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A classical solution is called universal if the quantum correction is a
multiple of the metric. Therefore, universal solutions play an important
role in the quantum theory. We show that in a spacetime which is universal
all scalar curvature invariants are constant (i.e., the spacetime is

In [

That is, if the spacetime is universal, then every symmetric conserved rank-2 tensor,

There are a number of related results we would like to investigate in this paper. We will state these in terms of a conjecture and will corroborate this conjecture by proving a number of subresults.

A Universal n-dimensional Lorentzian spacetime,

It is CSI.

It is a degenerate Kundt spacetime.

There exists a spacetime,

There exists a homogeneous isotropy-irreducible Riemannian spacetime

In low dimensions, this conjecture can be proven; in particular, dimension 2 is trivial as there is only one independent component, namely, the Ricci scalar R. In dimension 3, there are only Ricci invariants and the conjecture can be proven by brute force using symmetric conserved tensors. Most of our investigation will focus on dimension 4, and unless stated otherwise, we will assume that there is a 4 dimensional manifold.

We will use different methods to substantiate the above conjecture. This will consist of partial proofs and other arguments.

Let us first provide with results substantiating the claim that universal spacetimes are CSI. This is clearly the case in the Riemannian case where Bleecker [

Field theoretic calculations on curved spacetimes are nontrivial due to the systematic occurrence, in the expressions involved, of Riemann polynomials. These polynomials are formed from the Riemann tensor by covariant differentiation, multiplication, and contraction. The results of these calculations are complicated because of the nonuniqueness of their final forms, since the symmetries of the Riemann tensor as well as Bianchi identities can not be used in a uniform manner and monomials formed from the Riemann tensor may be linearly dependent in nontrivial ways. In [

In this paper, we actually use a slightly modified version of the FKWC-bases [

The most general expression for a scalar of order six or less in derivatives of the metric tensor is obtained by expanding it on the FKWC-basis for Riemann polynomials of rank 0 and order 6 or less [

Choosing

Thus to prove that the resulting spacetimes are CSI, we must show that all scalar contractions of the Weyl tensor and its derivatives are constants.

From (

Indeed, by choosing _{0}). We note that in higher dimensions, all Lovelock tensors are divergence free and consequently (by universality) proportional to the metric. However, we will not proceed in this way here.

The most general expression for a gravitational Lagrangian of order six in derivatives of the metric tensor is obtained by expanding it on the FKWC-basis for

In general, only 10 of these give rise to independent variations. The other 7 depend on these via total divergences (and Stokes theorem); the functional derivatives (i.e., conserved tensors) with respect to the metric tensor of the 7 remaining action terms can then be obtained in a straightforward manner.

In the case of an Einstein space satisfying the conditions (

Variations of the last four scalars in the list above give rise to 4 independent conserved rank-2 tensors (although

The functional derivatives of the ten independent action terms on the FKWC basis were expanded in [

Contracting (

We now proceed with the higher-order scalars: orders (8,10,12) were considered in [

Let

Assume that the spacetime under consideration is universal. If the spacetime is strongly universal, then all of these symmetric tensors are zero:

This means that we have a full set of invariants all of which has a zero variation:

For the degenerate Kundt metrics, there exists a one-parameter family of metrics

In the Riemannian case, the slice theorem was used by Bleecker [

Note that in the compact Riemannian case the slice theorem holds thus universality implies CSI. In the Riemannian case, CSI implies locally homogeneous, and thus this provides a slightly alternate proof to that of Bleecker [

This result depends crucially on the validity of the slice theorem and it is unclear to the authors for which Lorentzian spaces it holds. However, the result is important as one can see that there is a clear link between universality and CSI spaces and thus supports our conjecture.

In [

A Kundt

Equation (

The relation between CSI spacetimes and those that are universal is strong.

A universal spacetime of Petrov type D, II, or III, is Kundt CSI.

Consider type D first. Here, assuming the spacetime is Einstein, we have that the spacetime is necessarily CSI_{0} (which follows from previous discussion). This implies that the b.w. 0 components are constants also in the canonical frame. Using the Bianchi identities, it immediately follows that it is Kundt also. Since the previous analysis also implies that it is CSI_{1}, then we have that the spacetime is Kundt-CSI.

For type II, the analysis is almost identical to the type D analysis. For type III, it requires to calculate some conserved tensors. Using the Weyl type III canonical form, the Bianchi identities imply that _{1} implies CSI for Kundt spacetimes, the theorem follows.

Although the theorem does not include Weyl type N and I, it is believed that these are Kundt. For type I, the expressions are so messy for the conserved tensors to be manageable, and for type N it is necessary to compute a partricular order 16 conserved tensor.

Thus proves the first two statements in the Conjecture

Given a 4D Kundt CSI spacetime

The proof utilises the results from [

The opposite is not true namely, that for every Riemannian universal spacetime there is a Lorentzian spacetime with the same invariants. For example, the symmetric spaces

The authors would like to thank Gary W. Gibbons for helpful comments on the current paper. This work was supported in part by NSERC of Canada.