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Corrections and improvements to physics section of Warp drive

(new) Warp drives in general relativity

In theoretical physics, a description of warp drives as encountered in science fiction has been introduced in the context of general relativity by Miguel Alcubierre in 1994, which was later generalized by José Natário in 2002. In this description, a warp drive is a certain, Lorentzian metric g. While faithful to the portrayal in science-fiction, these metrics have features calling into question whether they actually describe anything physically meaningful: They can contain closed, time-like curves (a general relativistic notion of time travel), they are reverse-engineered and hence usually violate initial value formulations of general relativity, and they violate many energy conditions. Below, this will be explained in some more detail.

The Lorentz factor γ as a function of velocity v. For vanishing velocity, v = 0, its value is 1; as velocity approaches the speed of light, (v → c), γ increases without bound, (γ → ∞).

According to Newton's laws of motion, there would be no limitations on speeds achievable. However, at higher speeds, Einstein's theory of special relativity becomes relevant, which has been extensively confirmed by experiments. Most easily, the difference is seen when comparing the kinetic energies of particles moving at a given speed x: For Newtonian physics, this is $E={\tfrac {1}{2}}m_{0}v^{2}$. In special relativity, the energy instead is given by $E=(\gamma -1)m_{0}c^{2}$, with c the speed of light in vacuum, and $\gamma =1/{\sqrt {1-v^{2}/c^{2}}}$ the Lorentz factor. For low speeds small compared to the speed of light in vacuum, the relativistic formula comes close to the Newtonian one. For speeds approaching c from below, the Lorentz factor and so also the relativistic kinetic energy become arbitrarily large. This implies that in special relativity, an infinite amount of energy would be necessary to even achieve the speed of light c.

Einsteins's theory of general relativity extends special relativity so that one can formulate a relativistic theory of gravitation. Like special relativity, also general relativity has been tested successfully repeatedly. Special relativity is a kinematical theory describing geometric relations between physical quantities. General relativity, in contrast, is a dynamical theory: It describes the evolution of a specific physical object, the metric tensor. The metric tensor together with a (four-dimensional) manifold forms a mathematical model of spacetime, and its curvature is connected to gravitation. The dynamics of the spacetime metric are given by the Einstein field equations, $$G_{\mu \nu }+\Lambda g_{\mu \nu }={\frac {8\pi G_{\text{N}}}{c^{4}}}T_{\mu \nu },$$ where G_μν is the Einstein tensor, g_μν is the metric tensor, T_μν is the stress–energy tensor, Λ is the cosmological constant and $G_{\text{N}}$ is the gravitational constant. The Einstein tensor is a combination of partial derivatives of the metric tensor up to second order, so that the Einstein field equations are a (non-linear) system of partial differential equations. Usually, these are integrated by specifying the stress-energy tensor and solving for the unknown metric. Warp drive metrics, however, are solved in reverse: The metric is specified and the stress-energy tensor is initially unknown, and straightforwardly calculated by differentiation of the metric. This avoids the complicated mathematics involved in solving non-linear partial differential equations, but then the question is whether the calculated stress-energy tensor and the prescribed metric are physically meaningful.

The original metric of Alcubierre describes a bubble in spacetime, for simplicity's sake assumed to move in the x-direction, and of spherical shape from the center of the bubble. It is given as: $$dx^{2}=-c^{2}dt^{2}+\left(dx-v_{s}f(r_{s})dt\right)^{2}+dy^{2}+dz^{2}$$ where:

• $v_{s}$ is the speed of the warp bubble,
• $f(r_{s})$ is a shape function that determines the shape and size of the bubble,
• $r_{s}={\sqrt {(x-x_{s}(t))^{2}+y^{2}+z^{2}}}$is the radial distance from the bubble's center.

The shape function $f(r_{s})$ is demanded to be $0$ far away from the bubble's center as well as near the bubble's center. The form of the resulting metric is similar to that of a Painlevé–Gullstrand metric.

(old) Einstein's space warp and real-world physics

Lorentz factor γ as a function of velocity. Its initial value is 1 (when v = 0); and as velocity approaches the speed of light (v → c) γ increases without bound (γ → ∞).

Einstein's theory of special relativity states that speed of light travel is impossible for material objects that, unlike photons, have a non-zero rest mass. The problem of a material object exceeding light speed is that an infinite amount of kinetic energy would be required to travel at exactly the speed of light. Warp drives are one of the science-fiction tropes that serve to circumvent this limitation in fiction to facilitate stories set at galactic scales. However, the concept of space warp has been criticized as "illogical", and has been connected to several other rubber science ideas that do not fit into our current understanding of physics, such as the transporters and replicators in Star Trek, antigravity or negative mass.

Some argue that these effects mean that although it's not possible to travel faster than the speed of light, both space and time "warp" to allow travelling the distance of one light year, in less than a year. Although it is not possible to travel faster than the speed of light, the effective speed is faster than light. This warping of space and time is precisely mathematically specified by the Lorentz factor, which depends on velocity. Although only theoretical when published over 100 years ago, the effect has since been measured and confirmed many times. In the limit, at light speed time stops completely (relative to a certain reference frame) and it is possible to travel infinite distances across space with no passage of time.

The solution to Einstein’s field equations proposed by Alcubierre defines a spacetime metric—known as the Alcubierre metric—where spacetime itself is distorted in a controlled manner. The metric creates a region of compressed spacetime in front of the spacecraft and expanded spacetime behind it, forming a "warp bubble." The spacecraft resides within this bubble, moving with the local spacetime without experiencing relativistic time dilation or violating causality.

Mathematically, the Alcubierre metric is expressed as:

$dx^{2}=-c^{2}dt^{2}+\left(dx-v_{s}f(r_{s})dt\right)^{2}+dy^{2}+dz^{2}$

where:

• $v_{s}$ is the speed of the warp bubble,
• $f(r_{s})$ is a shape function that determines the smoothness and size of the bubble,
• $r_{s}={\sqrt {(x-x_{s}(t))^{2}+y^{2}+z^{2}}}$is the radial distance from the spacecraft center.

The creation of such a bubble requires exotic matter—substances with negative energy density (a violation of the Weak Energy Condition). Casimir effect experiments have hinted at the existence of negative energy in quantum fields, but practical production at the required scale remains speculative.

Similarly, a quantum drive might utilize quantum field theory to exploit vacuum energy, leveraging phenomena such as the zero-point energy of the quantum vacuum. The drive could theoretically manipulate virtual particle pairs or create localized energy gradients via quantum entanglement. The energy density of the quantum vacuum, often estimated as 10113J/m3 in the context of quantum field fluctuations, could provide immense power if harnessed effectively.

Although the concept of warp drive has originated in fiction, it has received some scientific consideration, most notably related to the 1990s concept of the Alcubierre drive. Alcubierre stated in an email to William Shatner that his theory was directly inspired by the term used in the TV series Star Trek and cites the "'warp drive' of science fiction" in his 1994 article.

In 2021, DARPA-funded researcher Harold White, of the Limitless Space Institute, claimed that he had succeeded in creating a real warp bubble, saying "our detailed numerical analysis of our custom Casimir cavities helped us identify a real and manufacturable nano/microstructure that is predicted to generate a negative vacuum energy density such that it would manifest a real nanoscale warp bubble, not an analog, but the real thing."