General formalism, classification, and demystification of the current warp-drive spacetimes

The human drive to explore and travel faster meets a fundamental barrier in Special Relativity (SR): the lightcone barrier. However, this has not stopped physicists to explore possible workarounds for fast and eventually faster-than-light (FTL) travels. This goes back to efforts within SR, started, to our knowledge, first by McMillan [1], 1 using the relativistic effects of the so-called hyperbolic motion resulted from a uniform acceleration. In his example, McMillan [1] shows that a uniformly accelerated observer (spaceship) can travel "1.2 billion light-years in only 21 years" as measured by an observer on, say, Earth (see also ruminations by Mumford [2]). This, later, motivated Rindler [3] to generalize the hyperbolic motion to the curved spacetime, after whom the Rindler coordinates are coined. However, the apparent superluminal travel is a consequence of time dilation of the uniformly accelerated observers, and not the breaking of the light-cone barrier, and both conclude that such travels are impossible because insurmountable energy costs and cosmological horizon render the prospect of such travel "slight" and "cosmologically quite insignificant" [1,3].

The idea of FTL travel entered General Relativity (GR) through the concept of wormholes, based on the works by Flamm [4], who first described a “tunnel” geometry joining two asymptotically flat surfaces of spacetime, and Einstein and Rosen [5], who proposed a “bridge” as a model for elementary particles. This early foundation led to the work of Wheeler [6], who explored multiplyconnected spacetimes through hypothesized geons, and the paper by Misner and Wheeler [7] that first coined the term “wormhole” to describe mass and electric charge as properties of empty curved space (geometry). These conceptual developments eventually led to the first specific models for traversability in 1973, such as Homer Ellis’s “drainhole” [8] and Bronnikov’s tunnel-like solutions [9]. A modern “renaissance” in the field was triggered by Morris and Thorne [10], who defined the formal requirements for traversable wormholes to be used in interstellar travel, specifically identifying the need for exotic matter that violates the null energy condition (NEC) to hold the throat open (see [11][12][13] for a more detailed account of the historical development).

Following the proposal of wormholes, the next step in conception of FTL travels in GR was the invention of the warp drive in 1994 by Alcubierre [14], 2 which allows for superluminal motion, supposedly, by contracting spacetime in front of a bubble and expanding it behind (see Section III H below). A motivation for Alcubierre’s idea can be drawn from cosmology, specifically the inflationary phase of the early universe that causally connects two observers who should not have been in causal contact in a standard Big-Bang scenario without inflation [14,16]. The main characteristic idea of Alcubierre is that, in contrast to the cosmological case, only a localized part of the universe would causing such effect. Because this original model required physically impossible amounts of exotic matter, Broeck [17] proposed a modification that significantly reduced the (supposed) necessary negative energy density (see Section III E 1 below). Natário [18] advanced the theory by first generalizing the one-component velocity to three-component one, and introducing a zeroexpansion warp drive, suggesting that the bubble could essentially “slide” through space by changing distances along the path of motion rather than relying on volume expansion (see Section III G below).

Several theoretical analyses highlight fundamental pathologies that suggest warp drives are physically or technologically improbable: from the violation of energy conditions indicated first by Alcubierre [14] himself, to application of quantum inequalities by Pfenning and Ford [19] showing that a superluminal warp bubble would require an absurdly large amount of negative energy (which was significantly reduced in the subsequent works with further considerations) while restricting bubble walls to nearly the Planck scale, and the “horizon” problem identified by Krasnikov [20] and Everett and Roman [21], demonstrating that a crew cannot create or control a superluminal bubble on demand, and finally the semiclassical analysis done by Finazzi et al. [22] revealing a fatal instability in which Hawking radiation accumulates at the bubble’s white horizon, leading to an exponentially increasing energy density that would likely destroy the structure (and more problems which will be addressed in this work; see also [12,13] for more details).

Despite all these pathologies, recent works starting with Lentz [23] followed by Bobrick and Martire [24] and Fell and Heisenberg [25] (and others) claim that “physically reasonable” warp-drive configurations can be constructed with positive energy density, and even more with respecting the energy conditions. Lentz [26] proposes using solitons in

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