2 and matter, a symmetry. Then, it is natural, but not necessary, to restrict the lapse function N to be space-independent, while the shift vector N i and the 3-dimensional metric g$_{ij}$ in general depend on both time and space, where g = det g$_{ij}$ , and This is the projectability condition, and clearly is pre- served by the foliation-preserving diffeomorphisms (1.3). However, due to these restricted diffeomorphisms, one more degree of freedom appears in the gravitational sec- tor - a spin-0 graviton. This is potentially dangerous, and needs to be highly suppressed in the IR regime, in order to be consistent with observations. Similar problems also raise in other modified theories, such as massive gravity [15]. Under the rescaling (1.1), the dynamical variables N, N i and g$_{ij}$ scale as, Here ζ 2 = 1 / 16 πG , and c denotes the speed of light. In the “physical” units, one can set c = 1 [17]. The covariant derivatives and Ricci and Riemann terms are all constructed from the three-metric g$_{ij}$ , while K$_{ij}$ is the extrinsic curvature, Note that in [7, 11], the constant c in the metric (1.2) was absorbed into N , so that there the lapse function scaled as ℓ − $^{2}$. where N$_{i}$ = g$_{ij}$N $^{j}$. The constants ξ, g$_{I}$ ( I = 2 , . . . 8) are coupling constants, and Λ is the cosmological constant. In the IR limit, all the high order curvature terms (with coefficients g$_{I}$ ) drop out, and the total action reduces when ξ = 0 to the Einstein-Hilbert action. So far most of the work on the HL theory has aban- doned the projectability condition but kept the detailed balance [4, 6, 12–14]. One of the main reasons is that the detailed balance condition leads to a very simple ac- tion, and the resulted theory is much easier to deal with, while abandoning projectability condition gives rise to lo- cal rather than global Hamiltonian constraint and energy conservation. However, with detailed balance a scalar field is not UV stable [5], and gravitational perturba- tions in the scalar section have ghosts [1] and are not stable for any given value of the dynamical coupling con- stant λ [16]. In addition, detailed balance also requires a non-zero (negative) cosmological constant, breaks the parity in the purely gravitational sector [17], and makes the perturbations not scale-invariant [18]. Breaking the projectability condition, on the other hand, can cause strong couplings [19] and gives rise to an inconsistency theory [20]. The SVW generalization seems to have the potential to solve the above mentioned problems [24], although it was found that gravitational scalar perturbations either have ghosts (0 ≤ ξ ≤ 2 / 3) or are not stable ( ξ < 0) [7, 25]. In order to avoid ghost instability, one needs to assume ξ ≤ 0. Then, the sound speed c 2 ψ = ξ/ (2 − 3 ξ ) becomes imaginary, which leads to an IR instability. Izumi and Mukohyama showed that this type of instability does not show up if | c$_{ψ}$ | is less than a critical value [26]. It is fair to say, in order to have a viable HL theory, much work needs to be done, and various aspects of the theory ought to be explored, including the renormaliza- tion group flows [27], Vainshtein mechanism [15, 28], so- lar system tests [29], Lorentz violations [30], and its ap- plications to cosmology [7, 11]. To resolve these problems, various modifications have been proposed [21]. In particular, Blas, Pujolas and Sibiryakov (BPS) [22] showed that the strong coupling problem can be solved without projectability condition (in which the lapse function becomes dependent on both t and x $^{i}$), when terms constructed from the 3-vector In this paper, we shall study another important issue in the HL theory - the problem of singularities, which is closely related to the issue of black holes in this the- ory [12]. Although we are initially interested in the case with projectability condition, our conclusions can be equally applied to the HL theory without projectabil- ity condition. The extrinsic curvature K$_{ij}$ and the 3- dimensional Riemann tensor R i jkl are not tensors under the 4-dimensional Lorentz transformations, are included. Contrary claims can be found in [23]. In addition, it is not clear how the inconsistency problem [20] is resolved in such a generalization. On the other hand, Sotiriou, Visser and Weifurtner (SVW) formulated the most general HL theory with pro- jectability but without detailed balance conditions [17]. The total action consists of three parts, kinetic, potential x µ → x ′ µ = ζ $^{µ}$( t, x $^{i}$) . (1.10) As a result, in GR one usually does not use them to construct gauge-invariant quantities. However, in the a$_{i}$ ≡ ∂$_{i}$N N , (1.6) K$_{ij}$ = 1 2 N ( − ˙ g$_{ij}$ + ∇ $_{i}$N$_{j}$ + ∇ $_{j}$N$_{i}$ ) , (1.9) N → N, N i → ℓ − $^{2}$N $^{i}$, g$_{ij}$ → g$_{ij}$. (1.5) L K = 1 c 2 [ K$_{ij}$K ij − (1 − ξ ) K 2 ] , L V = 2Λ − R + 1 ζ 2 ( g$_{2}$R 2 + g$_{3}$R$_{ij}$R $^{ij}$) + 1 ζ 4 ( g$_{4}$R 3 + g$_{5}$R R$_{ij}$R ij + g$_{6}$R i $_{j}$R j $_{k}$R k i ) + 1 ζ 4 [ g$_{7}$R ∇ $^{2}$R + g$_{8}$ ( ∇ $_{i}$R$_{jk}$ ) ( ∇ $^{i}$R$^{jk}$)] . (1.8) N = N ( t ) , N i = N $^{i}$( t, x ) , g$_{ij}$ = g$_{ij}$ ( t, x ) . (1.4) S = ζ 2 ∫ dtd $^{3}$xN √ g ( L K − L V + ζ − $^{2}$L M ) , (1.7)