Abstract
We consider a general class of scalar tensor theories in three dimensions whose action contains up to second-order derivatives of the scalar field with coupling functions that only depend on the standard kinetic term of the scalar field, thus ensuring the invariance under the constant shift of the scalar field. For this model, we show that the field equations for a stationary metric ansatz together with a purely radial scalar field can be fully integrated. The kinetic term of the scalar field solution is shown to satisfy an algebraic relation depending only on the coupling functions, and hence is constant while the metric solution is nothing but the BTZ metric with an effective cosmological constant fixed in terms of the coupling functions. As a direct consequence the thermodynamics of the solution is shown to be identical to the BTZ one with an effective cosmological constant, despite the presence of a scalar field. Finally, the expression of the semi-classical entropy of this solution is also confirmed through a generalized Cardy-like formula involving the mass of the scalar soliton obtained from the black hole by means of a double Wick rotation.
Highlights
Since the discovery of the Bañados-Teitelboim-Zanelli (BTZ) black hole solution [1], the study of three-dimensional gravity has received considerable attention to such an extent that it is considered an interesting laboratory to explore the many facets of the lower-dimensional physics at the classical level and at the quantum level
Enough, we will show that the integration of the equations of motion forces the scalar field to have a constant kinetic term while at the same time the metric functions turn out to be a BTZ-like spacetime with an effective cosmological constant expressed in terms of the coupling functions appearing in the action
We have shown that the equations of motion of a very general class of scalar tensor theories (1)– (3) can be fully integrated for a stationary metric ansatz together with a purely radial scalar field
Summary
Since the discovery of the Bañados-Teitelboim-Zanelli (BTZ) black hole solution [1], the study of three-dimensional gravity has received considerable attention to such an extent that it is considered an interesting laboratory to explore the many facets of the lower-dimensional physics at the classical level and at the quantum level. We will confirm this trend by showing that the equations of motion of a general class of scalar tensor theories, enjoying a shift symmetry of the scalar field, and involving up to second-order derivatives of the scalar field, can be fully integrated and solved by a BTZ-like metric. Enough, we will show that the integration of the equations of motion forces the scalar field to have a constant kinetic term while at the same time the metric functions turn out to be a BTZ-like spacetime with an effective cosmological constant expressed in terms of the coupling functions appearing in the action.
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