Gravitation and cosmology in generalized (1 + 1)-dimensional dilaton gravity

TL;DR

The paper generalizes (1+1)-dimensional dilaton gravity actions, including “R = T” and string-inspired theories, into a single action characterized by two functions \(H(\Phi)\) and \(D(\Phi)\). It studies two possible interpretations of the matter stress-energy tensor, yielding type-I and type-II field equations, and analyzes weak-field, post-Newtonian, stellar-structure, and cosmological consequences.

Key findings

• The generalized action \(S=S_g-S_s\) with \(H(\Phi)\), \(D(\Phi)\), and potential \(V\) yields two sets of field equations: type-I from defining \(T_{\mu\nu}\) as the metric variation of the matter action, and type-II under \(H(\Phi)=kD'(\Phi)\), which incorporates the dilaton variation. Type-II includes both the “R = T” and string-inspired theories; type-I and type-II have the same dynamical properties only if condition (10) holds and the dilaton source vanishes.
• In weak-field calculations, the roles of dilaton and matter sources are interchanged: for type-I, the dilaton perturbation depends only on the matter source while the trace of the metric perturbation depends on both dilaton and matter sources; the reverse occurs for type-II.
• Post-Newtonian expansions can be carried to arbitrary high order for both equation types. Stellar equilibrium reduces to Newton’s equation in the Newtonian limit: for type-I when \([D'(0)]^2/H(0)=2\), and for type-II when \(D'(0)=2k\). The case \(H=0\) is not appropriate for modeling one-dimensional stellar structure.
• Cosmological solutions are given for dust (\(\gamma=1\)) and radiation (\(\gamma=2\)) universes, with radiation-filled solutions generally more complicated. For inflationary exponential expansion, type-I requires \(H(\Phi)\) strictly negative; type-II yields \(D(\Phi)\) functions so complicated that the relevant differential equations appear analytically intractable, leaving numerical solutions as the only practical option.

Methods

• Theoretical analysis of a generalized (1+1)-dimensional dilaton gravity action, varying with respect to the metric and dilaton field, and deriving type-I and type-II field equations under different interpretations of the matter stress-energy tensor.
• Weak-field expansion about a background with conformal transformations and harmonic gauge; post-Newtonian expansion in powers of velocity \(v\); static metric and perfect-fluid stress tensor for stellar structure; Robertson-Walker metric with equation of state \(p=(\gamma-1)\rho\) for cosmology, considering \(\gamma=1,2,0\).
• Computations are carried out without restricting \(D\), \(H\), and \(V\) except for the constraint \(D'(\Phi)\neq 0\); Maple V was used for expansions.

Limitations

• The paper states that ambiguity in interpreting the matter action arises from lack of knowledge of the potential density \(V\), and can only be clarified once \(V\) is explicitly specified; many other interpretations are possible, but only type-I and type-II are considered.
• Type-II equations require \(H\neq 0\); if \(H=0\), only type-I equations are available. The paper also notes that \(H=0\) is not appropriate for modeling one-dimensional stellar structure.
• For inflationary type-II solutions, the derived \(D(\Phi)\) functions make the differential equations analytically intractable, so numerical solutions are the only practical option.
• Evident from the text: the analysis is confined to (1+1)-dimensional models, and many results depend on unspecified characteristic functions \(D\) and \(H\), so no direct (3+1)-dimensional observational predictions are established.