Black Holes and Warp Drive

The concept of warp drives was first introduced by Miguel Alcubierre in his seminal 1994 paper Alcubierre:1994tu , and elaborated upon by José Natario in Natario:2001tk . A warp drive is a solution of General Relativity that has the appearance of a ”bubble” propagating on some (flat or non–flat) spacetime background. The observers inside the bubble are in an inertial reference frame, which means warp drives do not require external energy sources to accelerate, and they may move at any speed (in principle including superluminal). This makes them a viable candidate for interstellar travel, but they have one significant downside: in order to sustain a bubble, one requires exotic matter with negative energy density.

As described by Homer Ellis in Ellis:2004aw , the Schwarzschild metric, which describes a black hole, can be mapped to a warp drive–type metric with the use of a coordinate system known as Painlevé–Gullstrand coordinates, which makes it possible to embed a warp drive in a black hole background. In section 2 of this paper, we generalize this result to an arbitrary static spherically symmetric metric and show that it corresponds to a generalization of Alcubierre–Natario warp drives with non–flat intrinsic metric. We also demonstrate that warp drives can traverse Schwarzschild horizons without ”feeling” them, while their embedding into more generic metrics without horizons produces a singularity. Even more crucially, we show that an external gravitational field can decrease the amount of negative energy required to sustain a warp drive, a fact which may be useful for the development of microscopic warp drives in a lab. Then, in section 3, we suggest an analogue gravity framework that could also help to study the physics of warp drives going through horizons in lab experiments. We conclude in section 4 with an overview of these key proposals of the paper.

A warp drive metric, as defined by Natario in Natario:2001tk , is given by

In ADM variables, one can define it by setting the lapse function $N$ to $1$, the shift vector $N^{i}$ to $\beta^{i}$, and the inner metric $h_{ij}$ to $\delta_{ij}$.

The warp drive itself is a localized perturbation of the metric moving on some (flat or non–flat) spacetime background. Assuming it’s moving along the x–axis, its velocity is given by

where $x_{s}(t)$ is the positon of the soliton’s center.

This means that the functions $\beta^{i}$ have the form

where $\beta^{i}_{out}(\vec{r},t)$ is the background metric, $\beta^{i}_{in}(t)$ is the perturbation, and $f(\vec{r}_{s}(t))$ is a bell–shaped function describing the shape of the warp, with $r_{s}$ given by

In the particular case when the background metric is spherically symmetric, $\beta^{i}_{out}$ are given by

In this case, the background metric can also be written in the more compact form in spherical coordinates

As shown by Painlevé and Gullstrand, the Schwarzshild metric

can be brought to the form (6) with

via a coordinate transformation

As suggested by Ellis in Ellis:2004aw , this relation can be used to embed an actual warp drive within the exterior of a black hole by replacing

In this paper, we are going to generalize the Ellis’ approach for every spherically symmetric metric, whose line element is given by

This metric is sufficiently general to represent a large variety of cases. We need also to include the following additional condition

A generic coordinate transformation has the form

however, since our metric background is time independent, then, given the condition

$\xi$ has to be a constant that can be set to $1$ by rescaling.

The transformation we seek is therefore given by

and it allows to represent the line element as

with

It differs from the standard Natario–type metric (6) by the form factor

Note that because of condition $\left(\ref{throat}\right)$, $g(r)$ has a singularity in $r\ =\ r_{0}$. In Cartesian coordinates (which have to be used to embed the warp drive, as it breaks the spherical symmetry) this metric has the form

with

and intrinsic metric

Since the lapse function $N=1$, the Eulerian observers with geodesics

are still in free–fall, and therefore the metric (19) can also be considered a kind of warp drive metric, albeit different from the Natario–type ones (a warp drive with non–flat spatial slices has previously been proposed in VanDenBroeck:1999sn ).

Because of the presence of $g(r)$, the induced metric $h_{ij}$ has a singularity too. Such a singularity can be eliminated if $g(r)$ is finite at $r=r_{0}$. Since

this means that the line element $\left(\ref{metric}\right)$ would have a horizon.

The simplest case

corresponds to the Schwarzschild metric. Thanks to Eq. (25), the metric is flat and the 3D scalar curvature vanishes. Therefore the energy density can be expressed as

with the extrinsic curvature tensor given by

in Cartesian coordinates.

Now, let us assume that the warp drive is moving along the x–axis:

and consider the limit in which the characteristic size of the warp drive (the support of the function $f$) is much smaller than the Schwarzschild radius of the black hole $R_{G}$. In this limit, $N_{y,z}\approx 0$, and the energy density reduces to

from which, once again neglecting terms $\propto\frac{y}{r}$ and $\frac{z}{r}$, we obtain

Hence, for negative $v$ (i. e. the warp drive moving towards the black hole) with

the amount of negative energy required to sustain a warp drive would be decreased by the black hole’s gravitational field (otherwise, or if the warp drive is moving in the opposite direction, it would be increased).

As shown in White:2021hip , it may be possible to create warp drive–like structures within Casimir cavities in a lab, so it appears a promising direction to investigate how they would be affected by an external gravitational field (and whether such a field can serve a practical purpose by making it possible to realize a warp drive with a lesser amount of negative energy).

Finally, let us note that, as the $00$–component of the metric tensor is approximately

the horizon would be effectively absent inside the warp bubble as long as it is subluminal ($v<1$).

While warp drives remain at the moment beyond the realm of experiment, one could, at least in principle, try to gain some understanding of the underlying physics through an analogue gravity set–up. The framework of analogue gravity is rooted in the observation that for a barotropic, inviscid, irrotational fluid, the perturbations of the velocity potential $\phi$ obey the equation Barcelo:2005fc

which is the equivalent of the Klein–Gordon equation with the effective metric

Here $c_{s}$ is the speed of sound in the medium, given by

and $\vec{v}$ is the background velocity of the fluid. For a radially symmetric system, it may also be written in the form

In particular, the Gross–Pitaevskii equation

which describes a Bose–Einstein condensate, can be written in the hydrodynamic form in the Madelung representation, so that the phase of the wave function would play the role of the velocity potential, and the speed of sound would be

where $n_{c}$ is the background density of the condensate, given by $|\psi|^{2}$.

The metric (34) is, up to the prefactor $\frac{\rho}{c_{s}}$, exactly the warp drive metric (1), and hence one could imitate a warp drive by introducing a perturbation in the velocity profile, as suggested in Fischer:2002jn  Finazzi . By performing the Painlevé–Gullstrand transformation (9) in reverse, one can map this metric to

which is exactly the Schwarzschild metric for the radial velocity profile

A 1+1–dimensional version of this setup was realized experimentally with a Bose–Einstein condensate of Rb atoms Lahav:2009wx , making it possible to observe the analogue Hawking radiation Steinhauer:2015saa . In principle, one can combine this setup with the one described in Finazzi to realize a condensate velocity profile given by

where $v_{0}$ is the background profile that has a step–like transition from subsonic to supersonic at some point $x_{0}$, and $\delta v$ is a solitonic wave in the velocity profile propagating at velocity $V$. By having $\delta v$ cross the point $x_{0}$, one can simulate the scenario described in Ellis:2004aw , in which a warp drive crosses a black hole horizon.

The more generic class of metrics of $g(r)\neq 1$ may be simulated by considering a variable speed of sound: in particular, it has been suggested in Mateos:2017sjv that one could modulate the speed of sound using the Feshbach resonance. In this case, we can make the identifications

to obtain from (39) a metric equivalent to (11), up to the conformal factor $\frac{\rho_{0}}{c_{s}}$ in front. In principle, it can be set to a constant if one chooses a fluid with the barotropic equation of state

By combining this setup with (41), it may be possible to simulate the passage of a warp drive through the horizon.

In this paper, we studied a class of coordinate transformations that generalizes Painlevé–Gullstrand coordinates to an arbitrary spherically symmetric metric and makes it possible to embed a warp drive into a given metric background. We found that the presence of a horizon is required to avoid a singularity. For the specific case of Schwarzschild black hole, the horizon would be effectively absent inside the warp bubble as long as it’s subluminal, making the warp drive a possible safe route for crossing horizons.

More importantly, we discovered that external gravitational field can decrease the amount of negative energy required to sustain a warp drive, which may be instrumental for creating microscopic warp drives with Casimir cavities, such as the one described in White:2021hip .

In the second part of the paper, we proposed another possible application of our results for tabletop experiments in the form of an analogue gravity set–up. For this purpose, we started with a known realization of analogue black holes with Bose–Einstein condensate and supplemented it with a warp drive analogue in the form of a solitonic wave. We also suggested that in order to incorporate the more generic non–Schwarzschild spherically symmetric metrics, one needs to introduce spatial variations in the condensate speed of sound.

In GarattiniZatrimaylov we shall explore the tangential topic of warp drive metrics with intrinsic curvature, generalizing the results of VanDenBroeck:1999sn . This class of models can possibly reduce the amount of exotic matter with negative energy density required to build a warp drive.