# Explicit Kundt type and solutions as gravitational waves in various type and universes

###### Abstract

A particular yet large class of non-diverging solutions which admits a cosmological constant, electromagnetic field, pure radiation and/or general non-null matter component is explicitly presented. These spacetimes represent exact gravitational waves of arbitrary profiles which propagate in background universes such as Minkowski, conformally flat (anti-)de Sitter, Edgar–Ludwig, Bertotti–Robinson, and type (anti-)Nariai or Plebański–Hacyan spaces, and their generalizations. All possibilities are discussed and are interpreted using a unifying simple metric form. Sandwich and impulsive waves propagating in the above background spaces with different geometries and matter content can easily be constructed. New solutions are identified, e.g. type pure radiation or explicit type electrovacuum waves in (anti-)Nariai universe. It is also shown that, in general, there are no conformally flat Einstein–Maxwell fields with a non-vanishing cosmological constant.

PACS: 04.20.Jb; 04.30.Nk

## 1 Introduction

The Kundt class, which is characterized by the geometrical property that it admits a non-expanding and non-twisting null congruence, is a well-known family of solutions to Einstein’s equations. Wolfgang Kundt was the first to introduce, emphasize and investigate this large class [1, 2, 3] (in the case of vacuum and pure radiation) although some of its important subclasses, in particular the pp -waves [4, 5, 6, 7] or the Nariai [8] and Bertotti-Robinson [9, 10, 11] universe, were discovered and studied previously. Since then a great number of papers have been devoted to the derivation of such spacetimes and an analysis of their properties, see e.g. [12] for the review. More recent articles which are related to the specific topic of this paper are mentioned below in the appropriate context.

Naturally, most works on exact non-diverging spacetimes investigate particular subclasses of the large Kundt family by restricting attention to a specific algebraic Petrov type and considering special matter contents (vacuum, cosmological constant, pure radiation, null or non-null electromagnetic fields). In many of these subcases, all solutions of the given type were explicitly obtained. On the other hand, at present it is still impossible to find a general solution which describes different Petrov types and matter fields in an explicit form. Even with an assumption of symmetries such solutions could in most cases only be given implicitly, see e.g. [12, 13].

This contribution focuses on the “gap” which lies between these “extreme” approaches. We present in section 2 a simple explicit metric which contains several arbitrary parameters and functions. For particular choices we recover, in sections 3 and 4 respectively, many Kundt spacetimes with various matter contents which are of the Petrov types , , and , . Moreover, the latter solutions can be understood as exact gravitational waves of arbitary profile which propagate in the corresponding “background” universes. All possibilities are discussed below in detail and are interpreted by analyzing the geodesic deviation. The paper also contains an appendix in which we prove that there are no conformally flat Einstein–Maxwell fields with .

## 2 General form of the solutions

Throughout the present paper we consider and discuss the following metric

(1) |

with

(2) | |||||

where , , and are real constants (without loss of generality we can assume or ), and are arbitrary functions of the null coordinate ( may be complex), and is an arbitrary function of the spatial coordinates , , and of . In the natural null tetrad

(3) | |||||

we obtain (using ) the following form of the only non-vanishing Weyl and Ricci scalars

(4) | |||||

(5) | |||||

(6) | |||||

(7) | |||||

(8) | |||||

The spin coefficients , and vanish identically so that the multiple principal null direction is non-expanding, twist-free, shear-free and geodesic. The spacetimes (1), (2) thus belong to the Kundt class and are algebraically special. By a simple transformation , these can be put into the Kundt canonical form [12]

(9) |

where

(10) |

It follows from (4) and (5) that, in general, the above spacetimes are of Petrov type . In special cases, these degenerate to types , , or can be conformally flat. It is convenient to introduce the following two functions,

(11) | |||||

(12) |

where and are arbitrary real functions and is an arbitrary complex function of . When the functions and in (2) take these special forms and , the scalars and vanish, respectively. We may thus easily summarize the possible types in the following table 1.

In general, the above spacetimes do not have a constant Ricci curvature . They also contain non-uniform pure radiation described by plus a non-null matter component (which may be associated to a fluid with anisotropic pressure, see [14], with the 4-velocity of the fluid , and being the vector responsible for the anisotropy), cf. (6)–(8). In particular cases, a constant scalar may correspond to the cosmological constant , and the matter components , could represent an electromagnetic field. In the latter case, the Maxwell equations (see, e.g. [12]) for the metric (1), (2) reduce to

(13) |

since because for ,
. The non-vanishing spin coefficients we need here are given by
and
.
Moreover, so that either or must vanish.
There are thus *only two* decoupled possibilities, and the
equations (13) can explicitly be integrated as

(14) | |||||

(15) |

where and are complex functions. However, for the purely electromagnetic case there are additional constraints since we have to satisfy the Einstein–Maxwell system. This requires that the Ricci scalar given by (6) is equal to , and the expressions for and given by (7), (8) have to satisfy

(16) |

respectively. In fact, it can be seen that these restrictions rule out the presence of a purely electromagnetic field in general spacetimes of the form (1), (2). Only for some special choices of , , , , and are the Einstein–Maxwell field equations satisfied.

In the next sections we shall describe all the particular possibilities in detail. We start with various backgrounds which may be either conformally flat or of the type . Then we shall introduce gravitational waves into all these background spacetimes by considering the corresponding type or solutions.

## 3 The possible backgrounds

### 3.1 Conformally flat spacetimes

The above solutions (1), (2) are conformally flat if and only if and , in which case . For this choice, the expressions (6)–(8) reduce to

(17) | |||||

(18) | |||||

(19) | |||||

In order to obtain only a pure radiation field, one has to set so that , , and

(20) |

Consequently, we obtain explicit solutions in the form (1) in which

(21) | |||||

where and are arbitrary functions of . (Note that in the expression (18) vanishes also for , which however gives just a subcase of (21).) This complete family of pure radiation, conformally flat spacetimes with was mentioned already in [15]. As discussed in [15, 16] there exist several invariant subclasses of this, namely (conformally flat) pp and Kundt waves with , and their generalizations to non-vanishing . For pp -wave spacetimes (, , so that ) or the solution of the Siklos type [17, 18] (, , ), one obtains which implies that is a Killing vector. In particular, this includes the only conformally flat pure radiation solution of the Einstein–Maxwell equations with in (15), namely the plane waves introduced by Baldwin, Jeffery [5] and Brdička [6] (see the appendix). Another special subclass of (2) arises for and , in which case . In the canonical coordinates (9) this corresponds to the metric coefficients (10)

(22) | |||||

For vanishing , this exactly reduces to the interesting class of conformally flat pure radiation metrics found by Edgar and Ludwig [19, 20] and discussed e.g. in [21, 22, 23].

The complementary situation in which there is no pure radiation () but only a non-vanishing component of the matter field, requires , , const., , and , so that . However, this function can always be removed by the coordinate transformation , , for a suitable choice of . Without loss of generality we may thus write the solutions of this subclass as the metric (1) in which

(23) |

By performing a simple transformation , we can put this into the form

(24) |

in which and are given by (23). (There is still a coordinate freedom given by , and being constants, and rescaling of and/or , which can be used to modify and .) The matter content in these spacetimes is described by which, in general, is a function of the spatial coordinate . However, it immediately follows from (18) that the (invariant) additional condition const. necessarily implies , , , . Thus, for a constant we obtain the Bertotti–Robinson universe [9, 10, 11, 12]

(25) |

This homogeneous space is the unique conformally flat solution of the Einstein–Maxwell equations with a (uniform) non-null electromagnetic field in (14) where (see the appendix).

Of course, conformally flat spacetimes of the above form exist which combine and . For example, one can introduce a pure radiation in the Bertotti–Robinson universe by adding the term to the metric (25).

Finally, the conformally flat spaces with no matter (Einstein spaces) are given by (21) with the additional constraint , see (20). In such a case, the function takes the form , which can be removed completely by a suitable coordinate transformation [16]. These constant curvature vacuum solutions with can then be put into the standard form of Minkowski, de Sitter or anti-de Sitter spacetimes [24]. For example, assuming , , we obtain

(26) |

For the case, this is obviously a flat space. For , the parametrization of the (anti-) de Sitter universe represented geometrically as a hyperboloid () imbedded in a flat five-dimensional space is

(27) | |||||

Analogous parameterizations can be found for other (canonical) choices of and .

### 3.2 Type spacetimes

In this case, the vanishing of requires , and the curvature scalars (4)–(8) reduce to

(28) | |||||

(29) | |||||

(30) | |||||

(31) | |||||

Notice that the component (31) has the same form as in the conformally flat case (19) but the other scalars differ since .

A pure radiation field requires , which implies that , , , and the metric takes the form ,

(32) |

the scalars being , and ,

(33) |

When () this solution represents the (anti-)Nariai universe [8] with pure radiation which can not be an electromagnetic field (see the general theorem in [25]). However, for this becomes the conformally flat electromagnetic plane wave mentioned in the previous section. Note also that the metric (32) is identical to the Kundt canonical form (9) since in the case of pure radiation solutions, and consequently , , , and thus seems to be a counterexample to a conjecture of [26] that the solutions exhibited there (for which ) are the only type pure radiation metrics of the Kundt class. . In particular, it belongs to the invariant subclass defined by

The situation in which there is no pure radiation (), but only the component of the matter field, admits two possibilities. If then , , , and so that . As in the conformally flat case, this function can always be removed by a coordinate transformation. Thus, all the solutions of this subclass can be written as the metric (1) in which

(34) |

With , where , we put the solution into the form

(35) |

(For the metric (34) reduces to the conformally flat spacetime (23), i.e. (35) gives (24).) Looking for the electrovacuum solutions we require which implies , , , and the scalars are , . In such a case, the metric (1) simplifies considerably to

(36) |

These are well-known electrovacuum spacetimes with the geometry of a direct product of two constant curvature 2-spaces [10], see also [27, 12]. This is obvious from the form (35) since , we recover the conformally flat solution (25). The second possibility for the case is which implies , i.e. . The corresponding scalars are , where , , . For electrovacuum solutions, so that , , , . We thus uniquely obtain the exceptional Plebański–Hacyan universe [28] . Of course, for

(37) |

for which (the function can be removed by a coordinate transformation [28]). When , this reduces to the form (36) of the direct product spacetimes.

Again, type spacetimes which combine and can be considered. For example, there exist solutions which represent pure radiation in the electrovacuum universes (36) and (37), see e.g. [29].

Finally, the type Einstein spaces with no matter are given by (32)-(33) with the constraints , . In such a case, the function can again be removed by a suitable coordinate transformation. Thus, without loss of generality one obtains

(38) |

which is the (anti-)Nariai universe [8], discussed recently in [30]. Obviously, this is also the direct product spacetime (36) for . There is no type vacuum solution of the form (1) for since (38) reduces to Minkowski space in such a case.

The main results presented above are summarized in the table 2.

## 4 Exact gravitational waves of arbitrary profile on the above backgrounds

By considering an arbitrary function in the metric (1), (2), different from
as introduced in (12), the scalar representing gravitational radiation becomes non-vanishing,
see (5). As the coordinate plays the role of the retarded time, gravitational waves of arbitrary
profiles
can thus be introduced into the above spacetimes. When the backgrounds are taken to be conformally flat,
exact radiative spacetimes of Petrov type are obtained. For the backgrounds of type one gets explicit
gravitational waves of type , see table 1. It is obvious from (1) that the wave-fronts const.
are *non-expanding* 2-spaces with a *constant curvature* equal to .

It can be observed from (4)–(8) that the function does not appear in the scalars , , and . In fact, introducing the additional term 31, 32]) of the background geometries^{1}^{1}1The authors are grateful to the referee for this observation..
Consequently, the radiative spacetimes (1) of type and have exactly the
same values of the Weyl component , the Ricci curvature , and the non-null matter component
as the corresponding backgrounds described in the previous section (see table 2). The only difference, apart from
the introduction of the component, may be in
the pure radiation field. Indeed, the component , which is linear in , can now be understood as
a superposition of the background term given by (19), identical to (31),
with the term related to the presence of gravitational radiation, in the metric corresponds to a generalised Kerr–Schild transformation (see e.g. [

(39) |

In general, a gravitational wave () is thus accompanied by the (additional) pure radiation field .

However, in special situations when , the gravitational wave is not related to the above pure radiation. For example when , i.e. for spacetimes (36) which are direct product of two constant curvature 2-spaces (Minkowski, Bertotti–Robinson, (anti-)Nariai, Plebański–Hacyan spaces), the relation is satisfied if and only if . This is just the Einstein equation for purely gravitational waves propagating in Einstein spaces, in which case .

Let us now discuss in some detail the particular subclasses which include gravitational waves.

### 4.1 Type spacetimes

A complete class of non-expanding type vacuum solutions with (possibly) non-vanishing cosmological constant was found by Ozsváth, Robinson and Rózga [15], and later studied also from a physical point of view [33]. These Einstein spaces represent exact pure gravitational waves which propagate in constant curvature backgrounds, i.e. in Minkowski, de Sitter or anti-de Sitter universe (for , or , respectively). For example, considering the background given by (26), one obtains radiative solutions of the form

(40) |

For vanishing cosmological constant we obtain exactly the well-known class of pp -waves [12]. Alternatively, with other forms of constant curvature backgrounds corresponding to different canonical choices of the parameters and other classes are obtained, namely the specific Kundt spacetimes for [1, 2, 3] or the Siklos family [17] for , see [15, 16]. When these are pure gravitational waves, otherwise they are accompanied by a pure radiation component.

Another possibility is to consider the conformally flat pure radiation (, ) backgrounds of the form (21) and introduce . Again the pp -waves and gravitational waves in the Edgar–Ludwig type backgrounds (22) with any are thus obtained. In fact, these radiative spacetimes are contained in the Ozsváth–Robinson–Rózga family [15].

On the other hand, considering the conformally flat backgrounds (24) without pure radiation (, ) gravitational waves (plus possibly pure radiation if ) are generated. In particular, one obtains the type solutions representing gravitational radiation in the Bertotti–Robinson electrovacuum universe

(41) |

where is given by (25).

### 4.2 Type spacetimes

All type solutions of the form (1), (2) which represent exact gravitational waves propagating in backgrounds with the matter field component vanishing can be written in the form (41) where is now the metric (32). These describe gravitational radiation in the (anti-)Nariai (type ) universe filled with a pure radiation field , where is given by (33) and by (39), which now reduces to a simple expression

As an important special subcase we easily obtain pure gravitational waves of the above form by considering, not only , but also , i.e. Einstein spaces of the Petrov type without matter. Obviously, these solutions can explicitly be written as

(42) |

where is the metric (38) of the (anti-)Nariai vacuum universe with (), and is an arbitrary function (holomorphic in ) which characterizes the profile of the gravitational wave. Note that the solution (42) of the Einstein vacuum equations with cosmological constant is included in the class of solutions that was investigated from a different point of view by Lewandowski [34].

The complementary situation in which , but , corresponds to introducing type waves into the background (35). In particular, considering only the non-null electrovacuum background universes we obtain gravitational waves in the spacetimes which are a direct product of two constant curvature 2-spaces (36). The second possibility for the electrovacuum case is the exceptional Plebański–Hacyan background universe (37). In both these cases the radiative metric has the form (42) with the corresponding form of .

Again, these gravitational waves are in general accompanied by a pure radiation contribution . As particular cases of the type spacetimes without pure radiation, , we obtain the special electrovacuum solutions with gravitational waves that were found by García and Alvarez [35]. For , , , this is their special II- solution with which can be written in the form (42) where is the Plebański–Hacyan metric (36). For , , , we obtain the (non-twisting subclass of) II- solution with in the form (42) where is the other direct product Plebański–Hacyan metric of the form (36). In both cases, , , . Note that all the above mentioned metrics representing pure gravitational waves in electromagnetic universes are in fact specific subcases of a solution presented by Khlebnikov [36].

### 4.3 Electrovacuum solutions

It may be useful to summarize all the possible cases in which the solution of the form (1), (2) represents a spacetime containing an electromagnetic field but no other matter fields. In section 3 we concentrated on the background spaces of type and . We now investigate the corresponding situations when gravitational waves are present, i.e. the spacetimes of type or , respectively.

We have demonstrated (see also the appendix) that the only conformally flat spacetimes (including ) which satisfy the Einstein–Maxwell equations are some special pp -waves with a null Maxwell field, and the Bertotti–Robinson universe with a non-null Maxwell field. For type pp -waves, corresponding to , in (1), it is well known [12] that there is a general combination of electromagnetic and gravitational waves when , where represents the null Maxwell field (15). On the other hand, one has type gravitational waves propagating in the Bertotti–Robinson electrovacuum universe, with , in (1), for , cf. [36, 30, 29].

For type solutions we may have only a non-null Maxwell field, represented by spacetimes which are the direct product of two constant curvature 2-spaces (36) plus the exceptional Plebański–Hacyan universe (37). Pure type gravitational waves in such type electrovacuum spacetimes arise for , see [35, 36, 30, 29].

Finally, we consider vacuum backgrounds and possible electromagnetic waves propagating on these. Conformally flat vacuum spacetimes are just spaces of constant curvature. Electromagnetic waves in the Minkowski spacetime are simply the pp -waves discussed above. Electromagnetic waves in the (anti-)de Sitter space were discussed in [15], and are necessarily accompanied by gravitational waves (see [15] for explicit formulae). The only remaining possibility is thus given by waves in vacuum backgrounds of type , i.e. in the (anti-)Nariai universe (38). Interestingly, also in this case the Einstein–Maxwell equation (16), (39) for the profile function can be integrated explicitly. Exactly as in the case of pp -waves, this admits the general solution . However, these electromagnetic waves are now necessarily accompanied by a gravitational wave component (see section 3 and [25]). Such an explicit solution for electromagnetic (plus gravitational) waves in the (anti-)Nariai universe seems to have remained unnoticed in the literature so far (whereas pure gravitational waves were already considered [36, 30]).

### 4.4 Effects on test particles

In order to analyze the effects of the gravitational and matter fields of the above solutions, it is natural to investigate the specific influence of various components of these fields on the relative motion of free test particles. Such a local characterization of spacetimes, based on the equation of geodesic deviation, was described in the pioneering works by Pirani [37, 38] and Szekeres [39, 40]. Following the notation introduced in [33] we set up an orthonormal frame , , such that the timelike vector coincides at a given event with the four-velocity of a geodesic test observer. By projecting the equation of geodesic deviation onto this frame we obtain

(43) |

where . The frame components of the displacement vector determine directly the distance between close test particles, and are their physical relative accelerations. From the standard decomposition of the curvature tensor (see, e.g., Eqs. (3.44)–(3.47) in [12]), we immediately obtain