A type N radiation field solution with \(\Lambda <0\) in a curved spacetime and closed timelike curves
Eur. Phys. J. C
< 0 in a curved
Faizuddin Ahmed 0
0 Ajmal College of Arts and Science , Dhubri, Assam 783324 , India
An antide Sitter background fourdimensional type N solution of the Einstein's field equations, is presented. The matterenergy content pure radiation field satisfies the null energy condition (NEC), and the metric is freefrom curvature divergence. In addition, the metric admits a nonexpanding, nontwisting and shearfree geodesic null congruence which is not covariantly constant. The spacetime admits closed timelike curves which appear after a certain instant of time in a causally wellbehaved manner. Finally, the physical interpretation of the solution, based on the study of the equation of the geodesics deviation, is analyzed.

1 Introduction
The energymomentum tensor of radiation field [1] is given
by
T μν = ρ kμ kν , kμ kμ = 0,
where ρ is the radiation energydensity and kμ is the tangent
vector field of geodesics null congruence. The tracefree and
covariant conservation of this tensor T μν
; ν = 0 implies that
the radiation propagates along geodesic i.e.,
kμ;ν kν = 0.
The cosmological constant Einstein’s field equations with
radiation field are given by
1
Rμν − 2 gμν R +
gμν = ρ kμ kν ,
where Rμν is the Ricci tensor, R is the scalar curvature, and
is the cosmological constant. Taking trace of the field
equations, one will get
(
1
)
(
2
)
(
3
)
(
4
)
(
5
)
(
6
)
R = 4 .
Substituting this into the field equations (
3
), one will get
Rμν =
gμν + ρ kμ kν , μ, ν = 0, 1, 2, 3.
In terms of traceless Ricci tensor Sμν , the field equations can
be written as
1
Sμν = Rμν − 4 gμν R = ρ kμ kν .
The type N solutions of the field equations play a
fundamental role in the theory of gravitational radiation. The various
algebraically special Petrov type have some interesting
physical interpretations in the context of gravitational radiation.
Particularly for type N spacetime, it has only one repeated
principal null direction (PND) of multiplicity 4, which means
that all four PNDs coincide. The nonvanishing components
of the Weyl scalar is 4 and this corresponds to transverse
gravitational wave propagate along geodesics null
congruence. A complete class of nontwisting type N vacuum
solutions with = 0 was obtained in [2], and a geometrically
different vacuum solution with < 0 in [3]. The
complete family of nonexpanding type N vacuum solutions with
= 0 was further analysed and classified in [4] (see also
[5,6]). These Einstein spaces represent exact pure
gravitational waves which propagate in Minkowski, deSitter or
antide Sitter background (for = 0, > 0 or < 0,
respectively). The algebraically special (type II, D, III or
N, conformally flat) nontwisting and shearfree pure
radiation spacetimes with or without cosmological constant are
known in literature (e.g. [7–15]) (see [1,16] for a
comprehensive review). In the present article, we attempt to
construct a type N radiation field solution with < 0. The
spacetime admits a nonexpanding, nontwisting and
shearfree geodesic null congruence which is not covariantly
constant null vector (CCNV). That means, the spacetime exhibit
geometrically different properties than planefronted
gravitational waves with parallel rays (ppwaves). In addition, the
spacetime display causality violation by admitting closed
timelike curves.
The presence of closed timelike curves (CTC) in a
spacetime violate the notion of causality in general relativity.
Hawking proposed the Chronology Protection Conjecture
[17] to counter the appearance of closed timelike curves.
However, the general proof of this Conjecture has not yet
been known. Spacetime with closed timelike curves
cannot be discard or rule out because such spacetimes are the
exact solutions of the field equations. On some physical
backgrounds, for examples, spacetime possesses a
curvature singularity or does not admit a partial Cauchy surface
and/or generate closed timelike curves which are come from
infinity are considered nonphysical solutions. A few
solutions content unrealistic/exotic matterenergy sources
violating one or more energy conditions. For CTC spacetime, the
matterenergy sources must be realistic, that is, the
stressenergy tensor must be known type of matter fields which
satisfy the different energy conditions. Many known CTC
spacetime, for examples the traversable wormholes [18,19],
and the warp drive models [20–22] violate the weak energy
condition (WEC), which states that Tμν U μ U ν ≥ 0 for a
timelike tangent vector field U μ, that is, the energydensity
must be nonnegative. The CTC spacetime in [23]
violate the strong energy condition (SEC), which states that
(Tμν − 21 gμν T ) U μ U ν ≥ 0 (see details in [24]). There
is another energy condition: the dominant energy
condition which directly implies the weak energy condition and
this implies the null energy condition, which states that
Tμν kμ kν ≥ 0 for any null vector kμ. The radiation field
solutions in curved spacetime without cosmological
constant (e.g. [25–28]) develops closed timelike curves. Thus
if the null energy condition is satisfied the three other energy
conditions are also satisfied. Therefore the null energy
condition appears to be the most fundamental among all the
energy conditions since it cannot be violated by the addition
of a suitably large vacuum energy contribution.
The present work comprises into four section: in Sect. 2,
a fourdimensional curved spacetime with negative
cosmological constant and pure radiation field, is analyzed, in
Sect. 3, the physical interpretation of the solution, will be
discussed, and finally conclusions in Sect. 4.
Our conventions are: Greek indices are taking values
0, 1, 2, 3 and Einstein’s summation convention is used. The
choice of signature is (−, +, +, +) and the units are chosen
c = 1 = 8 π G = h¯.
2 A radiation field spacetime with
Consider the following timedependent metric in (t, r, ψ, z)
coordinates given by
Rr r =
coth2
ds2 = grr dr 2 + gzz d z2 + 2 gtψ dt dψ
+ gψψ d z2 + 2 gzψ dψ d z,
where the different metric functions are
grr = coth2
r ,
< 0,
gψψ = − sinh t sinh2
1
gtψ = − 2 cosh t sinh2
gzψ = β0 z sinh2
gzz = sinh2
with β0 > 0 is a real number. The ranges of the coordinates
are
−∞ < t < ∞, 0 ≤ r < ∞,
−∞ < z < ∞,
(
9
)
and the coordinate ψ is chosen periodic. The validity of
imposing periodicity on ψ coordinate generally follows from
the regularity condition on the axis. A spacetime admitting
an axial Killing vector ημ, parameterized by a 2 π periodic
coordinate ψ is regular on the rotation axis (a set of fixed
points of ημ) if and only if the following condition holds:
(∇μ X) (∇μ X)
4 X
→ 1,
where the limit corresponds to the rotation axis [1]. In
our case, we find l.h.s. corresponds to the rotation axis is
(−1 − β02 z2 cscht ) in the region t < 0 where, ψ
coordinate is spacelike. Thus one can identify ψ ∼ ψ + ψ0 where,
ψ0 ∼ (2 π − δ) < 2 π and δ is the deficit angle. There are
cosmic string or conical singularity exist on the nonregular
axis. In the region t > 0, the coordinate ψ becomes timelike
and the spacetime generate closed timelike curves which we
shall discuss later in this article.
The determinant of the metric tensor gμν given by
1
det g = − 4 cosh2
(
11
)
degenerates at r = 0.
The nonzero components of the Ricci tensor Rμν are
(
7
)
(
8
)
(
10
)
The nonzero components of the Weyl tensor Cμνρσ and the
Riemann tensor Rμνρσ are
Rψψ = β0 −
sinh t sinh2
Rzψ =
β0 z sinh2
Rtψ = − 2 cosh t sinh2
r .
r ,
sinh t,
r ,
R0232 = β0 z R0332,
R0121 =
−
6
cosh2
R0332 = 6 sinh4
cosh t,
cosh t.
β02 z2 + sinh t
,
(
13
)
For the presented timedependent metric, there are
following Killing vector fields
ξ1μ = η
μ = (
0, 0, 1, 0
) ,
ξ2μ =
−2
tanht, tanh
ξ3μ = e−ψ secht, 0, 0, 0 ,
ξ4μ = e 21 (−1+√1−4 β0) ψ
r , 0, −
−
3
z ,
× z
−1 +
1 − 4 β0 + 2 β0 secht, 0, 0, 1 ,
The spacetime (
7
)–(
8
) satisfy the field equations (
5
)
provided the source energydensity
(
12
)
ρ = β0 > 0,
for the following null vector of the metric
Thus the radiation energydensity (ρ) which is a constant
satisfy the null energy condition (NEC) since the metric
function gψψ = 0. Noted that the null vector (
16
) satisfies the
condition (
2
) with
1 μ
= 2 k ; μ = 0, ω
2
1
= 2 k[μ ; ν] kμ ; ν = 0,
1
σ 2 = 2 k(μ ; ν) kμ ; ν −
The quantities , ω and σ are called the expansion, the twist
and the shear, respectively. Hence this null vector field can
be considered as the tangent vector field of geodesic null
congruence the radiation propagates along. But this null vector
field is not covariantly constant null vector (CCNV), that is,
kμ;ν = 0. Therefore the studied spacetime exhibit
geometrically different properties than the famous known ppwaves
spacetime.
To show the studied spacetime is freefrom curvature
divergence, we have calculated the following curvature
invariant constructed from the Riemann tensor as:
Rμν Rμν = 4 2, Rμνρσ Rμνρσ = 38 2,
16
Rμνρσ Rρσ λτ Rμλντ = 9
3,
R,μ R,μ = 0, Rμν;τ Rμν;τ = 0,
Rμνρσ ;τ Rμνρσ ;τ = 0.
Similarly, we have calculated the following curvature
invariant constructed from the Weyl tensor as:
I1 = Cμνρσ C μνρσ = 0, I2 = Cμνρσ C ∗ μνρσ = 0,
I3 = Cμνρσ ;τ C μνρσ ;τ = 0, I4 = Cμνρσ ;τ C ∗ μνρσ ;τ = 0,
where Cμνρσ is the Weyl tensor, and C μ∗νρσ its dual.
From the above analysis, it is clear that the curvature
invariants constructed from the Riemann tensor and the Weyl
tensor do not blow up which guaranteed that the studied
spacetime is freefrom curvature divergence.
Now we discuss closed timelike curves of the spacetime
which appear after a certain instant of time. Consider an
azimuthal closed curves γ defined by t = t0, r = r0, and
(
15
)
(
16
)
(
17
)
(
18
)
(
19
)
z = z0 where, t0, r0 > 0, z0 are constants and ψ is periodic,
ψ ∼ ψ + ψ0 where, ψ0 > 0. From the metric (
7
), we get
ds2 = − sinh t0 sinh2
There are timelike curves provided ds2 < 0 for t = t0 > 0,
spacelike provided ds2 > 0 for t = t0 < 0, and null curve
ds2 = 0 for t = t0 = 0. Therefore the closed curves defined
by (t, r, ψ, z) ∼ (t0, r0, ψ + ψ0, z0) being timelike in the
region t = t0 > 0, formed closed timelike curves. Noted
the Gott’s timemachine spacetime generated closed
timelike curves by imposing one of the coordinate ψ is periodic
identifying ψ ∼ ψ + ψ0 with period ψ0 < 2 π [29] (see
also [30,31]). These timelike closed curves evolve from an
initial spacelike t = const < 0 hypersurface [30,31]. We
find from metric (
7
) that the metric component g00 is given
by
g00 = 4 csch2
sech2t β02 z2 + sinh t .
(
21
)
Now we have chosen the constant zplanes defined by z = z0,
where z0, a constant equal to zero. Therefore, from (
21
) we
get
g00 =
sinh2
4 sinh t
−3 r
cosh2 t
.
(
22
)
A hypersurface t = const = t0 is spacelike (r = r0 > 0)
provided g00 < 0 for t < 0, and timelike provided g00 > 0
for t > 0. Therefore the spacelike t = const = t0 < 0
hypersurface can be chosen as initial conditions over which
the initial data may specified. There is a Cauchy horizon at
t = t0 = 0 for any such spacelike t = const = t0 < 0
hypersurface. The null curve at t = t0 = 0 serve as the
Chronology horizon (since g00 = 0) which divided the spacetime
a chronal region without CTC to a nonchronal region with
CTC. Hence, the spacetime evolves from an initial
spacelike hypersurface in a causally well behaved manner, up to a
moment, i.e., a null hypersurface t = t0 = 0, and the
formation of CTC takes place from causally well behaved initial
conditions in the z = const planes.
3 Further analysis of the spacetime
In this section, we first classify the presented spacetime
according to the Petrov classification scheme, and its
physical interpretation will be the subsequent part.
3.1 Classification of the metric
We construct a set of tetrad vectors (k, l, m,m¯ ) for the
presented metric. These are given by
The set of tetrad vectors are such that the metric tensor for
the line element (
7
)–(
8
) is
gμν = −kμ lν − lμ kν + mμ m¯ ν + m¯ μ mν ,
where the tetrad vectors are null and orthogonal except
kμ lμ = −1 and mμ m¯ μ = 1.
Using the above tetrad vectors, we calculate the five Weyl
scalars and these are
0 =
1 = 0 =
2 =
3,
4 = − β20 .
In addition, the Weyl tensor Cμνρσ satisfies the following Bel
criteria,
Cμνρσ kσ = 0.
Thus the metric (
7
)–(
8
) is of type N in the Petrov
classification scheme. One can calculate the Newmann–Penrose spin
coefficients [1] for the presented metric. These are given by
τ = −π = α = −
i β0 z
ν = − √2
sinh
−6 , γ = − 21 ,
where the symbols are same as in [1]. Thus the repeated
principal null direction k aligned with the radiative direction
is geodesic and shearfree.
The Riemann and Ricci tensor satisfies the following
relation
Rμνρσ kρ kσ = 0 = Rμν kμ kν .
Rμνρσ kσ
= 3
gμρ kν − gνρ kμ , Rμν kμ =
kν ,
(
28
)
,
.
(
23
)
(
24
)
(
25
)
(
26
)
(
27
)
The complex scalar quantities AB = ¯ AB , A, B = 0, 1, 2
associated with the tracefree Ricci tensor Sμν are
1
00 = 2 Sμν kμ kν = 0,
1
02 = 2 Sμν mμ mν
An orthonormal tetrad frame e(a) = {e(0), e(
1
), e(
2
), e(
3
)} in
terms of null tetrad vectors (
23
) can be express as
1
k = √2
1
m = √2
1
e(0) + e(
2
) , l = √
2
e(
1
) + i e(
3
) ,
where e(0) · e(0) = −1 and e(i) · e( j ) = δi j .
3.2 The relative motion of free test particles
e(0) − e(
2
) ,
In order to analyze the effects of the gravitational field and
matter field of the above solution, we used the technique
adopted in [32–35]. The equation of geodesic deviation frame
[6, 11] are given by
D2 Z μ
dτ 2
= − Rμνρσ uν Z ρ uσ ,
u = e(0),
where u · u = −1 is the fourvelocity of a free test particle
(observer), and Z μ(τ ) is the displacement vector
connecting two neighbouring free test particles. The equations of
geodesic deviation in terms of orthonormal tetrad frame (
30
)
are
Z¨ (i) = − R((i0))( j )(0) Z ( j ), i, j = 1, 2, 3,
where Z (i) ≡ eμ(i) Z μ are frame components of the
displacement vector and Z¨ (i) ≡ eμ(i) Dd2τZ2μ are relative accelerations.
Here we set Z (0) = 0 so that all test particles are
synchronized by the proper time.
From the standard definition of the Weyl tensor using (
6
)
one will get
1
R(i)(0)( j )(0) = C(i)(0)( j )(0) + 2
δi j S(0)(0) − S(i)( j ) − 3 δi j .
(
30
)
(
31
)
(
32
)
(
33
)
The nonvanishing Weyl scalars are given (
25
) so that
β0
C(
1
)(0)(
1
)(0) = − 4
= C(
1
)(
2
)(
1
)(
2
),
and rest are all vanish.
The equations of geodesic deviation (
32
) using (
33
)–(
34
)
are
Z¨ (
1
) = − R((
10
))( j )(0) Z ( j ) = 3
Z¨ (
2
) = − R((
20
))( j )(0) Z ( j ) = 3
Z¨ (
3
) = − R((
30
))( j )(0) Z ( j ) =
Z (
1
),
Z (
2
),
where Ai , Bi , i = 1, 2, 3 are arbitrary constants and
We presented a fourdimensional radiation field type N
solution of the Einstein’s field equations with negative
cosmological constant ( < 0). The presence of a negative
cosmological constant implies that the background space is
not asymptotically flat. The studied metric is nondiverging
(ρ = −(ω + i ) = 0), has a shearfree (σ = 0) geodesic
null vector field which is considered the principal null
direction aligned with radiative direction. This null vector field is
not a covariantly constant vector field, that means, the rays of
transverse gravitational wave are not parallel and therefore
the studied metric is geometrically different from the known
ppwaves. Furthermore, we shown the spacetime admits
closed timelike curves which appear after a certain instant
of time. These timelike closed curves evolve from an
initial spacelike t = const < 0 hypersurface in a causally well
behaved manner in the z = const planes. A reasonable
physical interpretation of a spacetime is possible if one
investigates the equation of geodesic deviation in a suitable frame.
We investigated the physical interpretation of the presented
solution, based on the equation of the geodesic deviation in
an orthonormal tetrad frame e(a). It was demonstrated that,
this spacetime can be understood as exact transverse
gravitational waves propagating in an everywhere curved antide
Sitter Universe, and the matterenergy sources radiation field
which affect the relative motion of the freetest particles.
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