UUITP-27/22

Geometry, conformal Killing-Yano tensors and conserved “currents”

Ulf Lindström a,b111email<EMAIL_ADDRESS>Leverhulme Visiting
Professor at Imperial College and Özgür Sarıoğlu a222email:
<EMAIL_ADDRESS>
aDepartment of Physics, Faculty of Arts and Sciences,

Middle East Technical University, 06800, Ankara, Turkey

bDepartment of Physics and Astronomy, Theoretical Physics, Uppsala University

SE-751 20 Uppsala, Sweden

and

Theoretical Physics, Imperial College London,

Prince Consort Road, London SW7 2AZ, UK

Abstract

In this paper we discuss the construction of conserved tensors (currents)
involving conformal Killing-Yano tensors (CKYTs), generalising the
corresponding constructions for Killing-Yano tensors (KYTs). As a useful
preparation for this, but also of intrinsic interest, we derive identities
relating CKYTs and geometric quantities. The behaviour of CKYTs under
conformal transformations is also given, correcting formulae in the
literature. We then use the identities derived to construct covariantly
conserved “currents”. We find several new CKYT currents and also include a
known one by Penrose which shows that “trivial” currents are also useful. We
further find that rank-$n$ currents based on rank-$n$ CKYTs $k$ must have a
simple form in terms of $dk$. By construction, these currents are covariant
under a general conformal rescaling of the metric. How currents lead to
conserved charges is then illustrated using the Kerr-Newman and the C-metric
in four dimensions. Separately, we study a rank-1 current, construct its
charge and discuss its relation to the recently constructed Cotton current for
the Kerr-Newman black hole.

###### Contents

  1. 1 Introduction
  2. 2 Rank-$n$ identities and transformation properties
    1. 2.1 Identities
    2. 2.2 CKYTs under conformal transformations
  3. 3 CKYT currents
  4. 4 Rank-$n$ currents from rank-$n$ CKYTs
  5. 5 Conserved charges
    1. 5.1 The Kerr-Newman metric
    2. 5.2 The C-metric
  6. 6 Discussion
  7. A Some special cases of trivial currents
    1. A.1 $n=1$
    2. A.2 $n=2$
  8. B Killing-Yano Cotton current

##  1 Introduction

In recent papers [1, 2, 3], we have investigated Killing tensors (KTs) and
Killing-Yano tensors (KYTs), identified new geometric identities constraining
the geometries that carry such tensors and shown the existence of covariantly
conserved antisymmetric tensors (that we call “currents”) as well as worked
out their related charges and, separately, asymptotic charges. So far, we have
only considered conformal Killing-Yano tensors (CKYTs) in connection with the
Cotton currents in [1], but a natural question to ask is which of the KYT
currents can be generalised to include CKYTs. In the present paper, we first
derive useful identities and relations for CKYTs, including the transformation
properties of a CKYT under a rescaling of the metric, and then discuss a
number of currents based on CKYTs. The existence of such currents and charges
can be quite rewarding; e.g. one can then look for applications that lead to
finding additional supersymmetries or construct asymptotic charges as already
done for KYTs.

For the special case of rank-$n$ currents based on rank-$n$ CKYTs we find the
unexpected result that the divergences of all currents can be expressed in a
trivial form reminiscent of a topological current. In the language of
differential forms, they only involve $dk$ of the $n$-form CKYT $k$. However
this is by no means obvious. We reach this conclusion from a general ansatz
making use of a number of identities derived in the preparatory section.

The outline of the paper is as follows: Section 2 contains a derivation of
identities of general interest, in particular (2.4), (2.9) and (2.10), some of
which are needed for the derivation of the rank-$n$ conserved currents in
section 4. In section 3 we present our CKYT currents relegating the
discussion of the rank-$n$ problems to section 4 with the simpler $n=1$ and
$n=2$ cases in appendix A. Section 5 consists of two subsections in which we
illustrate how to find charges from two of the currents discussed; the
nontrivial “Einstein current” utilizing the Kerr-Newman metric and the
“trivial current” deploying the C-metric. Our conclusions and a discussion are
contained in section 6. In appendix B, we review the Cotton current and
prove the important property of covariance under conformal transformations for
it as well.

Here we try to focus our attention on the discussion of currents, but a list
of publications containing useful general background information about (C)KYTs
and their applications can be found in [4]. Finally it should be noted that
throughout, the geometric setting is always curved (pseudo-)Riemannian
geometry in $D$ dimensions with the Levi-Civita connection.

##  2 Rank-$n$ identities and transformation properties

In this section we define a CKYT and derive identities involving CKYTs and
geometric tensors. These identities are nontrivial and allow us to identify
possible constituent terms of conserved currents and to arrive at a general
ansatz for such a current. This ansatz applied to a rank-$n$ current involving
a rank-$n$ CKYT is shown in section 4 to reduce to a trivial current, in the
sense mentioned in the introduction. We also examine the behaviour of CKYTs
under conformal transformations and correct the literature on this.

###  2.1 Identities

A CKYT $k$ generalises a KYT in the way a conformal Killing vector generalises
a Killing vector. A rank-$n$ CKYT $k$ can be defined [5] as an $n$-form that
satisfies

$\displaystyle\nabla_{a_{1}}k_{a_{2}\dots a_{n+1}}$ $\displaystyle=$
$\displaystyle\nabla_{[a_{1}}k_{a_{2}\dots
a_{n+1}]}+n\,g_{a_{1}[a_{2}}K_{a_{3}\dots a_{n+1}]}\,,$ (2.1) $\displaystyle
K_{a_{1}\dots a_{n-1}}$ $\displaystyle:=$
$\displaystyle\frac{1}{(D-n+1)}\,\nabla_{c}k^{c}{}_{a_{1}\dots a_{n-1}}\,.$
(2.2)

It follows that

$\displaystyle\nabla_{a_{1}}K^{a_{1}\dots a_{n-1}}=0\,.$ (2.3)

The definition reduces to that of a rank-$n$ KYT $f$ by setting $K^{a_{1}\dots
a_{n-1}}=0$. An important identity for KYTs [6] may be generalized to CKYTs as
follows111It has been pointed out that (2.4) is derivable from (1.4) of [5].
We find (2.4) more accessible.

$\displaystyle\nabla_{a}\nabla_{b}k_{c_{1}\dots c_{n}}$
$\displaystyle=(-1)^{n+1}\frac{(n+1)}{2}\,R^{d}{}_{a[bc_{1}}\,k_{c_{2}\dots
c_{n}]d}-(n+1)g_{a[b}\nabla_{c_{1}}K_{c_{2}\dots c_{n}]}$
$\displaystyle\quad+n\nabla_{a}\left(g_{b[c_{1}}K_{c_{2}\dots
c_{n}]}\right)\,.$ (2.4)

Using (2.1) and (2.4) in the commutator
$[\nabla_{a},\nabla_{b}]\nabla_{c}k_{c_{1}\dots c_{n}}$, and contracting the
index pairs $(a,c_{n})$ and $(b,c)$ on both sides of the resultant expression,
we get

$\displaystyle(n-1)\left(\nabla^{b}R^{a}{}_{[c_{1}}\right)k_{c_{2}\dots
c_{n-1}]ab}+\frac{1}{2}(\nabla^{a}R)k_{ac_{1}\dots c_{n-1}}+RK_{c_{1}\dots
c_{n-1}}+(D-n)\,\Box K_{c_{1}\dots c_{n-1}}$
$\displaystyle-\frac{1}{3}(n-1)(n-2)(D-n-1)\,K^{ab}{}_{[c_{1}\dots
c_{n-3}}\left(R_{c_{n-2}|a|c_{n-1}]b}+R_{c_{n-2}c_{n-1}]ab}\right)$
$\displaystyle+(n-1)(D-n-2)\,K_{[c_{1}\dots
c_{n-2}}{}^{a}R_{c_{n-1}]a}-(n-1)(D-n)\,\nabla_{a}\nabla_{[c_{1}}K^{a}{}_{c_{2}\dots
c_{n-1}]}=0\,.$ (2.5)

Since

$\displaystyle 2K^{ab}{}_{[c_{1}\dots
c_{n-3}}R_{c_{n-2}|a|c_{n-1}]b}=K^{ab}{}_{[c_{1}\dots
c_{n-3}}R_{c_{n-2}c_{n-1}]ab}\,,$ (2.6)

we have

$\displaystyle-\frac{1}{3}\,K^{ab}{}_{[c_{1}\dots
c_{n-3}}\left(R_{c_{n-2}|a|c_{n-1}]b}+R_{c_{n-2}c_{n-1}]ab}\right)=-\frac{1}{2}\,K^{ab}{}_{[c_{1}\dots
c_{n-3}}R_{c_{n-2}c_{n-1}]ab}\,.$ (2.7)

Moreover

$\displaystyle\nabla_{a}\nabla_{[c_{1}}K^{a}{}_{c_{2}\dots
c_{n-1}]}=[\nabla_{a},\nabla_{[c_{1}}]K^{a}{}_{c_{2}\dots
c_{n-1}]}=R_{a[c_{1}}K^{a}{}_{c_{2}\dots
c_{n-1}]}-(n-2)R_{b[c_{1}c_{2}|a|}K^{ab}{}_{c_{3}\dots c_{n-1}]}$
$\displaystyle\implies\nabla_{a}\nabla_{[c_{1}}K^{a}{}_{c_{2}\dots
c_{n-1}]}=K_{[c_{1}\dots
c_{n-2}}{}^{a}R_{c_{n-1}]a}-(n-2)R_{b[c_{n-2}c_{n-1}|a|}K^{ab}{}_{c_{1}\dots
c_{n-3}]}\,.$ (2.8)

These can be used for simplifying the second and third lines of (2.5) to
arrive at

$\displaystyle(n-1)\left(\nabla^{b}R^{a}{}_{[c_{1}}\right)k_{c_{2}\dots
c_{n-1}]ab}+\frac{1}{2}(\nabla^{a}R)k_{ac_{1}\dots c_{n-1}}+(D-n)\,\Box
K_{c_{1}\dots c_{n-1}}+RK_{c_{1}\dots c_{n-1}}$
$\displaystyle-2(n-1)\,K_{[c_{1}\dots
c_{n-2}}{}^{a}R_{c_{n-1}]a}+\frac{1}{2}(n-1)(n-2)\,K^{ab}{}_{[c_{1}\dots
c_{n-3}}R_{c_{n-2}c_{n-1}]ab}=0\,.$ (2.9)

On the other hand, by contracting the $(a,b)$ indices in (2.4), we also have

$\displaystyle\Box k^{c_{1}\dots c_{n}}=\frac{(n-1)}{2}k^{ab[c_{1}\dots
c_{n-2}}R^{c_{n-1}c_{n}]}{}_{ab}-R^{a[c_{1}}k_{a}{}^{c_{2}\dots
c_{n}]}+(2n-D)\nabla^{[c_{1}}K^{c_{2}\dots c_{n}]}\,.$ (2.10)

These may be combined to define the following current

$\displaystyle F^{c_{1}\dots c_{n}}=\Box k^{c_{1}\dots
c_{n}}-n\nabla^{[c_{1}}K^{c_{2}\dots
c_{n}]}=\nabla_{a}\left(\nabla^{[a}k^{c_{1}\dots c_{n}]}\right)\,.$ (2.11)

It is easy to show that this current is indeed covariantly conserved, i.e.

$\displaystyle\nabla_{c_{1}}F^{c_{1}\dots c_{n}}=0\,.$ (2.12)

This kind of current will be called “trivial”. It is conserved because it is
the covariant divergence of an antisymmetric tensor, not due to any other
property of the CKYT. The combination $\nabla_{a}T^{ac_{2}\dots c_{n}}$ will
be divergence-free for any completely antisymmetric tensor $T$.

###  2.2 CKYTs under conformal transformations

In this subsection we give the transformation properties of CKYTs under
conformal transformations and show that the covariant-conservation of the
trivial current (2.11) is left-invariant under such transformations.

Under conformal transformations $\tilde{g}_{ab}=\Omega^{2}g_{ab}$, a generic
rank-$n$ CKYT $k$ (2.1) and its rank-$(n-1)$ companion $K$ (2.2) transform as

$\displaystyle\tilde{k}^{a_{1}\dots a_{n}}=\Omega^{1-n}k^{a_{1}\dots
a_{n}}\quad\mbox{and}\quad\tilde{K}_{a_{1}\dots
a_{n-1}}=\Omega^{n-1}K_{a_{1}\dots
a_{n-1}}+\Omega^{n-2}(\nabla_{c}\Omega)k^{c}{}_{a_{1}\dots a_{n-1}}\,.$ (2.13)

Note that for all $n\neq 2$, the second part in the transformations of
$\tilde{K}$ differs significantly from the formulae given in appendix A of
[7].

With these, we find

$\displaystyle\tilde{\nabla}^{[c}\tilde{k}^{a_{1}\dots
a_{n}]}=\Omega^{-(n+1)}\nabla^{[c}k^{a_{1}\dots
a_{n}]}+(n+1)\Omega^{-(n+2)}(\nabla^{[c}\Omega)k^{a_{1}\dots a_{n}]}$ (2.14)

leading to the transformation rule

$\displaystyle\tilde{F}^{a_{1}\dots a_{n}}=\Omega^{-(n+1)}F^{a_{1}\dots
a_{n}}+(D-n-1)\Omega^{-(n+2)}(\nabla_{c}\Omega)\nabla^{[c}k^{a_{1}\dots
a_{n}]}$ (2.15)
$\displaystyle\qquad\qquad+(n+1)\Omega^{-(n+2)}\Big{(}\nabla_{c}\big{(}(\nabla^{[c}\Omega)k^{a_{1}\dots
a_{n}]}\big{)}+(D-n-2)\Omega^{-1}(\nabla_{c}\Omega)(\nabla^{[c}\Omega)k^{a_{1}\dots
a_{n}]}\Big{)}$

for a generic rank-$n$ current $F$. Since for any skew-symmetric rank-$(n+1)$
tensor $\tilde{T}$, one has

$\displaystyle[\tilde{\nabla}_{a_{1}},\tilde{\nabla}_{c}]\tilde{T}^{ca_{1}\dots
a_{n}}=2\tilde{\nabla}_{a_{1}}\tilde{\nabla}_{c}\tilde{T}^{ca_{1}\dots
a_{n}}=0\,,$ (2.16)

which can be shown by the expansion of the commutator, the symmetries of the
curvature tensors and the first Bianchi identity, it follows straightforwardly
that $\tilde{\nabla}_{a_{1}}\tilde{F}^{a_{1}\dots a_{n}}=0$ by the
identifications $\tilde{T}^{ca_{1}\dots
a_{n}}=\tilde{\nabla}^{[c}\tilde{k}^{a_{1}\dots a_{n}]}$ and
$\tilde{F}^{a_{1}\dots a_{n}}=\tilde{\nabla}_{c}\tilde{T}^{ca_{1}\dots
a_{n}}$. So the covariant-conservation of the trivial current $F$ is left
invariant under conformal transformations.

##  3 CKYT currents

In this section we present cases when CKYTs give rise to conserved currents.
The conserved currents are characterised by their rank, the rank of the CKYT
involved and the number of derivatives. One of the currents presented is
related to the Kastor-Traschen current (KT-current) [6] to which we return
below in the special case of a rank-2 CKYT and rank-2 current.

From the definitions (2.3) we see that a conserved current involving a rank-2
CKYT $k$ in $D=4$ dimensions is [8]

$\displaystyle j^{a}:=\epsilon^{abcd}\nabla_{b}k_{cd}\,.$ (3.1)

Here $\nabla_{a}j^{a}=0$ follows easily from (2.1). In this case the current
is rank-1. In fact, since the Hodge dual $\star{k}$ of a CKYT $k$ is again a
CKYT [9] $\hat{k}:=\star{k}$, one has

$\displaystyle j^{a}=\nabla_{b}\hat{k}^{ab}\,,$ (3.2)

so that this is a trivial current as described at the end of subsection 2.1.
It is used in [8] as the starting point for a discussion of KTs vs KYTs, and
quasilocal charges.

In [1], we showed that there are also other rank-1 currents constructed out of
the Cotton tensor $C$ and a rank-2 KYT $f$ or a rank-2 CKYT $k$, the Killing-
Yano Cotton currents

$\displaystyle J^{a}:=C^{abc}f_{bc}\,,\qquad\tilde{J}^{a}:=C^{abc}k_{bc}\,.$
(3.3)

These are covariantly conserved in arbitrary $D\geq 3$ dimensions. In showing
that the Cotton currents are divergence free, we use the fact that the Cotton
tensor is traceless on all index pairs and satisfies

$\displaystyle C_{abc}=C_{a[bc]}\,,\quad
C_{[abc]}=0\,,\quad\nabla^{a}C_{abc}=0\,.$ (3.4)

The conservation of the currents follow from an interplay between the geometry
and (C)KYT properties. It is therefore useful to explore such relations in
some detail, which we have done in subsection 2.1 and shall elaborate further
in section 4.

The Hodge dual of a KYT $f$ is a closed conformal Killing-Yano tensor222So a
CCKYT $h$ satisfies (2.1) with the first term on the right hand side vanishing
identically. (CCKYT) $h$ and vice versa [9], i.e.

$\displaystyle f=\star h\,,\qquad dh=0\,.$ (3.5)

Specifically when the rank of the CCKYT $h$ is $n=D-1$, one has

$\displaystyle\nabla_{a}h_{b_{1}\dots b_{D-1}}=(D-1)g_{a[b_{1}}H_{b_{2}\dots
b_{D-1}]}\,,$ (3.6)

so that e.g. the Hodge dual of the rank-1 current (3.1) in $D=4$ naturally
gives rise to another conserved current of rank-2, as mentioned in the context
of (3.2). Likewise the Hodge dualisation of the rank-1 Cotton currents (3.3)
give rise to two more conserved currents of rank-2 and rank-$(D-2)$,
respectively. Explicitly, the rank-$(D-2)$ dual of a rank-2 CKYT $k$ brings
about the current

$\displaystyle\mathscr{J}^{a}=C^{abc}\epsilon_{bcd_{1}\dots
d_{D-2}}k^{d_{1}\dots d_{D-2}}\,.$ (3.7)

In three dimensions this relates the Cotton current $\tilde{J}^{a}$ for a
rank-2 CKYT to a current for its rank-1 dual defined with the dual of the
Cotton tensor, the York tensor.

In a slight detour from the main thrust of this section, we also note that
given a CKYT $k$ (2.1), one may also construct a current from the associated
trace $K$ (2.2). For a rank-2 CKYT $k$, the associated rank-1 trace $K^{a}$ is
trivially conserved $\nabla_{a}K^{a}=0$ (2.3). Moreover, the “Einstein
current”

${\cal J}^{a}:=G^{ac}K_{c}\,,$ (3.8)

is also covariantly conserved (with a manifest potential, cf. (A.8)), thanks
to

$R^{ac}\nabla_{a}K_{c}=R^{ac}\nabla_{(a}K_{c)}=0\,,$ (3.9)

derived in [1]. When $K^{a}$ is itself a Killing vector333As is the case for
the trace part of a CKYT in $D=4$., we again find a dual $(D-1)$-form CCKYT
and an associated rank-1 current.

The duality relation (3.5) also makes it possible to find a version of the KT-
current involving CCKYTs. The KT-current [6] is defined for a rank-$n$ KYT $f$
as

$\displaystyle j^{a_{1}\dots a_{n}}=N_{n}\,\delta^{a_{1}\dots
a_{n}d_{1}d_{2}}_{b_{1}\dots b_{n}c_{1}c_{2}}\,f^{b_{1}\dots
b_{n}}\,R_{d_{1}d_{2}}{}^{c_{1}c_{2}}\,,$ (3.10)

where $\delta^{a_{1}\dots a_{m}}_{b_{1}\dots
b_{m}}=\delta^{[a_{1}}_{b_{1}}\cdots\delta^{a_{m}]}_{b_{m}}$ is the
generalised Kronecker delta, which is totally antisymmetric in all up and down
indices, $N_{n}:=-(n+1)(n+2)/4n$ and $R$ is the Riemann curvature tensor. The
covariant divergence of the KT-current vanishes thanks to the Bianchi
identities for $R$. Thus, dualising $f$ to $h$ via (3.5) in (3.10), we find a
new covariantly conserved current based on $h$

$\displaystyle j^{a_{1}\dots a_{n}}\sim\epsilon^{a_{1}\dots
a_{n}b_{1}b_{2}}\,R_{b_{1}b_{2}}{}^{c_{1}c_{2}}\,h_{c_{1}c_{2}}\sim\star
R^{a_{1}\dots a_{n},c_{1}c_{2}}\,h_{c_{1}c_{2}}\,,$ (3.11)

where star denotes the left dual.

The preceding construction involving CCKYTs in the KT-current prompts the
question of whether one can replace the KYTs in the KT-current by CKYTs,
modulo some modifications. This turns out not to give a conserved current, as
we discuss in the next section.

##  4 Rank-$n$ currents from rank-$n$ CKYTs

In this section, we use some of the formulae derived in section 2 to
investigate a rank-$n$ current constructed from a rank-$n$ CKYT. More
specifically, we look for a covariantly conserved rank-$n$ tensor $J$ linear
in the CKYT and its contraction, formed from geometric tensors and covariant
derivatives, such that $J$ is second-order in covariant derivatives. This
specification is in line with previous currents constructed using KYTs [6, 2].

It must be pointed out that there are many options for the terms in such a
current and our final result, that a large class of them lead to trivial
currents, is by no means obvious. It is only with the help of a number of
identities, some of which are derived in section 2 that we are able to show
this (somewhat disappointing) result.

We start by listing a number of relations between terms of the right kind:

$\displaystyle R^{[a_{1}|cd|a_{2}}k^{a_{3}\dots
a_{n}]}{}_{cd}=-{\textstyle{\frac{1}{2}}}R_{cd}{}^{[a_{1}a_{2}}k^{a_{3}\dots
a_{n}]cd}\,,$ (4.1) $\displaystyle\nabla_{c}\nabla^{[a_{1}}k^{a_{2}\dots
a_{n}]c}=\nabla_{c}\nabla^{[a_{1}}k^{a_{2}\dots
a_{n}c]}+(-1)^{n+1}\nabla^{[a_{1}}K^{a_{2}\dots a_{n}]}\,,$ (4.2)
$\displaystyle\nabla^{[a_{1}}\nabla_{c}k^{a_{2}\dots
a_{n}]c}=(-1)^{n+1}(D+1-n)\nabla^{[a_{1}}K^{a_{2}\dots a_{n}]}\,,$ (4.3)
$\displaystyle[\nabla_{c},\nabla^{[a_{1}}]k^{a_{2}\dots
a_{n}]c}=(-1)^{n}\frac{(n-1)}{2}R_{cd}{}^{[a_{1}a_{2}}k^{a_{3}\dots
a_{n}]cd}+R_{c}{}^{[a_{1}}\,k^{a_{2}\dots a_{n}]c}\,,$ (4.4)
$\displaystyle\Box k^{a_{1}\dots a_{n}}=\nabla_{c}\nabla^{[{c}}k^{a_{1}\dots
a_{n}]}+n\nabla^{[a_{1}}K^{a_{2}\dots a_{n}]}\,.$ (4.5)

Notice that the difference of (4.2) and (4.3) gives (4.4), and thus

$\displaystyle F^{a_{1}\dots a_{n}}+(D-n)\nabla^{[a_{1}}K^{a_{2}\dots
a_{n}]}=\frac{(n-1)}{2}R_{cd}{}^{[a_{1}a_{2}}k^{a_{3}\dots
a_{n}]cd}+(-1)^{n}R_{c}{}^{[a_{1}}\,k^{a_{2}\dots a_{n}]c}\,.$ (4.6)

These relations let us choose the following rank-$n$, skew-symmetric,
independent combinations

$\displaystyle R_{cd}{}^{[a_{1}a_{2}}\,k^{a_{3}\dots
a_{n}]cd}\,,\;R_{c}{}^{[a_{1}}\,k^{a_{2}\dots a_{n}]c}\,,\;R\,k^{a_{1}\dots
a_{n}}\,,\;\nabla^{[a_{1}}K^{a_{2}\dots a_{n}]}\,,\;$ (4.7)

as the building blocks of a possible current. For computational ease, without
loss of generality, we replace the first two terms in the list (4.7) by

$\breve{{\cal K}}^{a_{1}\dots
a_{n}}:=-\frac{(n-1)}{4}\,R^{[a_{1}a_{2}}\,_{bc}\,k^{a_{3}\dots
a_{n}]bc}+\frac{1}{2n}\,R\,k^{a_{1}\dots a_{n}}\,,$ (4.8)

and

$\hat{{\cal K}}^{a_{1}\dots a_{n}}:=R_{c}\,^{[a_{1}}\,k^{a_{2}\dots
a_{n}]c}+\frac{(-1)^{n}}{n}\,R\,k^{a_{1}\dots a_{n}}\,,$ (4.9)

modeled after two conserved currents in [2].

With these preambles, we make an ansatz for the conserved current:

$\displaystyle J^{a_{1}\dots a_{n}}=2\breve{{\cal K}}^{a_{1}\dots
a_{n}}+\alpha\hat{{\cal K}}^{a_{1}\dots a_{n}}+\beta Rk^{a_{1}\dots
a_{n}}+\gamma\nabla^{[a_{1}}K^{a_{2}\dots a_{n}]}\,.$ (4.10)

To this, we may add any amount of the conserved current $F^{a_{1}\dots a_{n}}$
in (2.11), its coefficient will remain arbitrary.

To proceed, we rewrite the identity (2.10) or (4.6) in terms of the tensors
(4.8) and (4.9):

$\displaystyle F^{a_{1}\dots a_{n}}=(-1)^{n}\hat{{\cal K}}^{a_{1}\dots
a_{n}}-2\breve{{\cal K}}^{a_{1}\dots a_{n}}+(n-D)\nabla^{[a_{1}}K^{a_{2}\dots
a_{n}]}\,.$ (4.11)

Using this identity in (4.10) yields for the divergence

$\displaystyle\nabla_{a_{1}}J^{a_{1}\dots
a_{n}}=(\alpha+{(-1)^{n}})\nabla_{a_{1}}\hat{{\cal K}}^{a_{1}\dots
a_{n}}+(\gamma+n-D)\nabla_{a_{1}}\nabla^{[a_{1}}K^{{a_{2}\dots
a_{n}}]}+\beta\nabla_{a_{1}}(Rk^{a_{1}\dots a_{n}})$
$\displaystyle\qquad\qquad\qquad+\nabla_{a_{1}}F^{a_{1}\dots a_{n}}\,.$ (4.12)

The last term is zero, of course, and the coefficients of the others give
$\alpha={(-1)^{n+1}}$, $\beta=0$ and $\gamma=D-n$. Plugging these values into
(4.10) and using (4.11), the rank-$n$ current is thus shown to be trivial,
i.e. proportional to the current $F^{a_{1}\dots a_{n}}$ in (2.11).

We thus see that we cannot construct a nontrivial rank-$n$ conserved current
from the ansatz (4.10). This ansatz is quite general and covers, e.g.,
possible generalisations of the KT-current, that is local, geometric, linear
in $k$ and second-order in $\nabla$s.

As an additional illustration of the problem, we further discuss the cases
$n=1$ and $n=2$ in appendix A.

##  5 Conserved charges

One obvious reason for constructing currents is to use them to find conserved
charges. In this section we illustrate the procedure for doing that on two
well-known solutions. For the case of the Kerr-Newman metric used in the
Einstein current (3.8), we are able to derive a closed-form charge expression
which reduces to a complicated integral on which we comment. Unfortunately
though, the corresponding expression for the C-metric with the trivial current
(2.11) diverges.

###  5.1 The Kerr-Newman metric

The current (3.8) is an example of a non-trivial current and it can be used
for defining a charge in the usual way444Recall what is meant by
“conservation” of charge in the classical field theory sense. For example,
consider Maxwell theory in flat Minkowski space. One defines charge $Q:=\int
d^{3}x\,J^{0}$, so that at $t=x^{0}=$ const. surfaces one has “conservation”
$dQ/dt=0$. So, it is only natural that the charge-integral hypersurface is
chosen as spacelike, with an everywhere timelike normal., e.g. as described in
Sec. 3 of [1]:

$\displaystyle{\cal Q}:=\int_{\Sigma_{t}}d^{3}x\,\sqrt{\gamma}\,n_{a}\,{\cal
J}^{a}\,.$ (5.1)

Let us see how things go on the example of the Kerr-Newman metric

$\displaystyle
ds^{2}=-\frac{Q(r)}{\rho^{2}}\left(dt-a\sin^{2}\theta\,d\varphi\right)^{2}+\frac{\rho^{2}}{Q(r)}dr^{2}+\rho^{2}d\theta^{2}+\frac{\sin^{2}{\theta}}{\rho^{2}}\left(adt-(r^{2}+a^{2})\,d\varphi\right)^{2}\,,$
(5.2)

where

$\displaystyle Q(r)$
$\displaystyle=\left(r-r_{+}\right)\left(r-r_{-}\right)\,,\quad\mbox{with}\quad
r_{\pm}:=m\pm\sqrt{m^{2}-a^{2}-e^{2}-g^{2}}\,,$ (5.3) $\displaystyle\rho^{2}$
$\displaystyle=r^{2}+a^{2}\cos^{2}\theta\,,$ (5.4)

with the rank-2 CKYT [10]

$\displaystyle
k=r\,dr\wedge\left[dt-a\sin^{2}{\theta}d\varphi\right]+a\cos{\theta}\sin{\theta}d\theta\wedge\left[a\,dt-\left(r^{2}+a^{2}\right)d\varphi\right]\,,$
(5.5)

that remarkably has $K^{a}=-(\partial_{t})^{a}$, the timelike Killing vector
of (5.2) at the same time. These give

$\displaystyle{\cal
Q}=2\pi\left(e^{2}+g^{2}\right)\int_{r_{+}}^{\infty}dr\int_{0}^{\pi}\frac{d\theta\,\sin{\theta}\,\left(a^{2}+2(r-r_{-})(r-r_{+})-a^{2}\cos{2\theta}\right)}{\left(a^{2}+2r^{2}+a^{2}\cos{2\theta}\right)\left((r-r_{-})(r-r_{+})-a^{2}\sin^{2}{\theta}\right)}\,.$
(5.6)

This integral is equivalent to the integral found in Sec. 5.3 of [1], but here
in a different context obviously. It is convergent and finite, but as
discussed in [1], even though the $\theta$ integral can be taken exactly, we
were unable to evaluate the $r$-integral. See Sec. 5.3 of [1] for details.

Finally, going back to the trivial rank-2 current (2.11) for the Kerr-Newman
metric, we immediately see that the CKYT (5.5) is a CCKYT, i.e.
$\nabla_{[a}k_{bc]}=0$ identically. So it is not suitable for defining a
conserved charge using the trivial current (2.11) for $n=2$.

###  5.2 The C-metric

The C-metric in “spherical-type coordinates” reads [11]

$\displaystyle
ds^{2}=\Omega^{2}(r,\theta)\left(-Q(r)\,dt^{2}+\frac{dr^{2}}{Q(r)}+\frac{r^{2}\,d\theta^{2}}{P(\theta)}+P(\theta)\,r^{2}\sin^{2}{\theta}\,d\phi^{2}\right)\,,$
(5.7)

where the conformal factor and the metric functions are

$\displaystyle\Omega(r,\theta)=\frac{1}{1+\alpha r\cos{\theta}}\,,\quad
Q(r)=\left(1-\frac{2m}{r}\right)(1-\alpha^{2}r^{2})\,,\quad
P(\theta)=1+2\alpha m\cos{\theta}\,.$ (5.8)

The analytic extension of the C-metric is thought to represent a pair of black
holes that accelerate from each other due to the presence of a string that is
represented by a conical singularity [12]. The C-metric (5.7) reduces to the
Schwarzschild black hole when $\alpha=0$ and has an obvious curvature
singularity at $r=0$. In fact, the parameter $m>0$ has to do with the mass of
the source, whereas the parameter $\alpha$, where $0<2\alpha m<1$, can be
interpreted as the acceleration of the black hole with an acceleration horizon
at $r=1/\alpha>2m$ [11].

In what follows, we will take the $t$-coordinate as temporal and the
$r$-coordinate as spatial, for which one needs $r\in(2m,1/\alpha)$. To stay
away from the coordinate poles, we also take $\theta\in(0,\pi)$. Finally, we
restrict $\phi\in(-C\pi,C\pi)$, where the parameter $C$ determines the balance
between the deficit/excess angles on the two halves of the symmetry axis of
the C-metric (with $t,r$ kept constant) [11]. Note also that the choice

$C=\frac{1}{1+2\alpha m}$ (5.9)

removes the conical singularity at $\theta=0$ [11].

The C-metric (5.7) admits two rank-2 CKYTs

$\displaystyle k_{1}$
$\displaystyle=\Omega^{3}(r,\theta)\,r^{3}\,\sin{\theta}\,d\theta\wedge
d\phi\quad\mbox{with}\quad K_{1}=\alpha\,\partial_{\phi}\,,$ (5.10)
$\displaystyle k_{2}$ $\displaystyle=\Omega^{3}(r,\theta)\,r\,dr\wedge
dt\quad\mbox{with}\quad K_{2}=-\partial_{t}\,.$ (5.11)

For the trivial current $F^{ac}$ (2.11), one may, in analogy to the discussion
in subsection 5.1 (see also further details given in Sec. 3 of [1]), define
the charge

$\displaystyle{\cal
Q}^{c}:=\int_{\Sigma_{t}}d^{3}x\,\sqrt{\gamma}\,n_{a}\,F^{ac}\,.$ (5.12)

Adapted to the C-metric (5.7), a $t=$ const. hypersurface $\Sigma_{t}$ has the
following unit normal vector $n^{a}$ and volume element $\sqrt{\gamma}$

$n^{a}=-\frac{1}{\Omega(r,\theta)\sqrt{Q(r)}}\left(\partial_{t}\right)^{a}\,,\quad
n_{a}dx^{a}=\Omega(r,\theta)\sqrt{Q(r)}\,dt\,,\quad\sqrt{\gamma}=\frac{\Omega^{3}(r,\theta)\,r^{2}\,\sin{\theta}}{\sqrt{Q(r)}}\,.$
(5.13)

We first find that the integrand in (5.12) vanishes identically for the CKYT
$k_{1}$ (5.10). Using the remaining CKYT $k_{2}$ (5.11) and identifying the
integrand in (5.12) as ${\cal P}^{c}:=\sqrt{\gamma}\,n_{a}F^{ac}$ for
convenience, we next find that

$\displaystyle{\cal P}^{r}$ $\displaystyle=\frac{\alpha
r^{2}\,\Omega^{3}\,\sin{\theta}}{2}\left(\cos{\theta}\left(7\alpha^{2}mr+4\right)+\alpha(2m+4r+6m\cos{2\theta}+\alpha
mr\cos{3\theta})\right)\,,$ $\displaystyle{\cal P}^{\theta}$
$\displaystyle=-2\alpha
r\,\Omega^{3}\,\sin^{2}{\theta}\,\left(1+2m\alpha\cos{\theta}\right)\,.$

It easily follows that the nontrivial $\theta$ integration of ${\cal P}^{r}$
over the $(0,\pi)$ interval vanishes and one is just left with ${\cal
P}^{\theta}$. We find

$\displaystyle\int_{-C\pi}^{C\pi}d\phi\int_{0}^{\pi}d\theta{\cal
P}^{\theta}=-\frac{2\pi^{2}C}{\alpha
r^{2}}\,\left(\frac{m\left(2\alpha^{2}r^{2}\left(2\sqrt{1-\alpha^{2}r^{2}}-3\right)-4\sqrt{1-\alpha^{2}r^{2}}+4\right)+\alpha^{2}r^{3}}{\left(1-\alpha^{2}r^{2}\right)^{3/2}}\right).$
(5.14)

The final step is the $r$ integration. which formally can be done, as

$\displaystyle\int\frac{m\left(2\alpha^{2}r^{2}\left(2\sqrt{1-\alpha^{2}r^{2}}-3\right)-4\sqrt{1-\alpha^{2}r^{2}}+4\right)+\alpha^{2}r^{3}}{r^{2}\left(1-\alpha^{2}r^{2}\right)^{3/2}}=\frac{2m\left(\alpha^{2}r^{2}+2\sqrt{1-\alpha^{2}r^{2}}-2\right)+r}{r\sqrt{1-\alpha^{2}r^{2}}}$
(5.15)

Unfortunately though, the right hand side clearly diverges as $r\to 1/\alpha$.

Finally, one may wonder what the Cotton charge of the C-metric (5.7) is. Since
the Ricci tensor identically vanishes for the C-metric (5.7), so does the
Cotton tensor. So the Killing-Yano Cotton current and, hence, charge is
trivially zero.

##  6 Discussion

After introducing a number of useful identities and relations for CKYTs,
including their correct transformations under conformal transformations, we
have discussed the construction of conserved “currents”, i.e. covariantly
divergence-free tensors, constructed out of geometric tensors and CKYTs. We
found a number of currents such as the Einstein current (3.8), the Cotton
current (3.3) and their related expressions involving duals of the (C)KYTs.
Based on the Einstein current for the Kerr-Newman metric and a rank-2 trivial
current for the $C$-metric, we illustrated the construction of charges for
General Relativity solutions.

Contrary to expectations, however, we found that naive generalisations of
rank-$n$ KYT currents, such as the KT-current, to rank-$n$ CKYT currents yield
trivial currents (2.11). To show this requires a number of the identities and
relations which we introduced in section 4.

In hindsight, it is worth mentioning the following regarding the charge ${\cal
Q}$ (5.6) of the Kerr-Newman metric that we found in subsection 5.1: As argued
in [1], the Cotton current $\tilde{J}^{a}$ (3.3) for the Kerr-Newman metric
(see Sec. 5.3 of [1] for details) can be written in terms of a 2-potential
$\ell$ as $\tilde{J}^{a}=\nabla_{a}\ell^{ac}$, which reduces to
$\ell^{ac}=8\nabla^{[a}K^{c]}$ in terms of the notation used in the present
work. On the other hand, since the curvature scalar $R=0$ and $K^{a}$ is
identical to the timelike Killing vector of the Kerr-Newman metric (5.2), the
Einstein current ${\cal J}^{a}$ (3.8) in fact reduces to $\tilde{{\cal
J}}^{a}=R^{ac}K_{c}$ in this case. Now it is a well-known fact that
$\nabla_{c}\left(\nabla^{[c}K^{a]}\right)=-R^{ac}K_{c}$ for a Killing vector,
so the Cotton current $\tilde{J}^{a}$ (3.3) and the current ${\cal J}^{a}$
(3.8) or $\tilde{{\cal J}}^{a}$ are indeed proportional to each other for the
Kerr-Newman metric. In that sense, it is interesting that the Einstein current
${\cal J}^{a}$ (3.8), $\tilde{{\cal J}}^{a}$ and the Cotton current
$\tilde{J}^{a}$ (3.3) [1] all define the same charge (up to some
proportionality constant) ${\cal Q}$ (5.6) for the Kerr-Newman metric. In
fact, the 3-surface integral of any one of these on a $t=$ const.
hypersurface, say $\Sigma_{t}$, should be proportional, up to some possible
constants, to the Komar “mass”555See e.g. section 11.2 of the seminal work
[13]., which is an integral over the “boundary” 2-surface,
$\partial\Sigma_{t}$ of constant $t$ and $r$, spanned by $\Sigma_{t}$.

In [1] the Cotton current was lifted to supergravity in three dimensions. An
interesting open problem is to find supergravity versions also of the trivial
currents (2.11).

As a final comment we note that the physical meaning of currents and charges
constructed from KTs, KYTs and CKYTs is not always obvious, unlike those from
Killing vectors which describe isometries. For this reason the charges based
on such tensors are often said to generate hidden symmetries and their study
is quite rewarding.

Acknowledgments
Ö.S. would like to thank D.O. Devecioğlu for help with xAct, and an austere
and grim first referee who forced us to forge this paper into a more resilient
form. The research of U.L. was supported in part by the 2236 Co-Funded Scheme2
(CoCirculation2) of TÜBİTAK (Project No:120C067)666However the entire
responsibility for the publication is ours. The financial support received
from TÜBİTAK does not mean that the content of the publication is approved in
a scientific sense by TÜBİTAK.. In addition, support from the Leverhulme trust
is gratefully acknowledged by U.L.

##  Appendix A Some special cases of trivial currents

Since we are considering rank-$n$ forms, we are restricted by the dimension
$D$ of the underlying manifold. Interesting cases are therefore
$n=1,\dots,D/2$ in even and $n=1,\dots,(D-1)/2$ in odd dimensions (modulo
their Hodge duals). Below we comment on the special cases $n=1$ and $n=2$.
Separately, we also introduce a rank-1 current, construct its charge and
discuss its relation to the recently constructed Cotton current for the Kerr-
Newman metric.

###  A.1 $n=1$

This is the special case when the CKYT is a CKV. Thus consider the following
contravariant vector

$\displaystyle J^{a}=R^{ab}k_{b}+\alpha Rk^{a}+\beta\nabla^{a}K~{}.$ (A.1)

Then

$\displaystyle\nabla_{a}J^{a}=k^{a}\nabla_{a}R\left(\frac{1}{2}+\alpha+\frac{\beta}{2(1-D)}\right)+KR\left(1+\alpha
D+\frac{\beta}{(1-D)}\right)\,.$ (A.2)

Demanding $\nabla_{a}J^{a}=0$ implies that either “$D=2\,,\beta=1+2\alpha$
with $\alpha$ left arbitrary” or “$D\neq 2\,,\alpha=0\,,\beta=D-1$”. However
note that when $D=2$, one automatically has $R_{ab}=(R/2)g_{ab}$. So one has a
conserved vector $J^{a}$ for

$\displaystyle D=2:$ $\displaystyle J^{a}=R^{ab}k_{b}+\alpha
Rk^{a}+(1+2\alpha)\nabla^{a}K=(1+2\alpha)(\nabla^{a}K+2Rk^{a})\,,$
$\displaystyle D\neq 2:$ $\displaystyle
J^{a}=R^{ab}k_{b}+(D-1)\nabla^{a}K=-\Box k^{a}+\nabla^{a}K=-F^{a}\,.$

This follows the pattern described for $D\neq 2$ in section 4.

Since we are dealing with a CKV, a natural question to ask is on the
transformation properties of $J^{a}$ under Weyl scaling of the metric
$\tilde{g}_{ab}=\Omega^{2}g_{ab}$. A CKV transforms as

$\displaystyle\tilde{k}^{a}=k^{a}\quad\mbox{and}\quad\tilde{K}=K+\Omega^{-1}(\nabla_{a}\Omega)k^{a}\,.$
(A.4)

The transformations of the Ricci and scalar curvature is given by

$\displaystyle\tilde{R}_{ab}$ $\displaystyle=$ $\displaystyle
R_{ab}-\Omega^{-1}\big{(}(D-2)\nabla_{a}\nabla_{b}\Omega+g_{ab}\Box\Omega\big{)}$
$\displaystyle+\Omega^{-2}\big{(}2(D-2)(\nabla_{a}\Omega)\nabla_{b}\Omega+(3-D)g_{ab}(\nabla_{c}\Omega)\nabla^{c}\Omega\big{)}\,,$
$\displaystyle\tilde{R}$ $\displaystyle=$
$\displaystyle\Omega^{-2}\big{(}R+2(1-D)\Omega^{-1}\Box\Omega+(D-1)(4-D)\Omega^{-2}(\nabla_{c}\Omega)\nabla^{c}\Omega\big{)}\,.$

For $D=2$, these give

$\displaystyle\tilde{J}^{a}=\Omega^{-2}J^{a}-(1+2\alpha)\Omega^{-2}\big{(}k^{a}\Box\ln{\Omega}+\nabla^{a}(k^{c}\nabla_{c}\ln{\Omega})\big{)}\,,$
(A.6)
$\displaystyle\tilde{\nabla}_{a}\tilde{J}^{a}=-(1+2\alpha)\Omega^{-2}\big{(}2K\Box\ln{\Omega}+k^{a}\nabla_{a}\Box\ln{\Omega}+\Box(k^{a}\nabla_{a}\ln{\Omega})\big{)}\,.$

So one has $\tilde{\nabla}_{a}\tilde{J}^{a}=0$ only when $\alpha=-1/2$, for
which $J^{a}=0$. The $D\neq 2$ case can be obtained from the discussion of the
Weyl scalings of a generic rank-$n$ current $F$ given in appendix 2.2, by
setting $n=1$, of course.

###  A.2 $n=2$

For $n=2$, following the general case we find the conserved (trivial) current

$\displaystyle J^{ab}=-\frac{1}{2}\nabla_{c}\nabla^{[a}k^{bc]}\,.$ (A.7)

As may be seen from the following relations, proven in [1]

$\displaystyle(D-3)G^{ac}K_{c}=\nabla_{c}\left(G^{b[c}k_{b}{}^{a]}+(D-2)\nabla^{[a}K^{c]}\right)$
(A.8)

and

$\displaystyle\nabla_{a}\breve{{\cal K}}^{ab}=\nabla_{a}\hat{{\cal
K}}^{ab}+\frac{1}{2}(D-3)G^{b}{}_{c}K^{c}\,,$ (A.9)

the case $D=3$ requires special attention.

When $D=3$, the Riemann tensor can be expressed in terms of the Ricci tensor
and the curvature scalar only, and one finds $\breve{{\cal K}}^{ab}=\hat{{\cal
K}}^{ab}$ identically. This implies

$\displaystyle\frac{1}{2}R^{abcd}k_{cd}+R_{c}{}^{[a}k^{b]c}=-\hat{{\cal
K}}^{ab}=-\nabla^{[a}K^{b]}+\Box\,k^{ab}\,,$ (A.10)

where the last equality follows from (2.10) evaluated for $n=2$. On the other
hand, when $D=3$ one has

$\displaystyle\nabla_{c}\left(\hat{{\cal K}}^{ac}+\nabla^{[a}K^{c]}\right)=0$
(A.11)

from (A.8). The discrepancy between (A.10) and (A.11) is accounted for by the
trivial current (2.11) for $n=2$ also in $D=3$.

##  Appendix B Killing-Yano Cotton current

In [1], we showed that the Killing-Yano Cotton current $J^{a}:=C^{abc}k_{bc}$,
constructed out of a rank-2 CKYT $k$ and the Cotton tensor $C$, is covariantly
conserved. It is known that the Cotton tensor transforms as

$\displaystyle\tilde{C}_{abc}=C_{abc}-(D-2)(\nabla_{d}\ln{\Omega})W^{d}{}_{abc}$
(B.1)

under conformal transformations $\tilde{g}_{ab}=\Omega^{2}g_{ab}$. Meanwhile,
one has777Here we correct an error in [1], where a factor of $(D-1)$ was
inadvertently forgotten in the second term on the right-hand-side of the
second equation in eqn. (2.12) in that paper, with no effect on the ensuing
discussion.

$\displaystyle\tilde{k}_{ab}=\Omega^{3}k_{ab}\quad\mbox{and}\quad\tilde{K}_{c}=\Omega
K_{c}+(\nabla_{a}\Omega)k^{a}{}_{c}\,.$ (B.2)

These give

$\displaystyle\tilde{J}^{a}=\Omega^{-3}\left(J^{a}-(D-2)(\nabla_{d}\ln{\Omega})W^{da}{}_{bc}k^{bc}\right)\,.$
(B.3)

Using these and the property $(D-3)C_{bdc}=(D-2)\nabla_{a}W^{a}{}_{bcd}$, it
is straightforward to show that

$\tilde{\nabla}_{a}\tilde{J}^{a}=\nabla_{a}\tilde{J}^{a}+D(\nabla_{a}\ln{\Omega})\tilde{J}^{a}=0\,,$

independent of the dimension $D$.

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