Take a look at this paper on "Warp Drives and Causality:" https://doi.org/10.1103/PhysRevD.53.7365

The author attempts to argue that the Alcubierre Drive spacetime could exhibit Closed Timelike Curves, if one is to consider a "Lorentz Transformation of the coordinates." In such coordinates, one finds that a light ray emitted on board the ship could travel backwards in time in such coordinates.

The author seems to confuse unphysical coordinate artifact with actual phenomenology. As far as I understand, in GR, one has the freedom to label points on the spacetime manifold by whatever $t, x, y, z$ they like, but the metric tensor remains a physical invariant, regardless of what you call your coordinates. However, its representation in a coordinate system will of course depend on your choice of coordinates.

To be mathematically precise, one cannot perform a Lorentz Transformation on the coordinates within the context of a curved spacetime as in GR. If we want to of course, we could transform Alcubierre coordinates, to coordinates which look like a Lorentz Boost, but then we must transform the representation of the metric as well, and in our case we don't have the Minkwoskian structure as we do in SR.

A light beam propagating inside the vessel will remain within its local light cone, regardless of what you call your coordinates, as specified in Alcubierre's original paper. The "Lorentz transformation" attempted by the author does not hold.

Generally speaking, to find a CTC, we are looking for a Timelike trajectory for a particle in the spacetime that returns to the same point on the spacetime manifold. In summary, I do not see that a CTC arises, but that the paper concludes the existence of CTC from a violation of causality in special relativity which is exactly what a CTC does not entail AND an attempt to apply special relativity beyond its domain of applicability. Lorentz Boosts could only be defined Locally at a point in GR, and not the entire spacetime.

I am only a student, and the author is surely more experienced in GR than me, but I can't see where I am wrong: could someone kindly explain to me where my attempted refutation fails? Sorry for the excessive text...


1 Answer 1


The paper is a bit confusing, but I think the author is trying to construct a setup similar to the classic tachyonic antitelephone in special relativity.

The argument for the tachyonic antitelephone is more or less the following: suppose you can send a tachyonic signal from $(0,0)$ to $(Δt,Δx)$ (where $|Δx| > Δt > 0$). Use Lorentz invariance to take that physical process and rotate (boost) it in the $tx$ plane, to obtain a physical process that lets you send a signal from $(0,0)$ to $(-Δt,Δx)$. Use rotational and translational invariance on that process to get a process that lets you send a signal from $(-Δt,Δx)$ to $(-2Δt,0)$. Now you can send a signal from the origin to a point in its causal past, in two legs.

You can do this with Alcubierre warp tubes if you make the assumption that something resembling Lorentz invariance applies to them. This assumption seems reasonable, since the tubes are localized phenomena in a Minkowski background and they resemble worldlines of tachyonic point particles if you zoom out enough.

The formal substitution of variables is just a way of obtaining a (actively) rotated warp tube. The rotation is physically meaningful because the tube is not alone in the universe, but shares it with another differently oriented tube.

In response to comments, I'll try to construct a spacetime geometry of this sort with an explicit closed timelike curve.

I would use the geometry (10) from the paper, but the paper is vague enough about the details that I'm not sure what the CTC would be or how to write it down, so I'll start from the beginning.

Start with the metric (1-3), which is just Alcubierre's original metric. Pick some $f(r)$ that is zero for all $r\ge\frac14$ so that the metric is precisely Minkowskian outside the tube. Let

$$x_0(t) = \begin{cases} 0 & t\le 0 \\ 2t & 0\le t\le 1 \\ 2 & t\ge 1 \end{cases}$$

but smoothed in small neighborhoods of $t=0$ and $t=1$ so it's differentiable everywhere. This describes a warp tube with its entrance at the origin and its exit at $t=1,x=2,y=z=0$.

Substitute $t=(5t'+4x')/3,\; x=(5x'+4t')/3$ in this metric (which is formally a Lorentz boost), then formally erase the primes. The result is a metric describing a warp tube with its entrance at the origin and its exit at $t=-1,x=2$.

In that metric, substitute $x = 2-x',\; y = y'-\frac12$ and then erase the primes to get a metric describing a warp tube with its entrance at $(t,x,y)=(0, 2, \frac12)$ and its exit at $(-1, 0, \frac12)$.

Glue the last two metrics together at the $y=\frac14$ hyperplane (where they are both exactly Minkowskian) to get a metric containing two warp tubes with entrances and exits at the aforementioned points.

The CTC has four straight segments, from $(0,0,0)$ to $(-1,2,0)$ to $(0,2,\frac12)$ to $(-1,0,\frac12)$ and back to $(0,0,0)$, but smoothed at the corners. The smoothing has to match the smoothing of $x_0$ in order for the curve to be timelike everywhere.

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    – rob
    Mar 4, 2021 at 17:12

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