A 1+5-dimensional gravitational-wave solution: curvature singularity and spacetime singularity

We solve a $1+5$-dimensional cylindrical gravitational-wave solution of the Einstein equation, in which there are two curvature singularities. Then we show that one of the curvature singularities can be removed by an extension of the spacetime. The result exemplifies that the curvature singularity is not always a spacetime singularity; in other words, the curvature singularity cannot serve as a criterion for spacetime singularities.


Introduction
We find a 1 + 5-dimensional cylindrical vacuum gravitational-wave solution of the Einstein equation. Such a spacetime has both the curvature singularity, the singularity of the Riemann tensor components R j ikl , and the scalar polynomial curvature singularity, the singularity of the scalars formed out of the curvatures such as R, R ij R ij , and R j ikl R ikl j . In this 1 + 5-dimensional spacetime, there is a singularity hypersurface formed by singular points behaving like a horizon.
The spacetime singularity is an important problem in gravity theory. Nevertheless, even the definition of the spacetime singularity is still an open question. Many attempts have been made to seek a general criterion for the spacetime singularity.
There is no general criterion to judge whether a spacetime is singularity-free [1]. By intuition, the singularity should involve infinite curvatures. However, Geroch constructs some singular spacetimes whose Riemann curvatures are bounded everywhere; that is to say, the singularity cannot be determined by the Riemann curvature [2]. Some efforts are made to define a singular boundary which is coordinate independent, such as the "g-boundary" [3] and the "b-boundary" [4]. These prescriptions and other similar procedures, however, produce pathological topological boundaries in simple examples [1,5,6]. Hence, the singular boundary cannot be generally defined. Up to now, a generally accepted definition of the spacetime singularity is based on timelike-or null-geodesic incompleteness [7]. Based on the geodesic incompleteness, Hawking and Penrose prove a series of singularity theorems [7]. Nevertheless, Geroch also constructs a singular spacetime whose geodesics are all complete, so that these singularities cannot be judged by the geodesic completeness [2]. In short, the reason why it is hard to present a general criterion for the singularity is that the spacetime may have an infinite variety of possible pathological behaviors [1].
On the contrary, one can easily judge that a spacetime has a singularity if the spacetime has one kind of pathological behaviors. It is a common impression that if the curvature is singular at some point, the spacetime is singular. It should be emphasized that an orthonormal frame is necessary. In an orthonormal frame, the metric of a 1 + n-dimensional spacetime reads ds 2 = − θ 0 2 + n i=1 θ i 2 with θ 0 and θ i the 1-form on the manifold, which are independent of the choice of coordinates. As a result, the curvature in an orthonormal frame is independent of the choice of the coordinates [8,9].
In this paper, we exemplify by the 1 + 5-dimensional gravitational-wave solution obtained in the present paper that a curvature singularity is not always a spacetime singularity. Concretely, we show that the curvature singularity can be removed by an extension of the solution to the maximal manifold. This result tells us that the curvature singularity is not always a spacetime singularity and thus cannot serve as a criterion for spacetime singularities.
We show that this 1 + 5-dimensional cylindrical vacuum solution describes a gravitational wave, through calculating energy-momentum pseudotensors [10]. The gravitational wave is always an important problem in gravity theory [11]. The existence of the gravitational wave is confirmed experimentally [12,13]. As the key source of the gravitational wave, the black-hole binaries are allowed stable, robust simulations of the merger process [14] based on the advance in numerical relativity [14]. New sources of the gravitational waves are discussed [15]. The application of the gravitational wave as a probe with the first order electroweak phase transition is also considered [16].
Moreover, we also give a brief discussion on the 1 + n-dimensional case. Higher dimensional gravity is an important issue in the theory of gravity. There are also some studies on the cylindrical spacetime, such as a cylindrical solution in the mimetic gravity theory [17] and higher dimensional cylindrical or Kasner type electrovacuum solutions [18]. For higher dimensional spacetime, there are some exact solutions of the Einstein equation, such as a rotating black ring solution in five dimensions [19], general Kerr-NUT-AdS metrics in arbitrary dimensions [20], and the exact solutions of Einstein-Maxwell gravity [21,22]. In Refs. [23,24], the property of the exact solution of the Einstein equation in higher dimensions are discussed. In Refs. [25,26], an inverse scattering method for solving the vacuum solution of five dimensional Einstein equation are provided. In the brane world model, the four-dimensional world is regarded as a brane embedded in higher-dimensional spacetime [27][28][29]. Beyond general relativity, there are also some modified gravity theories are considered, such as the Lanczos-Lovelock model [30], and the Einstein-Gauss-Bonnet theory [31].
In section 2, a 1 + 5-dimensional cylindrical vacuum solution of the Einstein equation is presented. In section 3, the singularity of this 1 + 5-dimensional cylindrical spacetime is analyzed. In section 4, we make an extension of this 1+ 5-dimensional cylindrical spacetime to remove the curvature singularity. In section 5, we show that in this 1 + 5-dimensional cylindrical spacetime there exists a horizon-like singularity hypersurface. In section 6, we show that the solution presented in this paper describes a gravitational wave. In section 7, we give a brief discussion on 1+n-dimensional cylindrical vacuum solutions. The conclusions are summarized in section 8.

1 + 5-dimensional cylindrical vacuum solution
In this section, we find a 1 + 5-dimensional cylindrical vacuum solution of the Einstein equation.
Choose the metric of a 1 + 5-dimensional cylindrical spacetime as By light-cone coordinates Eqs. (2.2)-(2.6) can be simplified as Solving Eq. (2.12), we have where c 1 (t) is an arbitrary function of t. Substituting Eq. (2.13) into Eq. (2.10) gives Taking the derivative with respect to ρ, we arrive at an equation of Φ, By Eq. (2.11), Φ can be expressed as Substituting Eq. (2.17) into Eq. (2.16) and separating variables, we have where a is the separation constant. Then we arrive at By Eq. (2.17) and then by Eq. (2.13), we obtain Then we obtain the solution Redefining the variables as α (t + a) → t, αρ → ρ, z/α → z, and 1 + M = 2e 2C 0 /α, we have The metric then reads (2.29) where t ≥ ρ ≥ 0 and the parameter M is an arbitrary constant.

Singularities of the spacetime
In this section, we analyze the singularity of the spacetime described by the metric (2.29). The metric (2.29) is singular at In order to show the curvature singularity and the scalar polynomial curvature singularity, we here calculate the Riemann tensor R j ikl and the scalar R j ikl R ikl j in the orthonormal frame.
An orthonormal frame (e 0 , e 1 , e 2 , . . . , e m ) and a natural frame ∂ ∂x 0 , ∂ ∂x 1 , ∂ ∂x 2 , . . . , ∂ ∂x m on a manifold are connected by where X k i is a nondegenerate matrix. Choosing the corresponding 1-form with X * i j the inverse matrix of X k i , we can express the metric of the manifold as In the orthonormal frame, the curvature components R j ikl can be obtained from the curvature components in the natural frame, R j ikl , by Then we arrive at [8] The curvature components R j ikl are independent of the coordinates. The components of the Riemann tensor R j ikl in the orthonormal frame can be calculated from the metric (2.29) directly: Here, the singularities (3.1) and (3.2) are embodied in the factors respectively. Recall that the singularity of the components of the Riemann tensor R j ikl is the curvature singularity and the singularity of the scalar R j ikl R ikl j is the scalar polynomial curvature singularity [1,7]. Now we see that (1)  t are both the curvature singularity and the scalar polynomial curvature singularity.

Extension of the spacetime: removing singularity
In this section, we make an extension of the spacetime described by the metric (2.29). It will be seen that the curvature singularity in the extended spacetime corresponding to ρ = t in the metric (2.29) is no longer singular; while the scalar polynomial curvature singularity in the extended spacetime corresponding to ρ = √ t is still singular. That is to say, some curvature singularities can be removed.
Introduce new coordinates (η, r) by where η ≥ r ≥ 0. In the coordinates (η, r), the metric (2.29) becomes The metric (4.2) with η ≥ 0 and r ≥ 0 describes a larger spacetime than the spacetime described by the metric (2.29). In fact, the spacetime described by the metric (2.29) is isometric to the area η ≥ r ≥ 0 in the metric (4.2). That is, the spacetime described by the metric (4.2) is an extension of the solution (2.29).
The metric (4.2) is singular at Similarly, we can calculate the Riemann tensor R j ikl and the scalar R j ikl R ikl j from the metric (4.2) in the orthonormal frame.
The Riemann tensor components in the orthonormal frame are the scalar is R j ikl R ikl j = 72 η 2 − r 2 2 W 2 (η, r) . Analyzing the components of the Riemann tensor R j ikl and the scalar R j ikl R ikl j , we see that η = r corresponding to ρ = t in the metric (2.29) is no longer singular, or, the curvature singularity at ρ = t is removed by the extension. Concretely, we extend the 1 + 5-dimensional solution (2.29) to a larger spacetime through the coordinates transform (4.1). This extended spacetime is described by the metric (4.2). The scalar R j ikl R ikl 6 Gravitational wave: energy-momentum pseudotensor In this section, we show that the 1 + 5-dimensional solution (2.29) is a gravitational-wave solution.
In the metric (2.29), the hypersurface ρ = t can be regarded as the boundary of the spacetime. It provides a picture that the scale of the spacetime expands with time going on. To illustrate that this solution describes a gravitational wave, we will show that the solution has a time-varying energy-momentum pseudotensor (energy-momentum complex). The energy-momentum pseudotensor is defined to describe the energy-momentum of the gravitational field. The reason why using pseudotensors is that according to the equivalence principle, the spacetime is flat at a given point in local geodesic coordinates; as a result the energy-momentum of a gravitational field in local geodesic coordinates vanishes. If a gravitational field has a time-varying energy-momentum pseudotensor, the solution may represent a gravitational wave [10,33,34]. More discussions on energy-momentum pseudotensors can be found in Ref. [35].
The 1 + n-dimensional cylindrical metric reads When n = 2, the metric becomes ds 2 = e 2Φ(t,ρ) −dt 2 + dρ 2 + e 2ψ(t,ρ) dz 2 and have only trivial solutions which can be transformed to a Minkowski solution. When n ≥ 3, the Einstein equation of the metric (7.1) in the light-cone coordinates u ≡ t + ρ and v ≡ t − ρ reads The 1 + 5-dimensional case is special. When n = 5, Eq. (7.4) becomes The solution of Eq. (7.7) then reads, as expressed in Eq. (2.17), Φ = F 1 (u) + F 2 (v). It can be shown that once Eq. (7.7) is satisfied, the spacetime can be extended by introducing new light-cone coordinates by Then in the new coordinates (ũ,ṽ), the metric (7.1) becomes The metric (7.9) is usually the extension of the metric (7.1).

Conclusions and outlook
In this paper, we show that the curvature singularity is not always a spacetime singularity. For exemplifying this, we first solve a vacuum solution of the Einstein equation, which is a 1 + 5-dimensional cylindrical gravitational-wave solution.
We first show that there are two singularities in the Riemann tensor R j ikl and the scalar R j ikl R ikl j of the 1+ 5-dimensional gravitational-wave solution obtained in the present paper. In other words, there are two curvature singularities in this spacetime. Then we show that one of these two singularities can be removed by an extension of the spacetime. That is to say, the curvature singularity which can be removed by the extension is not a real spacetime singularity, or, the curvature singularity cannot be used as a criterion for spacetime singularities. We also show that there is a horizon-like singularity hypersurface.
Moreover, in the above, we only consider the case of M > 0. When M < 0, the situation is different. In this case, when M = −2, the metric (4.2) reduces to a Minkowski spacetime described by ds 2 = dη 2 − dr 2 + η 2 dΩ 2 + r 2 dz 2 .
After the extension of the spacetime by the transformation of coordinates (4.1), when −2 < M < 0 (M = −1), there is no singularity. The metric normal form when −1 < M < 0 is diag (−, +, +, +, +, +) and the metric normal form when −2 < M < −1 is diag (+, −, +, +, +, +). When M < −2, there is a scalar polynomial curvature singularity at In future studies, we can start from the solution given in the present paper to consider the influence of the singularity to quantum effects by calculating, e.g., the partition function and the one-loop effective action based on the heat-kernel method [39,40].
The singularity problem is an important problem in gravity theory. Some authors consider the singularity theorem under more general conditions. The Raychaudhuri equation and the singularity theorem in the Finsler spacetime are studied in Ref. [41,42]. Some singularity theorems are proved in C 1,1 metrics [43,44]. There are also some analyses of the singularity starting from exact solutions of the Einstein equation [45][46][47][48][49]. It is shown that the central singularity can be replaced by a bounce by taking care of the quantum effects in the inhomogeneous dust collapse [50].