Warp Drives and Martel-Poisson charts
Abhishek Chowdhury













e1e-mail: achowdhury@iitbbs.ac.in



School of Basic Sciences 
Indian Institute of Technology Bhubaneswar, 
Jatni, Khurda, Odisha 752050 India 










Warp Drives and Martel–Poisson charts

    Abhishek Chowdhurye1,addr1



    Received: date / Accepted: date
======================================




We extend the construction of Alcubierre–Natário class of warp drives to an infinite class of spacetimes with similar properties. This is achieved by utilising the Martel–Poisson charts which closely resembles the Weak Painlevé–Gullstrand form for various background metrics (Mink, AdS, dS). The highlight of this construction is the non–flat intrinsic metric which in three dimensional spacetimes introduces conical singularities at the origin and in higher dimensions generates a non–zero Ricci scalar for the spatial hypersurfaces away from the origin. We analyse the expansion/contraction of space and (-ve) energy densities associated with this class of warp drives and find interesting deviations due to the global imprints of the conical defects. Several generalizations are also discussed.   







§ INTRODUCTION


The idea of “effective" superluminal travel is of perpetual interest focusing mainly on  “gedanken–experiments" pushing the foundations of general relativity and communication. Like wormhole geometries <cit.>,  warp drive spacetime first proposed by Alcubierre in 1994 <cit.> and then further elaborated upon by Natário in <cit.> allows for arbitrary large velocities within the framework of general relativity. It appears as a “bubble" propagating on some spacetime background such that the observers inside the bubble are in an inertial reference frame requiring no external energy sources to accelerate. These “reaction–less drives" have a severe limitation, sustaining them violates all known energy conditions even for non–relativistic velocities and would probably require exotic matter <cit.>. Another interesting limitation is the lack of control experienced by an observer inside a superluminal warp bubble which has been addressed by the Krasnikov tube <cit.>. 

In this paper we have generalized the construction of Alcubierre like warp drive in three dim Minkowski spacetime by leveraging the Martel–Poisson (MP) family of charts <cit.> of which the Painlevé–Gullstrand (PG) coordinates are a special case. This construction has been extended to AdS_3, dS backgrounds and their conical defect counterparts. A generic key feature is the appearance of a non–flat intrinsic metric with conical singularity which affect the global properties of the spacetime. Another important realization comes from the flow velocities of the background metrics where the expansion and (-ve) energy density of the warp drive is considerably reduced if the drive travels along the background flow velocities. This will be useful in analog gravity setups which imitate the warp metric <cit.> and can be studied experimentally <cit.>. Throughout the paper we have used natural units G=c=1 and metric signature (-,+,+, ⋯). 

The rest of the paper is organized as follows. In section <ref> we review and extend the construction of MP–charts for spacetimes of the form (<ref>) in D dimensions. Interpretation of the flow coordinates are discussed including the special case of PG–coordinates. Section <ref> highlights the conical singularities of spatial hypersurfaces in three dimensional Minkowski and AdS spacetimes and their conical defect cousins. Regularization of the singularities and embedding of the warp drive in these spacetimes are discussed. We conclude in section <ref> with comments on possible explorations to higher dimensions, analogue gravity, quantum effects and optimizations. The paper ends with the relevant references. 



§ MARTEL–POISSON CHARTS


For a large class of static spherically symmetric spacetimes in D≥ 3 dimensions the metric takes the form 

    ds^2=-f d t^2+d r^2/f+r^2 d Ω_(D-2)^2

where d Ω_(D-2)^2 is the line element on an unit (D-2) sphere and f(r)=(1+f̃(r)) [See later sections <ref> and <ref> for generalizations.]. The main idea is to describe the spacetime in terms of radially infalling or outgoing timelike geodesic observers with D–velocities u^μ=dx^μ/dτ=(ṫ,ṙ, 0, ⋯, 0) where τ is the proper time along the geodesic. For static spacetimes, the Killing vector ξ=(∂/∂ t) leads to a conserved energy per unit mass ϵ =-ξ_μ u^μ = f ṫ and utilizing the normalization of the D–velocities u_μ u^μ = -f ṫ^2 + (ṙ^2/f) = -1, we can finally write 

    u_μ =(-ϵ, ∓√(ϵ^2-f)/f, 0, ⋯, 0 )   ,

where the (-ve) and (+ve) signs are for the infalling and outgoing geodesics respectively. So far, we have not established that these D–velocities are indeed geodesics. We demand that this congruence of D–velocities are hypersurface orthogonal to timelike surfaces T(t, r) = constant i.e. u_μ =-(∂ T/∂ x^μ) where T is now the proper time for these free falling (geodesic) observers. Redefining the time coordinate as [Note that T is an exact differential i.e. ∂ϵ/∂ r=∂(√(ϵ^2 -f)/f)/∂ t.]

    dT = ϵ dt ±√(ϵ^2 -f)/f dr   ,

the metric in (<ref>) takes the form 

    ds^2 =-dT^2 +  1/ϵ^2(dr + v(r) dT )^2 +r^2  dΩ_(D-2)^2   ,

where v(r)=±√(ϵ^2-f) is the flow velocity. The coordinate charts are valid and cover regions for which v∈ℝ. The geodesic D–velocities of these Eulerian observers now take the form 

    u^μ =(Ṫ, ṙ, 0, ⋯, 0) = (1, -v, 0, ⋯, 0)   .

Note that except for the case ϵ =1, the induced metric on the hypersurface is not Ricci flat. In fact, as we will see later the scalar curvature and other curvature invariants of the induced metric blows up at r=0. 

At this point we can have two different interpretations for the parameter ϵ which we will now illustrate using the familiar example of Schwarzschild black holes in D dimensions where f(r)=(1-r_h/r^(D-3)), r_h being the location of the horizon and the Eulerian observers are infalling.


  §.§.§ Free fall from finite radius
 
From the D–velocity we have a conservation equation 

    1/2(dr/dT)^2-1/2( r_h/r^(D-3)) =ϵ^2-1/2= E   ,

where E is the total K.E. + P.E. of the observer. We can choose the observers to start at a finite radius r_0 with zero velocity, then E=-1/2r_h/ r_0^(D-3) but the coordinate chart is not valid beyond r>r_0 as v(r) becomes imaginary. However, if the observers start at infinity then E=0 or ϵ=1 which gives the PG–coordinates for the Schwarzschild black holes where the induced metric on T = constant hypersurfaces are flat.



  §.§.§ Free fall with initial velocity

The observers can start at infinity but now with some initial velocity v_∞ w.r.t. time coordinate  t such that 
ϵ=1/√(1-v_∞^2)≥ 1. Again, zero initial velocity gives the PG–coordinates for the Schwarzschild black holes.

There is another interesting limit where we can set v_∞=1 i.e. the speed of light. Scaling T̃ = (T/ϵ) in (<ref>) and then taking the limit ϵ→∞ gives the infalling Eddington–Finkelstein (EF) coordinates <cit.>.

Of course, it is also possible to have finite radius non–zero initial velocity observers but for the remainder of this paper we will focus on the initial velocity interpretation of the parameter ϵ.



§ WARP DRIVE IN VARIOUS SPACETIMES


While the original Alcubierre warp drive is a deformation of the Minkowski spacetime, we will now discuss embedding the drive in various spherically symmetric static spacetimes, see also <cit.>. Our focus will be on the minimal non–trivial dimension D=3 from which it is relatively clear how to generalize for higher dimensions. Three dimensions is also special as it admits Conical defect solutions which change the global properties of the spacetime retaining the local ones. 



 §.§ Minkowski spacetime


The MP–charts for the Minkowski spacetime in three dimensions with a tunable parameter ϵ takes the form,

    ds^2 =-dT^2 +  1/ϵ^2(dr + √(ϵ^2-1)  dT )^2 +r^2 dϕ^2   ,

where the radially infalling geodesic observers have the velocity v_∞=√(1-1/ϵ^2) at spatial infinity.
Though this metric has non–vanishing Christoffel symbols, it is just an artifact of the coordinate choice and the curvature tensors vanish. 


  §.§.§ Conical singularity

It is instructive to analyze the induced metric on the T = constant hypersurfaces,

    ds^2_C_ϵ =1/ϵ^2dr^2 +r^2 dϕ^2    0≤ϕ < 2 π  .

By carrying out the substitutions r= ϵr̃ and ϕ = ϵϕ̃ such that 0≤ϕ̃ < 2 πϵ it is straightforward to realize that this is a metric on a conical space C_ϵ with a deficit angle 2 π(1-ϵ). Like the plane, the cone is  everywhere flat except at the tip where its curvature R is singular. Calculations by means of the standard formulas
of the Riemannian geometry based on existence of tangent spaces would fail to reveal this delta–like singularity. A natural recipe to handle such singularities and their generalizations can be found in <cit.>. To summarize, we consider an embedding of 𝒞_ϵ in a three dimensional Euclidean space with the parametrization x=ϵr̃cos (ϕ̃ / ϵ),   y=ϵr̃sin (ϕ̃ / ϵ),  z=√(|1-ϵ^2|)r̃ defining the conical surface

    z^2-|1-ϵ^2|/ϵ^2(x^2+y^2)=0   ,    z ≥ 0    .

For ϵ≠ 1, there is a singularity at z=0 where we can't introduce a tangent space and calculate the curvature in the usual way.
 
We can however, introduce a regularization by rolling off the cone tip. The simplest one corresponds to changing 𝒞_ϵ to a hyperbolic space with a tunable regularization parameter a,

    z^2-|1-ϵ^2|/ϵ^2(x^2+y^2)=a^2   ,    z ≥ 0, 
    
    d s_H^2=r̃^2+a^2 ϵ^2/r̃^2+a^2 d r̃^2+r̃^2 d ϕ̃^2   .

To evaluate the curvature scalar, we first evaluate the integral curvature on the regularized hyperbolic space in the a→ 0 limit,

    lim _H → C_ϵ∫_H R=4 π(1-ϵ)+∫_C_ϵ -{r̃=0} R    ,

where the second integral is zero. In a more general setting one can embed the conical defect in a non Ricci flat two dimensional manifold and the second integral would collect the curvature around the defect. We note that as only the singular point r̃=0 can give rise to the first term in (<ref>), it is best to introduce a local representation for the curvature on C_ϵ,

    ^(ϵ) R=2(1-ϵ)/ϵδ(r̃)+R   .







  §.§.§ The warp drive

We can embed the Alcubierre warp drive in the infalling Minkowski spacetime background (<ref>). It is crucial to note that this embedding can't be achieved by a coordinate transformation of the warp drive in the usual ϵ =1 charts. Therefore, it represents an infinite class of different embeddings parameterized by ϵ or equivalently v_∞. 

We begin, by writing the metric in the standard ADM form in the Cartesian coordinates (warp drive breaks the spherical symmetry),

    d s^2=-N d t^2+ γ_ij(d x^i +N^i d t)(d x^j +N^j d t)   ,

where N is the lapse, N^i are the shift vectors and γ_ij are the components of the induced metric on the T = constant timelike hypersurfaces. For the metric as in (<ref>), the embedding is as follows,

    N    =1 
    
    N^i    = (1-f(r_s))√((ϵ^2-1)) x^i/r - δ^i_x  f(r_s)  v 
    γ_ij   = δ_ij +(1/ϵ^2-1) x^i x^j/r^2
    γ^ij   = δ^ij +(ϵ^2-1) x^i x^j/r^2  ,

where for three dimensions, r=√(x^2 +y^2) is the radial coordinate, r_s is the distance from the center of the drive which is moving along the (+ve) x–direction with constant velocity v. Here, the form function f(r_s) decides the size and shape of the wrap drive bubble, its exact form is of little consequence (unless one is optimizing) as long as it has the value f(0)=1 at the center of the bubble, f → 0 as r →∞ outside the bubble and drops sharply from 1 to 0 at the wall of the bubble at some distance R from the center of the bubble [One such specific function popular in the literature is 

    f(r_s)=tanh [σ(r_s+R)]-tanh [σ(r_s-R)]/2 tanh (σ R)  ,

where σ has the interpretation of being inversely proportional to the bubble wall thickness and can be taken to be large.]. The choice of shift vectors N^i are inspired by <cit.>, where the flow velocity is that of the warp drive inside the bubble and that of the background spacetime outside the bubble. The Eulerian observers with geodesics u_μ =(-1,0,0) are still in free fall and hence the above metric can be considered as a warp drive different from the Natário type <cit.>. We assume the warp drive to be a test particle moving along the timelike curve x=x_0(T)=vT, regardless of the value of v(T). One can verify that the proper time along this curve equals the coordinate time. Note that here, the induced metric is not flat and as discussed earlier will exhibit conical singularity at the origin. Warp drives with non flat γ_ij have been discussed earlier in <cit.>. As expected, for ϵ=1, we get back the original Alcubierre warp drive <cit.>. 

Few comments on the distance r_s and the induced metric γ_ij are in order. We can always choose r_s to be the coordinate distance r_s=√((x-x_0(T))^2+y^2) from the center of the warp drive. In general, the induced metric γ_ij is not δ_ij, therefore the size & shape of the bubble inside the boundary r_s=R will evolve over time. Even without the warp drive, the metric in (<ref>) has non–zero expansion/contraction and shear for a spacelike patch orthogonal to the congruence of Eulerian geodesic observers [Non zero rotation is not possible as Eulerian geodesics are hypersurface orthogonal.]. Our focus will be on the large x_0 and (ϵ-1) small limit where these background expansion and shear are at check. However, there are other possible choices, either r_s=√(γ_ij(x-x_0(T))^i (x-x_0(T))^j) or r_s=Geodesic(x,x_0(T)) where Geodesic(x,x_0(T)) is the geodesic distance between the center of the warp drive (x_0(T),0) and some point (x,y). Another possible choice for the induced metric is γ_ij = δ_ij +(1-f(r_s))(1/ϵ^2-1) x^i x^j/r^2, an interpolating metric which is δ_ij inside the bubble and γ_ij as  in (<ref>) outside [It is difficult to implement the other choices of r_s with this choice of induced metric as now the definition of r_s depends on r_s itself through the function f(r_s). This puts severe constraints on possible f(r_s), which we checked to be incompatible with the desired properties of f(r_s).]. We will discuss some of these cases in <ref>. 

With our first choice of r_s and the induced metric (<ref>) we shall now discuss the movement of the warp bubble due to the expansion and contraction of the space around it. The space expands behind the warp bubble and contracts in front of it, thereby pushing the bubble forward (and maybe a spaceship inside it) at arbitrary velocity v [The 
 same mechanism also applies to the the expanding universe, wherein
galaxies move faster than light w.r.t. each other due to the expansion of the space itself.]. We shall also discuss the energy density as measured by the Eulerian observers i.e. ρ =T_μνu^μ u^ν where T^μν is the stress energy tensor of the metric and u^μ=(1, -N^x,-N^y) are the 3–velocities.

To evaluate the expansion θ and energy density ρ associated with the warp drive spacetime, we shall compute at first the extrinsic curvature K_μν= ∇_μ u_ν = 1/2ℒ_u P_μν for the T = constant hypersurfaces embedded in the full spacetime. Here, the lapse is N=1, the timelike geodesics u^μ are hypersurface orthogonal and P_μν=g_μν +u_μ u_ν is the projector to the hypersurfaces. The pullback of K_μν to the above hypersurfaces are

    K_ij = 1/2ℒ_u γ_ij   = 1/2( ∂_T -ℒ_N )γ_ij
        = 1/2∂_T γ_ij -1/2(γ_kjD_i N^k + γ_ik D_j N^k )   ,

where D_i are the covariant derivatives w.r.t. the induced metric γ_ij. The expansion is the trace θ =K^i_i. The Einstein tensor G_μν = R_μν - 1/2 R  g_μν along with the Gauss–Codazzi relations which connects curvature tensors of the metric to curvatures on the hypersurface(s) give, 

    ^(2) R+(K^2-K^i j K_i j)= 2  G_μν u^μ u^ν = 16 πρ + 2Λ
    
    D_j K^j_i - D_i K = 8 π p_i   ,

where ρ is the matter energy density and p_i are the matter momentum density and Λ is the cosmological constant. 

For the background metric in (<ref>) without the warp drive, the extrinsic curvature and expansion away from the conical singularity at r=0 are given by [The regularization mentioned in (<ref>) needs modification as now we want ^(2) R+(K^2-K^i j K_i j) to vanish at r=0 as ρ=0. The regularized metric around r=0 with 0≤ϕ <2 π is given by

    ds^2 =-dT^2 +  1/ϵ^2(A(r)   dr + √(ϵ^2-1) B(r)  dT )^2 +r^2 dϕ^2  
    
    A(r) = a^2 ϵ ^4+ r^2/a^2 ϵ ^2+ r^2   
    B(r) = √(ϵ ^2 (1-log (A(r)))-1)/√(ϵ ^2-1)
  .
]

    K_i j= √(ϵ ^2-1)[ -y^2/r^3  x y/r^3;  x y/r^3 -x^2/r^3;          ]  ,   θ = -√(ϵ ^2-1)/r  .

Closed form expressions for the K_ij and ρ can be obtained for the metric with the warp drive centered at x_0. However, we are interested in the limit x_0 ≫ (x-x_0) ≈ y ≈ r_s, where the warp drive is sufficiently far way from the center r=0. We shall also take v_∞≪ 1 for the Eulerian observers. To 𝒪(1/r) and 𝒪(v_∞), the expressions are as follows [Here, ^(2) R =0 on the hypersurfaces, except at r=0. This is not true in D ≥ 4 dimensions, thereby opening up new possibilities to minimize the energy density.]

    θ =     (v+v_∞)(x-x_0)f'(r_s)/r_s -(1-f(r_s)) v_∞/r
        +y^2  v_∞/r_s r f'(r_s) 
    ρ =     -1/16 π [1/2 (v+v_∞)^2   y^2 f'(r_s)^2/r_s^2. 
       . +v  v_∞ (x-x_0) (2  r_s(1-f(r_s)) - y^2 f'(r_s)) f'(r_s)/r_s^2  r ].

At first, we note that for v_∞=0, the expressions falls back to the usual Alcubierre warp drive in Minkowski spacetime. As expected, the zeroth order expressions for both θ and ρ have the combination (v+v_∞) for velocity. In these river flow coordinates, the background metric is flowing like a fluid along the radially inward direction. However, the wrap drive is flowing against this “fluid” in the (+ve) x–direction, i.e. it is flowing with velocity (v+v_∞) w.r.t. the fluid. Therefore, it is best to move along the flow to minimize the peak expansions/contractions and (-ve) energy density. In fact, for a free falling warp drive to zeroth order they are zero. We also note that, the first 𝒪(1/r) term in θ is that of the background metric and maybe subtracted, the other one is proportional to (y/r) and hence very small. By construction f'(r_s) is non zero only along a very small width of the ring r_s=R. Hence, the effects of the warp drive in both θ and ρ are localized around this ring, the boundary of the warp bubble.


  §.§.§ Conical defect backgrounds

Ordinary gravity in three dimensions is special, it is dynamically trivial.  Effects of localized sources are on the global properties whereas the outside metric is flat. For a point source of mass M placed at the origin r=0, the spatial metric is a cone and the full metric is given by 

    ds^2 = -dt^2 +1/α^2 dr^2 + r^2 dϕ^2    0≤ϕ<2 π  ,

where α = 1-4M <cit.> [Owing to Birkhoff's theorem, in D≥4 dimensions, point mass solutions with spherical symmetry are Schwarzschild like. We can however break the spherical symmetry in solutions like the cosmic string or p–branes where the mass density is concentrated along a line or a membrane and the traverse spatial metric is still conical <cit.>.]. We would like α to be positive which restricts the mass M<(1/4). Negative mass or α>1 can be considered as  unphysical. It is instructive to scale t→α t and write the metric as MP–charts with the parameter ϵ,

    ds^2 = -dT^2 + 1/ϵ^2(dr +√((ϵ^2-α^2))  dT)^2 + r^2 dϕ^2  ,

such that v_∞ = α^2√((1-1/ϵ^2) ). Keeping in mind that the charts are invalid for (ϵ^2-α^2)<0, there are two natural choices for the parameter ϵ to realize an embedding of the warp drive in these defect backgrounds. 



  
Case ϵ =1: Though v_∞ is zero, the flow velocity of the background metric is non–zero. However, the spatial metric γ_ij=δ_ij and non–singular at the origin r=0. The expressions for the θ and ρ are as follows,

    θ =    (v+√(1-α^2))(x-x_0)f'(r_s)/r_s
    
     -     (1-f(r_s)) √(1-α^2)/r +√(1-α^2) y^2f'(r_s)/r_s r +𝒪(1/r^2) 
    ρ =     -1/16 π [1/2(v+√(1-α^2))^2f'(r_s)^2/r_s^2 + 𝒪(1/r) ]   .

Here, the first 𝒪(1/r) term in θ is the background metric contribution and may be subtracted. Again the (-ve) energy density is localized around the boundary of the warp bubble, even for the 𝒪(1/r) terms which we omit writing down. 


  
Case ϵ =α:  Here, v_∞= α√((α^2-1)). This is only possible if we allow the defect to be sourced by a (-ve) mass M. This is a particularly simple case where the flow velocity of the background metric vanishes. The expressions for the θ and ρ are as follows,

    θ =     v (x-x_0)f'(r_s)/r_s
    ρ=     -1/16 π [1/2 v^2y^2 f'(r_s)^2/α^2  r_s^2. 
       . +  v^2(x-x_0)y^2(α^2-1)f'(r_s)^2/α^2 r_s^2 r + 𝒪(r^-2)  ]   .

We note that the expansion θ is the same as the original Alcubierre warp drive whereas the (-ve) energy is suppressed by a factor of (1/α^2).


 §.§ AdS spacetime


As discussed in <cit.> the MP–charts for anti–de Sitter space covers only part of the spacetime. The D dimensional metric in global coordinates take the form 

    d s^2=-(1+H^2 r^2) d t^2+d r^2/1+H^2 r^2+r^2 d Ω^2_(D-2)

where H=1/ℓ is the Hubble constant, ℓ is the AdS length scale and Λ=-(D-1)(D-2)/2  ℓ^2 is the cosmological constant. Following the arguments in section <ref> one can construct the MP–charts with parameter ϵ, however now the Eulerian observers are outgoing and travels with initial velocity v_∞ from r=0 with proper time T defined as 

    dT = ϵ dt - √(ϵ^2 -(1+H^2 r^2))/(1+H^2 r^2) dr   .

Our focus is mainly on AdS_3 with the line element

    ds^2 =-dT^2 +  1/ϵ^2(dr - √(ϵ^2 -(1+H^2 r^2)) dT )^2 +r^2  dϕ^2 
    
    0 ≤ r ≤√(ϵ^2-1)/H≡ r_+  ,

where due to the (-ve) Misner–Sharp–Hernandez mass <cit.> of AdS spacetimes the usual ϵ=1,  v_∞=0, PG–coordinates doesn't exist [We note that for de–sitter dS spacetimes, the metric is as in (<ref>) with H^2→ -H^2 and PG–coordinates exists along with the MP–charts covering the full spacetime.].

Here too, the T = constant hypersurfaces exhibit conical singularity at r=0 as discussed in section <ref>. With the assumption that H≪ 1 and ϵ is some reasonable number greater than 1, we can push the  charts to cover large regions of spacetime except for regions near the AdS boundary r→∞. The extrinsic curvature tensor away from the singularity is as follows,

    K_ij = - 
       [    H^2 r^2 (x^2+y^2 ϵ ^2)-y^2 ϵ ^2 (ϵ ^2-1)/r^3 ϵ ^2 √(ϵ^2 -(1+H^2 r^2))              x y (ϵ
       ^2-1) (ϵ ^2-H^2 r^2)/r^3 ϵ ^2 √(ϵ^2 -(1+H^2 r^2));                 x y (ϵ ^2-1) (ϵ ^2-H^2 r^2)/r^3 ϵ ^2 √(ϵ^2 -(1+H^2 r^2)) H^2 r^2 (x^2 ϵ ^2+y^2)-x^2
       ϵ ^2 (ϵ ^2-1)/r^3 ϵ ^2 √(ϵ^2 -(1+H^2 r^2));                                                                          ]
       θ = ϵ^2 -(1+2 H^2 r^2)/r √(ϵ^2 -(1+H^2 r^2))  .

We shall now embed the warp drive centered at x_0 in this background. Closed form expressions for the K_ij and ρ can be obtained but as in the Minkowski case, we are mostly interested in the limit x_0 ≫ (x-x_0) ≈ y ≈ r_s, where the warp drive is sufficiently far way from the singularity at the center r=0. As we have finite range charts, we have to treat the order of limits carefully. The radial distance r in these MP–charts is cutoff at r_+(ϵ, H) therefore it is best to scale it as r= (1-s_r) r_+ such that 0≤ s_r ≤ 1 and then focus on the region near s_r ≈ 0. To 𝒪(s_r) and 𝒪(H), the expansion and energy density are as follows, 

    θ =   (v-√(2(ϵ^2-1) s_r))(x-x_0)f'(r_s)/r_s
        -(1-f(r_s)/√(2)(1/√(s_r)-11/4√(s_r)) H 
        -√(2) y^2 √(s_r)f'(r_s)/r_s H 
    ρ =    -1/16 π [1/2(v-√(2 (ϵ^2-1)s_r))^2 y^2 f'(r_s)^2/r_s^2 ϵ^2.  
        +v^2 y^2(x-x_0)√((ϵ^2-1))f'(r_s)^2/r_s^2 ϵ^2 H 
       . +𝒪(√(s_r)) H +𝒪(H^2) ]   .

Again, we notice the combination (v-√(2(ϵ^2-1) s_r)) appearing at the zeroth order, however for AdS the background flow velocity and the warp drive velocity are both along (+ve) x–direction. The first 𝒪(H) term in (<ref>) is the contribution from the background AdS and maybe subtracted and the second term is very small. Similarly, in (<ref>) the background contribution from the cosmological constant has been subtracted in the 𝒪(H^2) terms. After the appropriate subtractions remaining terms are again proportional to f'(r_s)^# such that the change in θ and ρ are localized around the boundary of the warp bubble. Interestingly, the zeroth order term in the (-ve) energy density is suppressed by (1/ϵ^2) thereby reducing the energy as ϵ>1 [This is also true for the Minkowski space. It is not visible in (<ref>) as we truncated the expressions to 𝒪(v_∞).]. 

The analysis for warp drive embedded in dS spacetime is similar. However, the MP–charts now cover the full spacetime. Even crossing the  apparent horizon at r=(1/H) is regular. To study the warp drive near this horizon, we can use the scaling r=((1-s_r)/H) and follow the steps as discussed above. the results are similar and as expected, nothing special happens because of the horizon.


  §.§.§ Conical defect backgrounds

Similar to the case of conical defect solutions in three dimensional Minkowski space, AdS_3 too have conical defect solutions for a point mass M placed at the origin r=0. The spatial metric is a cone and the full metric is given by

    ds^2 =-(α^2 +r^2H^2)dt^2 +dr^2/(α^2 +r^2H^2) +r^2 dϕ^2   ,

where α =1-4M and 0≤ϕ<2 π <cit.> [ dS_3 is  bit tricky. In all three spacetimes, there is a second source at r=∞. In Minkowski and AdS_3 spacetimes the geodesic distance between the source at r=0 and r=∞ is infinite, so the second source can be ignored but for dS_3 it is finite and contributes.]. The MP–charts with ϵ=α doesn't exist. Therefore, we focus on the ϵ=1,  v_∞=0 charts given by

    ds^2=-dT^2+(dr -√((1-α^2)-r^2H^2) dT)^2 + r^2 dϕ^2 
    
    0≤ r ≤√(1-α^2) H^-1≡^c r_+ .

We note that for physical particles, we have 0<α≤1. We shall now embed the warp drive in this conical AdS_3 background and do a similar analysis as done earlier. The important changes are that the spatial metric is now δ_ij and scaling of the radial coordinate is r=(1-s_r) ^cr_+ such that 0≤ s_r≤ 1 and then focus on the region s_r≈ 0. The expansion θ is as given in (<ref>) with the replacement (ϵ^2-1) → (1-α^2). Parts of the 𝒪(H) not proportional to f'(r_s) are from the background metric and should be subtracted. For the energy density ρ, the zeroth order 𝒪(H^0) terms is the same as in (<ref>) with the replacement (ϵ^2-1) → (1-α^2) and omitting the ϵ^2 in the denominator. From 𝒪(H), terms are different starting at 𝒪(√(s_r)). Again the background contribution from the cosmological constant needs to be subtracted from the 𝒪(H^2) terms. Throughout, we also don't have the (1/ϵ^2) suppression as ϵ=1.



§ DISCUSSION


In this paper we have extended the Alcubierre warp drive spacetime to an infinite class of warp drives utilizing the Martel–Poisson charts for Minkowski and AdS spacetimes and their conical defect cousins in three dimensions. The Minkowski spacetime admits Painlevé–Gullstrand coordinates but for AdS spacetime the usual PG charts vanish and the warp drive spacetimes discussed in this paper are the only possibility. Though we have analyzed the warp drives away from the conical singularities at the origin, the fact that spatial hypersurfaces are cones have global consequences which have been captured in the expressions for the expansion and (-ve) energy densities localized around the warp drive boundary. 

There are several possibilities for generalizations. We can look for warped spherically symmetric background spacetimes 

    ds^2=-e^2Φ d t^2+d r^2/f+r^2 d Ω_(D-2)^2

which admits MP–charts with partial coverage of the spacetime, 

    ds^2=    -dT^2 +e^2Φ/ϵ^2 f(dr +ϵ√(f)/e^Φ√(1-e^2Φ/ϵ^2) dT)^2 
       +r^2 d Ω_(D-2)^2   ,   (1-e^2Φ/ϵ^2)≥ 0   .

Another direction would be to look at higher dimensions D≥ 4 and find spacetimes whose MP–charts has spatial hypersurfaces which near the singularity at the origin takes the form, 

    d s_C^2=    d r̃^2+r̃^2 d ϕ̃^2 
        +∑_i, j=1^D-3(w_i j(θ)+h_i j(θ) r̃^2) d θ^i d θ^j+𝒪(r̃^4)

or generalize the definition of a cone from two to higher dimensions,

    ds_C^2= dr̃^2 +r̃^2 d Ω̃_(D-2)^2   ,

where unless d Ω̃_(D-2)^2 is the metric of the unit sphere S^(D-2), the generalized cone ds_C^2 is singular. For example, MP–charts of various black hole backgrounds would come under its purview. For most of these generalizations, the Ricci scalar of the spatial hypersurface will be non zero thereby opening up new possibilities for the energy density ρ. If we focus on the spherically symmetric cases, the expressions involving the extrinsic curvatures would generalize easily from the three dimensional expressions in the previous sections upto some coefficients. We have to symmetrize the expressions in all spatial coordinates except x, the direction of travel for the warp drive.  

While warp drives are beyond the realm of current experiments, various analog gravity setups imitate the metric <cit.> and can be studied experimentally <cit.>. It might be possible to generalize these setups to accommodate the Martel–Poisson versions of the warp drive. 

Various other properties of the wrap drive are of interest, for example the zero–expansion drives of Natário <cit.>, the formation of event horizon and various quantum effects  <cit.> to optimization of various parameters including the form functions and the idea of a warp bubble inside another bubble <cit.>. We hope to study some of these aspects in our future works. 


The work A.C. is supported by IIT Bhubaneswar Seed Grant SP–103.





§ DIFFERENT CHOICES FOR  AND 


In this appendix we shall explore, two combinations of choices for the warp drive radial coordinate r_s and the spatial induced metric γ_ij as mentioned in section <ref>. Our focus is on the warp drives embedded in the MP–charts of Minkowski spacetime but similar analysis goes through for the conical defect background, AdS_3 & dS_3 spacetimes and their conical defects cousins.


  
Case I :We take r_s=√((x-x_0)^2+y^2) and γ_ij=δ_ij+(1-f(r_s))(1/ϵ^2-1)x^i x^j/r^2. There are two important deviations from the expressions in (<ref>). Firstly, the effect of (1-f(r_s)) term in γ_ij starts to show only at 𝒪(v_∞^2), so if we are truncating to 𝒪(v_∞), all results will match. Secondly, the spatial curvature tensors, particularly the Ricci scalar ^(2) R is not zero at finite r. Its contribution to the energy density ρ starts at order 𝒪(v_∞^2), 

    (y^2-r_s^2)f'(r_s)-r_sy^2f”(r_s)/r_s^3v_∞^2 + 𝒪(1/r)

Similarly, the K_ij in (<ref>) now picks up contributions from the 1/2∂_T γ_ij term, again modifying expressions from 𝒪(v_∞^2) onward. In all, this choice complicates things but after background subtraction θ and ρ are proportional to f'(r_s)^# and f”(r_s)^#, therefore localized around the wrap drive bubble. A lot now depends on the choice of f(r_s), tweaking it might give a (+ve) ^(2)R thereby reducing the overall (-ve) energy density required to operate the drive.


  
Case II :We take r_s=√(γ_ij(x-x_o(T))^i(x-x_0(T))^j) and γ_ij=δ_ij+(1/ϵ^2-1)x^i x^j/r^2. Here, as we are not changing the spatial metric, there is no contribution from ^(2)R, neither 1/2∂_T γ_ij contributes to K_ij. Extra contributions as compared to results in section <ref> are due to the changes in the derivatives of f(r_s) or equivalently r_s w.r.t. (x, y) coordinates. Again, the changes are from 𝒪(v_∞^2) onward and yet again the contributions to θ and ρ are localized to the boundary of the warp drive.







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