The Horndeski theory is known as the most general scalar-tensor theory with second-order field equations. In this paper, we explore the bi-scalar extension of the Horndeski theory. Following Horndeski's approach, we determine all the possible terms appearing in the second-order field equations of the bi-scalar-tensor theory. We compare the field equations with those of the generalized multi-Galileons, and confirm that our theory contains new terms that are not included in the latter theory. We also discuss the construction of the Lagrangian leading to our most general field equations.

We show that strictly stationary spacetimes cannot have nontrivial configurations of form fields/ complex scalar fields. This means that the spacetime should be exactly Minkowski or anti-deSitter spacetime depending on the presence of negative cosmological constant. That is, self-gravitating complex scalar fields and form fields cannot exist.

This work studies wave propagation in the most general scalar-tensor theories, particularly focusing on the causal structure realized in these theories and also the shock formation process induced by nonlinear effects. For these studies we use the Horndeski theory and its generalization to the two scalar field case. We show that propagation speeds of gravitational wave and scalar field wave in these theories may differ from the light speed depending on background field configuration, and find that a Killing horizon becomes a boundary of causal domain if the scalar fields share the symmetry of the background spacetime. About the shock formation, we focus on transport of discontinuity in second derivatives of the metric and scalar field in the shift-symmetric Horndeski theory. We find that amplitude of the discontinuity generically diverges within finite time, which corresponds to shock formation. It turns out that the canonical scalar field and the scalar DBI model, among other theories described by the Horndeski theory, are free from such shock formation even when the background geometry and scalar field configuration are nontrivial. We also observe that gravitational wave is protected against shock formation when the background has some symmetries at least. This fact may indicate that the gravitational wave in this theory is more well-behaved compared to the scalar field, which typically suffers from shock formation. CONTENTS

We prove that static black holes in n-dimensional asymptotically flat spacetime cannot support non-trivial electric p-form field strengths when (n+1)/2<= p <= n-1. This implies in particular that static black holes cannot possess dipole hair under these fields.Comment: 5 pages; minor changes, typos corrected, reference added, accepted for publication in PR

This paper studies a class of D = n + 2(≥ 6) dimensional solutions to Lovelock gravity that is described by the warped product of a two-dimensional Lorentzian metric and an n-dimensional Einstein space. Assuming that the angular part of the stress-energy tensor is proportional to the Einstein metric, it turns out that the Weyl curvature of an Einstein space must obey two kinds of algebraic conditions. We present some exact solutions satisfying these conditions. We further define the quasilocal mass corresponding to the Misner-Sharp mass in general relativity. It is found that the quasilocal mass is constructed out of the Kodama flux and satisfies the unified first law and the monotonicity property under the dominant energy condition. Making use of the quasilocal mass, we show Birkhoff's theorem and address various aspects of dynamical black holes characterized by trapping horizons.1 The replacement to the Einstein space has a significant impact upon the linear instability of black holes [33]. 2 The solutions to special cases of Lovelock gravity with a more general base space were also studied in [38][39][40][41]. 3 Note that some literatures have employed the convention k = ⌊D/2⌋ different from ours. Since the D/2th term in even dimensions amounts to the topological invariant, it fails to contribute to the field equation. Hence, both of these conventions do not make any physical difference.