Obtaining a class of type O pure radiation metrics with a negative cosmological constant, using invariant operators

Abstract

Using the generalised invariant formalism we derive a special subclass of conformally flat spacetimes whose Ricci tensor has a pure radiation and a Ricci scalar component. The method used is a development of the methods used earlier for pure radiation spacetimes of Petrov types O and N, respectively. In this paper we demonstrate how to handle, in the generalised invariant formalism, spacetimes with isotropy freedom and rich Killing vector structure. Once the spacetimes have been constructed, it is straightforward to deduce their Karlhede classification: the Karlhede algorithm terminates at the fourth derivative order, and the spacetimes all have one degree of null isotropy and three, four or five Killing vectors.

Type O pure radiation metrics with a cosmological constant

“…On the other hand, it is emphasised that when we fail to obtain a unique intrinsic second spinor we cannot take this short cut to the GHP formalism. The absence of a second unique intrinsic spinor is linked to the presence of null isotropy freedom, and we have recently considered such a situation -a subclass of conformally flat pure radiation spacetimes with a cosmological constant -and demonstrated how such a problem is solved by the GIF method [11].…”

“…So now we will further develop the GIF operator method by generalising the earlier derivation [13] of the metric for conformally flat pure radiation spaces to include the case of a non-zero cosmological constant. In [11] we looked at the subclass of these spacetimes for which we were unable to find a second unique intrinsic spinor due to the presence of one degree of null isotropy freedom; in this paper we look at the other subclass where there is no null isotropy freedom, and a second unique intrinsic spinor I A is quickly generated within the GIF. This means that we can quickly transfer to the GHP formalism and so minimise the calculations.…”

“…These spacetimes will illustrate further refinements of our method, and they will also be shown o have a richer Killing vector structure. More details of the philosophy and techniques of the GIF operator integration procedure has been given in [13], [10], [11] so we will not repeat these discussions in this paper, but rather we will only summarise the relevant parts of the GIF which are needed in this paper. In Section 2 we describe the differential operators, and the equations for the class of spaces under consideration are given in Section 3.…”

“…We are concerned with the Petrov type O pure radiation spaces with non-zero Ricci scalar, and in fact we begin with identical equations to those in [11], but we shall repeat them here for easy reference. In the usual way, we choose o o o A to be aligned with the propogation direction of the radiation, so that the Ricci spinor takes the form…”

“…In fact, in a particular calculation such as [13] and in the present paper, once we have used the GIF to obtain the table for the second spinor I, and applied the GIF commutators to this table, we can then identify I with the second dyad spinor ι ι ι in the GHP formalism and immediately translate any existing tables into the GHP formalism, which means that we can then complete the calculations for the remaining tables in the GHP formalism. For the subsequent calculations we will need the GHP commutator equations, which can be obtained from the GIF commutator equations by projecting on the appropriate number of ι ι ι A ,ῑ ι ι A ′ spinors and using (8), (9), (10), (11). An alternative method, which will be quicker for our purpose, is to make use of the GHP commutator equations as quoted in [15] (specialised to this class of spacetimes),…”