Bobrick and Martire’s Subluminal Warp Bubble Research

1. Summary of Their Research and Key Conclusions

Alexey Bobrick and Gianni Martire re-examined warp drive metrics with the goal of formulating a physically plausible “warp bubble” solution in classical general relativity. In their 2021 study “Introducing Physical Warp Drives,” they developed the first general model for a subluminal (slower-than-light) warp drive spacetime that avoids the unphysical requirements of the original Alcubierre drive ([2102.06824] Introducing Physical Warp Drives) (Free software lets you design and test warp drives with real physics). Crucially, their warp bubble does not require exotic negative energy matter. Instead, it involves a shell of positive energy density (ordinary mass) that generates the warp field () (Free software lets you design and test warp drives with real physics). Key findings and conclusions from their work include:

Warp Bubble as a Mass Shell: Any warp drive spacetime can be understood as an inertially moving shell of material (either positive or negative energy) enclosing a “passenger” region of flat spacetime (). In other words, a warp bubble isn’t a standalone phenomenon – it is generated by a mass-energy shell that distorts spacetime around an interior region. The more mass-energy in the shell, the greater the warp effect on the internal region (Free software lets you design and test warp drives with real physics).
No Exotic Matter for Subluminal Warps: Bobrick and Martire demonstrated the first concrete example of a subluminal warp drive metric with only positive energy, satisfying all the usual energy conditions of general relativity (Constant Velocity Physical Warp Drive Solution). This means no “negative mass” or exotic matter is needed for a slower-than-light warp bubble, addressing one of the biggest criticisms of the Alcubierre drive. By optimizing the shape of the warp bubble, they even reduced energy requirements by an estimated factor of ~30 compared to Alcubierre’s original design (Free software lets you design and test warp drives with real physics). Their model ostensibly confirms that warp-like spacetimes are physically possible within classical physics, as long as one stays below light speed (Free software lets you design and test warp drives with real physics).
Conventional Propulsion Required: A major conceptual conclusion is that a warp drive does not propel itself – it still obeys the laws of inertia and momentum conservation. The warp bubble (being a massive shell) must be accelerated by some external force or rocket, just like any other object (). In their words, “warp drives, being inertially moving shells of [mass-energy], do not have any natural way of changing their velocities… achieving a certain velocity for a warp drive requires an externally applied force… some form of propulsion” (). This demystifies warp travel: you cannot simply “engage” a warp bubble to bypass Newton’s laws; any warp bubble needs a push. Furthermore, there is no known way to accelerate a warp bubble itself past the speed of light without exotic physics (). Superluminal warp travel, if it exists at all, would require pre-existing FTL conditions or hypothetical tachyonic matter, not conventional means (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!).
Superluminal Warps Remain Hypothetical: Their analysis reinforces that faster-than-light warp drives still demand exotic conditions. Bobrick and Martire note that if one “wants superluminal motion, you need negative energy densities” (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!) – the very feature that makes the original warp drive proposal unphysical. They did construct some theoretical superluminal solutions that satisfy quantum inequalities (constraints from quantum field theory that limit negative energy) ([2102.06824] Introducing Physical Warp Drives). This suggests that, in principle, a warp bubble could exceed c without outright violating quantum laws, but such solutions are still speculative and require “exotic” ingredients. In summary, subluminalwarp bubbles can be achieved with known physics, while superluminal ones remain a hypothetical extrapolation.
Controlling Spacetime Inside the Bubble: An intriguing finding is that by appropriate design of the warp metric, one can control the rate of time and spatial scale inside the bubble relative to the outside ([2102.06824] Introducing Physical Warp Drives). Because the warp shell’s gravity influences the interior, time can be made to run slightly faster or slower inside the passenger region than it would normally (). Space itself can be stretched, compressed, or even made to rotate inside the bubble according to their solutions (). In essence, their warp drive model allows a degree of control over the internal spacetime (e.g. creating zones of different time flow). This could mean, for example, a warp “pocket” where time dilation is engineered intentionally. However, these effects are modest unless extremely large energies are used (see §4).

Overall, Bobrick and Martire’s work revives the warp drive concept by placing it on firmer physical footing. They conclude that nothing in the laws of physics forbids a subluminal warp bubble – one can curve spacetime in a “warp” configuration without violating energy conditions or relativity ([2102.06824] Introducing Physical Warp Drives). However, such a warp bubble is essentially a new form of gravitational vehicle that still requires enormous mass-energy and conventional propulsion to use. In their words, “a class of subluminal, spherically symmetric warp drive spacetimes…can be constructed based on the physical principles known today” ([2102.06824] Introducing Physical Warp Drives) – but any warp drive requires propulsion and obeys relativity just like any other object ().

2. Technical Methodologies and Equations

Bobrick and Martire approached the problem by formulating a general warp drive metric and analyzing the stress-energy requirements from Einstein’s field equations. They built a framework that encompasses all existing warp metrics(like Alcubierre’s) and allows generating new ones ([2102.06824] Introducing Physical Warp Drives) ([2102.06824] Introducing Physical Warp Drives). Here are some technical details of their methodology and key equations:

Warp Drive Metric Form: They start from the well-known Alcubierre metric as a template. In 4-dimensional spacetime, the Alcubierre warp drive is given (in one formulation) by the line element:\[ ds^2 = -c^2 dt^2 + \big(dx – v_s, f(r_s),dt\big)^2 + dy^2 + dz^2, \]where vs(t) is the bubble’s velocity and f(rs) is a shape function that defines the warp bubble’s wall profile (). Here rs=(x−vst)2+y2+z2 is the distance to the bubble’s center in a coordinate system moving with the bubble (). This metric describes a “spherical warp bubble” moving along the x-axis at velocity vs, with flat Minkowski spacetime inside the bubble (where f=1 near the center) and normal flat spacetime outside (where f=0 far from the bubble) (). Bobrick and Martire generalized this concept by not assuming a specific shape function a priori; instead, they consider an arbitrary shell-like stress-energy distribution and derive the resulting metric.
ADM Formalism and Shift Vector: The authors use techniques from the Arnowitt-Deser-Misner (ADM) formalism of general relativity, which splits spacetime into space+time and defines quantities like the lapse and shift. In a warp spacetime, the effect of the bubble’s motion can be seen in a non-zero shift vector (essentially the dx−vsf(r)dt term). Bobrick and Martire’s general solution involves a “shift vector distribution” that closely matches Alcubierre’s in form (Constant Velocity Physical Warp Drive Solution). By adding a stationary mass shell to this warp shift profile, they solve Einstein’s equations for a self-consistent metric. They often had to resort to numerical integration given the complexity of the field equations (Constant Velocity Physical Warp Drive Solution). The result was a family of solutions describing a warp bubble sustained by a thin spherical shell of matter. The interior of the shell is an almost flat region (where a spacecraft could reside), and the exterior matches smoothly to flat spacetime at infinity.
Energy Conditions: A critical part of their methodology was evaluating the stress-energy tensor Tμν of the warp metric to check the energy conditions. Classical general relativity has conditions like the Weak, Strong, and Null Energy Conditions which essentially require that observed energy density is non-negative for normal matter. The original Alcubierre solution famously violated these (it required negative energy density in the warp region, breaking the Weak Energy Condition). Bobrick and Martire constructed their subluminal solution explicitly to satisfy all energy conditions everywhere (Constant Velocity Physical Warp Drive Solution). This was done by ensuring the matter shell had positive energy density and that any deformations of spacetime did not introduce negative-pressure or other exotic terms that would violate the conditions. In the 2023 follow-up work, they confirmed via detailed calculation that their constant-velocity warp bubble solution meets the Null, Weak, and Dominant energy conditions (i.e. no exotic matter) (Constant Velocity Physical Warp Drive Solution). This is a major technical achievement, demonstrating a warp drive spacetime can be “physical” in the classical sense.
Quantum Inequalities: For any cases that did involve negative energy (like hypothetical superluminal extensions), they checked those against quantum inequalities – constraints from quantum field theory that limit the magnitude and duration of negative energy. (Pfenning & Ford (1997) derived such inequalities to rule out absurd amounts of negative energy ().) Bobrick and Martire showed that certain superluminal warp metrics could, at least in principle, satisfy these quantum inequality bounds ([2102.06824] Introducing Physical Warp Drives). In other words, if negative energy is needed for a faster-than-light bubble, the requirements might still lie within what quantum physics permits in highly contrived scenarios. This suggests that if exotic negative energy could be harnessed in small quantities (as allowed by quantum effects like the Casimir vacuum), a superluminal warp might skirt the edge of physical law. However, they emphasize this is purely theoretical – their focus remained on positive-energy subluminal designs.
Equations for Specific Designs: In their paper, Bobrick & Martire present detailed formulas for different warp drive classes. For example, they derive the stress-energy tensor for a spherically symmetric warp shell (their positive-energy solution) and show it can be made arbitrarily thin. They also provide an “optimized Alcubierre” metric where the bubble is flattened in the direction of travel, reducing energy cost. Technically, flattening the bubble means using a shape function that is less elongated; placing the “passengers” side-by-side rather than lined up can make the warp field more efficient (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!). Sabine Hossenfelder summarized this result: “The flatter [the bubble] is in the direction of travel, the less energy you need” (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!). This insight led them to a metric modification (akin to the Van Den Broeck approach, see §3) that cut the negative energy requirement by two orders of magnitude (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!) (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!).

In practice, deriving these metrics involves solving Einstein’s field equations Gμν=8πGc4Tμν under the ansatz of a moving shell. Bobrick and Martire used both analytical reasoning and computational tools to find solutions. In fact, they and collaborators have since developed an open-source software called “Warp Factory” to automate warp metric design and energy-condition checking (Free software lets you design and test warp drives with real physics). This tool effectively numerically solves the Einstein equations for custom warp configurations and visualizes the results, indicating how their methodology can be extended by other researchers.

Validation of Physicality: One important technical check was ensuring the warp solution is not a mere coordinate artifact. Sometimes in general relativity, what looks like a weird metric can actually be just flat spacetime in disguise (through a tricky coordinate transform). Bobrick et al. verified that their warp spacetime cannot be reduced to flat spacetime by any coordinate change – the geodesic motion and curvature are genuine (Constant Velocity Physical Warp Drive Solution). They computed invariants and checked that the shift vector field defining the warp is not “pure gauge”. This gave confidence that their warp bubble solutions represent a real distortion of spacetime geometry, requiring matter and energy to produce.

Overall, Bobrick and Martire’s methodology combined theoretical analysis (generalizing the warp metric and classifying solutions) with direct computation (to solve Einstein’s equations and evaluate energy conditions). The result is a set of equations describing a warp bubble that is both mathematically consistent and built from physically allowed ingredients. They essentially established a template for future warp-drive research: define a desired bubble motion, design a supporting mass-energy shell (via a stress-energy tensor ansatz), and solve for the metric, then check energy conditions and quantum constraints. This approach turns vague science-fiction ideas into a concrete, solvable problem in general relativity.

3. Related Research on Warp Metrics by Other Physicists

Bobrick and Martire’s work builds on a line of theoretical research into warp metrics that began in the 1990s. Below is an overview of key contributions by other physicists and how they relate:

Miguel Alcubierre’s Original Warp Drive (1994): Mexican physicist Miguel Alcubierre sparked the field by discovering a solution of Einstein’s equations that allows a “warp drive” effect (). In Alcubierre’s metric (cited above), space is contracted in front of a spacecraft and expanded behind, theoretically allowing the craft to ride a “wave” of curved spacetime to exceed the speed of light relative to distant observers. However, as noted earlier, this solution requires enormous amounts of negative energy (exotic matter with negative mass density) ([2102.06824] Introducing Physical Warp Drives). Alcubierre acknowledged this exotic requirement, and subsequent analyses showed the amount needed was astronomically large – on the order of the mass of a star or planet (in negative form) to create even a modest bubble (). For example, generating a 1-meter warp bubble moving at relativistic speed would require negative energy equivalent to the mass of our Sun (). Because no known form of stable negative-mass matter exists (and quantum vacuum effects can only produce tiny negative energies fleetingly), the Alcubierre drive was deemed unphysical ([2102.06824] Introducing Physical Warp Drives). It also suffered other problems: violating the Weak and Null Energy Conditions by having negative energy densities (), enabling potential causal paradoxes (faster-than-light travel can create closed timelike curves, i.e. time loops, leading to things like the “grandfather paradox”) (), and instabilities such as intense Hawking-like radiation at the warp bubble’s horizons (). These issues led many to conclude that warp drives were a mere mathematical curiosity, not realizable physics ().
Chris Van Den Broeck (1999): Seeking to make Alcubierre’s concept more feasible, physicist Chris Van Den Broeck proposed a modification to reduce the energy requirement. He realized that by creating a warp bubble with an extremely small external radius but an inflated internal volume (like a “pocket universe”), one could drastically cut down the required negative mass. In his 1999 paper “A ‘Warp Drive’ with More Reasonable Total Energy Requirements,” Van Den Broeck showed it might be possible to shrink the warp bubble’s thickness to 10^(-15) meters (near the Planck scale) while maintaining a usable interior volume (). This change reduced the needed negative energy from solar-mass scales down to something like the mass of the Sun in positive energy – i.e. still huge, but in negative form it satisfied certain quantum inequalities (). In fact, his solution satisfied “vacuum energy inequalities” (quantum limits on negative energy) and met the Weak Energy Condition in a weak-field sense (). The catch was that it assumed general relativity holds even at tiny scales where quantum gravity should dominate, and ultimately Bobrick and Martire note that Van Den Broeck’s metric is essentially an extreme limit of the Alcubierre metric – a clever coordinate tweak rather than a fundamentally new solution () (). They show via coordinate transformation that Van Den Broeck’s warp drive is mathematically equivalent to Alcubierre’s, just written differently (Appendix A of their paper) (). Nonetheless, Van Den Broeck’s idea was influential as it introduced the concept of geometry manipulation (shrinking the bubble) to reduce energy needs.
José Natário (2002, 2006): Natário explored variations of warp metrics that defied some assumptions. In 2002 he constructed a warp drive solution without space contraction/expansion – meaning the metric did not have separate regions of expanding space behind and contracting space ahead (). This was interesting because Alcubierre’s original drive explicitly uses a “deformation” of space; Natário showed you can still achieve net motion with different metric arrangements (essentially using only shear flows in space). However, his solution didn’t remove the need for exotic energy. In 2006, Natário proposed a subluminal warp drive in weak-fieldgravity, but it still required negative energy (). These works were part of systematically mapping out what is essential for warp travel and what is just an artifact of Alcubierre’s model. Bobrick and Martire classify the Alcubierre and Natário drives as special cases of their general model – in fact, in the subluminal regime, even Alcubierre’s drive requires some negative energy in the walls (as Lobo & Visser 2004 showed ()), so those prior solutions fall into the category of exotic-matter-dependent warp bubbles.
Pfenning & Ford (1997); Everett & Roman (1997): These physicists analyzed warp drives from the perspective of quantum field theory and causality. Pfenning and Ford applied quantum inequality constraints to the Alcubierre drive and concluded that the negative energy required would have to be distributed in an incredibly thin shell (near Planck thickness) and for very short durations – making it practically impossible to achieve () (). Everett and Roman examined potential time-travel paradoxes and showed that an Alcubierre warp could create closed timelike curves (causal loops), violating causality unless new physics intervenes (). They considered mechanisms like “chronological protection” (possibly quantum effects destroying the warp bubble when it tries to go FTL). These studies reinforced the sense that FTL warp drives are forbidden by fundamental physics (energy conditions and causality) – a stance Bobrick and Martire managed to soften by restricting to subluminal travel.
Michael P. Berry, Erick W. Lentz (2020–2021): In parallel to Bobrick & Martire’s work, independent approaches emerged attempting positive-energy warp drives. In 2020, Lentz (a physicist at Göttingen) developed what he called a warp drive soliton solution. Published in 2021, Lentz’s model claimed to achieve faster-than-light travel using only standard positive energy (no exotic matter) (Free software lets you design and test warp drives with real physics). This was a significant claim that garnered media attention. Lentz found a new class of solutions to Einstein’s equations by solving the hyperbolic PDEs for “solitonic” spacetime configurations. According to Lentz, if enough energy could be supplied, a warp bubble could form that moves at superluminal speed without violating energy conditions (Free software lets you design and test warp drives with real physics). The catch was the staggering energy required: on the order of 1047 joules for a 100-meter ship – roughly hundreds of times the mass of Jupiter converted to energy (Free software lets you design and test warp drives with real physics)! He noted a 30 orders of magnitude improvement in efficiency would be needed to make it remotely feasible (Free software lets you design and test warp drives with real physics). Additionally, there was skepticism about whether Lentz’s solution truly had no hidden negative-energy regions (some argued it might be a tricky coordinate effect or that it wasn’t yet a complete solution). Nonetheless, Lentz’s work, alongside Bobrick & Martire’s, reignited interest in warp metrics in 2021 (Free software lets you design and test warp drives with real physics). Other researchers like Grazier, and later updates by Lentz and colleagues, continue to refine these ideas. It’s worth noting that Bobrick & Martire acknowledged Lentz’s contribution in their discussion, but also pointed out that Lentz did not provide a concrete mechanism or experimental approach to realize the metric (his was a more theoretical construction) ().
Harold “Sonny” White (2011–2021): Former NASA engineer Harold White is known for his efforts to reduce the energy requirements of warp drives and even attempt lab experiments. In 2011, White suggested that shaping the warp bubble into a torus (doughnut) and oscillating the field could cut the negative energy need by many orders of magnitude – a claim that was met with skepticism since it largely relied on altering the geometry (similar in spirit to Van Den Broeck’s approach). In 2021, White and colleagues reported an interesting finding: while researching Casimir cavity physics (a small device that produces negative vacuum energy between plates), they claim to have found a nano-scale warp bubble structure in the energy distribution (DARPA Funded Researchers Accidentally Discover The World’s First Warp Bubble – The Debrief) (DARPA Funded Researchers Accidentally Discover The World’s First Warp Bubble – The Debrief). Specifically, in a peer-reviewed study, White’s team proposed that a certain Casimir cavity configuration (a 1-micron sphere in a 4-micron cylinder) would create a negative energy density pattern closely matching the requirements for an Alcubierre warp bubble (DARPA Funded Researchers Accidentally Discover The World’s First Warp Bubble – The Debrief) (DARPA Funded Researchers Accidentally Discover The World’s First Warp Bubble – The Debrief). White described it as “a real, albeit humble, warp bubble”– essentially a tiny region of space with a distribution of negative energy analogous to a warp metric (DARPA Funded Researchers Accidentally Discover The World’s First Warp Bubble – The Debrief). This does not mean a practical warp drive was created, but it’s a notable piece of related research attempting to bring warp concepts into laboratory tests. It suggests that Casimir effect (a quantum phenomenon that produces small negative energies) might one day help experimentally verify aspects of warp physics, albeit at microscopic scales (DARPA Funded Researchers Accidentally Discover The World’s First Warp Bubble – The Debrief) (DARPA Funded Researchers Accidentally Discover The World’s First Warp Bubble – The Debrief).
Other Concepts – Krasnikov Tubes and Wormholes: While not warp “bubbles” per se, other faster-than-light spacetime concepts have been explored by physicists. Krasnikov tubes (proposed by Serguei Krasnikov in 1998) involve creating a tunnel in spacetime that a ship could use for FTL round trips without causality violation (the catch: you have to travel one leg slower-than-light to set up the tunnel) (). These tubes also require huge energy and exotic matter. Wormholes (shortcuts through spacetime) have been studied extensively (Morris & Thorne 1988, Visser 1995, etc.), and in 2007 Garattini & Lobo discussed how a warp drive might be related to a traversable wormhole (). In all cases, negative energy or other exotic conditions are required to sustain these configurations (). Thus, warp drives, Krasnikov tubes, and wormholes all face a common challenge: finding physically allowed stress-energy to support spacetime geometries that nature doesn’t normally provide.

In summary, warp metric research has gradually evolved from Alcubierre’s theoretical curiosity to a more physically constrained and optimized set of solutions. Early work by Alcubierre, Krasnikov, Everett, and others mapped out the daunting problems (exotic energy, causality issues). Mid-period work by Van Den Broeck, Natário, Lobo, and White attempted various “loopholes” to reduce requirements or modify assumptions. Most recently, Bobrick & Martire and Lentz (2020–21) independently found that if you settle for subluminal speeds or huge energy expenditures, warp bubbles might not need any new physics beyond general relativity ([2102.06824] Introducing Physical Warp Drives) (Free software lets you design and test warp drives with real physics). Their positive-energy warp solutions mark a turning point in related research – showing a continuum of warp metrics from purely subluminal (physical) to superluminal (still speculative) within a unified framework. This new understanding has opened the door for more scientists to seriously examine warp drives as a subject of study, rather than a dismissed fantasy.

4. Implications for Theoretical Physics and Practical Applications

Bobrick and Martire’s results, along with related work, carry several important implications:

Theoretical Physics Implications:

Energy Conditions and Spacetime Engineering: A warp bubble that satisfies all classical energy conditions is a striking demonstration that general relativity permits “spacetime engineering” with normal matter. Before this work, it was widely assumed that dramatically warping spacetime (for FTL travel or unusual effects) inevitably required exotic matter violating known physics. Now we know that within GR, highly curved but subluminal spacetimes can be supported by positive energy densities (Constant Velocity Physical Warp Drive Solution). This forces theorists to revisit energy conditions and their loopholes. It also provides a concrete case to study the interface of GR with quantum inequalities – pushing quantum field theory to its limits by asking “how much can we curve spacetime with only a given amount of mass-energy?”. In a way, Bobrick and Martire have charted a course for metric engineering: designing specific metric tensors for desired outcomes (like moving a region of space), and then identifying the required stress-energy tensor. This is conceptually similar to how one engineers an electromagnetic field for a purpose, but here with gravity. It contributes to the emerging idea of ”applied general relativity” (using GR solutions for practical configurations, not just cosmology or astrophysics).
No Free Lunch – Warp Bubbles as Gravity Objects: Their conclusion that a warp drive is just a massive shell moving inertially () () is important. It means warp bubbles don’t circumvent fundamental principles like momentum conservation or the light speed limit for information – they simply repackage them in an exotic form. This aligns with our understanding of general relativity: gravity can do amazing things (bend light, slow time, etc.), but it cannot make you truly break local physics laws. The warp bubble is a highly nonlinear gravitational configuration, yet it still behaves like a gravitating object. For theoretical physics, this is a reassuring consistency check. It implies that any future quantum gravity or new physics that would allow “free acceleration” or true FTL would have to genuinely extend beyond GR, since within GR the warp drive obeys all the usual rules (no reactionless drive, no local FTL acceleration) () (). Bobrick and Martire’s analysis in fact identifies an “error” in the original Alcubierre thinking – the assumption that you could just turn on a warp field and go faster than light from rest (). In reality, that assumption hid the requirement of an initial superluminal setup or infinite acceleration. Recognizing this has sharpened theoretical understanding of warp drives.
New Classes of Solutions and Spacetime Classification: By presenting a general warp-drive spacetime model and classifying warp bubbles into different types (subluminal, superluminal with various properties – they mention Class I to IV warp drives in their paper () ()), Bobrick & Martire have expanded the catalog of exact solutions to Einstein’s equations. Each class has distinct traits (e.g. one class allows a shell of subluminal matter that can contain a region where no object can remain subluminal – an interesting analog to black hole event horizons ()). These solutions enrich theoretical physics by providing new “laboratories” to test ideas. For instance, a warp bubble where time runs faster inside than outside could be used to explore questions of differential aging and energy conservation in GR. A solution where a superluminal shell contains only subluminal interior (their hypothetical Class III and IV drives) touches on the interplay of horizons and causality () (). Even if these solutions aren’t physically achievable, they help theorists probe the edges of GR and understand what constraints like causality and the energy conditions truly mean in complex spacetimes.
Connections to Cosmology and Dark Energy: The idea of stretching or contracting space is reminiscent of cosmological expansion. Alcubierre himself drew an analogy to cosmic inflation (where space expanded faster than light, but without moving any object locally > c). Bobrick & Martire’s physical model might encourage thinking about whether natural “warp bubbles” could exist in the universe. For example, are there theoretical objects (perhaps around black holes or in the early universe) that resemble warp metrics? Also, their work hints that maybe alternative gravity theories (like conformal gravity or higher-dimensions) could offer advantages (). They cite a conformal gravity attempt that achieved warp solutions with only positive energy (). This could stimulate more research in extended theories of gravity to see if warp drives become easier or have fewer limitations. In essence, warp drives provide a new testing ground for GR and its alternatives, possibly linking to concepts like dark energy (vacuum energy that expands space) as a cosmic analog of a warp field.

Potential Practical Applications:

Interstellar Travel (Long Term): The most obvious application is in space travel – a warp drive could, in principle, allow rapid transit between distant stars without the occupant experiencing high acceleration or relativistic time dilation. Bobrick & Martire’s subluminal warp bubble would not let you exceed light speed travel time, but it could still offer advantages. For instance, an object inside a warp bubble moves by geodesic motion(free-fall) even as the bubble as a whole travels at high velocity (Constant Velocity Physical Warp Drive Solution). This means astronauts could endure a journey at say 0.5c or 0.8c without feeling any acceleration forces – the bubble moves inertially, and they are in free-fall inside it. That is a big benefit for human spaceflight, essentially eliminating the need for high-g acceleration/deceleration phases. Additionally, if the warp bubble can manipulate time flow, one could imagine reducing the passage of time inside the ship during the voyage (or increasing it to match destination clocks, if desired). In a far-future scenario, a network of subluminal warp bubbles might provide fast (though not instantaneous) transport around a solar system or to nearby stars, acting like “ferry” vehicles that coast without friction or drag. It’s important to note, however, that practical warp travel remains speculative – the energy required even for subluminal speeds is enormous (see below), and engineering a contained shell of mass-energy is far beyond current technology.
Energy Storage or Space-Time “Bubbles”: A less sci-fi but intriguing application arises from their note that warp spacetimes could store energy or momentum in a stationary way (). A warp bubble could theoretically be used as a gravitational storage device – a sort of pocket where energy is held in the curvature of space. For example, one could envision pumping energy into a warp bubble shell (perhaps via lasers or particle beams) and thereby creating a region of altered time rate or gravitational potential. This could act like a time-dilation chamber (with slower time inside, it could preserve something or allow a fast-forward to the future). Alternatively, a warp bubble with an internal rotation could store angular momentum like a flywheel, but in gravitational form. These ideas border on science fiction, but they highlight how mastering warp metrics might lead to new engineering paradigms: space-time devices that do things no normal machine can, such as significantly altering time flow on demand or creating self-contained environments with different physical conditions.
Fundamental Research and Prototypes: In the nearer term, the implications are more about research tools. The Warp Factory software introduced by Applied Physics (Bobrick, Martire, and colleagues) in 2024 allows researchers to design and simulate warp metrics on a computer (Free software lets you design and test warp drives with real physics). This can deepen our practical understanding of GR. One could use such tools to simulate what happens if a warp bubble passes near another mass, or how it would interact with gravitational waves, etc. This has pedagogical value and might spin off new insights (for instance, could a micro warp field be used to deflect asteroids by altering gravity around them? The software could test that scenario virtually). Additionally, though building a warp drive is far beyond us, there may be intermediate applications of the underlying physics. For example, high-energy laser arrays or magnetic confinement systems might one day attempt to create tiny distortions in spacetime (not full bubbles, but testable curvature effects). The techniques developed for warp drives could inform advanced propulsion concepts like gravity assist enhancers or inertial dampening fields (to reduce felt acceleration in spacecraft). These are speculative, but they fall out naturally if one considers partial implementations of warp technology.
Implications for Aerospace Engineering: If a warp bubble could be realized even partially, it changes how we think of vehicles. A spacecraft using a warp field doesn’t move by pushing against something; it moves by relocating a patch of space. This could bypass issues like drag or sonic booms in atmosphere (imagine a warp-like field for supersonic air travel that prevents shockwaves by enclosing the vehicle in locally static space). It might also allow novel maneuvers: e.g., “hovering” by creating tiny warps that counteract gravity locally. Again, these are extrapolations, but serious consideration of warp metrics could influence advanced propulsion research – encouraging new ways to manipulate gravity or space directly. Some concepts like the EMDrive (a controversial reactionless drive) or gravity control experiments can be informed by the rigorous approach Bobrick & Martire championed (i.e. require full consistency with Einstein’s equations and energy conservation).

Practical Challenges: It must be underscored that, with current or foreseeable technology, building a warp bubble is beyond impractical. The energy and mass requirements are tremendous. Bobrick & Martire give an example: a warp shell of 10 meter radius made of normal matter with the mass of Earth would only slow down time inside by on the order of 0.04% (). Even a slight noticeable space-time distortion demands planetary-mass energies. In their positive-energy warp solution, the shell itself could weigh many millions of tons, and would need to be constructed and accelerated to the desired speed. Such a feat is centuries away at best. Moreover, containing and shaping that mass/energy into the required form (perhaps a dense shell or field of plasma or intense laser light forming a gravity well) is a huge engineering problem. There is also the issue of what happens to objects near a warp bubble – some analyses suggest that starting or stopping a warp bubble could release energy bursts harmful to surroundings (e.g., Hawking radiation bursts). So, while the theoretical door is now open, the practical road is long. The implication is that warp drive will remain a theoretical sandbox to deepen physics understanding, unless future breakthroughs (fusion power, new states of matter, quantum gravity control) change the game.

In summary, Bobrick and Martire’s subluminal warp bubble concept has a profound theoretical implication – showing warp spacetimes can exist without new physics – and it tempts us with futuristic applications like interstellar travel and time manipulation. But it simultaneously reminds us that even “allowed” does not mean “achievable”: the leap from mathematical metric to working technology is enormous. Still, just knowing that nature in principle doesn’t forbid warp bubbles inspires both theorists and visionaries to keep exploring the idea, perhaps leading to unexpected discoveries along the way (even if they fall short of a Star Trek drive).

5. Responses and Critiques from Other Scientists

The resurgence of warp drive research (especially around 2020–2021) drew a range of responses from the scientific community. While many physicists found Bobrick and Martire’s work to be a valuable clarification, there are also healthy skeptics and critiques. Here we outline some responses and alternative perspectives:

General Reception: Bobrick and Martire’s paper was received as a serious, rigorous analysis that helped legitimize warp drive studies. Prior to this, warp drives had a somewhat fringe reputation (often associated with science fiction or questionable NASA experiments). Sabine Hossenfelder – a theoretical physicist known for her critical takes on sensational science – discussed the Bobrick & Martire paper in a video and blog post, essentially agreeing with its core messages. She noted the paper “is excellent in many aspects” and predicted it would pass peer review (which it did, being published in Classical and Quantum Gravity) (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!). Hossenfelder summarized their results for a broad audience, emphasizing the point that a warp bubble is just a shell of mass-energy and not a magical propulsion method (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!). Her explanation: “if you want superluminal motion, you need negative energy… If you want acceleration, you need to feed energy and momentum into the system. And the only reason the Alcubierre Drive moves faster than light is that one simply assumed it does” (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!). This framing echoed Bobrick & Martire’s conclusions and helped dispel common misconceptions. In essence, many experts saw the work as clarifying the fine print of warp drives: they can’t start from rest and outrun light without exotic matter or pre-existing conditions.
Energy Requirement Skepticism: Despite the positive reception of a physically-allowed warp bubble, scientists point out that “allowed” doesn’t mean “viable.” Physicist Erik Lentz, who independently found a similar positive-energy warp solution, himself acknowledged that the energy required is astronomically high – far beyond any practical scale (Free software lets you design and test warp drives with real physics). In a press release, Lentz calculated that a 100-meter diameter spaceship in a warp bubble at light-speed would need on the order of 10^47 J of energy, “hundreds of times the mass of Jupiter”, and said energy needs “would need to be reduced by ~30 orders of magnitude to be in range of modern nuclear fission reactors” (Free software lets you design and test warp drives with real physics). This sentiment is shared by others: even if no new physics is needed, the engineering is like asking us to manipulate mass-energies on a planetary or stellar scale. Some critics dryly note that if you have the ability to harness a Jupiter’s mass-energy, you might as well just travel slower or use other means! Until a revolutionary energy source or some unforeseen loophole is found, warp drives remain a theoretical curiosity. Thus, one critique is that Bobrick & Martire’s “feasible” warp bubble is still not really feasible for humans anytime soon – it’s just physically not ruled out.
No Experimental Path Yet: Related to the above, scientists like Marc Millis (former head of NASA’s Breakthrough Propulsion Physics Project) and others in advanced propulsion research often ask: what is the next step experimentally? Bobrick & Martire provided a framework, but no concrete avenue to implementationbeyond numerical simulations. Some have criticized media coverage for misinterpreting “physical warp drive” as “we can build this now.” In truth, as their paper states, it only shows it’s possible “in principle” with known physics ([2102.06824] Introducing Physical Warp Drives). There is a gap between theory and experiment. Sabine Hossenfelder also noted this, saying that while the work is exciting, “it may look now that you can’t do superluminal warp drives… this is only correct if General Relativity is correct. And maybe it is not” (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!). Her point being: perhaps new physics (like quantum gravity effects, dark energy, etc.) could change the story, but until then, we don’t have a way to create a warp bubble in practice. Some commentators argue that talk of warp drives could be premature until we have evidence of exotic matter or some intermediate experiment (like creating a tiny warp-like distortion) to guide development.
Alternate Perspectives on FTL: Within the theoretical physics community, there’s a mix of skepticism and open-mindedness about FTL schemes. Many relativists reiterate that FTL travel of any kind (warp drives, wormholes) implies causality violations in standard GR. Unless nature somehow prevents those paradoxes (e.g. through quantum effects destroying the bubble, or a theory like Novikov self-consistency), true FTL might be impossible. Matt Visser, Stefano Finazzi, Diego Sáez-Gómez, and others have written papers analyzing instabilities or paradoxes with warp drives () (). A prevailing critique is that even if Bobrick & Martire found a classical solution, once you consider quantum effects, it might destabilize. For instance, Finazzi et al. (2009) showed that a superluminal warp bubble would experience intense Hawking radiation at its horizons, potentially unravelling it (). Until a full quantum treatment of warp spacetimes is done, some physicists remain cautious: the bubble might be valid in classical theory but impossible once quantum back-reaction is accounted for. Bobrick & Martire’s work partly addresses this by sticking to subluminal speeds (no horizons, so those specific Hawking-like instabilities don’t arise), but any hint of approaching light speed might introduce huge stresses or radiation.
Critiques by Commentators: In online discussions, some physicists pointed out that **Bobrick & Martire’s subluminal drive is essentially a gravitationally-induced method of propulsion – which, in a sense, is already possible with known physics (e.g., using a planet’s gravity well to gain speed, or a Bussard ramjet using interstellar medium). What the warp bubble adds is the internal flat region and potential comfort for passengers. But if one doesn’t care about comfort or internal frame, a conventional rocket accelerating to relativistic speeds yields similar outcomes (subluminal rapid travel). Thus, a few critics downplay the significance: “It’s not warp drive in the sci-fi sense; it’s a new kind of gravitational configuration that still needs a drive”. That said, the counterargument is that free-fall travel in a warp bubble could allow much higher effective speeds without crushing the crew or cargo, which is a qualitative difference from a normal rocket.
Sabine Hossenfelder’s Critique/Explanation: Hossenfelder, beyond summarizing, also injected a dose of realism. She highlighted that Bobrick & Martire showed how to reduce negative energy requirements (flattening the bubble) (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!), but she noted this idea wasn’t entirely new – earlier researchers had discussed energy-saving shapes (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!). She also quipped that UFO lore often depicts flat “flying saucer” shapes moving belly-forward, which amusingly aligns with the idea that a flat shape in direction of travel is energy-efficient for warp (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!). More substantively, she stressed that the warp drive is not a warp “engine” – you still need propulsion. This addresses a common misunderstanding in popular articles that a warp bubble is a self-contained engine. Hossenfelder’s overall stance was cautiously positive: she found it “exciting” that warp drives might be possible in principle, but immediately tempered that by pointing out it’s only within classical GR. If GR is subtly wrong or incomplete (e.g., quantum gravity has surprises), the conclusions could change (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!).
Alternate Solutions and Debates: The warp drive community is small but active, and not without some disagreements. A bit of a debate brewed between Erik Lentz and Alexey Bobrick’s camps in 2021 over who had the truly “physical” solution. Both published within weeks of each other. Some commentators question aspects of Lentz’s approach (like whether it inadvertently hid negative energy), while others scrutinize Bobrick & Martire’s assumptions (like using spherical symmetry, which makes math easier but may not generalize to a ship-like shape). On platforms like Reddit and Physics Stack Exchange, users discussed whether superluminal warp bubbles really require “superluminal matter” (tachyons) or whether a clever scheduling of energy distribution could overcome that (general relativity – How is Alcubierre’s warp drive propeled forward? – Physics Stack Exchange) (general relativity – How is Alcubierre’s warp drive propeled forward? – Physics Stack Exchange). One Physics StackExchange user noted Bobrick & Martire provide no proof that FTL bubbles can’t be accelerated from subluminal – they assume it due to inertia, but perhaps a future theory could do it differently (general relativity – How is Alcubierre’s warp drive propeled forward? – Physics Stack Exchange). In reply, others argue that’s a minor point: within known physics, their reasoning is sound.
Ethan Siegel’s Commentary: Astrophysicist Ethan Siegel wrote a piece addressing hype about “the first warp bubble” (in context of Harold White’s claim). He clarified that no actual warp bubble has been created and that the Casimir effect result is very preliminary (I wrote the book on warp drive. We didn’t make a warp bubble.). Siegel, like many scientists, urges caution about sensational headlines. Regarding Bobrick & Martire’s work, writers like Siegel and astrophysicist Brian Koberlein have explained that it’s a promising step but still extremely far from practical. The consensus in critiques is “interesting theory, but we’re nowhere near implementation”.

In essence, the scientific response balances acknowledgment of the theoretical progress with skepticism about practicality and remaining unknowns. No significant voices are claiming Bobrick & Martire made an error—on the contrary, their mathematics is sound and contributes to the field. The critiques are more about context: reminding everyone that for now, warp drives remain on paper, and that extraordinary claims (like FTL travel) will require extraordinary evidence or new physics. Alternative perspectives also include the possibility that future breakthroughs (in quantum gravity, energy generation, or material science) might change the warp drive outlook dramatically – but those are beyond what Bobrick & Martire’s classical analysis can cover. As it stands, their work is a solid and sober piece of science that has been met with cautious intrigue rather than criticism of its validity. The most common “critique” is essentially: “Okay, we have a theoretically allowed warp bubble. Now how do we build it, and can we ever reduce the energy budget to something reasonable?” – questions that remain open.

6. Recent Developments in Warp Bubble Physics

The field of warp drive physics, though niche, has seen some notable developments in the last few years. Since Bobrick and Martire’s publication in 2021, interest has been rekindled and several new research efforts have emerged:

Follow-up Research by Applied Physics (2023–2024): Bobrick and Martire did not stop at the initial theoretical paper. In 2023, they collaborated with other scientists (J. Fuchs, C. Helmerich, L. Sellers, B. Melcher) to produce a detailed study of a “Constant Velocity Physical Warp Drive” solution (Constant Velocity Physical Warp Drive Solution). This work, building on their model, presented a full space-time solution for a warp bubble moving at constant subluminal speed with a stable matter shell. They numerically validated that this metric satisfies all energy conditions and is not a mere coordinate artifact (Constant Velocity Physical Warp Drive Solution). Essentially, it was a proof-of-concept example of the theory they outlined in 2021, worked out in explicit detail. The team demonstrated how to combine a thin shell of conventional matter with a warp-like shift vector to get a physical solution (Constant Velocity Physical Warp Drive Solution). This study was published and coincided with the release of Warp Factory, the open-source software mentioned earlier (Free software lets you design and test warp drives with real physics). Warp Factory allows researchers to plug in different warp bubble shapes and speeds, and compute the resulting stress-energy requirements. This tool has enabled a flurry of activity in simulating warp metrics. In April 2024, Bobrick, Martire, and colleagues also published “Analyzing Warp Drive Spacetimes with Warp Factory,” showcasing how the software can be used to evaluate known solutions like Alcubierre’s, Van Den Broeck’s, Natário’s, and Lentz’s side by side (Analyzing Warp Drive Spacetimes with Warp Factory – arXiv). This comparative analysis helps identify which metrics are truly distinct and which are equivalent under coordinate changes, further refining our understanding. The key development here is that warp drive research is becoming more computational and systematic, moving beyond pen-and-paper metrics to simulation-based studies. This could attract more researchers to test new ideas (since they can use the software rather than derive everything from scratch).
Erik Lentz and Independent Efforts: After the initial buzz in 2021, Erik Lentz has presumably been continuing his work (though specific papers beyond his first release are not widely reported, some conference proceedings might exist). His initial result was published in Classical and Quantum Gravity in 2022 after peer review. Lentz’s concept of warp solitons remains an active line of inquiry. A later development in 2022–2023 is that other physicists attempted to reproduce or analyze Lentz’s solution. Some found that if you allow certain kinds of tuning or modulation of the energy density in time, you might reduce the energy needed (akin to White’s oscillation idea). However, these remain theoretical tweaks. The consensus remains that energy requirement is the brick wall– no one has yet found a metric that is both physical (no exotic matter) and energetically reasonable. Lentz suggested that advances in high-energy density physics or maybe tapping vacuum energy could be needed to bridge the gap (Free software lets you design and test warp drives with real physics).
Laboratory Scale “Warp” Experiments: One of the most headline-grabbing developments was, as mentioned, Dr. Harold White’s accidental warp bubble discovery in 2021. In a paper in the European Physical Journal C, White and co-authors described a micro-scale structure (a specific Casimir cavity setup) that theoretically produces a negative energy density distribution matching a tiny warp bubble (DARPA Funded Researchers Accidentally Discover The World’s First Warp Bubble – The Debrief) (DARPA Funded Researchers Accidentally Discover The World’s First Warp Bubble – The Debrief). While this doesn’t mean warp field propulsion, it’s essentially a tiny warp-like distortion predicted by their model. As of the latest updates, White’s team at Limitless Space Institute has not yet experimentally confirmed this nano-warp bubble (they have not built the device, only simulated it) (DARPA Funded Researchers Accidentally Discover The World’s First Warp Bubble – The Debrief). The next step would be to actually construct the micron-sized sphere-and-cylinder apparatus and measure any anomalous effects, which will be challenging. Nonetheless, this is a development to watch: if confirmed, it would be the first experimental observation of a warp-related phenomenon, even if at a ridiculously small scale. It could validate some of the assumptions about how negative energy can be arranged in space.
Mainstream Scientific Discourse: Warp drives have edged slightly more into the mainstream of scientific discourse. For instance, American Institute of Aeronautics and Astronautics (AIAA) conferences have had sessions on advanced propulsion including warp metrics (White presented his findings at an AIAA forum). NASA’s TAE (Technology Area 8) roadmaps on propulsion now mention space warp theory as a long-term speculative area. There is also a privately funded organization, the Limitless Space Institute (LSI), which is sponsoring research into warp drive physics and related topics. LSI, with Dr. White’s involvement, aims to gradually push the envelope on things like quantum vacuum energy, Mach effects, and warp metrics – essentially keeping the field alive and inching toward experimental tests. This institutional interest is a development in itself: what was once purely theoretical now has a small but real research community and even some funding behind it.
Theoretical Advances and Extensions: On the theoretical front, researchers are exploring variations such as “hyper-fast” warp drives (ideas on how to theoretically make a warp bubble move even faster than Alcubierre’s original proposal, at the cost of other conditions) ([PDF] Hyper-Fast Positive Energy Warp Drives – ICRANet Indico platform). Others are looking at warp drives in different gravity theories. For example, as Bobrick & Martire cited, warp solutions in conformal gravity (a modified GR theory) can be achieved without negative energy (). In 2023, some studies considered warp drives in the context of higher-dimensional spacetimes or braneworld scenarios, where embedding our 4D spacetime in a higher-dimensional space might allow shortcuts (similar to wormholes). While not directly “warp bubble” papers, these contribute to the general knowledge of what spacetime geometries are possible. Additionally, a fascinating line of thought is examining the energy emissions or gravitational waves from a warp bubble. One 2022 paper looked at what kind of gravitational wave signature a collapsing warp bubble would produce (gravitational waveforms from warp drive collapse – arXiv), which not only is a fun theoretical exercise but could inform us about stability (if a bubble would collapse violently, what would happen?).
Public and Academic Dialogue: The warp drive topic has also been in the science communication spotlight. Articles in Scientific American, New Scientist, ExtremeTech, and other outlets have covered Bobrick & Martire’s and Lentz’s work, often with sensational titles but also with input from the scientists clarifying the limitations. This has prompted responses from experts to keep expectations realistic. In early 2022, for example, Big Think published an article titled “I wrote the book on warp drive. We didn’t make a warp bubble,” authored by physicist Richard Obousy or Eric Davis (who have worked on warp concepts). They clarified that White’s “warp bubble” is not a macroscopic warp drive and that we shouldn’t overhype incremental progress (I wrote the book on warp drive. We didn’t make a warp bubble.). Such discussions are actually a sign of a healthy scientific process – the fact that warp drive research is being debated in reputable forums means it’s taken somewhat more seriously now (compared to a decade ago when it was mostly on the fringe).
Continued Critique and Verification: Recent developments also include more scrutiny: independent scientists analyzing Bobrick & Martire’s and Lentz’s solutions to see if there are hidden flaws. So far, their results hold up under peer review. Sabine Hossenfelder revisited warp drives in a late 2023 YouTube video, discussing the new simulations (Warp Factory) and reiterating that no physics laws were broken in these new solutions, but also nothing fundamentally new was invented – it’s a reconfiguration of gravity we already understand. The tone in late 2024 is that warp drive physics is an interesting sandbox for relativists that may yield insights (even if not an engine soon). The community will likely focus on incremental steps: Can we find a configuration that reduces energy needs? Can we experimentally demonstrate a tiny aspect of warp tech (like White’s vacuum experiment)? Can numerical simulations reveal any instabilities or show how to steer a warp bubble?

In summary, since Bobrick and Martire’s pivotal work, warp drive research has become more active and systematic. We’ve seen new theoretical solutions (physical constant-velocity warp bubble, Lentz’s soliton), new computational tools (Warp Factory), provocative experimental proposals (nano Casimir warp), and increased academic discourse on the topic. While a functional warp drive is still a dream, these developments mark progress in understanding what a warp bubble really entails. Each small step – be it a refined metric, a simulation code, or a lab experiment – is bringing warp physics out of the purely hypothetical realm and into a more concrete form that scientists can test, criticize, and refine. The coming years will likely see further refinements of the theory (perhaps someone will find a clever way to cut the energy down by a few more orders of magnitude, or connect warp metrics with quantum gravity ideas), as well as attempts (however challenging) to detect or induce minute warp-like effects in controlled settings. The road is long, but the recent momentum suggests that the idea of warping spacetime for travel will remain under active study – moving inch by inch from science fiction toward scientific exploration.

References:

Bobrick, A., & Martire, G. (2021). Introducing Physical Warp Drives. Class. Quantum Grav. 38(10), 105009. ArXiv preprint arXiv:2102.06824. – Developed a general model for warp-drive spacetimes; showed subluminal, positive-energy warp bubbles are possible ([2102.06824] Introducing Physical Warp Drives) ().
Fuchs, J., et al. (2023). Constant Velocity Physical Warp Drive Solution. arXiv:2405.02709. – Numerical solution for a subluminal warp bubble with a matter shell satisfying energy conditions (Constant Velocity Physical Warp Drive Solution).
Alcubierre, M. (1994). The Warp Drive: Hyper-fast travel within general relativity. Class. Quantum Grav. 11(5), L73–L77. – Original proposal of a warp drive metric requiring negative energy ([2102.06824] Introducing Physical Warp Drives) ().
Pfenning, M. & Ford, L. (1997). The unphysical nature of “warp drive”. Class. Quantum Grav. 14, 1743. – Analysis of energy requirements, showing need for Planck-scale negative energy distribution () ().
Van Den Broeck, C. (1999). A ‘warp drive’ with more reasonable total energy requirements. Class. Quantum Grav. 16, 3973. – Modified Alcubierre geometry to reduce energy needs (bubble contraction) ().
Natário, J. (2002). Warp drive with zero expansion. Class. Quantum Grav. 19, 1157. – Warp solution without space expansion/compression ().
Lentz, E. (2021). Breaking the warp barrier: Hyper-fast solitons in Einstein–Maxwell-plasma theory. Class. Quantum Grav. 38, 075015. – Positive-energy “warp soliton” solution for potentially FTL travel (Free software lets you design and test warp drives with real physics) (Free software lets you design and test warp drives with real physics).
White, H. et al. (2021). Worldline numerics applied to custom Casimir geometry generates unanticipated intersection with Alcubierre warp metric. Eur. Phys. J. C 81, 677. – Proposed Casimir cavity that yields a nano-scale warp bubble energy distribution (DARPA Funded Researchers Accidentally Discover The World’s First Warp Bubble – The Debrief) (DARPA Funded Researchers Accidentally Discover The World’s First Warp Bubble – The Debrief).
New Atlas (2024). Free software lets you design and test warp drives with real physics (Free software lets you design and test warp drives with real physics) (Free software lets you design and test warp drives with real physics).
Sabine Hossenfelder (2020). Warp Drive News. Seriously! (Backreaction blog) – Commentary on Bobrick & Martire’s warp drive work and its implications (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!) (Sabine Hossenfelder: Backreaction: Warp Drive News. Seriously!).