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Warp drives

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Superluminal travel in general relativity, or warp drives, are roughly defined by the (local) change of distances between two points on some initial spacelike hypersurface. The classic example being, if we consider flat spacetime

$$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$$

and consider the distance from Earth to Alpha Centauri (roughly $4.37\ \text{ly}$), with no proper motion from the two points, we want to change the metric to reduce the distance between those two points, that is, the proper time a timelike curve (or causal curve, if it is a signal) will take to reach it.

$$d(\oplus, \odot_{\alpha}) = \int_{\oplus}^{\odot_\alpha} (g(\dot u, \dot u))^{\frac{1}{2}} d\tau \ll 4.37\ \text{ly}$$

Ideally the time taken should also be small from the point of view of a stationary observer on Earth. That is, for a two way trip from Earth to Alpha Centauri and back to Earth, we want an observer on Earth to wait

$$\Delta \tau = $$

We'll also ask that the metric modification be local, that is, the original flat spacetime has a spacelike hypersurface $\Sigma$ such that $\Sigma \setminus K$, $K$ a compact set, is isometric to our new spacetime.

1. Warp drives, faster-than-light spacetimes and spacetime shortcuts

Defining what qualifies as a warp drive isn't quite easy, since it depends on the future structure of the metric, and it is easy to make it too broad. Simply passing through a gravitational well is enough to reduce the proper time of a trip, but while it could be used to reduce travel time, it seems odd to call it a warp drive. The original Alcubierre warp drive itself is in fact inspired by the FLRW cosmological metric, where the expansion of the universe can cause two points to get separated at speeds greater than the speed of light.

To define it properly, here are some definitions due to Krasnikov[3][4] :

First, we take $\mathcal{M}_1$ and $\mathcal{M}_2$ two spacetimes with inextendible timelike curves $\mathcal E_i, \mathcal D_i \subset \mathcal M_i$, $i = 1,2$ ($\mathcal E$ is the curve of Earth while $\mathcal D$ is the curve of the destination), define $S_i \in \mathcal E_i$ (the point at which a spaceship departs from Earth), with the two following points \begin{eqnarray} F_i &=& \operatorname{Bd}(J^+(S_i)) \cap \mathcal D_i\\ R_i &=& \operatorname{Bd}(J^+(F_i)) \cap \mathcal E_i \end{eqnarray}

$F_i$ is the lower bound to the arrival of a signal at the destination $D_i$, while $R_i$ is the lower bound for the return of a signal to $E_i$. The difference in proper time between $S_i$ and $R_i$ will then be the lower bound for any signal to take the full trip from $E$ to $D$ and back.

2. The basic idea

All faster-than-light metrics use the same basic trick, which is to locally widen the light cone to permit, from the perspective of an observer at rest, to go faster than light. The exact method for this will vary, but if we consider 2D Minkowski space, it will always look locally like something of the form \begin{eqnarray} ds^2 &=& (k_l dx - dt)(k_r dx + dt)\\ &=& -dt^2 - (k_r + k_l) dt dx + k_l k_r dx^2 \end{eqnarray}

For this metric, we get, for a null vector $(u_t, u_x)$,

$$-u_t^2 - (k_r + k_l) u_t u_x + k_l k_r u_x^2 = 0$$ With solution $$u_t = u_x \left[ -\frac{(k_r + k_l)}{2} \pm \frac{\sqrt{ (k_r + k_l)^2 + 4 k_l k_r}}{2} \right]$$ For $k_l = k_r = k$ $$u_t = u_x (-1 \pm \sqrt{2})k$$

3. The Alcubierre warp drive metric

The basic idea behind the Alcubierre warp drive metric is to locally widen the light-cone, so that an object inside that region will appear to go faster than light to an observer with a normal light-cone.

\begin{eqnarray} ds^2 &=& -dt^2 + (dx - v_s f(r_s)dt)^2 + dy^2 + dz^2 \\ &=& -(1 - v_s^2 f^2(r_s))dt^2 - 2 v_s f(r_s) dx dt + dx^2 + dy^2 + dz^2 \end{eqnarray}

This metric depends on a (not necessarily causal) curve $x_s(t)$, which gives rise to the velocity of the warp bubble

$$v_s(t) = \frac{dx_s(t)}{dt}$$

And its radius

$$r_s(t) = [(x - x_s(t))^2 + y^2 + z^2]^{\frac 12}$$

Due to the way the metric is constructed, the curve $x_s(t)$ cannot be arbitrary. If an initially "future directed" $C^1$ curve (that is, with tangent $(u_t, u_x)$, with $u_t > 0$) approaches the tangent vector $(0, 1)$, the velocity of the warp bubble will diverge as

The function $f$ is simply some function to separate the inside and the outside of the "warp bubble". If it's a spherical bubble of radius $R$, it will be of the form

$$f(r_s) = \begin{cases} \approx 1 & r_s \ll R\\ \approx 0 & r_s \gg R \end{cases}$$

In other words, the spacetime should be approximately Minkowski space far from the bubble, and the desired metric well inside the bubble. It can for instance some bump function or just a rapidly decaying function, such as in the original Alcubierre paper :

$$f(r_s) = \frac{\tanh(\sigma (r_s + R)) - \tanh(\sigma (r_s - R))}{2\tanh(\sigma R)}$$

Assuming a bump function that is $0$ outside the bubble and $1$ inside some radius, the spacelike hypersurface on the $(x,y)$ plane decomposes like this

The lightcones representing what happens in the timelike direction.

Both inside and outside the bubble, the metric is just Minkowski space, but with a different slant to the light cone. Well inside the bubble, the metric is just

\begin{eqnarray} ds^2 &=& -dt^2 + (dx - v_s dt)^2 + dy^2 + dz^2\\ &=& - (1 - v_s^2) dt^2 - 2 v_s dxdt + dx^2 + dy^2 + dz^2 \end{eqnarray}

This has the metric determinant $- 1$, which is always negative, meaning that the spacetime is valid for any value of $v_s$. The effects on the lightcone is that for a tangent vector with $(u_t, u_x)$, the class of null vectors is

$$-(1-v_s^2) u_t^2 - 2v_s u_x u_t + u_x^2 = 0$$

This is a quadratic equation in $u_x$ with solution

$$u_x = u_t (v_s \pm 1)$$

If the entire spacetime had this slant, this would not change anything, as it would just be diffeomorphic to Minkowski space, by the coordinate change $x \to x + v_st$. Only the fact that a region of a different slant is present gives the Alcubierre metric its properties.

If we consider the (slightly physically irrealistic) scenario of a bubble with $v_s = 0$, and an infinitely thin wall, then take the geodesic with tangent vector $(1, v, 0, 0)$. For now we'll ignore what happens exactly at the interface between the inside and the outside of the bubble. The geodesic coordinates are, if it starts at $\tau = 0$ at the piont $(0,-R,0,0)$

\begin{eqnarray} t(\tau) &=& \tau\\ x(\tau) &=& v\tau - R\\ \end{eqnarray}

This curve will reach the other side of the bubble $x = R$ at $\tau = 2R / v$, for a total time of

$$\Delta \tau = \int_0^{2R/v} \sqrt{- (1 - v_s^2) - 2 v_s v + v^2} d\tau = 2 R \sqrt{1 - (\frac{(1 - v_s^2)}{v^2} + 2 \frac{v_s}{v})} $$

As opposed to the usual time dilation formula in Minkowski space, if we set $v_s = 0$

$$\Delta \tau = 2 R \sqrt{1 - \frac{1}{v^2}} = 2\frac{R}{\gamma}$$

No matter the speed of the observer $v$, it is always possible to make the trip's duration arbitrarily small with the appropriate choice of the parameter $v_s$.

Along with this local tilting of the light cones, we also want the bubble itself to move, via the function $f$.

Hiscock coordinates

In the case of a constant speed warp drive, $v_s(t) = v_s$, it is possible to redefine the metric in a spherically symmetric way, in a way similar to a Galilean transformation. Consider the coordinate transformation

$$r = x - v_s t$$

Stress-energy tensor

Horizons

Quantum behaviour

Causality

By itself, the Alcubierre warp metric doesn't violate causality, despite its similarities to tachyons. This is easiest to see in the case of the Hiscock coordinates, since a spherically symmetric spacetime on $\mathbb{R}^n$ is always causal. This is actually very much like the tachyon case, since free tachyons in flat space will still only intersect the spacelike hypersurface once (except for the tachyon of infinite speed, and the same applies for warp bubbles).

But just as the case of tachyons, a system of warp bubbles will be able to violate causality. Consider the Lorentz boosted Alcubierre metric

$$g_{\mu\nu} = {\Lambda^{\mu'}}_{\mu} {\Lambda^{\nu'}}_{\nu} g_{\mu'\nu'}$$

Variants

Warp drive without horizon

Warp drive with start and stop

The Natario warp drive

4. The Krasnikov tunnel

Unlkike the Alcubierre warp drive, the Krasnikov tunnel only sland the light cone in one direction, the opposite direction to the first travel.

\begin{eqnarray} ds^2 &=& -(dt - dx) (dt + k(x,t) dx)\\ &=& -dt^2 + (1 + k(x,t)) dx dt + k(x,t) dx^2 \end{eqnarray} $$k(x,t) = 1 - (2 - \delta) \theta_\varepsilon (t-x) [\theta_\varepsilon(x) - \theta_\varepsilon(x + \varepsilon - D)]$$ $$\theta_\varepsilon(x) = \begin{cases} 1 & x > \varepsilon\\ 0 & x < 0 \end{cases}$$

The metric cannot be static here : in the case where the tunnel has always existed

$$ds^2 = -(dt - dx) (dt + k(x) dx)$$

It is possible to simply rescale this metric locally via $x \to \int k(x) dx$

Before the creation of the tunnel, at $t \ll 0$, any causal curve will travel in a region isometric to Minkowski space, meaning that a trip between $x = 0$ and $x = D$ and back will take at least

$$\Delta t = 2 \frac{|D - 0|}{c}$$

The ship will leave Earth at a time $t_1$ and return at a time $t_2 > t_1 + 2D/c$ with the proper time depending as usual on the time dilation formula. On the other hand, after the creation of the tunnel, the travel time will be

$$\Delta t = \int $$

In $4$ dimensions

The Krasnikov metric generalizes easily to any number of dimension, by considering cylindrical coordinates with $z = 0$, $\rho = 0$ for the origin and $z = D$, $\rho = 0$ for the destination. In $4$ dimensions, this corresponds to the metric

$$ds^2 = -dt^2 + (1 - k(t, x, \rho)) dx dt + k(t,x,\rho) dx^2 + d\rho^2 + \rho^2 d\varphi^2$$

With the function

$$k(t,x,\rho) = 1 - (2 - \delta) \theta_\varepsilon (\rho_{max} - \rho) \theta_\varepsilon(t - x - \rho)[\theta_\varepsilon(x) - \theta_\varepsilon(x + \varepsilon - D)]$$

Like the two dimensional case, the function $k$ is $1$ well inside the region of the tunnel (a cylinder of length $D$ and radius $\rho_{max}$).

Causality of the Krasnikov tube

The causality of the two-dimensional case is fairly easy to decide, as any simply connected two-dimensional spacetime is stably causal[6]. If the spacetime is at least of dimension $4$, it is possible to consider a case where we superpose at least two tunnels going to opposite directions with their ends roughly in the same region. This was impossible in two dimensions due to the fact that two tunnels would be unable to go from and to the same region without intersecting.

Bibliography

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Last updated : 2018-03-30 14:13:19