These are not experimentally verified, but they're mathematically consistent within their own assumptions -FTL [ faster than light ] motion / Tachyon energy–momentum relation derivation. Credit CoPilot

Within special relativity, faster-than-light (FTL) travel is forbidden because it would require infinite energy as velocity approaches 

$cc$. But outside that framework, speculative physics and mathematics have proposed ways to describe FTL motion. These are not experimentally verified, but they are mathematically consistent within their own assumptions.

🚀 Examples of FTL Equations Beyond Special Relativity

1. Tachyonic Energy–Momentum Relation

For hypothetical particles called tachyons (always moving faster than light), the energy–momentum relation is written as:

$$E^2=p^2{c}^2-m^2{c}^{4}E^2\; =\; p^2\; c^2\; -\; m^2\; c^4$$

• Here (imaginary mass).

• Unlike normal matter, increasing energy reduces velocity toward $cc$ from above.

2. Alcubierre Warp Metric (General Relativity)

Miguel Alcubierre proposed a spacetime geometry where a “warp bubble” contracts space in front and expands it behind:

$$d{s}^2=-c^{2}d{t}^2+(dx-v_{s}f(r_s)dt{)}^2+d{y}^2+d{z}^{2}ds^2\; =\; -c^2\; dt^2\; +\; \backslash big(dx\; -\; v\_s\; f(r\_s)\; dt\backslash big)^2\; +\; dy^2\; +\; dz^2$$

• $v_{s}v\_s$: bubble velocity (can exceed $cc$)

• $f(r_s)f(r\_s)$: shaping function defining the bubble wall

• Locally, nothing moves faster than light, but the bubble can move superluminally.

3. de Broglie Phase Velocity

For quantum waves, the phase velocity can exceed $cc$:

$$v_p=\frac{\omega }{k}=\frac{c^2}{v_g}v\_p\; =\; \backslash frac\{\backslash omega\}\{k\}\; =\; \backslash frac\{c^2\}\{v\_g\}$$

• $v_{g}v\_g$: group velocity (signal speed, always $\le c\backslash leq\; c$)

• $v_{p}v\_p$: phase velocity (can be $>c>\; c$, but carries no information).

4. Hypothetical Superluminal Lorentz Transformation

If one extends Lorentz transformations to $v>cv\; >\; c$, one gets:

$$t^{\mathrm{\prime }}=\gamma (t-\frac{vx}{c^2}),x^{\mathrm{\prime }}=\gamma (x-vt)t\text{'}\; =\; \backslash gamma\; \backslash left(t\; -\; \backslash frac\{vx\}\{c^2\}\backslash right),\; \backslash quad\; x\text{'}\; =\; \backslash gamma\; (x\; -\; vt)$$

with

$$\gamma =\frac{1}{\sqrt{\frac{v^2}{c^2}-1}}\backslash gamma\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash frac\{v^2\}\{c^2\}\; -\; 1\}\}$$

• This requires $v>cv\; >\; c$, making $\gamma \backslash gamma$ real.

• It flips causal structure, leading to paradoxes.

⚖️ Key Takeaway

• Special relativity forbids FTL for real particles with real mass.

• Outside it, equations exist (tachyons, warp drives, extended Lorentz transforms), but they are speculative and often require exotic matter or imaginary mass.

Tachyon energy–momentum relation derivation

Setup and assumptions

• Start from relativistic action: For a free particle, the invariant mass-shell condition is

$$p^{\mu }{p}_{\mu }=-m^2{c}^{2}p^\backslash mu\; p\_\backslash mu\; =\; -m^2\; c^2$$

where $p^{\mu }=(E\mathrm{/}c,\mathbf{p})p^\backslash mu\; =\; (E/c,\; \backslash mathbf\{p\})$ and the metric signature is $(-+++)(-+++)$. This is the standard relativistic relation we will analytically continue to tachyonic mass.

• Key idea: For tachyons, take . Write $m^2=-{\mu }^{2}m^2\; =\; -\backslash mu^2$ with $\mu >0\backslash mu\; >\; 0$. The “mass” becomes imaginary, $m=i\mu m\; =\; i\backslash mu$, but $m^{2}m^2$ is the physically relevant invariant.

Step 1: Expand the mass-shell condition

• Four-momentum norm:

$$p^{\mu }{p}_{\mu }=-\frac{E^2}{c^2}+{\mathbf{p}}^{2}p^\backslash mu\; p\_\backslash mu\; =\; -\backslash frac\{E^2\}\{c^2\}\; +\; \backslash mathbf\{p\}^2$$

• Set equal to invariant:

$$-\frac{E^2}{c^2}+{\mathbf{p}}^2=-m^2{c}^2-\backslash frac\{E^2\}\{c^2\}\; +\; \backslash mathbf\{p\}^2\; =\; -m^2\; c^2$$

Step 2: Solve for energy and substitute tachyonic mass

• Solve for $E^{2}E^2$:

$$\frac{E^2}{c^2}={\mathbf{p}}^2+m^2{c}^2\Rightarrow E^2={\mathbf{p}}^2{c}^2+m^2{c}^4\backslash frac\{E^2\}\{c^2\}\; =\; \backslash mathbf\{p\}^2\; +\; m^2\; c^2\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; E^2\; =\; \backslash mathbf\{p\}^2\; c^2\; +\; m^2\; c^4$$

• Analytic continuation to $m^2=-{\mu }^{2}m^2\; =\; -\backslash mu^2$:

$$E^2={\mathbf{p}}^2{c}^2-{\mu }^2{c}^{4}E^2\; =\; \backslash mathbf\{p\}^2\; c^2\; -\; \backslash mu^2\; c^4$$

Step 3: Interpret the result

• Energy–momentum relation (tachyon):

$$E^2=p^2{c}^2-{\mu }^2{c}^{4}E^2\; =\; p^2\; c^2\; -\; \backslash mu^2\; c^4$$

• Consequences:

  • Real energy requires: $p^2{c}^2\ge {\mu }^2{c}^{4}p^2\; c^2\; \backslash ge\; \backslash mu^2\; c^4$, i.e., sufficiently large momentum.

  • Speed relation: Using $\mathbf{v}=\mathrm{\partial }E\mathrm{/}\mathrm{\partial }\mathbf{p}\backslash mathbf\{v\}\; =\; \backslash partial\; E\; /\; \backslash partial\; \backslash mathbf\{p\}$ and the above dispersion, one finds $\mathrm{\mid }\mathbf{v}\mathrm{\mid }>c|\backslash mathbf\{v\}|\; >\; c$. As energy increases, speed asymptotically approaches $cc$ from above.

  • No rest frame: Setting $\mathbf{p}=0\backslash mathbf\{p\}=0$ makes $EE$ imaginary, so tachyons cannot be at rest.

Alcubierre warp metric construction (outline)

Goal and intuition

• Goal: Construct a spacetime metric where a “warp bubble” moves with coordinate velocity $v_s(t)v\_s(t)$, compressing space ahead and expanding behind, allowing superluminal effective transport without locally exceeding $cc$.

• Intuition: Keep local light cones intact; move the bubble by shaping the spatial metric with a function that is sharply peaked at the bubble wall.

Step 1: Choose a comoving bubble center

• Bubble center worldline:

$$x_s(t)={\int }^t{v}_s(\tau )d\tau x\_s(t)\; =\; \backslash int^t\; v\_s(\backslash tau)\backslash ,\; d\backslash tau$$

• Radial coordinate around bubble:

$$r_s=\sqrt{(x-x_s(t){)}^2+y^2+z^2}r\_s\; =\; \backslash sqrt\{(x\; -\; x\_s(t))^2\; +\; y^2\; +\; z^2\}$$

Step 2: Define a shaping function

• Shaping function properties: Smooth, compact-like support around the bubble wall; $f(0)\approx 1f(0)\; \backslash approx\; 1$, decays to $00$ away from the bubble.

• Example (one of many):

$$f(r_s)=\frac{\tanh (\sigma (r_s+R))-\tanh (\sigma (r_s-R))}{2\tanh (\sigma R)}f(r\_s)\; =\; \backslash frac\{\backslash tanh\backslash left(\backslash sigma\; (r\_s\; +\; R)\backslash right)\; -\; \backslash tanh\backslash left(\backslash sigma\; (r\_s\; -\; R)\backslash right)\}\{2\; \backslash tanh(\backslash sigma\; R)\}$$

• Parameters: $RR$ sets bubble radius; $\sigma \backslash sigma$ controls wall thickness.

Step 3: Write the metric ansatz

• Alcubierre line element:

$$d{s}^2=-c^{2}d{t}^2+{(dx-v_s(t)f(r_s)dt)}^2+d{y}^2+d{z}^{2}ds^2\; =\; -c^2\; dt^2\; +\; \backslash left(dx\; -\; v\_s(t)\backslash ,\; f(r\_s)\backslash ,\; dt\backslash right)^2\; +\; dy^2\; +\; dz^2$$

• Meaning: Spatial slices are deformed so that points inside the bubble are effectively carried along at speed $v_{s}v\_s$, while distant regions remain Minkowski.

Step 4: Check local light cones

• Local speed bound: For any local inertial observer, null condition $d{s}^2=0ds^2=0$ keeps the local signal speed at $cc$.

• Global motion: The bubble’s center moves at $v_{s}v\_s$, which can exceed $cc$ without locally violating causality in the bubble interior.

Step 5: Compute stress–energy (implications)

• Einstein equations: Insert the metric into $G_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }G\_\{\backslash mu\backslash nu\}\; =\; \backslash frac\{8\backslash pi\; G\}\{c^4\}\; T\_\{\backslash mu\backslash nu\}$ to find the required $T_{\mu \nu }T\_\{\backslash mu\backslash nu\}$.

• Outcome: The solution generally requires regions with negative energy density relative to some observers (violating classical energy conditions), implying “exotic matter” is needed.

Notes on physical viability

• Tachyons: The dispersion is mathematically consistent but leads to causality issues and instability in interacting field theories; modern physics treats tachyonic mass terms as signals of symmetry breaking rather than real FTL particles.

• Warp metric: It is an exact GR solution given the chosen stress–energy, but the exotic matter requirement, enormous energy scales, and potential quantum instabilities make it speculative.