Eur. Phys. J. Plus (2018) 133: 381
https://doi.org/10.1140/epjp/i2018-12220-x

Regular Article

Einstein-scalar field equation in LTB space-time: General scheme and special solutions

LTB gravity interacting with scalar field

¹ Dipartimento di Fisica dell’Università degli Studi di Milano, Via Celoria, 16, 20133, Milano, Italy
² GNFM, Gruppo Nazionale per la Fisica Matematica, Milano, Italy

^* e-mail: antonio.zecca@mi.infn.it

Received: 29 May 2018
Accepted: 8 August 2018
Published online: 24 September 2018

Abstract

A complex massive scalar field minimally coupled to the Einstein field equation in the Lemaître-Tolman-Bondi space-time is considered. The energy momentum tensor of the scalar field is assumed to be the source of the Einstein equation. The spherical symmetry implies that the scalar field is allowed to depend only on the radial and time coordinate. In turn the validity of the scalar field equation ensures the consistency of the Einstein field one. The spherical symmetry of the field also ensures a further symmetry condition required by the spherical symmetry of the Ricci scalar. The explicit equations of the scheme are studied in the case the field depends only either on the radial or on the time coordinate. To that end a first general integration step of the system of equations is useful. In the case of purely time-dependent massless scalar field, the solution results in a Robertson-Walker-like space-time. This is an already known homogeneization effect. Such solution also admits an initial inflationary phase and a late accelerated expansion. In the massive time-dependent field case, a very simple but non-trivial solution is given. Such solution is possible because the field can take complex values. The purely static field case is also considered. It has no non-trivial factorized solutions of the physical radius both in the massive and in the massless case.