# Absence of gyratons in the Robinson–Trautman class

###### Abstract

We present the Riemann and Ricci tensors for a fully general non-twisting and shear-free geometry in arbitrary dimension . This includes both the non-expanding Kundt and expanding Robinson–Trautman family of spacetimes. As an interesting application of these explicit expressions we then integrate the Einstein equations and prove a surprising fact that in any the Robinson–Trautman class does not admit solutions representing gyratonic sources, i.e., matter field in the form of a null fluid (or particles propagating with the speed of light) with an additional internal spin. Contrary to the closely related Kundt class and pp-waves, the corresponding off-diagonal metric components thus do not encode the angular momentum of some gyraton. Instead, we demonstrate that in standard general relativity they directly determine two independent amplitudes of the Robinson–Trautman exact gravitational waves.

PACS class: 04.20.Jb, 04.30.–w, 04.50.–h, 04.40.Nr

Keywords: gyratons, Robinson–Trautman class, Kundt class, gravitational waves

## 1 Introduction

The Robinson–Trautman class of spacetimes, discovered more than fifty years ago [1, 2], is one of the most fundamental families of exact solutions to Einstein’s field equations. Geometrically, it is defined by admitting a geodesic, shear-free, twist-free but *expanding* null congruence. This group of spacetimes contains many important vacuum solutions, in particular Schwarzschild-like static black holes, accelerating black holes (-metric) and radiative spacetimes of various algebraic types. It also admits a cosmological constant, electromagnetic field or pure radiation, as in the case of the Vaidya metric or Kinnersley photon rockets. More details and a substantial list of references can be found in chapter 28 of [3] or chapter 19 of [4].

In [5] the Robinson–Trautman family of solutions was extended to higher dimensions in the case of empty space (with any value of the cosmological constant) and for aligned pure radiation. Interestingly, there are great differences with respect to the usual case (see also [6]). Aligned electromagnetic fields were subsequently also incorporated into the Robinson–Trautman higher-dimensional spacetimes within the Einstein–Maxwell theory [7], and an additional Chern–Simons term for odd dimensions was also considered. The results were recently summarized in the review work [8] on algebraic properties of spacetimes of higher dimensions.

The complementary *non-expanding* Kundt class of twist-free and shear-free geometries also admits explicit vacuum solutions with an arbitrary cosmological constant, electromagnetic fields and pure radiation (null fluid), see chapter 31 of [3] or chapter 18 of [4] for summary concerning the Einstein theory in . The corresponding extensions to higher dimensions were presented in the work [9]. Interestingly, the whole Kundt class also admits spacetimes representing null fields of *gyratonic matter* with internal spin/helicity. It turns out that the angular momentum of such rotating sources is encoded in the non-diagonal metric functions.

This observation was made by Bonnor already in 1970 [10, 11]. He studied both the interior and the exterior field of a “spinning null fluid” in the class of axially symmetric *pp*-wave spacetimes which are the simplest representatives of the Kundt family.
In the natural coordinates of non-twisting geometries (see section 2), the energy-momentum tensor in the interior region is phenomenologically described by the radiation energy density and by the components representing the spinning character of the source (its non-zero angular momentum density). Spacetimes with such localized spinning sources moving at the speed of light were independently rediscovered and investigated (in four and higher dimensions) in 2005 by Frolov and his collaborators who called them gyratons [12, 13]. These *pp*-wave-type gyratons were later studied in greater detail and generalized to include a negative cosmological constant [14], electromagnetic field [15],
and various other settings including non-flat backgrounds. An extensive summary can be found in [16]. This recent work presents and investigates gyratons in a fully general class of Kundt spacetimes in any dimension.

In fact, all the so far known spacetimes with gyratonic sources belong to the Kundt class. The question thus arises: Is it possible to find gyratons in other geometries? The most natural candidate is clearly the Robinson–Trautman family because it shares the non-twisting and shear-free property and in it admits a similar algebraic structure. It differs only in having a non-vanishing expansion of the geometrically privileged null congruence.

This is the purpose of the present paper: We systematically study the possible existence of Robinson–Trautman gyratonic solutions (in any dimension) which would be analogous to those known in the Kundt class. First, in section 2 we present the general form of the non-twisting shear-free line element and all its components of the Christoffel symbols, the Riemann and the Ricci tensors. In subsequent section 3 we derive the explicit solutions to Einstein’s equations in such a setting by performing their step-by-step integration. We summarize the obtained spacetimes and discuss them in section 4. Appendix A contains the proof of some useful identities.

## 2 General Robinson–Trautman and Kundt geometry

In the most natural coordinates the line element of a general non-twisting -dimensional spacetime is given by [5]

(1) |

where is a shorthand for spatial coordinates .^{1}^{1}1Throughout this paper the indices label the spatial coordinates and range from to .
The nonvanishing contravariant metric components are (an inverse matrix to ), , and , so that

(2) |

The geometrically privileged null vector field generates a geodesic and affinely parameterized congruence. A direct calculation for the metric (1) immediately shows that the covariant derivative of is given by , so that . The optical matrix [8] defined as where are unit vectors forming the orthonormal basis in the transverse Riemannian space is thus simply given by

(3) |

This can be decomposed into the antisymmetric twist matrix , symmetric traceless shear matrix and the trace determining the expansion such that with , i.e., . From (3) we immediately see that which confirms that the metric (1) is *non-twisting*. If we impose the additional condition that the metric is *shear-free*, , we obtain the relation . Using we thus infer

(4) |

The first expression can be integrated as

(5) |

When the expansion vanishes, , this effectively reduces to so that the spatial metric is independent of the affine parameter . It yields exactly the *Kundt class* of non-expanding, twist-free and shear-free geometries [3, 4, 9, 8]. The other case gives the expanding *Robinson–Trautman class* which we will study in this contribution.

The Christoffel symbols for the general non-twisting spacetime (1) after applying the shear-free condition (4) are

(6) | |||

(7) | |||

(8) | |||

(9) | |||

(10) | |||

(11) | |||

(12) | |||

(13) | |||

(14) | |||

(15) | |||

(16) | |||

(17) | |||

(18) | |||

(19) | |||

(20) | |||

(21) |

where -dimensional Riemannian space. are the Christoffel symbols with respect to the spatial coordinates only, i.e., the coefficients of the covariant derivative on the transverse

The Riemann curvature tensor components are then obtained (after straightforward but lengthy calculation) in the form

(22) | |||

(23) | |||

(24) | |||

(25) | |||

(26) | |||

(27) | |||

(28) | |||

(29) | |||

(30) | |||

(31) |

Finally, the components of the Ricci tensor are

(32) | |||

(33) | |||

(34) | |||

(35) | |||

(36) | |||

(37) |

and the Ricci scalar is

(38) |

In the above expressions, , and are the Riemann tensor, Ricci tensor and Ricci scalar for the transverse-space metric , respectively. The symbol denotes the covariant derivative with respect to :

(39) | |||||

(40) | |||||

(41) | |||||

(42) | |||||

(43) | |||||

(44) |

and , , are convenient shorthands defined as

(45) | |||||

(46) | |||||

(47) |

where, of course, . It will also be useful to rewrite the following -derivatives of the metric functions in terms of the contravariant components, see (2), namely

(48) | |||||

(49) | |||||

(50) | |||||

(51) |

The expressions (32)-(37) of the Ricci tensor enable us to write explicitly the gravitational field equations for any non-twisting and shear-free geometry of an arbitrary dimension , that is for any Kundt or Robinson–Trautman spacetime.

## 3 Einstein’s field equations with gyratons and their complete integration

General Einstein’s equations for the metric have the form and an arbitrary matter field given by its energy momentum-tensor with the trace . By substituting their trace we obtain , where we admit a nonvanishing cosmological constant

(52) |

Our main aim here is to solve the Einstein field equations (52) in the case of expanding Robinson–Trautman geometry with a *gyratonic matter*, which is a natural generalization of a pure radiation field to admit a spin of the null source [10, 12, 16]. We thus assume that the only nonvanishing components of the energy-momentum tensor are corresponding to the classical pure radiation component and which encodes inner gyratonic angular momentum. We immediately observe from (1), (2) that the trace of such energy-momentum tensor vanishes, .

Moreover, the condition which follows from Bianchi identities, after a straightforward manipulation, gives the constraints

(53) |

These can be explicitly rewritten using (6)–(21) as

(54) | |||

(55) |

We can now perform a step-by-step integration of the Einstein field equations (52).

### 3.1 The equation

From (32) we get the explicit form of this equation

(56) |

which obviously determines the -dependence of the expansion scalar . Its general solution can be written as . However, the metric (1) is invariant under the gauge transformation and we can thus, without loss of generality, set the integration function to zero. The expansion simply becomes

(57) |

The integral form (5) of the shear-free condition (4) with the expansion given by (57) completely determines the -dependence of the -dimensional spatial metric , namely

(58) |

so that , where is the inverse matrix of . The -independent metric part will be constrained by the next Einstein’s equations.

### 3.2 The equation

Using (33), (48) and (49) we rewrite the Ricci tensor component in a more compact way

(59) |

Employing now the restriction given by , i.e., the explicit form of expansion (57) the equation becomes

(60) |

We easily find its general solution in the form

(61) |

where and are arbitrary integration functions of and . In view of (2) and (58), the corresponding covariant components of the Robinson–Trautman metric are

(62) |

where and .

### 3.3 The equation

It is convenient to rewrite the general Ricci tensor component (34) using the contravariant metric components,

(65) |

Employing the previous results (57), (58) and (61) the corresponding Einstein equation becomes

(66) |

Its homogeneous solution is , where and are integration functions. The particular solution can be obtained as a superposition of terms on the right hand side of (66). The general solution with an explicit -dependence of the metric component thus becomes corresponding to all terms of the form

(67) |

Notice that is then simply obtained using (2) as

(68) |

### 3.4 The equation

Using (50), (57), (58), (61) and (62) the general Ricci tensor component (35) becomes

(69) |

in which, employing (67),

(70) |

The corresponding Einstein equations (52) thus take the form

(71) |

The trace of this equation explicitly determines the function introduced in (67), namely

(72) |

where is the Ricci scalar curvature of the spatial metric which is the -independent part of . Notice that due to (58) the corresponding Ricci tensor is , while . Decomposing the equation (71) into the terms with different powers of we obtain the following constraints on the metric functions:

(73) | |||||

(74) | |||||

(75) | |||||

(76) |

Now, if we multiply both sides of (76) by we obtain

(77) |

whenever . If