METRIC-POLYNOMIAL STRUCTURES FROM GRAVITATIONAL LAGRANGIANS

ANDRZEJ BOROWIEC

Abstract

We study these metric-polynomial structures on manifold which arises as extremals of the Palatini variational principle for some class of gravitational Lagrangians. They can be described in the following way.

Let (M, g, Γ) be a n-dimensional (pseudo-) Riemannian manifold (M, g) equipped with a symmetric (i.e. torsion-free) connection Γ. Define a (1, 1) tensor field concomitant

S_{ν}^{μ} ≡ S_{ν}^{μ}(g, Γ) ≐ g^{μα}R_{(αν)}(Γ)

where R_{(αν)}(Γ) denotes the symmetric part of the Ricci tensor of Γ. Consider a family of scalar-valued concomitant

s_{k} ≡ s_{k}(g, Γ) ≐ tr S^{k}

the so-called Ricci scalar of order k, k = 1, ⋯, n. For F being an arbitrary (differentiable) real-valued function of n-variables one can define the corresponding Lagrangian of the Ricci type

L_{F} ≐ \sqrt{g} F (s_{1}, ⋯, s_{n})

Applying now the Palatini variational principle we arrive to the following results:

• the connection Γ is a Levi-Civita connection for some pseudo-Riemannian Einstein metric h,

• the (1, 1) tensor field S satisfies a polynomial equation

w_{F} (S) = 0

for some polynomial function w_{F} (t) of constant coefficients,

• the metric h and polynomial structure S are compatible in a sense

h(SX, Y) = h(X, SY)

for each pair of tangent vector fields (X, Y) on M.

In particular, for f being a function of one variable the Lagrangian L_{f} = \sqrt{g} f(s_{1}) reconstructs the Einstein theory. For L_{f} = \sqrt{g} f(s_{2}), besides the Einstein equation, one gets a pseudo-Riemannian almost product structure (S^{2} = I) and/or an almost-complex anti-Hermitian structure (S^{2} = −I). Some other examples will be also considered.

Institute of Theoretical Physics, Wrocław University pl. M. Borna 9, 50-204 WROCŁAW, Poland e-mail: borow@ift.uni.wroc.pl