No Hyperfast Travel

As a corollary of Special Relativity and Quantum Field Theory, superluminal propagations and tachyonic particles are forbidden. More technically correct, no momentum flux can be of a greater magnitude than its energy density (ρ² ≥ P²). This rule is referred to as the Flux Energy Condition (FEC), which prohibits all hyperfast travel metrics. This can be intuitively grasped in that, these metrics offload the work of passing through spacetime faster than is allowed by Special Relativity onto gravity (still violating the FEC).

Shifting the burden of a momentum flux that is greater than one's energy density onto gravity (i.e., spacetime curvature) will always be unsuccessful. General Relativity is mathematically deduced from Special Relativity by adopting the equivalence principle (i.e., that inertial and gravitational masses are equivalent). General Relativity respects the energy-momentum relation of Special Relativity, from where the FEC is deduced. So, spacetime geometries and energy-momentum fluxes are two sides of equivalent phenomena.

So, even with the acquisition of gravity control technologies by using magnetic monopoles from the Iota Persei star system, wormholes and warp drives remain unphysical.

Terminology

Throughout this article the phrase 'hyperfast travel' is used instead of the phrase 'faster-than-light travel'. This is to reflect the original inspiration behind spacetime shortcuts rooted in General Relativity: they provide effective but not genuine FTL travel. Hence the term 'hyperfast' is used, as coined by Miguel Alcubierre in 1994.

Physics of Hyperfast Travel

No Superluminal Tachyons

Just by looking at the math of Special Relativity, while nothing with real positive mass (like ourselves) can reach or surpass lightspeed locally, if one envisions a particle with a negative mass squared term (-m², or an imaginary invariant mass √-1) these "tachyons" always travel FTL or have spacelike worldlines. While the mathematics of Special Relativity allow for such a bizarre phenomenon, Quantum Field Theory does not. But before delving into Quantum Field Theory, let's look at some of the bizarre behavior of superluminal tachyons anyway, relying on Special Relativity by itself.

Let's make use of the energy-momentum relation from Special Relativity (E² - p² = m², where p is momentum), where for particles with positive real mass always include the (spatial) momentum as part of their total energy density (E² = p² + m²). For massless particles, their energy is equal to their momentum (E² = p²). What of superluminal particles, or spacelike tachyons with a negative mass squared term (-m²)? Well, the momentum is greater than the total energy density (E² - p² < 0). This again confirms the spacelike nature of tachyons, since there is a spacelike or superluminal flow of momentum. So, we can map these relationships in terms of a space-time diagram, where E represents the temporal axis, and p represents a spatial axis. Causality is bounded into the past and future "light-cones" (the top and bottom shaded regions of the diagram below), with the superluminal "elsewhere" outside of the light-cones (white right and left regions). Let's summarize:

Timelike or Real Mass: E² > p²
Lightlike or Massless: E² = p²
Spacelike or Tachyonic: p² > E²

by Hokon

The sequence of spacelike separated events is frame-dependent. Meaning, what appears as a positive energy tachyon moving forward in time, in another frame of reference, appears as a negative energy tachyon moving backwards in time. So, what is a tachyon emitter in one frame of reference is a tachyon receiver in another. Causality only holds within light-cones (where E² ≥ p²). This "double-truth" concerning tachyons will be important later.

Before moving onto Quantum Field Theory, let's examine a key component of the math of Special Relativity: the Stress Energy Tensor (T^μν). The Stress Energy Tensor mathematically "maps" the energy-momentum distribution in spacetime. The four-by-four matrix below accounts for this distribution across time and our three spatial dimensions (0, 1, 2, 3), such that one can translate between different frames of reference while preserving the same properties and laws of physics (i.e., Lorentz Invariance).

It is typical to interpret the components of the Stress Energy Tensor in the terms of a perfect fluid, allowing one to import the relations above regarding energy and momentum. So, 'ρ' is the relativistic energy density (the time-time component or T⁰⁰) and each 'P_n' is the pressure or the spatial momentum flux perpendicular to each spatial dimension:

Timelike Flux: ρ² > P_n²
Lightlike Flux: ρ² = P_n²
Spacelike Flux: P_n² > ρ²

by Hokon

One should now be able to see that there are certain "energy conditions" implied by these correlations within the mathematics of Special Relativity. By invoking the equivalence of inertial mass and gravitational mass, the Stress-Energy Tensor becomes the source of spacetime curvature in General Relativity. It must be noted, that the relationship between the metric tensor (g_μν) and the Stress-Energy Tensor (T^μν) is highly nonlinear. This relation between spacetime geometry and energy-momentum distribution imputes relativistic energy conditions to General Relativity. Now, to Quantum Field Theory's understanding of tachyons.

Quantum fields with a negative mass squared term, tachyonic fields, do not have localized excitations that propagate at superluminal speeds (i.e., no superluminal particles). Instead, tachyonic fields are an instability of the vacuum, and have an exponential cascade of subluminal energies (i.e., timelike energies: ρ² > P²) into a (more) stable vacuum. This aspect of Quantum Field Theory was confirmed in 2013 with the discovery of the Higgs Boson at CERN, since tachyon condensation is the mechanism by which the Higgs field potential reached its current vacuum expectation value (VEV) of 246 GeV. Since every particle is an excitation of a quantum field, and not even tachyonic fields have superluminal excitations, we can say with confidence that Quantum Field Theory forbids superluminal behavior.

Flux Energy Condition (FEC)

We can finally express our first energy condition, which acts as a guiding rule when creating spacetime metrics in General Relativity. The Flux Energy Condition (FEC) can be expressed in terms of perfect fluids such that ρ² ≥ P². That is, should this inequality be violated, one then has an energy-momentum distribution where pressure (or momentum flux) is traveling at superluminal velocities. This can be physically expressed as asserting that the speed of sound cannot be superluminal. That is, pressures and their disturbances must obey relativistic limits.

Ultimately, the FEC is a corollary of Special Relativity and Quantum Field Theory, and hence belongs to the scientific canon. That is, the FEC ensures that no energy or momentum flows outside of one's light cones (i.e., into one's elsewhere), preserving causality.

Hyperfast Travel: Wormholes

That traversable wormholes require the dominance of repulsive gravitation can be easily understood with the following analysis. Normally, positive mass-energy warps spacetime such that light-rays converge after passing near a massive object (e.g., a star). However, light-rays initially parallel inside a spherical wormhole's throat must diverge upon exiting: repulsive gravity is what keeps a wormhole open or traversable (i.e., Raychaudhuri's Theorem). That is, a wormhole is made traversable by the dominance of a repulsive gravitational effect along their radial (r) coordinate. Without this negative radial pressure or radial tension (-P = τ), wormholes will close or "pinch off" faster than lightspeed particles could ever traverse them.

by Kip Thorne

It's important to note that there were already regions of repulsive gravity within all black hole metrics except the Schwarzschild metric, but these were sourced and offset by an equal positive energy density (e.g., the angular momentum that exerts a repulsive gravity below the inner Cauchy horizon of the Kerr metric has a positive energy density, same for the electric charge of the Reissner–Nordström metric). Hence, black holes were never traversable wormholes. Truly traversable wormholes need an overwhelmingly negative radial pressure (P < 0 and P² > ρ²), which in some accelerated frames of reference the energy density also becomes negative (ρ < 0).

Now, the magnitude of the pressure can be significantly reduced in volume and easily traversable if the energy density is allowed to be negative (ρ < 0) in the wormhole's rest frame. However, the magnitude of the radial pressure remains larger than the energy density's, violating the FEC. This is exactly what we should expect, since the negative energy density is doing most of the work in generating a dominance of repulsive gravity that keeps a wormhole traversable.

Visser Planar Wormholes

Matt Visser, a mathematical physicist of the late 20^th / early 21^st centuries, came up with a clever method to mathematically construct wormhole geometries, hoping to reduce the amount of negative energy and negative pressure. While these mathematically work, and even satisfy the FEC, the "cut and paste" procedure used to produce the metrics is entirely abstract. That is, there is no physical mechanism by which one can "cut out" segments of spacetime and "glue" them together. Either one makes a wormhole at the macroscopic level by inducing and fulfilling the "flare-out" condition (which requires FEC violating material), or by enlarging a wormhole "out of the quantum foam" (which also requires FEC violating material).

Visser planar wormholes are interesting thought experiments, since they exhibit geometries for hyperfast travel without violating the FEC. Yet, they ultimately ignore the dynamics of wormhole construction: they bypass violating the FEC by a priori abstracting away the physical obstacles to traversable wormhole formation.

Hyperfast Travel: Warp Drives

The first warp drive metric, by Alcubierre, works by relocating a "bubble" of spacetime by expansion at the back of the bubble and contraction at the front of the bubble. The negative energy density in the bubble's walls are perpendicular to the direction of motion. That is, perpendicular to the expansion and contraction. This means that where the magnitude of the energy density is largest, the metric obeys the Flux Energy Condition. Yet, where the expansion and contraction become their strongest in the direction of motion, there is a minimal energy density and an overwhelming positive or negative pressure (P² > ρ²), the sign of the pressure depending on whether it is the expansive or contractive end.

by Miguel Alcubierre

While warp drive metrics have been constructed that avoid the expansion/contraction dynamic, the momentum flux at the front and back ends still exceeds the energy densities' magnitudes there. Which is what we should expect: the warp drive does its work by using a negative energy density to achieve superluminal propulsion, by getting spacetime to behave in a spacelike manner (through gravitational masses possessing superluminal momentum fluxes).

Physical Subluminal Drives

What if we remove the requirement that a warp drive is used for superluminal travel? Not only can one avoid the use of negative energy densities, but one also has a FEC-compliant Stress-Energy Tensor! Such a subluminal warp drive is entirely physical, even when appealing to the archaic Dominant Energy Condition (DEC) of the 20^th century (ρ ≥ |P|). Sadly, such drives require astronomical quantities of mass-energy and pressure (≈ 4.49 × 10²⁷ kg). So, no implementation of these physical and subluminal warp drives has taken place, and will not for the foreseeable future.