# A simple test for spacetime symmetry

###### Abstract

This paper presents a simple method for investigating spacetime symmetry for a given metric. The method makes use of the curvature conditions that are obtained from the Killing equations. We use the solutions of the curvature conditions to compute an upper bound on the number of Killing vector fields, as well as Killing-Yano tensors and closed conformal Killing-Yano tensors. We also use them in the integration of the Killing equations. By means of the method, we thoroughly investigate Killing-Yano symmetry of type D vacuum solutions such as the Kerr metric in four dimensions. The method is also applied to a large variety of physical metrics in four and five dimensions.

###### pacs:

02.40.Hw,02.40.Ky,04.20.-q,4.20.CvKOBE-TH-14-08

OCU-PHYS 411

## 1 Introduction

Spacetime symmetry has played an important role in general relativity, particularly isometry that is described by Killing vector fields

(1) |

The analysis of the equation leads to the well-known result that if the spacetime dimension is given by , the metric admits at most linearly independent Killing vector fields. Only Minkowski, de Sitter and anti-de Sitter spacetimes are maximally symmetric, which admit the maximum number of Killing vector fields. At the same time, it is not always easy to find Killing vector fields for a given metric because one needs to solve coupled partial differential equations obtained from (1). Alternatively, a Killing vector field can be identified if all metric components are independent of a coordinate . However, finding such a coordinate is still difficult.

Meanwhile, Killing-Yano (KY) tensors have been recognised to be describing hidden symmetry of spacetimes because thanks to such symmetry many complicated physical problems become tractable. For instance, the Kerr spacetime admits a nondegenerate rank-2 KY tensor that guarantees integrability of the Hamilton-Jacobi equation for geodesics [5, 2, 3, 4]. KY tensors were originally introduced in [1] as a generalisation of Killing vector fields to higher-rank antisymmetric tensors

(2) |

In four dimensions, if a spacetime admits a nondegenerate rank-2 KY tensor, the Hamilton-Jacobi equation for geodesics, the Klein-Gordon and the Dirac equations can be solved by separation of variables [6, 7, 8]. The similar properties can be seen also in higher dimensions (see reviews [9, 10] and references therein). However, it is as difficult to find KY tensors as Killing vector fields.

The purpose of this paper is to present a simple method for investigating KY tensors for a given metric. Since rank-1 KY tensors are 1-forms dual to Killing vector fields, the method can be applied to Killing vector fields, too. We also deal with the Hodge duals of KY tensors, which are known as closed conformal Killing-Yano (CCKY) tensors. The idea is based on the work of U. Semmelmann [11]. It was shown that one can introduce a connection on the vector bundle , known as a Killing connection, whose parallel sections are one-to-one corresponding to rank- KY tensors. In this paper, using the Killing connection, we calculate the curvature on the vector bundle . From the curvature and its covariant derivative, we obtain some curvature conditions that provide necessary conditions for the parallel sections. Solving the curvature conditions, the number of linearly independent solutions puts an upper bound on the number of KY tensors.

A feature of the method is that the curvature conditions are obtained as algebraic equations, which enables us to compute the upper bound for any metric. In Sec. 3, we actually compute the upper bound on the number of Killing vector fields on the Kerr spacetime. We will see that the Kerr spacetime admits exactly two Killing vector fields without solving the Killing equation. Another feature is that the solution of the curvature conditions gives an ansatz for solving the original differential equations (1) and (2). As we will see in Sec. 3, the Killing equation for the Kerr metric becomes tractable with such an ansatz.

This paper is organised as follows: In Sec. 2, we begin with the familiar discussion on the maximum number of Killing vector fields. Following [11], we introduce the Killing connection and calculate its curvature. Hence, we obtain the curvature conditions that provide an upper bound on the number of Killing vector fields. Sec. 3 shows how to exploit the obtained curvature conditions. As an example, we actually investigate Killing vector fields on the Kerr spacetime. In Sec. 4, we extend the discussion in Sec. 2 to higher-rank KY tensors. Similar to Killing vector fields, we introduce a Killing connection. Calculating its curvature, we derive curvature conditions that provide an upper bound on the number of KY tensors. We also discuss the curvature conditions on CCKY tensors. In Sec. 5, only using type D vacuum conditions, we thoroughly investigate KY symmetry of type D vacuum solutions in four dimensions. Sec. 6 also investigates KY tensors for some physical metrics in four and five dimensions, where we only list the results (see Table 1, 2 and 3). Sec. 7 is devoted to discussion and conclusions.

## 2 Curvature conditions on Killing vector fields

As is well-known, the maximum number of linearly independent Killing vector fields on an -dimensional spacetime with a Lorentzian metric is obtained by the following discussion (e.g., see the Wald’s book [12]). The Killing equation (1) is written as

(3) |

where and is the Levi-Civita connection. Taking the covariant derivative, eq. (3) leads to

(4) |

where is the Riemann curvature. Eqs. (3) and (4) show that a Killing vector field is determined by the initial values of and at a point on . Since is antisymmetric, one can provide at most data at each point, which give the maximum number of Killing vector fields.

We consider the covariant derivative of eq. (4). Then, we obtain the curvature condition

(5) |

where we have used eq. (3). Since the condition is no longer differential equations but linear algebraic equations for and , it provides restrictions on the values of and at each point on . In four dimensions, there are 20 equations for 10 functions. In dimensions, there are equations for functions.

Furthermore, taking the covariant derivative of eq. (5), we obtain the curvature condition

(6) |

where the derivative terms and have been replaced by and with the help of eqs. (3) and (4). It provides further restrictions on the values of and at each point on .

One can take further derivatives of the curvature condition and, in principle, one could take the infinite number of the derivatives. However, in this paper, we only deal with the curvature condition (5) and its first derivative (6). Those conditions are strong enough to restrict the values of and at least for metrics that we investigate in this paper, as we will see later. If the curvature conditions only have the trivial solution, , one can conclude that the metric admits no Killing vector field.

## 3 Killing vector fields on the Kerr spacetime

We investigate Killing vector fields on the Kerr spacetime by use of the curvature conditions (5) and (6). For the Kerr metric, we begin with the metric form [5],

where

(7) |

For later calculation, we introduce an orthonormal frame

(8) |

where

(9) |

In such a frame, the metric is written as . Since all the metric components are independent of the coordinates and , and are Killing vector fields. The dual 1-forms are given by

(10) | |||

(11) |

which are known Killing 1-forms on the Kerr spacetime.

For the metric, we first solve the curvature condition (5) for and . Suppose that and

(12) | |||

(13) |

hence, and take the form

(14) | |||||

(15) | |||||

where , , and are arbitrary functions. Since the solution of the curvature condition (5) is parametrised by four parameters at each point, the Kerr metric admits at most four Killing vector fields.

Furthermore, in addition to the curvature condition (5), we solve the curvature condition (6). Then, we find the solution

(16) | |||

(17) |

Since the independent parameters have reduced to two parameters and , the Kerr metric admits at most two Killing vector fields. As we already have two Killing vector fields, we can conclude that the Kerr metric admits exactly two Killing vector fields.

Finally, we attempt to solve the Killing equation (1). We already know of course that the two Killing 1-forms are given by (10) and (11). However, even if we did not know the Killing 1-forms, we can obtain them as follows. Since the Killing 1-forms on the Kerr spacetime must take the form (14), we make use of the form as an ansatz for solving the Killing equation. Then, it comes to be easy to solve the Killing equation. The solution is given by

(18) |

hence, we have

(19) |

where and are constants. This is indeed the linear combination of the known Killing 1-forms (10) and (11).

## 4 Generalisation

The discussion in Sec. 2 can be interpreted as the following. From eqs. (3) and (4), one can introduce a connection on the vector bundle ,

(20) |

where is a 1-form and is a 2-form. The connection is known as a Killing connection. It is manifest that if a section of is given by a Killing vector field and its exterior derivative , then satisfies

(21) |

which means that is a parallel section of . Conversely, one can demonstrate that if is a parallel section of , is a Killing vector field and is its exterior derivative, . It follows that Killing vector fields on are in one-to-one correspondence with parallel sections of . Since the number of linearly independent parallel sections is bound by the rank of , the number of linearly independent Killing vector fields is also bound by the rank of , which is given by .

Calculating the curvature of the Killing connection , we obtain some conditions for the parallel sections of . Since we have eq. (21), the parallel sections satisfy the curvature condition

(22) |

where is called a Killing curvature. The Killing curvature gives linear maps for any choice of and at each point on . Moreover, taking the covariant derivative of the curvature condition (22) on , we obtain further condition

(23) |

where is defined by (40). Since those conditions are algebraic equations for , they give restrictions on the values of at each point on . Hence, investigating the linearly independent solutions of the curvature conditions, we obtain an upper bound on the number of linearly independent Killing vector fields.

### 4.1 Curvature conditions on Killing-Yano tensors

The discussion about Killing vector fields can be naturally generalised to higher-rank KY tensors and CCKY tensors. For the purpose, we slightly change our notation. Let be an -dimensional Riemannian or Lorentzian manifold and be the Levi-Civita connection. We work in a local orthonormal frame of denoted by and its dual frame of denoted by . Namely, they satisfy where is the interior product. The Latin indices range from to . To deal with Riemannian and Lorentzian metrics simultaneously, we define the matrix which is diagonal with entries . The signature is for Riemannian or for Lorentzian metrics. We also define and , where is the inverse of . For a vector field , we introduce the dual 1-form . In other words, .

A rank- KY tensor (or a KY -form) is defined as a -form satisfying

(24) |

for any vector field . Covariantly differentiating (24), we obtain

(25) |

where

(26) |

The Riemann curvature is defined by

(27) |

In accordance with eqs. (24) and (25), one can introduce a connection on the vector bundle [11],

(28) |

where is a section of , is a section of and

(29) |

If a section of is given by a KY -form and its exterior derivative , then it satisfies the parallel equation

(30) |

Conversely, if is a parallel section of then is a KY -form and is its exterior derivative, . It follows that KY -forms on are in one-to-one correspondence with parallel sections of . Hence, the maximum number of KY p-forms is bound by the rank of [11], which is given by

(31) |

The equality is attained if a spacetime is maximally symmetric. Note that when we take , eqs. (24) and (25) are equivalent to eqs. (3) and (4). Eqs. (28) and (30) correspond to eqs. (20) and (21), respectively. The maximum number (31) becomes for .

As before, we calculate the curvature of the Killing connection (28) by

(32) |

on the vector bundle . A straightforward calculation leads to the Killing curvature written by

(33) |

The entries are given by

(34) |

where

(35) |

Since we have eq. (30), the parallel sections of satisfy the curvature condition

(36) |

which is equivalent to the conditions

(37) | |||

(38) |

### 4.2 Curvature conditions on closed conformal Killing-Yano tensors

Similarly, for a CCKY p-form , we have

(44) | |||

(45) |

where

(46) |

Hence, this time, one can introduce a connection on the vector bundle ,

(47) |

where

(48) |

Similar to KY tensors, we can demonstrate that CCKY tensors on are in one-to-one correspondence with parallel sections of . Hence, the number of CCKY p-forms is bound by the rank of , which is given by

(49) |

Note that the number of rank-p CCKY tensors is same as that of rank-(n-p) KY tensors because CCKY tensors are given as the Hodge duals of KY tensors.

Calculating the curvature of the Killing connection (48), we obtain the Killing curvature

(50) |

with the entries

(51) |

where

(52) |

Hence, we obtain the curvature conditions

(53) | |||

(54) |

Furthermore, as before, the covariant derivatives of the curvature conditions lead to further conditions. Now, they are given by

(55) | |||

(56) |

## 5 Killing-Yano tensors on type D vacuum spacetimes

By use of the method that was shown in previous sections, we shall reconsider KY symmetry of type D vacuum solutions. The results in this section cover the previous works [13, 14, 15, 16, 17, 18] (see propositions 5.1 to 5.4). This section also aims to illustrate how simply the method enables us to investigate KY symmetry of type D vacuum solutions.

We work in the Newman-Penrose formalism, which introduces the complex null tetrad that satisfies

(57) |

The basis and are real vector fields, whereas and are complex. The complex conjugate of is denoted by . The 1-forms , which satisfy , are given by . Using the matrix , we define 1-forms by . For the null tetrad, the spin coefficients are defined as usual:

(58) |

where the all spin coefficients are complex in general.

Firstly, we consider type D conditions. There are two principal null directions of multiplicity two in type D spacetimes. When the and are chosen along the two principal null directions, type D conditions are given by a complex scalar function ,

(59) |

where is the Weyl curvature. The other components are vanishing. In addition, we consider vacuum condition, under which the Riemann curvature is equal to the Weyl curvature. From the Goldberg-Sachs theorem, type D vacuum conditions lead to

(60) |

Furthermore, making the boost and the rotation of the basis, we can set while preserving (60).

All type D vacuum solutions in four dimensions were obtained by W. Kinnersley [19]. He classified the solutions into four cases (case I–IV). However, we do not use the explicit expressions of the solutions. Only using type D vacuum conditions on the Weyl curvature and its covariant derivatives, we compute the Killing curvature (33) on the vector bundle (see appendix A for details). Solving the curvature conditions (36) and (39), the following propositions 5.1 to 5.4 are obtained.

###### Proposition 5.1

Every type D vacuum solution, with the exception of case III of Kinnersley’s classification [19], admits exactly one rank-2 Killing-Yano tensor. Case III solution does not admit any rank-2 Killing-Yano tensor.

#### Proof.

The Killing curvature on are given by (128)-(145). Solving the curvature conditions (126), the solution is given at each point by

(61) | |||||

(62) | |||||

where and are free complex parameters and

(63) |

Imposing the reality conditions and , the parameters must satisfy the conditions , , , and . The reality conditions together with (63) lead to the conditions

(64) | |||

(65) | |||

(66) |

where is a real and is a pure imaginary parameter. For cases I-III (), the spin coefficients , , and satisfy the relations (e.g., see [20])

(67) |

where is a certain real function. Then, eq. (65) is equivalent to (64). Eq. (66) is written as

(68) |

For cases I and II, is pure imaginary
and hence eq. (68) reduces to eq. (64).
This implies that the case I and II solutions admit at most one rank-2 KY tensor.
On the other hand, we have two independent conditions (64) and (68) for case III,
to which the nonzero solution for and does not exist.
Thus, the case III solution does not admit any rank-2 KY tensor.
For case IV, we have [19].
Eqs. (64) and (65) become identities.
Eq. (66) remains as the only equation to solve, so that
case IV solution admits at most one rank-2 KY tensor.
Since it is known that every type D vacuum solution, except for the case III solution,
admits (at least) a rank-2 KY tensor [13, 14, 15, 16, 17],
we arrive at the statement of the present proposition.

Remark. Proposition 5.1 is consistent with the result of [18].
In [18], spacetimes admitting at least two rank-2 KY tensors were discussed in four dimensions.
Since type D vacuum solutions admit just one rank-2 KY tensor,
they are outside the latter calss of metrics.
When considering the Euclidean counterparts of the solutions,
we can construct self-dual Ricci-flat (hyper-Kähler) metrics for particular choice of the parameters.
In those cases, we obtain additional rank-2 KY tensors that are hyper-Kähler forms (see Sec. 6.3 for details).

###### Proposition 5.2

Case II and III solutions admit exactly two Killing vector fields, whereas case I and IV admit exactly four Killing vector fields.

#### Proof.

The Killing curvature on is given by (114)-(125). Solving the curvature condition (112) obtained from the Killing curvature, the parallel sections of are necessarily written at each point on as

(69) | |||||

(70) | |||||

where and are free complex parameters with the constraints

(71) |

The reality conditions and
imply that , , ,
, , and .
Together with (71), the remaining degrees of freedom are given by four real parameters
for case II and IIIB, or given by five real parameters for case I, IIIA and IV.
In addition, solving the condition (43) [cf. (6)]
that is obtained from the covariant derivative
of the curvature condition (112), we obtain the result of the proposition.

###### Proposition 5.3

All type D vacuum solutions admit no rank-3 Killing-Yano tensor.