Doubly torqued vectors and a classification of doubly twisted and Kundt spacetimes

Abstract

The simple structure of doubly torqued vectors allows for a natural characterization of doubly twisted down to warped spacetimes, as well as Kundt spacetimes down to PP waves. For the first ones the vectors are timelike, for the others they are null. We also discuss some properties, and their connection to hypersurface orthogonal conformal Killing vectors, and null Killing vectors.

1 Introduction

Recently, we introduced timelike doubly torqued vectors [15]. They provide a simple characterization of $1+n$ doubly twisted spacetimes, and its subcases of twisted, doubly warped, generalized Robertson-Walker spacetimes. Remarkably, the same definition of doubly torqued vectors fits in the characterization of Kundt spacetimes: a Kundt spacetime is precisely defined by the existence of a null doubly torqued vector, and special cases as the Walker and Brinkmann metrics are naturally identified. The purpose of this paper is to present such characterizations, that are summarized in the tables of this introduction.

An important variety of spacetimes are foliations with totally umbilical spacelike Riemannian hypersurfaces of dimension n, parametrized by time [20]. In proper coordinates, the metric tensor has a $1+n$ block-diagonal structure. Depending on the arguments of the two scale functions $a^2$ and $b^2$, the spacetimes bear different names (Table 1).

There is a vast literature about them, since the paper by Yano [23] in 1940, who introduced doubly twisted manifolds. Warped $1+n$ spacetimes are also known as generalized Robertson-Walker [1, 6, 13]. The table includes spacetimes without name, that naturally emerge in this classification.

The same spacetimes have a tensor characterization, independent of the choice of coordinates, through the existence of a timelike-unit vector field $u_i$ that is vorticity-free and shear-free. Besides this description, preferred by physicists, we recently identified another one in terms of a timelike doubly torqued vector [15]:

$$\begin{aligned} \nabla _j \tau _k = \kappa g_{jk} +\alpha _j \tau _k + \tau _j\beta _k \end{aligned}$$

(1)

where $\alpha _k\tau ^k=0$ and $\beta _k\tau ^k=0$. Despite being $u_i=\tau _i/\sqrt{-\tau ^2}$, where $\tau ^2=\tau _k\tau ^k$, the vector $\tau _i$ offers a straightforward classification of the spacetimes (Table 2). In some cases, $\alpha _i$ and $\beta _i$ are gradients of scalar functions. In parallel, the vector field $u_i$ gets more and more specialized through requirements on the expansion parameter $\varphi$ and the acceleration $\dot{u}_i=u^k\nabla _k u_i$.

Timelike doubly-torqued vectors extend the characterizations by Bang-Yen Chen of twisted spacetimes in terms of torqued vectors ($\beta _i=0$) and of warped spacetimes in terms of concircular vectors ($\alpha _i=\beta _i =0$). They also identify other spacetimes, that do not have simple description in terms of $u_i$. The special case $\alpha _i+\beta _i=0$ identifies doubly torqued vectors with hypersurface orthogonal conformal Killing vectors, making contact with literature.

Surprisingly, null doubly torqued vectors exactly match the Newman-Penrose characterization of Kundt spacetimes. Since $\tau ^2=0$ it is $\kappa =0$ in Eq. (1), and a proper rescaling gives a vector $\tau '$:

$$\begin{aligned} \nabla _i \tau '_j = \theta \tau '_i\tau '_j +\beta '_i\tau '_j + \tau '_i\beta '_j \end{aligned}$$

(2)

with $\beta '$ the non-null component of $\beta$. Conditions on $\theta$ and $\beta '$ give special cases, as the Walker anf Brinkmann metric of PP waves (Table 3).

Table 1 1+n doubly twisted spacetimes

Table 2 Characterizations with timelike doubly torqued and unit vectors

Table 3 Kundt class spacetimes, and null doubly torqued vectors

2 Timelike doubly torqued vectors

We obtain properties for timelike doubly torqued vectors and revisit the relations among $\tau _i, \kappa , \alpha _i, \beta _i$ and the scale functions $a,b>0$ of the metric, discussed in [15], to obtain new results. We refer to the coordinate frame where the space components $\tau _\mu$ and $u_\mu$ vanish, as the “comoving” frame.

Timelike doubly torqued vectors satisfy the Frobenius condition $\tau _{[i}\nabla _j\tau _{k]}=0$ and are hypersurface orthogonal.

This symmetry is useful:

Proposition 2.1

If $\tau _i$ is a timelike doubly torqued vector with $(\kappa , \alpha _i, \beta _i)$ in Eq. (1), then $\mu \tau _i$ is doubly torqued with $(\mu \kappa , \alpha _i+\partial _i\mu /\mu , \beta _i)$ provided that $\tau ^k\partial _k \mu =0$.

In the comoving frame ($\tau _\mu =0$) the condition means that $\partial _t\mu =0$.

If $\alpha _i=\partial _i \alpha$ (orthogonal to $\tau _i$), then a rescaling of $\tau _i$ brings it to $\alpha _i=0$.

Let us enquire when $\alpha _i$ is a gradient, i.e. is closed. Contraction of (1) with $\tau ^k$ gives:

$$\begin{aligned} \alpha _j = \nabla _j \log \sqrt{ -\tau ^2} - \kappa \frac{\tau _j}{\tau ^2} \end{aligned}$$

(3)

The evaluation of $\nabla _i\alpha _j$ gives the useful identity

$$\begin{aligned} (\nabla _i \alpha _j -\nabla _j\alpha _i) \tau ^2 =\tau _i(\nabla _j \kappa -\kappa \alpha _j -\kappa \beta _j) - \tau _j (\nabla _i \kappa -\kappa \alpha _i -\kappa \beta _i) \end{aligned}$$

(4)

Proposition 2.2

$\alpha _j$ is closed if and only if $\nabla _j \kappa -\kappa \alpha _j -\kappa \beta _j$ is parallel to $\tau _j$.

In the comoving frame $\tau _\mu =0$, $\alpha _0 =\beta _0=0$, with the Christoffel symbols listed in appendix, Eq. (1) for doubly torqued vectors becomes ($\mu =1,...,n$):

$$\begin{aligned} \begin{array}{ll} &{}\partial _t \tau _0 -\tau _0\partial _t \log b = -\kappa b^2, \\ &{}\partial _\mu \tau _0- \tau _0\partial _\mu \log b =\tau _0\alpha _\mu \\ &{}-\partial _\mu \log b =\beta _\mu , \\ &{}- \tau _0\partial _t\log a = \kappa b^2 \end{array} \end{aligned}$$

The following propositions concern the two unnamed spacetimes, respectively, and their subcases:

Proposition 2.3

In a doubly twisted spacetime, if $\alpha _i=0$ (or $\alpha _i$ is a gradient orthogonal to $\tau$) then $a^2(t)$ only depends on time.

Proof

If $\alpha _\mu =0$ the second equation gives $\tau _0 (t,\mathbf{q}) = F(t) b(t,\mathbf{q})$ with some function F. The first and last equations give $\partial _t \log a = (\partial _t F)/F(t)$. $\square$

Proposition 2.4

In a doubly twisted spacetime, $\kappa =0$ if and only if $a^2$ only depends on $\mathbf{q}$ (and may be included in $g^\star _{\mu \nu } (\mathbf{q})$).

Then $\alpha _i$ is a gradient (and can be absorbed to zero) and $\tau ^2$ is independent of time.

Proof

The last equation gives $a^2$ that only depends on $\mathbf{q}$ if and only if $\kappa =0$. The first one gives $\tau _0=C(\mathbf{q})b(t,\mathbf{q})$, and the second one results in $\alpha _\mu =\partial _\mu \log C(\mathbf{q})$. Then $\alpha _i$ is a spacetime gradient. Equation (3) gives $\alpha _i =\nabla _i \log \sqrt{-\tau ^2}$. In the comoving frame $\alpha _0 =0$ so that $\tau ^2$ is independent of time. $\square$

3 Timelike hypersurface orthogonal conformal Killing vectors

We show that timelike doubly-torqued vectors with $\alpha _i+\beta _i=0$ coincide with hypersurface orthogonal conformal Killing vectors ( [9] Ch.11, [22] pp.69, 564). We revisit in this light some theorems, and give new ones.

Definition 3.1

$\xi _i$ is a conformal Killing vector if $\nabla _i\xi _j + \nabla _j \xi _i = 2\kappa g_{ij}$ or, equivalently, $\nabla _i \xi _j = \kappa g_{ij} + F_{ij}$ with $F_{ij}=-F_{ji}$. It is a Killing vector if also $\kappa =0$.

Lemma 3.2

A timelike conformal Killing vector $\xi _i$ is hypersurface orthogonal if and only if: $F_{jk} = \alpha _j \xi _k - \xi _j \alpha _k$, $\alpha _k\xi ^k=0$, i.e.

$$\begin{aligned} \nabla _i \xi _j =\kappa g_{ij} + \alpha _j \xi _k - \xi _j \alpha _k \end{aligned}$$

(5)

Proof

By the Frobenius theorem, a vector is hypersurface orthogonal if and only if $0=\xi _{[i}\nabla _j \xi _{k]} = \xi _i (\nabla _j \xi _k -\nabla _k \xi _j)+$cyclic permutations i.e. $\xi _i F_{jk} + \xi _j F_{ki} + \xi _k F_{ij} =0$. A contraction with $\xi ^i$ gives $\xi ^2 F_{jk}+\xi _j (F_{ki}\xi ^i) - \xi _k (F_{ji}\xi ^i)=0$. It is always possible to choose $\alpha _k\xi ^k =0$, as $\alpha _k - \alpha _j\xi _k\xi ^j/\xi ^2$ does the job. $\square$

Proposition 3.3

Doubly torqued vectors with $\alpha _i =-\beta _i$ are hypersurface orthogonal conformal Killing vectors. They are hypersurface orthogonal Killing vectors if also $\kappa =0$.

In Ref. [15] we showed that a doubly twisted spacetime is doubly warped if and only if $\alpha _i=\partial _i \alpha$ and $\beta _i =\partial _i\beta$ in (1) (see Table 2). Since they are both orthogonal to $\tau$ we may rescale $\tau$ such that $\alpha _i=-\partial _i \beta$ and obtain $\nabla _i\tau _j =\kappa g_{ij} -(\partial _i \beta )\tau _j + \tau _i(\partial _j\beta )$, a conformal Killing vector. Therefore:

Proposition 3.4

A spacetime is doubly warped if and only if it is equipped with a hypersurface orthogonal conformal Killing vector with closed vector $\alpha _i$.

With $\alpha _i =-\beta _i$ in (4), we read that $\alpha _j$ is closed if and only if $\nabla _j \kappa$ is proportional to $\tau _k$. Therefore, we have the statement (Theorem 1 in [21]): A spacetime is doubly warped if and only if it is equipped with a hypersurface orthogonal conformal Killing vector with $\partial _i\kappa$ parallel to $\xi _i$.

Moreover, if $\tau$ is closed ($\alpha _i =\beta _i$) then $\alpha _i=\beta _i=0$: the spacetime is generalized Robertson-Walker (Cor. 2 in [21]).

A doubly torqued vector with $\kappa =0$, $\alpha _i=-\beta _i$ is a hypersurface orthogonal Killing vector. Since $\alpha _i$ and $-\alpha _i$ are gradients (Prop.2.4), the spacetime is doubly warped. Then $a^2$ is a function of t and $b^2$ is a function of $\mathbf{q}$. $\kappa =0$ means that $\partial _t a=0$ i.e. a is a constant. The metric $ds^2 = -b^2(\mathbf{q})dt^2 + a^2 g_{\mu \nu }^\star (\mathbf{q}) dq^\mu dq^\nu$ has the form of a static spacetime [22] p.283.

4 Null doubly torqued vectors and Kundt spacetimes

A Kundt spacetime is defined by the presence of a null geodesic congruence that is expansion-free, shear-free, and twist-free [22] Ch.31, [3, 11, 17, 18]. We show that it precisely means that it admits a doubly torqued null vector field.

We begin with some facts on null doubly torqued vectors.

The contraction of $\nabla _i\tau _j =\kappa g_{ij} + \alpha _i\tau _j + \tau _i\beta _j$ with $\tau ^j$ gives $\kappa =0$. Then:

$$\begin{aligned} \nabla _i \tau _j = \alpha _i \tau _j + \tau _i\beta _j, \quad \alpha _k\tau ^k=0, \; \beta _k\tau ^k=0. \end{aligned}$$

(6)

Contraction with $\tau ^i$ gives that $\tau$ is geodesic: $\tau ^i\nabla _i \tau _j =0$.

For null vectors one considers the optical scalars [19]:

$$\begin{aligned} \Theta = \frac{1}{d-2} \nabla _k \tau ^k , \quad \omega ^2 = -\nabla _{[k}\tau _{j]} \nabla ^k\tau ^j , \quad \sigma ^2 = \nabla _{(k}\tau _{j)}\nabla ^k \tau ^j - (d-2)\Theta ^2 \end{aligned}$$

(7)

where d is the dimension of spacetime. It is simple to prove that all the three optical scalars vanish for null doubly torqued vectors. In particular, the vanishing of the twist ($\omega ^2=0$) is the condition for $\tau$ to be hypersurface orthogonal.

Since $\tau ^2=0$, $\alpha _i= a\tau _i +\alpha '_i$ where $\alpha '$ is a spacelike vector orthogonal to $\tau$, and $\beta =b\tau _i+\beta _i'$. Then, for a null doubly torqued vector, with $\theta =a+b$, it is

$$\begin{aligned} \nabla _i\tau _j = \theta \tau _i \tau _j + \alpha '_i\tau _j + \tau _i \beta '_j \end{aligned}$$

(8)

We now turn to Kundt spacetimes and show that (8) is precisely the equation for the congruence. Let $\ell _i$ be the geodesic null congruence, and $n_i$ a second null vector field with $n_i \ell ^i=-1$. ${\hat{h}}_{ij} = g_{ij} + \ell _i n_j + n_i\ell _j$ is the projection on the space orthogonal to $\ell$ and n. Consider the decomposition

$$\begin{aligned} \nabla _i \ell _j&= ({\hat{h}}_i^l - \ell _in^l - n_i \ell ^l)({\hat{h}}_j^m -\ell _jn^m - n_j \ell ^m) \nabla _l \ell _m \\&=({\hat{h}}_i^l - \ell _in^l)({\hat{h}}_j^m -\ell _jn^m) \nabla _l \ell _m \\&={\hat{h}}_i^l {\hat{h}}_j^m \nabla _l \ell _m + \ell _i\ell _j (n^ln^m \nabla _l \ell _m) - {\hat{h}}_i^l \ell _k n^m \nabla _l \ell _m - \ell _in^l {\hat{h}}_j^m \nabla _l \ell _m \end{aligned}$$

The omitted terms contain $\ell ^l\nabla _l \ell _m=0$ (the field is geodesic) and $\ell ^m\nabla _l \ell _m=0$. The first term is the projection onto the subspace of dimension $d-2$ orthogonal to $\ell _i$ and $n_i$, and is decomposed into expansion, shear and twist:

$$\begin{aligned} {\hat{h}}_i^l {\hat{h}}_j^m \nabla _l \ell _m = \frac{\nabla _l \ell ^l}{d-2} {\hat{h}}_{ij} +{\hat{\sigma }}_{ij} +{\hat{\omega }}_{ij} \end{aligned}$$

For Kundt spacetimes these terms are zero, and we have the known statement (we shift to the letter $\tau _i$):

$$\begin{aligned} \nabla _i \tau _j = (n^ln^m \nabla _l \tau _m) \tau _i\tau _j - ({\hat{h}}_i^l n^m \nabla _l \tau _m)\tau _j -\tau _i ({\hat{h}}_i^m n^l \nabla _l \tau _m) \end{aligned}$$

Theorem 4.1

A spacetime is Kundt if and only if there is a doubly torqued null vector field, Eq. (6) or (8).

The property $\lambda \tau _i =\nabla _i f$ (hypersurface orthogonality) offers a rescaling of $\tau$ that makes it a closed vector:

Proposition 4.2

The vector $\tau '_i =\lambda \tau _i$ is null doubly torqued, closed, and

$$\begin{aligned} \nabla _i\tau '_i = \theta \tau '_i \tau '_j + \beta '_i \tau '_j + \tau '_i \beta '_j \end{aligned}$$

(9)

where the vector $\beta '$ is the component of $\beta$ not aligned with $\tau$.

Proof

The evaluation gives: $\nabla _i\tau '_j = (\alpha _i +\partial _i \lambda /\lambda )\tau '_j + \tau '_i \beta _j$. Since $\tau '_i$ is closed, it is $(\alpha _i -\beta _i +\partial _i \lambda /\lambda )\tau '_j = (\alpha _j -\beta _j +\partial _j \lambda /\lambda )\tau '_i$. Then: $\alpha _i +\partial _i \lambda /\lambda =\beta _i+\gamma \tau '_i$ and $\nabla _i\tau '_j = \gamma \tau '_i \tau '_j + \beta _i\tau '_j + \tau '_i \beta _j$. Next, being $\beta _i\tau ^i=0$ and $\tau$ null, it is $\beta =b\tau _i +\beta '_i$. The expression is obtained. $\square$

The metric of a Kundt spacetime in coordinates adapted to the null vectors is:

$$\begin{aligned} ds^2 = H(u,v,\mathbf{q}) du^2 -2dudv + 2W_\mu (u,v,\mathbf{q}) du dq^\mu +g_{\mu \nu } (u,\mathbf{q})dq^\mu dq^\nu \end{aligned}$$

(10)

The coordinates u and v refer to the subspace spanned by $\tau _i$ and $n_i$, where $\tau _u=-1$, $\tau _v=0$, $\tau _\mu =0$, $\alpha '_u=\beta '_u=0$. Equation (8) gives the following relations:

$$\begin{aligned} \theta = \frac{1}{2}\frac{\partial H}{\partial v}, \quad \alpha '_v=\beta '_v =0, \quad \alpha '_\mu =\beta '_\mu = -\frac{1}{2}\frac{\partial W_\mu }{\partial v} \end{aligned}$$

(11)

It turns out that the metric is evaluated with the vector (9).

We have three special cases:

• (i) $\partial H/\partial v=0$ corresponds to $\theta =0$

• (ii) $\partial W_\mu /\partial v=0$, i.e. $\alpha '_i=\beta '_i=0$. It is $\nabla _i \tau _j = \theta \tau _i\tau _j$. This recurrent case gives the Walker metric [12].

• (iii) $\partial H/\partial v=0$ and $\partial W_\mu /\partial v=0$ equivalent to $\theta =0$, $\alpha '_i=\beta '_i=0$. This case gives the Brinkmann metric (PP wave, i.e. plane-fronted waves with parallel propagation) [2, 17].

Another special case is $\beta '$ closed. The equation $\nabla _i\beta '_j = \nabla _j \beta '_i$ gives: 1) $\partial _\mu \beta '_\nu =\partial _\nu \beta '_\mu$ i.e. $W_\mu = \partial _\mu \Phi (u,v,\mathbf{q})$ for some potential; 2) $\partial _u \beta '_\mu =0$, then $\Phi$ does not depend on u; 3) $\partial _v \beta '_\mu = 0$, then $\Phi$ is a linear function of v. In summary: $\beta '$ closed implies $W_\mu (v,\mathbf{q}) = \partial _\mu \Phi _0 (\mathbf{q}) + v \partial _\mu \Phi _1 (\mathbf{q})$, (in Table 3).

This case is realized in the solutions of the Einstein-Maxwell equations in vacuo, or with electromagnetic field aligned to $\tau$ ($F_{ij}\tau ^j \propto \tau _i$), or with the cosmological constant. For this problem H is a quadratic function of v (Eqs. 77 and 112 in [18]).

5 Null hypersurface orthogonal Killing vectors

In analogy with timelike vectors, we consider null doubly torqued vectors with $\alpha _i=-\beta _i$. They coincide with (hypersurface orthogonal) null Killing vectors, and describe a subclass of Kundt spacetimes [7].

Proposition 5.1

A null hypersurface orthogonal Killing vector is a doubly torqued vector with $\alpha _i =-\beta _i$.

A null doubly torqued vector $\nabla _i\tau _j=\alpha _i\tau _j -\tau _i\alpha _j$ is a Killing vector.

Proof

The hypothesis are: $\nabla _i \tau _j =F_{ij}$ ($F_{ij}=-F_{ji}$) and $\tau _i=\lambda \nabla _i f$. Then: $F_{ij}= (\nabla _i\lambda )\nabla _j f + \lambda \nabla _i\nabla _j f$. Subtraction of $F_{ji}$ gives $F_{ij} = \frac{1}{2}\frac{\nabla _i\lambda }{\lambda } \tau _j - \frac{1}{2}\frac{\nabla _j\lambda }{\lambda } \tau _i$. Since $F_{ij}\tau ^j=0$, the vector $\tau$ is doubly torqued with $\alpha _i=-\beta _i$.

A doubly torqued vector is hypersurface orthogonal and, if $\beta _i=-\alpha _i$ it is $\nabla _i\tau _j + \nabla _j \tau _i=0$ i.e. $\nabla _i\tau _j = F_{ij}=-F_{ji}$. $\square$

The metric in $d=4$ is given in [22] p.380. If $\tau _i$ is also closed, then $\nabla _i\tau _j=0$ and PP waves are obtained.

6 Curvature tensors

The integrability conditions for a null or timelike doubly torqued vector are:

$$\begin{aligned} R_{jklm}\tau ^m&=g_{kl}(\nabla _j \kappa -\kappa \alpha _j) - g_{jl}(\nabla _k \kappa -\kappa \alpha _k) +(\nabla _j \alpha _k -\nabla _k\alpha _j) \tau _l \nonumber \\&\quad +\tau _k(\nabla _j\beta _l -\beta _j\beta _l)- \tau _j(\nabla _k\beta _l -\beta _k\beta _l) \end{aligned}$$

(12)

The contraction of the Ricci tensor with $\tau ^m$ is obtained:

$$\begin{aligned} R_{km}\tau ^m = -(n-1)\nabla _k \kappa + \kappa (n\alpha _k +\beta _k) +\tau ^j\nabla _j \alpha _k +\tau _k(\alpha ^j\beta _j +\nabla _j\beta ^j) \end{aligned}$$

(13)

Then, a null $\tau$ is eigenvector if and only if $\tau ^j\nabla _j \alpha _k \propto \tau _k$.

Lemma 6.1

For null doubly torqued vectors:

$$\begin{aligned}&\tau _i \nabla _j (\alpha _k-\beta _k) + \tau _j \nabla _k (\alpha _i-\beta _i) + \tau _k \nabla _i (\alpha _j -\beta _j) =0\\&\tau ^k \nabla _k (\alpha _i -\beta _i) = \tau _i (\alpha ^k\beta _k -\beta ^2) \end{aligned}$$

Proof

The first Bianchi identity $R_{jklm}+R_{kljm}+R_{ljkm}=0$ is contracted with $\tau ^m$ and the expressions (12) are inserted, with $\kappa =0$.

Contraction with $\tau ^k$ gives the other identity. $\square$

The property of Weyl or Riemann compatibility for vectors and symmetric tensors is presented in [16]. Riemann compatibility implies Weyl compatibility.

Theorem 6.2

A timelike doubly torqued vector is Weyl compatible:

$$\begin{aligned} \tau _i C_{jklm}\tau ^m + \tau _j C_{kilm}\tau ^m + \tau _k C_{ijlm}\tau ^m =0 \end{aligned}$$

(14)

A null doubly torqued vector with $\alpha _i$ closed or with $\beta _i = C\alpha _i$ with $C\ne 1$ a constant, is Riemann compatible

$$\begin{aligned} \tau _i R_{jklm}\tau ^m + \tau _j R_{kilm}\tau ^m + \tau _k R_{ijlm}\tau ^m =0 \end{aligned}$$

(15)

and is an eigenvector of the Ricci tensor.

Proof

Multiplication of (12) by $\tau _i$ and a cyclic sum give:

$$\begin{aligned}&\tau _i R_{jklm}\tau ^m + \tau _j R_{kilm}\tau ^m + \tau _k R_{ijlm}\tau ^m \\&\quad = [\tau _i (\nabla _j \alpha _k-\nabla _k\alpha _j) +\tau _j (\nabla _k \alpha _i-\nabla _i\alpha _k) +\tau _k (\nabla _i \alpha _j-\nabla _j\alpha _i)]\tau _l \\&\qquad - g_{il}[\kappa (\tau _j \alpha _k - \tau _k\alpha _j) - (\tau _j \nabla _k \kappa - \tau _k \nabla _k \kappa )] +\\&\qquad - g_{jl}[\kappa (\tau _k \alpha _i - \tau _i\alpha _k) - (\tau _k \nabla _i \kappa - \tau _i \nabla _k \kappa )] +\\&\qquad - g_{kl}[\kappa (\tau _i \alpha _j - \tau _j\alpha _i) - (\tau _i \nabla _j \kappa - \tau _j \nabla _i \kappa ] \end{aligned}$$

If $\tau _i$ is null it is $\kappa =0$. If also $\nabla _j \alpha _k=\nabla _k\alpha _j$ or if $\beta _i=C\alpha _i$ then the cyclic sum is zero (in the second case, use the Lemma).

The contraction of (15) with $g^{jl}$ gives $\tau _i R_{km}\tau ^m = \tau _k R_{im}\tau ^m$. Then $\tau$ is an eigenvector of the Ricci tensor.

Let $\tau _i$ by timelike. The contraction of the Weyl tensor with $\tau$ is:

$$\begin{aligned} C_{jklm}\tau ^m&= R_{jklm}\tau ^m + \frac{1}{n-2} [\tau _j R_{kl}-\tau _k R_{jl}] \\&\quad + \frac{1}{n-2}g_{kl}\left[ R_{jm}\tau ^m - \frac{R\tau _j}{n-1}\right] - \frac{1}{n-2}g_{jl}\left[ R_{km}\tau ^m - \frac{R\tau _k}{n-1}\right] \end{aligned}$$

Multiplication by $\tau _i$ and a ciclic sum give:

$$\begin{aligned}&\tau _i C_{jklm}\tau ^m + \tau _j C_{kilm}\tau ^m + \tau _k C_{ijlm}\tau ^m \\&\quad = [\tau _i (\nabla _j \alpha _k-\nabla _k\alpha _j) +\tau _j (\nabla _k \alpha _i-\nabla _i\alpha _k) +\tau _k (\nabla _i \alpha _j-\nabla _j\alpha _i)]\tau _l \\&\qquad +\tfrac{1}{n-2} g_{kl}\{(\tau _i R_{jm}-\tau _j R_{im})\tau ^m - (n-2) [\kappa (\tau _i \alpha _j - \tau _j\alpha _i) - (\tau _i \nabla _j \kappa - \tau _j \nabla _i \kappa ]\} \\&\qquad +\tfrac{1}{n-2} g_{jl}\{(\tau _k R_{im}-\tau _i R_{km})\tau ^m -(n-2) [\kappa (\tau _k \alpha _i - \tau _i\alpha _k) - (\tau _k \nabla _i \kappa - \tau _i \nabla _k \kappa )]\}\\&\qquad +\tfrac{1}{n-2} g_{il}\{(\tau _j R_{km}-\tau _k R_{jm})\tau ^m -(n-2) [\kappa (\tau _j \alpha _k - \tau _k\alpha _j) - (\tau _j \nabla _k \kappa - \tau _k \nabla _k \kappa )]\} \end{aligned}$$

The contraction of the Ricci tensor with $\tau$ is (13). The cyclic sum for the Weyl tensor simplifies:

$$\begin{aligned}&\tau _i C_{jklm}\tau ^m + \tau _j C_{kilm}\tau ^m + \tau _k C_{ijlm}\tau ^m \\&\quad = [\tau _i (\nabla _j \alpha _k-\nabla _k\alpha _j) +\tau _j (\nabla _k \alpha _i -\nabla _i\alpha _k) +\tau _k (\nabla _i \alpha _j-\nabla _j\alpha _i)]\tau _l \\&\qquad +\tfrac{1}{n-2} g_{kl}[\tau _i (-\nabla _j\kappa +\kappa (2\alpha _j+\beta _j)+\tau ^m\nabla _m\alpha _j) -\tau _j (-\nabla _i\kappa +\kappa (2\alpha _i+\beta _i)+\tau ^m\nabla _m\alpha _i)]\\&\qquad +\tfrac{1}{n-2} g_{jl} [\tau _k (-\nabla _i\kappa +\kappa (2\alpha _i+\beta _i)+\tau ^m\nabla _m\alpha _i) -\tau _i (-\nabla _k\kappa +\kappa (2\alpha _k+\beta _k)+\tau ^m\nabla _m\alpha _k)] \\&\qquad +\tfrac{1}{n-2} g_{il} [\tau _j (-\nabla _k\kappa +\kappa (2\alpha _k+\beta _k)+\tau ^m\nabla _m\alpha _k) -\tau _k (-\nabla _j\kappa +\kappa (2\alpha _j+\beta _j)+\tau ^m\nabla _m\alpha _j)] \end{aligned}$$

For timelike vectors, contraction of (12) by $\tau ^l\tau ^k$ gives:

$$\begin{aligned} 0&=\tau ^2 (\nabla _j \kappa -\kappa \alpha _j) - \tau _j\tau ^k \nabla _k \kappa +\tau ^2 \tau ^k(\nabla _j \alpha _k -\nabla _k\alpha _j) +\tau ^2 \tau ^l \nabla _j\beta _l - \tau _j\tau ^l\tau ^k\nabla _k\beta _l \\&=\tau ^2 [\nabla _j \kappa -\kappa (2\alpha _j+\beta _j) - \tau ^k\nabla _k\alpha _j] - \tau _j(\tau ^k \nabla _k \kappa +\tau ^2 \alpha ^k\beta _k) \end{aligned}$$

With this identity and (4) the cyclic sum is zero. $\square$

Some remarks:

• For a timelike doubly torqued vector: $C_{jklm}\alpha ^j \beta ^k\tau ^m =0$.

• Weyl compatibility (14) guarantees that all doubly twisted spacetimes are purely electric [10].

• Null hypersurface orthogonal Killing vectors are Riemann compatible.

• A Kundt spacetime with Weyl compatible vector $\tau$ is type II(d) in the Bel-Debever classification (Table 4 in [17]).

7 Conclusions

We showed that the structure of doubly torqued vector is the necessary and sufficient condition for the spacetime to be doubly twisted (timelike vector) or a Kundt spacetime (null vector). A simple classification of relevant subcases follows, with connection to other characterizations in terms of Killing or conformal Killing vectors.

8 Appendix

The Christoffel symbols for the doubly-twisted metric:

$$\begin{aligned} \Gamma _{0,0}^0= & {} \frac{\partial _t b}{b}, \quad \Gamma _{\mu ,0}^0=\frac{b_\mu }{b},\quad \Gamma _{0,0}^\mu = \frac{bb^\mu }{a^2},\quad \Gamma ^\rho _{\mu ,0} = \frac{\partial _t a}{a} \delta ^\rho _\mu , \quad \Gamma ^0_{\mu ,\nu } = \frac{a\partial _t a}{b^2} g^*_{\mu \nu }, \\ \Gamma ^\rho _{\mu ,\nu }= & {} \Gamma ^{*\rho }_{\mu ,\nu } + \frac{a_\nu }{a} \delta ^\rho _\mu + \frac{a_\mu }{a} \delta ^\rho _\nu - \frac{a^\rho }{a} g^*_{\mu \nu } \end{aligned}$$

where $a_\mu = \partial _\mu a$ and $a^\mu = g^{*\mu \nu } a_\nu$, and the same is for b.

The Christoffel symbols for the Kundt metric (that are needed in this paper. Taken from [18]) :

$$\begin{aligned} \Gamma _{u,u}^u=\frac{1}{2}\frac{\partial H}{\partial v}, \quad \Gamma ^u_{\mu ,u} = \frac{1}{2} \frac{\partial W_\mu }{\partial v}\quad \Gamma _{u,v}^u=\Gamma _{v,v}^u=\Gamma _{\mu ,v}^u=\Gamma ^u_{\mu ,\nu } = 0 \end{aligned}$$