Graviton Propagation in an Asymmetric Warped Background: Lorentz Violation and the Null Energy Condition

The graviton propagation in an asymmetric background is studied. The background is a configuration in the six-dimensional Salam-Sezgin model, in which a 3-form H-field turned on [JHEP 0910(2009)086]. The compact dimensions form a cylindrical space with branes as boundaries. The background gets asymmetry due to the H-field and violates the Lorentz symmetry. We derive the graviton equation in this background and show that it gets massless mode traveling with superluminal speed. A tower of K-K modes exists with a mass gap. On the other hand, it is known that breaking the Lorentz symmetry on an asymmetric background is constrained by the null energy condition. This no-go theorem doesn't work well in six-dimensional space-times and by this model we provide a counterexample for which the null energy condition is satisfied while the Lorentz symmetry is gravitationally violated.


I. INTRODUCTION
Higher dimensional theories have been at the center of interest in recent decades. They emerge as necessary ingredients of string theory and when utilized with branes and warped compactification, provide great phenomenological implications such as hierarchy problem in field theories [1][2][3] or cosmological constant problem in gravitational theories [4,5]. Embedding our 4-dimensional world as a brane in a higher dimensional spacetime brings us more chance to capture higher dimensions in accessible energy scales in high energy accelerators such as LHC.
Including H field was firstly done in [27] where a static model obtained and searching for a dynamical metric was followed in [28,29]. In [27] an axially symmetric internal space was introduced, where the radial direction was cut by two 4-branes which wrapped over the azimuthal circle. Smeared 3-branes and zero-branes were also introduced to satisfy Israel junction conditions.
Since H is a 3-form, turning it on presumably violates the Lorentz symmetry. Indeed in the presence of the H field, an asymmetry shows up in the metric as the warp factors for time and space are different. This asymmetric warping had been studied before in different models, sometimes known as time warp [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. However, it was shown that the warp factor can be made to be symmetric at the physical brane which restores the Lorentz symmetry on the brane. This is interesting for standard model fields which are confined on the brane, but doesn't save the Lorentz invariance for gravity modes which inherently propagate in all directions including off the brane.
The Lorentz violation is claimed to be one of the most efficient way to explore new physics and important to those who are curious in the relation of gravitational and quantum phenomena [47].
On the other hand, the importance of the model in [27] is bypassing a no-go theorem originated from the null energy condition [39,46,48]. The no-go theorem states that the internal space for any asymmetric warp compactification in D = 6, indeed can not be compact, unless the null energy condition violated. So any Lorentz violation scenario based on higher dimensional gravity is restricted by the no-go theorem [48]. The silence of the no-go theorem in D = 6 dimension makes the model [27] a candidate for the gravitational Lorentz violation in higher dimensions without violating the null energy condition.
In this article, we follow H field model in [27], consider the spatial tensor perturbation of metric and derive gravitational wave equation. The equation is accompanied by boundary conditions at branes. Since it is too complicated to be solved analytically, we perform numerical analysis to find a solution. Results involve the graviton spectrum including a massless mode with a mass gap for higher modes. Positive definiteness of the spectrum indicates the stability of the model as long as tensorial perturbation is concerned. Phenomenologically, finding a massless state with a mass gap is interesting and shows that the effective four dimensional gravity can be obtained in this model.
As expected, the graviton propagation generates an energy-momentum dispersion relation which violates the Lorentz symmetry. Our numerical results show that the phase velocity as c = E/P in some range of energy exceeds the limit 1. This says that while the electromagnetic wave speed is already 1 on the brane-world, the gravitational wave speed limit is over 1 due to the asymmetric warp factor. This is an explicit example of gravitational Lorentz violation while the null energy condition is satisfied.

II. THE SET-UP
In this section we give a brief introduction to the model in [27]. Let us begin by the bosonic part of Salam-Sezgin Lagrangian as where φ, F and H are respectively dilaton, 2 and 3-form fields. The constant g may also be recognized as the cosmological constant. Equation of motion governing each field is obtained as follows, We also take the space-time described by For later convenience we take z = η/l z and θ = θ ′ /l θ . Now (z, θ) are dimensionless cylindrical coordinates and (l z , l θ ) stand for compactification radii of extra dimensions. Inserting the metric ansatz into field equations (2.2), a natural gauge condition for fixing parameter z seems to be w ′ + 3a ′ − v ′ + b ′ = 0 that leads to following solution: where q, λ 3 , λ 4 and z 3 are some constants, and auxiliary functions x and y satisfy with λ being another constant,q = κq/l z andg = gl z /κ are now dimensionless. The absolute value of extra dimension originates from the fact that to avoid any singularity, one needs to cut the geometry, say between 0 and L, then double it to find a periodic solution between −L and L.
where z i 's are integration constants. Notice that solutions to (2.5) in the limitq andg → 0 are x = ±λz + c 1 and y = ±z + c 2 . However the hyperbolic functions in Eqs. (2.6) could not essentially reduce to these limiting solutions, unless the constants z i 's are chosen properly. This can be done by rewriting, for example, e −x in (2.6) as and then taking logarithm of both sides and letq √ 2λ e λz 1 = 1, the limiting solution x = ±λz + c 1 can be achieved asq → 0. In the same way,ge −z 2 = 1. The solutions to Eqs. (2.5) are therefore: that reduces number of independent constants by one.
Introducing boundaries and including absolute value in the solution suggests some branes as delta function singularities which arise as second derivative of absolute values. A suitable configuration of branes in the closed interval [0, L] is [27]: where T p (T Lp ) stands for tension of p−brane located at z = 0(z = L) and tilde denotes density of tension. In this configuration, 4-branes are boundaries of the space and 3 and zero branes are smeared over 4-branes. Inclusion of 3 and zero branes is essential for matching the energy-momentum and the Einstein tensors. The Israel junction conditions then read 1 from which brane tensions can be derived and [f (z)] z 0 is defined as Since metric functions are even function of z and we are working in the interval [0, L], then on The first condition of Eqs. (2.10) gives where α = 1+κ 2q2 1−κ 2q2 . Using (2.8), we then get the following relations between (λ 3 , λ 4 ) and (λ, q): The only remaining constant to be noted is z 3 in (2.4) that is essentially unimportant and can be absorbed by rescaling coordinates. However, we keep this constant for further simplification.

III. THE NULL ENERGY CONDITION
Before study the gravitational perturbation in the above background, it is worth to pause for a while and consider the null energy condition. This condition appears as a constraint for a matter distribution to be physical in the context of classical general relativity. It simply states that for any null vector ξ M , the following inequality holds for the energy momentum tensor, Since ξ is a null vector using the Einstein equation one finds, To be specific, let us choose ξ M = (e −w , e −a , 0, 0, 0, 0), so (3.2) turns to −R 0 0 + R 1 1 ≥ 0. This condition is satisfied in the bulk as in the following [27], where the gauge condition is used. It is easy to verify that the last inequality x ′′ ≥ 0 is true.

IV. SMALL SPACE-TIME FLUCTUATIONS
To understand behaviour of graviton in this space-time, we consider small fluctuations around the background metric. Recalling the Palatini identity, the small fluctuation δg M N implies a variation in Ricci tensor as, to leading order in δ, We take the tensorial fluctuations in the spatial sector on brane, i.e. δg M N = δg ij δ i M δ j N , and also adopt the conventional transverse-traceless gauge in which δg i i = 0 and ∂ i δg i k = 0. The immediate consequence of this gauge is that the last term in (4.1) vanishes identically. The other terms simply show that just the components δR zi and δR ij may be non-zero. Keeping in mind that the background metric depends only on z coordinate, δR zi is obtained to be zero as well. The only remaining possibility is therefore where ≡ g AB ∇ A ∇ B stands for the d'Alembert operator. We now consider the right-hand side of the Einstein equation in (2.2) as and then change the metric tensor as g M N → g M N + δg ij δ i M δ j N to give in which we have used the (ii)−component of Einstein field equations. The only remaining contribution to the energy-momentum tensor to be taken into account is that of branes. Recalling Einstein equation in (2.2) and energy-momentum tensor on the branes (2.9), then δS brane M N is obtained as The branes contribution finally becomes where we used Eqs. (2.10) and definition (2.11). We now gather Eqns. (4.2), (4.5) and (4.8) to get the equation governing fluctuations: Notice that the function a ′′ here should be written as a ′′ sign(z) + 2a ′ δ(z) because of absolute value in its argument. Since we have previously chosen the gauge ∂ z (g zz √ −g) = 0 in fixing coordinate z, the d'Alembertian operator reduces to g M N ∂ M ∂ N which simplifies (4.9). To recast this equation in the form of a Schrödinger-like one, we perform the transformation δg ij = δg ij e −a and take the Fourier decomposition of the form δg ij = exp(iη µνp µ x ν )ψ(z) to get where we used g zz = l 2 z e 2v and defined dimensionless energy E :=Ẽl z and momentum p :=pl z . Also, c(z) := e w−a that is, in terms of metric functions (2.4), (4.11) We now fix z 3 such that c(0) 2 = 1 implying e αλz 3 = 1 −q 2 . This choice also imposes a restriction onq 2 to be smaller than unity for which c 2 (z) > 0. Then we rewrite c 2 (z) as, In eq. (4.10), a factor e 2(λ 3 +λ 4 )z 3 is included in the function e 2v−2w that can be absorbed in E and p by rescaling.
To find boundary conditions, we integrate Eqn. Having found boundary conditions we now proceed to find a solution in the bulk. However, the complication in the potential of (4.10) leads us to numerical methods.
Before restricting ourselves to any special values of constants, it is worth to make sense of dispersion relation by rewrite Eq. (4.10) in the bulk as − ψ ′′ +q(z)ψ =λŵ(z)ψ, (4.14) where we have defined the eigenvalueλ = E 2 , weight functionŵ(z) = e 2v−2w > 0, andq(z) = p 2 c 2ŵ + a ′′ − 3a ′ 2 . The weight function suggests to adopt the normalization of wave function as L 0 ψ ⋆ŵ ψdz = 1, and consequently define the expectation value of a given function f (z) as f := L 0ŵ ψ ⋆ f ψdz. We now multiply Eq. (4.14) by ψ ⋆ , complex conjugate of wave function, and integrate the result from z = 0 to L to obtain the well-known Green's first identitŷ Inserting corresponding quantities and functions in this identity, one finds This is an energy-momentum dispersion relation for which the group velocity v g = dE/dp times the phase velocity v ph = E/p reads as, To find extrema of c 2 (z) defined by (4.12), we notice that this function is strictly increasing meaning that its derivative is positive for all z in the domain 0 ≤ z ≤ L. This observation ensures us that the extrema occur at endpoints z = 0 or z = L. We therefore can safely write min{c 2 } = c 2 (0) = 1 and max{c 2 } = c 2 (L). Inserting these values in the inequality (4.18), it becomes The constantsq and λ here refer to contributions of electric H field and dilation to the dispersion relation while the effect of cosmological constant does not appear explicitly. This relation deter-mines the most speed violation from speed of light for a given set of constants. In particular, the problem becomes non-dispersive if eitherq = 0 or λ = 0, and the r.h.s approaches to infinity for large L limit.
We can now solve the equation (4.10) for dimensionless quantities (z, E, p) and thereafter interpret them as (η/l z ,Ẽl z ,pl z ). To find 6D Planck mass, we integrate over extra dimensions of the action (2.1) as with M 2 (4) := 1 . As a result of this relation, the 6D Planck mass is obtained as M 4 (6) = M 2 (4) /V 2 with M (4) = 2 × 10 18 GeV. We use this relation in the next section.

V. NUMERICAL RESULTS
To solve equation (4.10) numerically, we firstly study the constraints on constants involved. The charge q and coupling constant g seem to be arbitrary everywhere, as expected from a physical point of view. Returning to metric functions (2.4), one finds that a real metric tensor implies that both of λ 3 and λ 4 in (2.13) are real. This condition imposes a constraint on λ as The final constant to be specified is the distance separating the branes, L. Notice that L is not fixed in this model. Instead it is chosen phenomenologically to fit experimental bounds as explained below. It is possible to study this radial mode and its spectrum as well. Since the radion field propagates in the bulk, we expect that its massless mode, if any, violates the Lorentz symmetry.
However in this article we focus on the tensorial perturbation and postpone the radial one for future works.
Once the set of constants (λ,q,g, L) is fixed, for every value of momentum p, boundary conditions are satisfied just for some special values of energy E. Then the mass spectrum of graviton can be obtained by finding energies correspond to zero momentum limit. Especially, the massless graviton is of great interest and it does exist provided the smallest energy approaches to zero when momentum does so. This statement may be considered as a criteria for fixing either L or λ, given other constants.
Among all possible configurations, we are interested in the case that all tensions are nonnegative. As said before, the choice (λ 3 , λ 4 ) = (λ − 3 , λ − 4 ) guarantees that T L0 , T L3 and T 3 are non-negative and consequently null energy conditions are satisfied. Furthermore, the inequality T 4 ≥ 0 reads the following condition which imposes a lower bound ong 2 , provided the left-hand side itself is non-negative that is so if This inequality now implies that α ≥ √ 2 orq 2 ≥ 0.17. We now have two conditions (5.1) and (5.3) on λ that reduces to (5.3). Therefore, the constant λ can be written as for 0 < µ < 1 being a fine-tuned parameter satisfying the criteria above. Finally T L4 can be checked easily to be non-negative where T 4 does so. In this manner we firstly fixq 2 and putg 2 twice of that obtained from equality sign of (5.2) and then search for suitable µ in (5.4). This strategy leaves L unconstrained and the violation of speed from unity, ε, may take every value due to inequality (4.19).
However, there have been reported some constraint on the size of violation of graviton's propagation speed by general relativity tests in solar system and binary pulsar [50] that is about ε ≤ 10 −6 .
Recalling equation (4.19), this upper bound of ε is translated as a constraint on λL. It is easy to check that for small λL ≪ 1, this equation reads c 2 (λL) = 1 + O(λ 2 L 2 ) + ..., in which the ellipses indicates higher orders of λL. Here we change both λ and L under the criteria that massless graviton does exist and the upper bound λL ≈ 10 −3 that gives ε ≈ 10 −6 .
Inserting these values, we chose momentum in the interval [0, 7 × 10 5 ] and changed energy, by the increment δE, from zero to the value satisfying boundary conditions. To be more accurate, the energy increment was chosen in two regimes: δE = 10 −8 for p ∈ [0, 1], δE = 10 −6 for remaining part of interval. Since momentum varies in a wide range of, we used logarithmic scale for momentum.
For each set of constants, the violation from speed of light ( E p − 1) is shown in figure (1). and v g |q2 =0.7 < 2.112 × 10 −5 that is verified by figure (1). The tension of each brane is also shown in the Table (I) which ensures us that null energy conditions are satisfied.
Similar to the case of Randall-Sundrum model, the positive T 4 guarantees that Newtonian gravity can be recovered on the 4−brane located at z = 0. It is also worth to find the mass gap between zero mode and some lowest massive modes that are listed in the following table:   Taking l z ∼ TeV −1 turns the mass spectrum into TeV units.
The appearance of a mass gap would be interesting phenomenologically. To make sense of order of magnitude of energy levels, we notice that the dimensionless factor E here is in factẼl z .
Hence, the energies are of order l −1 z . Taking l z ∼ TeV −1 turns the mass spectrum into TeV units, so phenomenologically consistent with observation bounds on massive gravitons. Notice that we have ignored the θ-direction KK modes of graviton in equation (4.10), so we expect l θ to be much smaller than l z .
As the last quantity we consider 6D Planck mass. The graviton contribution to the Loop corrections to standard model particles gives a bound on the graviton dispersion relation [51]. This loop correction bound depends on M (6) and would be stronger than ε ≤ 10 −6 by the solar system observation, only if M (6) is not far above TeV scale.

VI. CONCLUSION
We have considered the dispersion relation for gravitational wave in the six-dimensional space compactified to 4D, in the presence of dilaton and an electric H field. The dispersion relation seems to depend on the charge and the dilaton coupling constant as well as an additional integration constant to be fine-tuned in the model. We have determined the constant under the condition that the model contains a massless graviton, beside massive modes which are high enough to satisfy experimental bounds. The compactified lengths order of magnitude were chosen such that the graviton dispersion relation to be consistent with direct observations bounds as well as its contribution to the standard model particle propagator loop corrections. Any radial perturbation of the background which may fix the separation of two branes and presumably show a Lorentz violating behavior is left for future studies.
We take two numerical examples and found that the graviton moves at speed of unity for small momenta. As the momentum increases the speed experiences a rapid change and get to a maximum greater than unity, the speed of light. For a large interval of momenta, the speed remains approximately constant at the maximum, and finally it approaches to unity asymptotically. On the other hand, since standard model fields are confined on the brane at z = 0 where c = 1, they don't expertise any Lorentz violating dispersion relation. This model provides an example of asymmetric time warp compactification which presents Lorentz violation for gravitational waves while the standard model fields well behaved with Lorentz symmetry. This is achieved despite of a no-go theorem according to which in D = 6, no compactification with asymmetric time warping exists unless violates the null energy condition. Hereby we presented a model in which the null energy condition is satisfied and the speed limit is exceeded 1 for gravitational waves as a sign of gravitational Lorentz violation. This model can be an example (or candidate) for any situation where the Lorentz violation is interesting either theoretically or experimentally.