[Paper Review] Comment on "Scalar Einstein-Aether theory"

Ted Jacobson, Antony J. Speranza|arXiv (Cornell University)|May 25, 2014

Cosmology and Gravitation Theories6 references4 citations

TL;DR

This comment establishes that the scalar Einstein-aether theory (S-theory) proposed in Ref. [1] is dynamically equivalent to the projectable version of the infrared limit of Hoðava gravity when the scalar potential V(S) is constant. The equivalence arises because S-theory enforces a projectable lapse function N(T), reducing the symmetry to S → S + const, and the theory only matches projectable Hoðava gravity when V(S) is constant, otherwise breaking the symmetry and breaking the equivalence.

ABSTRACT

A recent paper studies a modification of Einstein-aether theory in which the aether vector is restricted, at the level of the action, to be the gradient of a scalar. In this comment we note that this scalar version of Einstein-aether theory is equivalent to the projectable version of the IR limit of Hořava gravity when the potential for the scalar is constant. This provides a generally covariant formulation for projectable Hořava gravity.

Motivation & Objective

• To clarify the dynamical equivalence between scalar Einstein-aether theory (S-theory) and the projectable limit of Hoðava gravity.
• To identify the conditions under which S-theory matches projectable Hoðava gravity, particularly regarding the scalar potential V(S).
• To analyze the role of the unit norm constraint and the implications of the S-shift symmetry in the context of Hoðava gravity.
• To distinguish S-theory from nonprojectable Hoðava gravity and T-theory, where the lapse function depends on a scalar field T.

Proposed method

• The authors compare the action of S-theory, where the aether vector is u_a = ∇_a S with S a timelike unit scalar, to the action of T-theory, where u_a = N∇_a T with N ensuring unit norm.
• They show that S-theory corresponds to T-theory with the additional constraint that N dT = dS, implying N depends only on T, which enforces the projectability condition.
• The unit norm constraint in S-theory is implemented via a Lagrange multiplier λ(∇^a S ∇_a S - 1), rather than by solving for N from the metric.
• The symmetry under dS = N dT is reduced to S → S + const in S-theory, corresponding to the shift symmetry of projectable Hoðava gravity.
• The authors examine the effect of a non-constant potential V(S) on the symmetry and equivalence to projectable Hoðava gravity.
• They use known results from prior work [4,5] to show that nonprojectable Hoðava gravity arises from Einstein-aether theory with hypersurface-orthogonal aether, and that S-theory is a further restriction of this.

Experimental results

Research questions

• RQ1 Is S-theory dynamically equivalent to projectable Hoðava gravity, and under what conditions?
• RQ2 How does the inclusion of a non-constant potential V(S) affect the symmetry and equivalence of S-theory to projectable Hoðava gravity?
• RQ3 What is the role of the scalar S in enforcing the projectability condition on the lapse function?
• RQ4 How does S-theory differ from T-theory and nonprojectable Hoðava gravity in terms of constraints and symmetries?
• RQ5 Why does the S-shift symmetry reduce to a global constant shift when V(S) is constant?

Key findings

• S-theory is dynamically equivalent to the projectable version of the infrared limit of Hoðava gravity when the potential V(S) is constant.
• The projectability condition N = N(T) is enforced in S-theory by requiring dS = N dT, which restricts the lapse function to depend only on the scalar T.
• The S-shift symmetry S → S + const arises from the invariance of dS = N dT under constant shifts, and this symmetry is preserved only when V(S) is constant.
• A non-constant potential V(S) breaks the S-shift symmetry and thus breaks the dynamical equivalence between S-theory and projectable Hoðava gravity.
• The unit norm constraint in S-theory cannot be solved by choosing N from the metric; it must be imposed via a Lagrange multiplier term λ(∇^a S ∇_a S - 1).
• When V(S) is constant, S-theory reduces to a cosmological constant term, and the theory matches projectable Hoðava gravity exactly as formulated in prior work [4].