefore, we are interested to consider the geodesic motion of a massive particle with unit mass in this spacetime i.e., ds2_¼
20ð
2þ 1Þ2 4 dt 2_þ20ð
2þ 1Þ2 4
2 d
2 þ 4 2 0
2 ð
2_{þ 1Þ}2d’2: (48)

The Lagrangian is given by

L ¼
20ð
2þ 1Þ2 8 _t 2_þ20ð
2þ 1Þ2 8
2 _
2 þ 2 2 0
2 ð
2_{þ 1Þ}2 _’2; (49)

in which an overdot shows the derivative with respect to the affine parameter . The conservation of energy and angular momentum is obvious such that

@L @_t ¼
2 0ð
2þ 1Þ2 4 _t ¼
E (50) and @L @ _’¼ 42 0
2 ð
2_{þ 1Þ}2 _’ ¼ ‘: (51) Having gdx  d dx

d ¼
, where  ¼ 0=1 (for unit mass) yields the null or timelike geodesics, implies

20ð
2þ 1Þ2 4 _t 2_þ20ð
2þ 1Þ2 4
2 _
2_{þ 4} 2 0
2 ð
2_{þ 1Þ}2 _’ 2_¼ (52)

or upon using the conserved quantities one finds

_
2 _¼ 16
2E2 4 0ð
2þ 1Þ4
4
2 2 0ð
2þ 1Þ2
‘2_{: (53)}

1. Radial motion of massive particle

Let us consider, as the first case, the motion with zero angular momentum of a massive particle, i.e., ‘ ¼ 0 and  ¼ 1. These in turn yield

_
2 _¼ 16
2E2 4 0ð
2þ 1Þ4
4
2 2 0ð
2þ 1Þ2 (54)

which after getting help from

_
2_¼ _@
@ ₂ ¼@
@t ₂_@t @ ₂ ¼@
@t ₂ 16E2 4 0ð
2þ 1Þ4 (55) one finds _@
@t ₂ ¼
2
20
2ð
2þ 1Þ2 4E2 : (56)

Nevertheless, one may set the affine parameter to be the proper time  and therefore

 @
@ ₂ ¼ 16
2E2 4 0ð
2þ 1Þ4
4
2 2 0ð
2þ 1Þ2 : (57)

Now suppose the particle starts from rest at
¼
₀ where @
@t ¼ @
@¼ 0, which gives E2 _¼20ð
20þ 1Þ2 4 : (58)

Hence, the equations of motion become  @
@t ₂ ¼
2

2ð
2þ 1Þ2 ð
2 0þ 1Þ2 (59) and

_@
@ ₂ ¼ 4
2ð
20þ 1Þ2 2 0ð
2þ 1Þ4
4
2 2 0ð
2þ 1Þ2 : (60)

In Fig.2we plot
versus t (a) and  (b). It is very clear that the motion is periodic which means that the particle is attracted by the origin and while it approaches the origin it gains energy, and this energy causes it to pass the origin and in the other direction slows down to rest, and in the same way repeats the motion. The difference between the period of motion measured by an observer on the particle and an observer in the lab is also manifested in the figures.

2. Radial motion of a massless particles

In the same way, one may study the motion of a null particle with  ¼ 0 and ‘ ¼ 0. The equation of motion [Eq. (53)] then reads

_
2_¼ 16
2E2 4

0ð
2þ 1Þ4

(61)

which after using the chain rule we find _d

dt ₂

¼
2_{; (62)}

whose explicit solution is given by

¼
₀et; (63)

where  refers to the direction of motion.

3. Circular motion

To work out the circular motion of a particle on the plane z ¼ 0 we use the chain rule in (53) to find

_d
d’ ₂ ¼ 162
6E2 8 0ð
2þ 1Þ8‘2
64
6 6 0ð
2þ 1Þ6‘2
16
4 4 0ð
2þ 1Þ4 : (64) As usual we introduce u ¼1

to change the equation of motion in the form of

 du d’ ₂ ¼ 256u14E2 8₀ðu2þ 1Þ8‘2
64u10 6₀ðu2þ 1Þ6‘2
16u8 4₀ðu2þ 1Þ4 ¼ AðuÞ: (65)

Having a photon ( ¼ 0) or a massive particle ( ¼ 1) moving on a circular orbit means AðuÞjuc ¼ 0 and having an equilibrium path needs an additional condition dAðuÞ

du juc ¼ 0. Herein
c¼ 1

uc is the radius of the equilib-rium circular orbit. For the massive particle ( ¼ 1) these conditions yield E2 _¼ðu2c
1Þðu2cþ 1Þ220 4ðu2 c
3Þu4c (66) and ‘2 _¼ 8u2c 2 0ðu2cþ 1Þ2ðu2c
3Þ (67)

and therefore the radius of the circular path is found to be the positive root of the following equation:

ðu4 c
1Þðu2cþ 1Þ4 ¼ 32 E2 4 0‘2 u6 c: (68)

From the latter equation we see that for E ¼ 0 a circular path with uc ¼ 1 is possible. This is in fact the maximum value of the possible radius for a circular motion. Particles with higher energy may be able to orbit about the origin with a radius less than 1.

For a massless particle the same conditions dictate a single circular path with

u_c ¼pffiffiffi3 (69)

and energy satisfying E2

‘2 ¼ 1627 4

0: (70)

4. Stability of the circular motion

To see whether the circular path of the particles found above are stable or not we go back to Eq. (54) and rewrite it in the form of one-dimensional motion

1 2 _d
d ₂ þ V_eff ¼ 0; (71) V_eff ¼
8
2_E2 4 0ð
2þ 1Þ4 þ 2
2 2 0ð
2þ 1Þ2 þ‘2 2 : (72) An expansion of V_eff about
¼
_c yields (we note that at the equilibrium circular path both V_eff and its first deriva-tive vanish) _dx d ₂ þ V00 effð
cÞx2 ¼ 0; (73)

FIG. 2. Radial fall from
¼ 1 through
¼ 0 (in the z ¼ 0 plane) for a fixed angle as a function of the coordinate time [Fig.2(a)]/proper time [Fig.2(b)]. The particle crosses
¼ 0 freely since
¼ 0 ¼ z is not singular in the chosen intervalB0

where x ¼


c and V_eff00 ð
_cÞ ¼ 16ð2u 4 c
3u2cþ 3Þu4c ðu2 c
3Þðu2cþ 1Þ420 : (74)

A second derivative with respect to  from (71) admits _d2_x

d2 

þ V00

effð
cÞx ¼ 0 (75) which has an oscillatory motion of x with respect to  (stable motion) if V_eff00 ð
_cÞ > 0. Figure3displays V_eff00 ð
_cÞ versus
c. As it is clear those orbits whose radius is less than 1ffiffi

3

p _{are stable. A similar argument can be repeated for} the massless particles. The effective potential and its first derivative at
¼
c¼p1ffiffi₃are zero while

V_eff00 ð
cÞ ¼ 243 E2 324

0

(76)

which is clearly positive. Therefore the orbit of a photon is stable which is unlike the Schwarzschild and Reissner-Nordstro¨m spacetime.

5. Null geodesics in Kundt form

The Lagrangian of an uncharged particle moving in the spacetime identified by (37) reads as

L ¼ _u _v þH _u2_{þ e}2ðK
UÞ_{ð _}
2_{þ _z}2_{Þ (77)} in which ( _dd ). The first equation_dd ð@L_{@ _v}Þ ¼@L_@vyields

€u ¼ 0 (78)

which in turn implies _u ¼ constant. This basically suggests that our affine parameter  is u. The second equation

d dð @L @_uÞ ¼ @L @ugives dv duþ 2H ¼ 0; (79)

where ₀ is an integration constant. The other two equations are also given by

€

þ ðK
UÞ
ð _
2
_z2Þ þ 2 _
_zðK
UÞz¼ _
U
e2ð2U
KÞ (80) and

€z þ ðK
UÞzð _z2
_
2Þ þ 2 _
_zðK
UÞ
¼ _zUze2ð2U
KÞ (81) in which herein ( _dud ). For ’ ¼ constant, one finds du ¼ dt, and the equation (79) is satisfied if ₀ ¼ 0. For null geodesics we find from (77) that

ð
2

uþ z2uÞ ¼ e2ð2U
KÞ (82) and upon the symmetry between
and z we set
¼ z with  ¼ constant to get (we choose also ₀ ¼ 1)

dz e2U
K ¼ du ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2 p (83)

in which du ¼ dt. A substitution and integration admits

z ¼  B2 0
1 k₀ ðt
t0Þ  1 B2 0
1 ðB2 0
1Þ; (84) where k₀¼ ffiffiffiffiffiffiffiffiffi2B0þB20 1þ2 p ð1þpffiffiffiffiffiffiffiffiffi1þ2_Þ2B0 and t0 is an integration constant. For B2 0¼ 1 we find k₀ln z ¼ t
t0: (85) This brief analysis of Kundt’s null geodesics recovers the equivalent results of the previous analysis. Namely, that the exact integrals of geodesics in a section of the ð
; zÞ plane does not cover the whole plane. We conclude therefore that null geodesic incompleteness remains intact irrespective of the representation of the metric.

IV. CONCLUSION

Being inspired by the superposed solutions in colliding wave spacetimes which unfortunately received no atten-tion, we show here in a similar manner that BR and ML spacetimes can be combined in a single metric. The dis-tinction between the two problems, i.e., colliding waves and axial symmetry, is that in the latter case superposition worked in the more familiar cylindrical ð
; zÞ coordinates rather than the prolate/oblate ones. The obtained metric inherits the imprints of both solutions. It is not conformally flat for instance, and regularity at the origin, i.e., at
¼ z ¼ 0, holds provided in 0 B0

0 1. For an arbitrary ML parameter, however, our solution becomes singular on the symmetry axis. Due to the fractional powers of
our solution is neither smooth nor flat on the symmetry axis. The exact solution of geodesics reveals that null geodesics

in the singular manifold are not complete whereas BR and ML spacetimes separately are known to admit complete geodesics. One drawback of our solution is that ₀! 0 limit, i.e., the ML limit, does not exist. In a single coor-dinate patch the large type-D Einstein-Maxwell family of Plebanski and Demianski (PD) also suffers a similar prob-lem. In this regard our overall impression is that our nonsmooth solution does not belong to the class of PD.

Finally we add that this simple example may serve to pave the way for further ‘‘superposed’’ spacetimes in general relativity, including the higher dimensional ones.

ACKNOWLEDGMENTS

We wish to thank the anonymous referee for valuable suggestions.

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