# A type N radiation field solution with \(\Lambda <0\) in a curved space-time and closed time-like curves

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## Abstract

An anti-de Sitter background four-dimensional type N solution of the Einstein’s field equations, is presented. The matter-energy content pure radiation field satisfies the null energy condition (NEC), and the metric is free-from curvature divergence. In addition, the metric admits a non-expanding, non-twisting and shear-free geodesic null congruence which is not covariantly constant. The space-time admits closed time-like curves which appear after a certain instant of time in a causally well-behaved manner. Finally, the physical interpretation of the solution, based on the study of the equation of the geodesics deviation, is analyzed.

## 1 Introduction

*R*is the scalar curvature, and \(\Lambda \) is the cosmological constant. Taking trace of the field equations, one will get

*pp*-waves). In addition, the space-time display causality violation by admitting closed time-like curves.

The presence of closed time-like curves (CTC) in a space-time violate the notion of causality in general relativity. Hawking proposed the Chronology Protection Conjecture [17] to counter the appearance of closed time-like curves. However, the general proof of this Conjecture has not yet been known. Space-time with closed time-like curves cannot be discard or rule out because such space-times are the exact solutions of the field equations. On some physical backgrounds, for examples, space-time possesses a curvature singularity or does not admit a partial Cauchy surface and/or generate closed time-like curves which are come from infinity are considered nonphysical solutions. A few solutions content unrealistic/exotic matter-energy sources violating one or more energy conditions. For CTC space-time, the matter-energy sources must be realistic, that is, the stress-energy tensor must be known type of matter fields which satisfy the different energy conditions. Many known CTC space-time, for examples the traversable wormholes [18, 19], and the warp drive models [20, 21, 22] violate the weak energy condition (WEC), which states that \(T_{\mu \nu }\,U^{\mu }\,U^{\nu }\ge 0\) for a time-like tangent vector field \(U^{\mu }\), that is, the energy-density must be non-negative. The CTC space-time in [23] violate the strong energy condition (SEC), which states that \((T_{\mu \nu }-\frac{1}{2}\,g_{\mu \nu }\,T)\,U^{\mu }\,U^{\nu }\ge 0\) (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 \(T_{\mu \nu }\,k^{\mu }\,k^{\nu }\ge 0\) for any null vector \(k^{\mu }\). The radiation field solutions in curved space-time without cosmological constant (e.g. [25, 26, 27, 28]) develops closed time-like 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 four-dimensional curved space-time 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 \(c=1=8\,\pi \,G=\hbar \).

## 2 A radiation field space-time with \(\Lambda <0\)

*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, \(k_{\mu ;\nu }\ne 0\). Therefore the studied space-time exhibit geometrically different properties than the famous known

*pp*-waves space-time.

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 space-time is free-from curvature divergence.

*z*-

*planes*defined by \(z=z_0\), where \(z_0\), a constant equal to zero. Therefore, from (21) we get

## 3 Further analysis of the space-time

In this section, we first classify the presented space-time according to the Petrov classification scheme, and its physical interpretation will be the subsequent part.

### 3.1 Classification of the metric

### 3.2 The relative motion of free test particles

*R*(or cosmological constant \(\Lambda \)) responsible for overall background isotropic motions ; the influence of local free gravitational field \(\Psi _{4}\), and the stress-energy tensor \(T_{(a)(b)}\) terms describing interaction of matter-content both of their amplitudes depend on the real number \(\beta _0\).

## 4 Conclusions

We presented a four-dimensional radiation field type N solution of the Einstein’s field equations with negative cosmological constant (\(\Lambda <0\)). The presence of a negative cosmological constant implies that the background space is not asymptotically flat. The studied metric is non-diverging (\(\rho =-(\varvec{\omega }+i\,\varvec{\Theta })=0\)), has a shear-free (\(\varvec{\sigma }=0\)) 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 *pp*-waves. Furthermore, we shown the space-time admits closed time-like curves which appear after a certain instant of time. These time-like 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 space-time 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 \(\mathbf{e}_{(a)}\). It was demonstrated that, this space-time can be understood as exact transverse gravitational waves propagating in an everywhere curved anti-de Sitter Universe, and the matter-energy sources radiation field which affect the relative motion of the free-test particles.

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