Warp drives and Martel–Poisson charts

Warp drives and Martel–Poisson charts

Abhishek Chowdhury — School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Jatni, Khurda, Odisha 752050, India.

The European Physical Journal C (2025) 85:112. Regular Article — Theoretical Physics. Received 8 July 2024; accepted 19 January 2025.

(This paper is a chain of tensor equations that did not survive text extraction cleanly. The displayed equations are given here as named results and in words; the exact forms, with every equation number, are at the source. Inequalities are written out in words. Throughout, natural units with the gravitational constant and the speed of light set to one are used, and the metric signature is minus, plus, plus, and so on.)

Abstract

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 introduce conical singularities at the origin and in higher dimensions generates non-zero Ricci scalar for the spatial hypersurfaces away from the origin. We analyse the expansion and contraction of space and the NEC violations associated with these warp drives and find interesting scalings due to the global imprints of the conical defects. Other properties like tilting of light cones, event horizons and several generalisations are also discussed.

1 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, warp drive spacetime first proposed by Alcubierre in 1994 and then further elaborated upon by Natário 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. 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.

In this paper we have generalised the construction of Alcubierre–Natário like warp drive in three dimensional Minkowski spacetime by leveraging the Martel–Poisson (MP) family of charts of which the Painlevé–Gullstrand (PG) coordinates are a special case. This construction has been extended to AdS3 and dS backgrounds along with their conical defect counterparts. A generic key feature is the appearance of non-flat intrinsic metric with conical singularities which affect the global properties of the spacetimes. Another important realisation comes from the flow velocities of the background metrics where the expansion and negative 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 and can be studied experimentally.

The rest of the paper is organised as follows. In Sect. 2 we review and extend the construction of MP-charts for spacetimes of the standard static spherically symmetric form in D dimensions. Interpretation of the flow coordinates are discussed including the special case of PG-coordinates. Section 3 highlights the conical singularities of spatial hypersurfaces in three dimensional Minkowski and AdS spacetimes and their conical defect cousins. Regularisation of the singularities and embedding of the warp drive in these spacetimes are discussed. Various properties, like tilting of light cones, event horizons and the violation of NEC along with total negative energy estimates are explored. We conclude in Sect. 4 with comments on possible explorations to higher dimensions, analogue gravity, quantum effects and optimisations. The paper ends with the relevant references.

2 Martel–Poisson charts

For a large class of static spherically symmetric spacetimes in D of three or more dimensions the metric takes the standard static form: minus the metric function times the square of the time element, plus the radial element squared divided by that same metric function, plus the radius squared times the line element on a unit sphere of dimension D minus two.

The main idea is to describe the spacetime in terms of radially infalling or outgoing timelike geodesic observers whose D-velocity is the derivative of position with respect to the proper time along the geodesic. For static spacetimes, the Killing vector associated with time translation leads to a conserved energy per unit mass — this is the chart parameter of the construction — and utilising the normalisation of the D-velocities we can write the velocity components explicitly, with the negative and positive signs standing 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 spacelike surfaces of constant T, so that the D-velocity is minus the gradient of T, where T is now the proper time for these free falling (geodesic) observers. Redefining the time coordinate accordingly, the metric takes the flow form: minus the square of the proper-time element, plus one over the chart parameter squared times the square of the radial element added to the flow velocity times the proper-time element, plus the radius squared times the line element on the unit sphere. The flow velocity is the square root of the chart parameter squared minus the metric function. The coordinate charts are valid and cover regions for which the flow velocity is real. The geodesic D-velocities of these Eulerian observers now take the simple form of unit time component and minus the flow velocity in the radial direction.

Note that except for the case where the chart parameter equals one, the induced metric on the hypersurface is not Ricci flat. In fact, as we shall see later the scalar curvature and other curvature invariants of the induced metric blows up at the origin.

At this point we can have two different interpretations for the chart parameter which we will now illustrate using the familiar example of Schwarzschild black holes in D dimensions, where the metric function is one minus the horizon radius divided by the radial coordinate raised to the power D minus three, and the Eulerian observers are infalling.

Free fall from finite radius. From the D-velocity we have a conservation equation whose right-hand side is the total kinetic plus potential energy of the observer. We can choose the observers to start at a finite radius with zero velocity, which fixes that energy, but the coordinate chart is not valid beyond that radius as the flow velocity becomes imaginary. However, if the observers start at infinity then the energy is zero, equivalently the chart parameter is one, which gives the PG-coordinates for the Schwarzschild black holes where the induced metric on constant-T hypersurfaces is flat.

Free fall with initial velocity. The observers can start at infinity but now with some initial velocity with respect to the time coordinate, such that the chart parameter is the corresponding Lorentz factor, which is one or greater. Again, zero initial velocity gives the PG-coordinates for the Schwarzschild black holes. There is another interesting limit where we can set the initial velocity to one, that is the speed of light. Rescaling the proper time and then taking the chart parameter to infinity gives the infalling Eddington–Finkelstein coordinates. Of course, it is also possible to have finite radius non-zero initial velocity observers but for the remainder of this paper we shall focus on the initial velocity interpretation for the chart parameter.

3 Warp drive in various spacetimes

While the original Alcubierre warp drive is a deformation of the Minkowski spacetime, we shall now discuss embedding the drive in various spherically symmetric static spacetimes. Our focus will be on the minimal non-trivial dimension of three, from which it is relatively clear how to generalise for higher dimensions. Three dimensions is also special as it admits conical defect solutions which change the global properties of the spacetime while retaining the local ones.

3.1 Minkowski spacetime

The MP-charts for the Minkowski spacetime in three dimensions with a tuneable chart parameter take the flow form given above, where the radially infalling geodesic observers have a velocity at spatial infinity equal to the square root of one minus the reciprocal of the chart parameter squared. Though this metric has non-vanishing Christoffel symbols, it is just an artefact of the coordinate choice and the curvature tensors vanish.

Conical singularity

It is instructive to analyse the induced metric on the constant-T hypersurfaces: one over the chart parameter squared times the radial element squared, plus the radius squared times the angular element squared, with the angle running from zero to two pi.

By carrying out a rescaling of the radius and the angle it is straightforward to realise that this is a metric on a conical space with a deficit angle of two pi times one minus the chart parameter. Like the plane, the cone is everywhere flat except at the tip where its curvature is singular. Calculations by means of the standard formulas of the Riemannian geometry based on the existence of tangent spaces would fail to reveal this delta-like singularity. A natural recipe to handle such singularities and their generalisations can be found in the literature. To summarise, we consider an embedding of the cone in a three dimensional Euclidean space with an explicit parametrisation defining the conical surface. For any chart parameter other than one, there is a singularity at the tip where we cannot introduce a tangent space and calculate the curvature in the usual way.

We can however, introduce a regularisation by rolling off the cone tip. The simplest one corresponds to changing the cone to a hyperbolic space with a tuneable regularisation parameter. To evaluate the curvature scalar, we first evaluate the integral curvature on the regularised hyperbolic space in the limit where the regularisation parameter goes to zero: the result is four pi times one minus the chart parameter, plus the integral of the curvature over the cone away from the tip, and that 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 since only the singular point at the tip can give rise to the first term, it is best to introduce a local representation for the curvature on the cone, in which the singular piece is a delta function at the origin weighted by twice one minus the chart parameter divided by the chart parameter.

The warp drive

We can embed the Alcubierre warp drive in the infalling Minkowski spacetime background. In the literature, various warp drives are shown to be related to each other by coordinate transformations. It is crucial to note that the embedding discussed in this paper is not a coordinate transformation of the warp drive in the usual charts with chart parameter one. Therefore, it represents an infinite class of different embeddings parameterised by the chart parameter, or equivalently by the velocity at infinity.

We begin by writing the metric in the standard ADM form in the Cartesian coordinates, since the warp drive breaks the spherical symmetry: minus the lapse times the square of the proper-time element, plus the induced spatial metric contracted with the coordinate elements each shifted by the shift vector times the proper-time element. Here the lapse, the shift vectors and the components of the induced metric on the constant-T spacelike hypersurfaces are given explicitly in the paper. For the MP-chart metric the lapse is one; the shift vector is built from the background flow velocity outside the bubble and the warp velocity inside it; and the induced metric is the flat metric plus a term proportional to the outer product of the radial direction with itself, weighted by one over the chart parameter squared minus one.

Here the radial coordinate is the usual distance from the origin, and the bubble coordinate is the distance from the centre of the drive, which is moving along the positive x-direction with constant velocity. The form function decides the size and shape of the warp drive bubble; its exact form is of little consequence — unless one is optimising — as long as it has the value one at the centre of the bubble, tends to zero far outside the bubble, and drops sharply from one to zero at the wall of the bubble at some distance R from the centre. One such specific function popular in the literature is built from the difference of two hyperbolic tangents, normalised by twice the hyperbolic tangent of the wall parameter, where that parameter is inversely proportional to the bubble wall thickness and can be taken to be large.

The choice of shift vectors is inspired by earlier work, 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 are still in free fall and observe no time dilation. Hence, the above metric can be considered as a warp drive different from the Natário type. We assume the warp drive to be a test particle moving along the timelike curve whose position is the warp velocity times the proper time, regardless of the value of the flow velocity. 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 induced metrics have been discussed earlier. As expected, for chart parameter equal to one, we get back the original Alcubierre warp drive.

For the MP-chart Minkowski metric we see that the light cones in the radial direction are tilted by an amount proportional to the velocity at infinity. This carries over to the generic form of the warp drives, only now the light cone along the x-direction has the tilt which can be a complicated function of the spatial metric and the lapse vectors, but at large radius takes a simplified form combining the form function, the velocity at infinity and the warp velocity. The warp drive travels along a timelike path inside these locally tilted light cones without violating anything in relativity, as the bubble is just a specific choice of the curvature of spacetime itself and not a massive object obeying the geodesic equation. Globally, the path taken by the warp drive can be spacelike for high enough warp velocity.

We can also analyse the event horizons for these warp drives in a similar fashion. In order to modify the geometry of the warp bubble from inside, that is to accelerate or to stop the spaceship, the crew must be able to causally influence the bubble wall in front of them by sending say a photon forward towards the bubble wall. In the reference frame of an observer on the spaceship the forward photon has a definite trajectory. At the centre of the bubble the form function is one and the photon travels with the speed of light, but if the combination of the velocity at infinity and the warp velocity exceeds one, then at some point inside the bubble the photon stops and remains forever at that point. Thus, there is a horizon there such that the outer edge of the bubble is outside the causal future of the spaceship. A warp drive beyond a certain velocity cannot be controlled, though in principle it is possible that the entire trajectory of the bubble may be constructed beforehand. It is relatively straightforward to extend these arguments to the conical defect backgrounds and the warp drives in AdS and dS spacetimes discussed in the later sections.

A few comments on the bubble distance and the induced metric are in order. We can always choose the bubble distance to be the coordinate distance from the centre of the warp drive. In general, the induced metric is not the flat metric, therefore the size and shape of the bubble inside its boundary will evolve over time. Even without the warp drive, the MP-chart metric has non-zero expansion and 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 limit of large drive displacement and small departure of the chart parameter from one, where these background expansion and shear are held in check. However, there are other choices: the bubble distance measured with the induced metric, or the geodesic distance between the centre of the warp drive and the field point. Another possible choice for the induced metric is an interpolating metric which is flat inside the bubble and takes the chart form outside. We will discuss some of these cases in Appendix A.

With our first choice of bubble distance and induced metric 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. The same mechanism also applies to the expanding universe, wherein galaxies move faster than light with respect to each other due to the expansion of the space itself. We shall also discuss the energy density as measured by the Eulerian observers, that is the stress–energy tensor contracted twice with their velocity.

To evaluate the expansion and energy density associated with the warp drive spacetime, we shall compute at first the extrinsic curvature for the constant-T hypersurfaces embedded in the full spacetime. Here, the lapse is one, the timelike geodesics are hypersurface orthogonal, and the projector to the hypersurfaces is the metric plus the outer product of the velocity with itself. The pullback of the extrinsic curvature to the above hypersurfaces is one half the Lie derivative of the induced metric along the observer congruence, which expands into a proper-time derivative term and two shift-vector terms built with the covariant derivatives of the induced metric. The expansion is the trace of the extrinsic curvature. The Einstein tensor along with the Gauss–Codazzi relations, which connect curvature tensors of the metric to curvatures on the hypersurfaces, give the Hamiltonian constraint — the intrinsic Ricci scalar plus the square of the trace of the extrinsic curvature minus the square of the extrinsic curvature equals sixteen pi times the matter energy density plus twice the cosmological constant — and the momentum constraint, which sets the divergence of the extrinsic curvature equal to eight pi times the matter momentum density.

For the background metric without the warp drive, the extrinsic curvature and expansion away from the conical singularity at the origin are given in closed form; the expansion is minus the square root of the chart parameter squared minus one, divided by the radius.

Closed form expressions for the extrinsic curvature and the energy density can be obtained for the metric with the warp drive centred at a given point. However, we are interested in the limit where the warp drive is sufficiently far away from the centre, and we shall also take the velocity at infinity to be small for the Eulerian observers. To first order in the reciprocal radius and in the velocity at infinity, the expansion and the energy density are given explicitly in the paper. The regularisation of the cone tip needs modification here, because we now want the Hamiltonian constraint combination to vanish at the origin, where both the energy density and the cosmological constant are zero; the regularised metric around the origin is given in the paper's footnote with two explicit profile functions. Note also that in three dimensions the intrinsic Ricci scalar vanishes on the hypersurfaces except at the origin. This is not true in four or more dimensions, thereby opening up new possibilities to minimise the energy density.

At first, we note that for zero velocity at infinity, the expressions fall back to the usual Alcubierre warp drive in the Minkowski spacetime. As expected, the leading expressions for both the expansion and the energy density carry the sum of the warp velocity and the velocity at infinity. In these river flow coordinates, the background metric is flowing like a fluid along the radially inward direction. However, the warp drive is flowing against this "fluid" in the positive x-direction, that is, it is flowing with the sum of the two velocities with respect to the fluid. Therefore, it is best to move along the flow to minimise the peak expansions and contractions and the negative energy density. In fact, for a freely falling warp drive to zeroth order they are zero but pick up a negative contribution at second order in the reciprocal radius.

We also note that the first reciprocal-radius term in the expansion is that of the background metric and may be subtracted; the other one is proportional to the ratio of the transverse coordinate to the radius and hence very small. As the derivative of the form function is negative, behind the bubble the expansion is positive and the space is expanding, while at the front the expansion is negative and the space is contracting. By construction the form function varies only along a very small width of the ring at the bubble radius. Hence, the effects of the warp drive in both the expansion and the energy density are localised 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 localised sources are on the global properties whereas the outside metric is flat. For a point source of mass M placed at the origin, the spatial metric is a cone and the full metric is the flat metric with the radial part divided by the square of a defect parameter equal to one minus four times the mass. We would like that parameter to be positive, which restricts the mass to less than one quarter. Negative mass, or a defect parameter greater than one, can be considered as unphysical. It is instructive to rescale the time and write the metric as MP-charts with the chart parameter, in which case the velocity at infinity is fixed by the chart parameter and the defect parameter together. Keeping in mind that the charts are invalid when the chart parameter squared falls below the defect parameter squared, there are two natural choices for the chart parameter to realise an embedding of the warp drive in these defect backgrounds.

Case: chart parameter equal to one. Though the velocity at infinity is zero, the flow velocity of the background metric is non-zero. However, the spatial metric is flat and non-singular at the origin. The expansion and the energy density then carry the warp velocity in combination with the square root of one minus the defect parameter squared. Here, the first reciprocal-radius term in the expansion is the background metric contribution and may be subtracted. Again the negative energy density is localised around the boundary of the warp bubble, even for the reciprocal-radius terms which we omit writing down.

We also note that in this conical defect background the light cone tilt is modified, and the forward horizon is now controlled by a modified equation. At some point inside the bubble a horizon develops even for warp velocities below one — specifically for warp velocities at or above one minus the square root of one minus the defect parameter squared. This should be contrasted with the original Alcubierre warp drive in Minkowski space where the horizon forms only when the warp velocity reaches one.

Case: chart parameter equal to the defect parameter. Here the velocity at infinity is set by the defect parameter, and this is only possible if we allow the defect to be sourced by a negative mass. This is a particularly simple case where the flow velocity of the background metric vanishes. We note that the expansion is the same as the original Alcubierre warp drive, whereas the negative energy is suppressed by a factor of one over the defect parameter squared. In this case the light cone tilt is set by the defect parameter times the warp velocity times the form function, and a horizon forms for all warp velocities at or above the defect parameter.

Energy considerations

So far, we have shown that the warp drives in the three dimensional Minkowski spacetime and its conical defect cousins in MP-charts carry negative energy densities localised around the warp bubble, which means they violate the weak energy condition (WEC). Matt Visser and collaborators argued that for almost all Natário type warp drives discussed in the literature with a generic line element of the standard flow form, including the ones that claim positive energy densities observed by the Eulerian observers, either the strong energy condition (SEC), the dominant energy condition (DEC), the WEC or the null energy condition (NEC) is violated for a generic observer. However, their analysis does not cover the warp drives discussed in this paper as the spatial metric is not flat. In general, the violation of the energy conditions follows the linkage: NEC implies WEC implies DEC, and NEC implies SEC. As we shall see, for the warp drives discussed here, NEC is violated which leads to the violation of the other energy conditions.

In order to explore the violation to the NEC it is best to work with the natural co-moving orthonormal frame (triad) attached to the Eulerian observers of the warp drive metric with unit lapse. For the spatial metric, we can choose the spatial dyad such that the induced metric is the flat metric in the dyad basis. Then the triads for the full metric are given explicitly in the paper.

Usually it is assumed that the stress–energy tensor is of the Hawking–Ellis type I, which means it is diagonal in an orthonormal basis for the observer. Unfortunately, for the warp drive solutions we do not know a priori whether or not the stress–energy is Hawking–Ellis type I in general, nor do the Eulerian observers diagonalise the stress–energy tensor. This requires modification to the usual approach to testing energy conditions.

The null energy condition demands that for all null vectors, the stress–energy tensor contracted twice with the null vector is non-negative. In the orthonormal frame, let us take any two oppositely oriented null vectors built from an arbitrary unit spatial two-vector. Then the NEC would demand two conditions, which upon averaging imply the following one-way relation: the energy density plus the spatial stress contracted with the unit vector twice is non-negative. For a dyad of two mutually orthogonal unit vectors, the NEC implies the same for each. Noting that by construction the sum over the dyad of the outer products is the flat spatial metric, and averaging over the two members of the dyad, we get the energy density plus one half the trace of the spatial stress is non-negative.

Even if the two-dimensional stress tensor is not diagonal, we can define the average pressure as usual, one half the trace of the spatial stress. Finally, we have the one-way relation: NEC implies the energy density plus the average pressure is non-negative. Similarly, for the other energy conditions the implications are:

• WEC: the energy density is non-negative, and the energy density plus the average pressure is non-negative, coming from the demand that the stress–energy contracted twice with any timelike vector is non-negative.
• SEC: the energy density plus twice the average pressure is non-negative, and the energy density plus the average pressure is non-negative, coming from the demand on the trace-reversed stress–energy tensor contracted twice with any timelike vector.
• DEC: the magnitude of the average pressure is at most the energy density, coming from the demand that the stress–energy contracted with any two future-pointing timelike vectors is non-negative.

There are other energy conditions like the trace energy condition and the flux energy condition, and various weaker average energy conditions of which the averaged null energy condition is widely applicable.

To compute the average pressure for the warp drives discussed in this paper we can use the Hamiltonian constraint along with the other Gauss–Codazzi relations for the spatial Ricci tensor, the time–time Ricci component and the mixed component, to evaluate the Einstein tensor and hence the stress tensor. For the warp drive described by the ADM metric with the lapse, shift and induced metric given above, the energy density is already evaluated. To check for the violation of NEC to the leading order in radius, we note that the form function is a monotonically decreasing function with negative first and second derivatives, taking values between zero and one. Therefore, NEC is violated for the MP-chart warp drives: the sum of the energy density and the average pressure is a manifestly non-positive expression, given in full in the paper, built from the warp velocity plus the square root of the chart parameter squared minus one, the transverse coordinate, and the first and second derivatives of the form function.

We can do similar exercises for the warp drives embedded in the conical defect backgrounds; again NEC is violated as expected. The NEC violation for these warp drive solutions leads to violations of other energy conditions, WEC, DEC and SEC.

We now turn our focus to give a reasonable estimate of the total energy measured by the Eulerian observers. For the MP-chart warp drives, shifting the origin to the location of the warp drive and to the leading order in radius, the total energy is given by integrating the energy density over a spatial slice. The result is negative and depends on the choice of the form function. Instead of choosing the usual hyperbolic-tangent form function, we shall use the piecewise-continuous form function suggested by Pfenning and Ford as it provides better estimates for the total energy: it is one inside the bubble radius less half the wall thickness, falls linearly across the wall, and is zero beyond the bubble radius plus half the wall thickness. Plugging this in, the total energy is negative and proportional to the square of the warp velocity plus the square root of the chart parameter squared minus one, divided by the chart parameter squared, times the ratio of the bubble radius to the wall thickness.

We can choose a reasonable size for the bubble, say a radius of 100 metres, and make a choice for the background flow velocity, but the crucial ingredient is to estimate the maximum thickness of the bubble wall so as to reduce the overall negative energy to its minimum.

The restrictions to the thickness of the bubble can be obtained from quantum inequalities (QI) which do allow negative energies but place serious limitations on its magnitude and duration. It would be very difficult to quantise a scalar field on a warp drive background and arrive at the exact quantum inequality, but up to corrections in the inverse powers of the local radius of curvature, the flat spacetime inequality can be applied to curved spacetimes with the additional restriction that the negative energy be sampled on timescales smaller than the minimum local radius of curvature. Most of the literature on quantum inequalities is in four dimensions but the warp drives in this paper are in three dimensions. Redoing the computations, we arrive at the three-dimensional bound: the sampled energy density, integrated against a Lorentzian sampling function of width equal to the sampling time, is bounded below by minus one over sixteen pi times the cube of the sampling time.

For the MP-chart warp drives in consideration we notice that the largest component of the Riemann tensor is proportional to the square of the warp velocity plus the square root of the chart parameter squared minus one, which yields a minimum radius of curvature near the boundary of the bubble. As mentioned earlier, the sampling time must be smaller than this length scale, so we take the sampling time to be a small fraction of it, choosing that fraction to be one tenth. Following the steps of Pfenning and Ford, the quantum inequality puts a bound on the thickness of the bubble: the wall thickness is at most of order ten to the third Planck lengths, times the warp velocity plus the square root of the chart parameter squared minus one, divided by the chart parameter. In three dimensions the Planck length is fixed by the three-dimensional Newton constant; if we assume that we reach three dimensions by compactifying one spatial direction of order the Planck length in four dimensions, the Planck length is 1.6 by ten to the minus thirty-five metres.

We can assume the mass of a Milky-way galaxy in a three dimensional universe to be two thirds of the actual Milky-way, that is 1.3 by ten to the thirty-third Planck masses. Putting it all together, the total energy is bounded above by minus three Milky-way masses times the warp velocity plus the square root of the chart parameter squared minus one, divided by the chart parameter. It is about ten to the twentieth times better than the four dimensional result but still a stupendously large amount of negative energy is required to operate the warp drive in three dimensions.

The energy considerations for the warp drives embedded in the conical defect backgrounds are similar to the above analysis. If we compare with the original Alcubierre warp drive in three dimensions, the energy density and total energy of the conical defect warp drives with chart parameter one scale by a factor of at least one, assuming that the warp drive is travelling outwards. This is expected as the conical defect background has a flow inwards towards the origin. Similarly, for the other case where the chart parameter equals the defect parameter, the suppressing ratio is one over the defect parameter squared, which is at most one — that is, it has the potential to lower the energy requirements, but unfortunately it also requires negative mass to create the defect itself. The analysis also goes through for the warp drives embedded in the AdS and dS spacetimes and their conical defect cousins discussed in the next section in the small Hubble constant limit.

The finite Hubble constant computations are challenging. Firstly the metric is complicated in the Cartesian coordinates which is necessary to correctly embed the warp drives, and secondly the quantum inequalities for AdS and dS spaces would require quantising scalar fields in curved spacetimes. We shall attempt a more complete analysis for these cases in a future work.

3.2 AdS spacetime

The MP-charts for anti-de Sitter space cover only part of the spacetime. The D dimensional metric in global coordinates takes the standard static form with metric function one plus the Hubble constant squared times the radius squared, where the Hubble constant is the reciprocal of the AdS length scale and the cosmological constant is negative. Following the arguments in Sect. 2 one can construct the MP-charts with the chart parameter; however now the Eulerian observers are outgoing and travel with the velocity at infinity from the origin.

Our focus is mainly on three-dimensional AdS, where the radial coordinate is bounded above by the square root of the chart parameter squared minus one, divided by the Hubble constant. Here, due to the negative Misner–Sharp–Hernandez mass of AdS spacetimes, the usual PG-coordinates with chart parameter one and zero velocity at infinity do not exist. We note that for de Sitter spacetimes the metric is the same with the sign of the Hubble constant squared reversed, and PG-coordinates exist along with the MP-charts covering the full spacetime.

Here too, the constant-T hypersurfaces exhibit conical singularity at the origin as discussed in Sect. 3.1. With the assumption that the Hubble constant is small and the chart parameter is some reasonable number greater than one, we can push the charts to cover large regions of spacetime except for regions near the AdS boundary. The extrinsic curvature tensor away from the singularity, and the expansion, are given explicitly in the paper.

We shall now embed the warp drive centred at a given point in this background. Closed form expressions for the extrinsic curvature and the energy density can be obtained but as in the Minkowski case, we are mostly interested in the limit where the warp drive is sufficiently far away from the singularity at the centre. As we have finite range charts, we have to treat the order of limits carefully. The radial distance in these MP-charts is cut off at the maximum radius, therefore it is best to scale it and then focus on the region near that cutoff. To leading order in the scaled radius and in the Hubble constant, the expansion and energy density are given explicitly in the paper. Very near to the MP-charts cutoff, the light cone tilt and the location of the forward photon horizon take simple forms in terms of the warp velocity and the velocity at infinity.

We notice the combination of the warp velocity and the square root of twice the chart parameter squared minus one times the scaled radius appearing at the zeroth order; however for AdS the background flow velocity and the warp drive velocity are both along the positive x-direction. The first Hubble-order term in the expansion is the contribution from the background AdS and may be subtracted, and the second term is very small. Similarly, in the energy density the background contribution from the cosmological constant has been subtracted in the terms of second order in the Hubble constant. After the appropriate subtractions the remaining terms are again proportional to powers of the form function and its derivatives, such that the change in expansion and energy density are localised around the boundary of the warp bubble. Interestingly, the zeroth order term in the negative energy density is suppressed by one over the chart parameter squared, thereby reducing the energy as the chart parameter exceeds one. This is also true for the Minkowski space; it is not visible in the earlier expression as we truncated to first order in the velocity at infinity.

The analysis for a warp drive embedded in dS spacetime is similar. However, the MP-charts now cover the full spacetime. Even crossing the apparent horizon at the reciprocal Hubble radius is regular. To study the warp drive near this horizon, we can use the same scaling of the radial coordinate 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, three-dimensional AdS too has conical defect solutions for a point mass placed at the origin. The spatial metric is a cone and the full metric has metric function the defect parameter squared plus the radius squared times the Hubble constant squared, where the defect parameter is one minus four times the mass. The MP-charts with chart parameter equal to the defect parameter do not exist. Therefore, we focus on the charts with chart parameter one and zero velocity at infinity, where the radial coordinate is bounded by the square root of one minus the defect parameter squared, divided by the Hubble constant. We note that for physical particles the defect parameter lies between zero and one. In the conical AdS background, very near to the MP-charts cutoff, the light cone tilt is one plus or minus the warp velocity and the forward photon horizon forms for warp velocities at or above one; this of course gets corrected at first order in the scaled radius.

We shall now embed the warp drive in this conical AdS background and do a similar analysis as done earlier. The important changes are that the spatial metric is now flat and the scaling of the radial coordinate uses the new cutoff. The expansion is as before with the chart parameter squared minus one replaced by one minus the defect parameter squared. Parts of the Hubble-order terms not proportional to the form function are from the background metric and should be subtracted. For the energy density, the zeroth order terms are the same as before with the same replacement, while omitting the chart parameter squared in the denominator. From the Hubble-order terms onwards the expressions differ, in this case starting at the square root of the scaled radius. Again the background contribution from the cosmological constant needs to be subtracted from the terms of second order in the Hubble constant. Throughout, we also do not have the suppression by one over the chart parameter squared, as that parameter is one here.

There is a subtlety with three-dimensional de Sitter space. In all three spacetimes, there is a second source at infinite radius. In Minkowski and three-dimensional AdS spacetimes the geodesic distance between the source at the origin and the source at infinity is infinite, so the second source can be ignored, but for three-dimensional de Sitter it is finite and contributes.

4 Discussion

In this paper we have extended the Alcubierre warp drive spacetime to an infinite class of warp drives utilising 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 analysed 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 negative energy densities localised around the warp drive boundary. To the leading order, energy requirements either scale up or down compared to the original Alcubierre warp drive depending upon the choice of the embedding. All the warp drives discussed in the paper violate the null energy condition. Some other physical attributes like the tilting of the light cones and formation of forward horizon are also affected by the conical defect backgrounds.

There are several possibilities for generalisations. We can look for warped spherically symmetric background spacetimes with an additional exponential factor multiplying the time part, which admit MP-charts with partial coverage of the spacetime. Another direction would be to look at higher dimensions and find spacetimes whose MP-charts have spatial hypersurfaces which near the singularity at the origin take a cone-plus-correction form, or generalise the definition of a cone from two to higher dimensions: unless the angular part is the metric of the unit sphere, the generalised cone is singular. For example, MP-charts of various black hole backgrounds would come under its purview. For most of these generalisations, the Ricci scalar of the spatial hypersurface will be non zero thereby opening up new possibilities for the energy density. If we focus on the spherically symmetric cases, the expressions involving the extrinsic curvatures would generalise easily from the three dimensional expressions in the previous sections up to some coefficients. We have to symmetrise the expressions in all spatial coordinates except 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 and can be studied experimentally. It might be possible to generalize these setups to accommodate the Martel–Poisson versions of the warp drive. Various other properties of the warp drive are of interest, for example the zero-expansion drives of Natário, the exact analysis of the formation of event horizon and various quantum effects, optimisation of various parameters including the form functions and the idea of a warp bubble inside another bubble. Another tangent line of investigation would be to explore the AdS/CFT dictionary and its consequences in the warp drive perturbed AdS metrics. We hope to study some of these aspects in our future works.

Acknowledgements

A.C. thanks Manirujjaman Chowdhury for useful discussions.

Funding

I am funded through my salary from MHRD, Govt. of India. The work is also supported by IIT Bhubaneswar Seed Grant SP-103.

Data availability statement

This manuscript has no associated data. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Code availability statement

This manuscript has no associated code or software. Code and software sharing not applicable to this article as no original code or software was generated or analysed during the current study.

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Appendix A: Different choices for the bubble distance and the induced metric

In this appendix we shall explore two combinations of choices for the warp drive radial coordinate and the spatial induced metric as mentioned in Sect. 3. 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, three-dimensional AdS and dS spacetimes and their conical defect cousins.

Case I. We take the bubble distance to be the coordinate distance and the induced metric to be the interpolating one, flat inside the bubble and the chart metric outside. There are two important deviations from the earlier expressions. Firstly, the effect of the interpolating term in the induced metric starts to show only at second order in the velocity at infinity, so if we are truncating at first order, all results will match. Secondly, the spatial curvature tensors, particularly the intrinsic Ricci scalar, are not zero at finite radius. Its contribution to the energy density starts at second order in the velocity at infinity. Similarly, the extrinsic curvature now picks up contributions from the proper-time derivative of the induced metric, again modifying expressions from second order onwards. In all, this choice complicates things but after background subtraction, the expansion and the energy density are proportional to powers of the form function and its derivatives, therefore localised around the warp drive bubble. A lot now depends on the choice of form function; tweaking it might give a positive intrinsic Ricci scalar thereby reducing the overall negative energy density required to operate the drive.

Case II. We take the bubble distance measured with the induced metric, and the induced metric itself as in the main text. Here, as we are not changing the spatial metric, there is no contribution from the intrinsic Ricci scalar, neither does the proper-time derivative of the induced metric contribute to the extrinsic curvature. Extra contributions as compared to results in Sect. 3 are due to the changes in the derivatives of the form function, or equivalently of the bubble distance, with respect to the Cartesian coordinates. Again, the changes are from second order in the velocity at infinity onwards and yet again the contributions to the expansion and the energy density are localised to the boundary of the warp drive.

(The 34-item reference list is omitted for length; the complete list is at the source.)