Spatially covariant gravity with nonmetricity

Abstract

Scalar fields play an important role in constructing modified gravity theories. In the case of a single scalar field with timelike gradient, the corresponding Lagrangian in the unitary gauge takes the form of spatially covariant gravity (SCG), which is proved useful in analyzing and extending the generally covariant theories. In this work, we apply the SCG method to the scalar-nonmetricity theory, of which the Lagrangian is built of the nonmetricity tensor and a scalar field. We derive the 3+1 decomposition of the geometric quantities and especially covariant derivatives of the scalar field up to the third order in the presence of a nonvanishing nonmetricity tensor. By fixing the unitary gauge, the resulting Lagrangian takes the form of a SCG with nonmetricity, in which all the ingredients are spatial tensors. We then exhaust the scalar monomials of SCG with nonmetricity up to $d=3$ with $d$ the total number of derivatives. Since the disformation tensor plays as an auxiliary variable, we take the Lagrangian with $d=2$ as an example to show that after solving the disformation tensor, we can get an effective SCG theory for the metric variables but with modified coefficients. Our results provides a novel approach to extending the scalar-nonmetricity theory.

26 pages, no figure; v3: match epjc version

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• Scalar induced gravitational waves in metric teleparallel gravity with the Nieh-Yan term
• Constraining parity and Lorentz violations in gravity with future ground- and space-based gravitational wave detectors
• Scalar-induced gravitational waves in spatially covariant gravity
• Parity-violating corrections to the orbital precession of binary system
• Degrees of freedom of a quadratic scalar-nonmetricity theory
• Slowly rotating black hole in chiral scalar-tensor theory
• Spatially covariant gravity with two degrees of freedom in the presence of an auxiliary scalar field: Hamiltonian analysis