Abstract
In this chapter we focus on \(n\)-DBI gravity and uncover its theoretical aspects. To begin with, in Sect. 3.1 we show that solutions of \(n\)-DBI gravity include any solutions of general relativity with a specific property of the curvature and effects of the space-time foliation appear in the solutions which cannot be removed by allowed coordinate transformations. Next, in Sect. 3.2 we reveal the existence of a scalar graviton originated with a preferred space-time foliation based on Dirac’s theory of constrained systems and investigate its possible pathological behaviors. As it turns out, the scalar mode does not propagate and there is no evidence of the pathology related to the scalar graviton such as vanishing lapse, instabilities and strong self-coupling at low energy scales.
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Notes
- 1.
- 2.
For convenience, we have rescaled the scalar fields by the factor of \((4\pi G_4)^{1/2}\).
- 3.
In non-projectable HL gravity, the lapse \(N\) is a Lagrange multiplier as in GR, but the Hamiltonian constraint is second class. Moreover, the time flow of the Hamiltonian constraint yields an additional constraint that depends on the lapse \(N\). Thus, there are 3 second class constraints; the conjugate momentum \(p_N\) of the lapse, the Hamiltonian constraint \({\fancyscript{H}}\), and its time flow \(\dot{\fancyscript{H}}\). Together with 6 first class constraints, the number of degrees of freedom is counted as \((2\times 10-2\times 6-3)/2=2+1/2\). Note, however, that linear perturbations about flat space-time yield a misleading result. There appear four second and two first class constraints, implying incorrectly that the number of scalar degrees of freedom is zero. In projectable HL gravity, there is an additional primary constraint \(\partial _iN=0\) on top of \(p_N=0\). In this case, the time flow of \(p_N=0\) does not yield the Hamiltonian constraint. Instead, it determines the Lagrange multiplier of \(\partial _i N\). Hence \(p_N\) and \(\partial _iN\) are the only second class constraints, and the total physical degrees of freedom is \(2+1\).
- 4.
A more common gauge is to set the shift \(B=0\). We can go from the \(E=0\) to the \(B=0\) gauge by choosing the gauge parameter \(L(t,x)=-B_0(x) t-{1\over 2}B_1(x) t^2\). This yields the conformal mode \(E(t,x)=\varDelta B_0(x) t +{1\over 2}\varDelta B_1(x) t^2\). Note that in either gauge the lapse \(n\) alone only accounts for a half degree of freedom of the scalar graviton.
- 5.
In the full theory, the nonlinear lapse dependence is somewhat obscured by the introduction of the auxiliary field \(e\) which linearizes the lapse dependence.
- 6.
- 7.
Most easily seen by redefining \(\phi \rightarrow \sqrt{\lambda }\phi \).
- 8.
Here we assume that the IR divergences can be properly regularized.
- 9.
As an illustration, consider a simple mechanical model. For canonical kinetic term, a linear time growth is the behavior seen in a flat potential. The flat potential does not indicate an instability and is marginally stable. However, any potentials which are flat to quadratic order all lead to the linear time growth for perturbations in the linearized approximation. The simplest examples are cubic and (convex) quartic potentials. The former is clearly unstable (inverse-squared blow-up), whereas the latter is stable (oscillation).
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Sato, Y. (2014). \(n\)-DBI Gravity. In: Space-Time Foliation in Quantum Gravity. Springer Theses. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54947-5_3
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