Step-by-step display, a few equations: mathematical frameworks for FTL ( Faster than light ) travel / A concrete, worked-out numerical example using the tachyonic energy–momentum relation.

 



Alright — here’s the key: outside the framework of special relativity, we’re no longer bound by its core postulate that the speed of light 

cc is the maximum speed for information or matter.
So if we ignore that postulate, we can freely write equations that allow v>cv > c.

Let’s go step-by-step and display a few such equations clearly.

1️⃣ Simple Newtonian-style velocity equation

In classical (pre-relativistic) physics, velocity has no upper limit.
So you can simply write:

v=dxdtv = \frac{dx}{dt}

and nothing prevents v>cv > c.
That’s the most basic “faster-than-light” expression outside relativity.

2️⃣ “Superluminal factor” version

We can define a dimensionless ratio:

β=vc\beta = \frac{v}{c}

In relativity, β<1\beta < 1.
Outside it, we may allow:

β>1\beta > 1

That’s a compact mathematical way of saying “faster than light”.

3️⃣ Hypothetical “tachyonic” energy–momentum relation (non-relativistic form)

If we abandon relativity but keep the same structure of energy–momentum relations, we might write:

E2=p2c2+m2c4E^2 = p^2 c^2 + m^2 c^4

Then, for a hypothetical superluminal particle (tachyon), one can flip the sign of m2m^2:

E2=p2c2−μ2c4E^2 = p^2 c^2 - \mu^2 c^4

where μ\mu is called the imaginary mass parameter.
That yields:

v=dEdp=pc2E>cv = \frac{dE}{dp} = \frac{pc^2}{E} > c

Note that this equation formally predicts v>cv > c — but it only makes sense outside special relativity or in speculative extensions.

4️⃣ “Warp metric” form (general-relativistic extension, but outside SR constraints)

In models like Alcubierre’s warp drive, local light speed is never broken, but globally you can have FTL-like motion.
Mathematically:

ds2=−c2dt2+[dx−vs(t)f(rs)dt]2+dy2+dz2ds^2 = -c^2 dt^2 + [dx - v_s(t) f(r_s) dt]^2 + dy^2 + dz^2

If the “warp velocity” vs(t)>cv_s(t) > c, the apparent motion of the bubble exceeds light speed — again, beyond SR limits.


A concrete, worked-out numerical example using the tachyonic energy–momentum relation.

We’ll use the equation (a speculative “tachyon” form obtained by flipping the sign of m2m^2):

E2  =  p2c2  −  μ2c4E^2 \;=\; p^2 c^2 \;-\; \mu^2 c^4

where:

  • EE is the energy (Joules),

  • pp is the momentum (kg·m/s),

  • cc is the speed of light, c=3.00×108 m/sc = 3.00\times10^8\ \mathrm{m/s},

  • μ\mu is the tachyonic mass parameter (kg). For tachyons μ\mu is often thought of as an imaginary rest mass in SR language — here we treat μ\mu as a positive parameter so the sign flip yields superluminal behaviour.

Velocity (group velocity) for this dispersion is

v  =  dEdp  =  pc2E.v \;=\; \frac{dE}{dp} \;=\; \frac{p c^2}{E}.

We will pick a tiny μ\mu so the algebra is simple but the effect is visible:

μ=1.00×10−27 kg.\mu = 1.00\times 10^{-27}\ \mathrm{kg}.

Precompute constants (digit-by-digit style):

  1. c=3.00×108 m/sc = 3.00\times 10^{8}\ \mathrm{m/s}.

  2. c2=(3.00×108)2=9.00×1016 m2/s2.c^2 = (3.00\times10^8)^2 = 9.00\times10^{16}\ \mathrm{m^2/s^2}.

  3. c4=(c2)2=(9.00×1016)2=81.00×1032=8.10×1033.c^4 = (c^2)^2 = (9.00\times10^{16})^2 = 81.00\times10^{32} = 8.10\times10^{33}.

  4. μ2=(1.00×10−27)2=1.00×10−54 kg2.\mu^2 = (1.00\times10^{-27})^2 = 1.00\times10^{-54}\ \mathrm{kg^2}.

  5. μ2c4=1.00×10−54×8.10×1033=8.10×10−21 (J2).\mu^2 c^4 = 1.00\times10^{-54}\times 8.10\times10^{33} = 8.10\times10^{-21}\ \text{(J}^2\text{)}.

(Units: p2c2p^2 c^2 and μ2c4\mu^2 c^4 both have units of J2^2.)

We will evaluate three sample momenta (all > μc\mu c, to keep E2>0E^2>0):

  • p1=1.00×10−18 kg⋅m/sp_1 = 1.00\times10^{-18}\ \mathrm{kg\cdot m/s}

  • p2=1.00×10−17 kg⋅m/sp_2 = 1.00\times10^{-17}\ \mathrm{kg\cdot m/s}

  • p3=1.00×10−16 kg⋅m/sp_3 = 1.00\times10^{-16}\ \mathrm{kg\cdot m/s}

Now compute for each pp: p2c2p^2 c^2, E2E^2, EE, and v=pc2Ev = \dfrac{p c^2}{E}.

Case 1 — p1=1.00×10−18p_1 = 1.00\times10^{-18}

  1. p12=(1.00×10−18)2=1.00×10−36.p_1^2 = (1.00\times10^{-18})^2 = 1.00\times10^{-36}.

  2. p12c2=1.00×10−36×9.00×1016=9.00×10−20 J2.p_1^2 c^2 = 1.00\times10^{-36}\times 9.00\times10^{16} = 9.00\times10^{-20}\ \mathrm{J^2}.

  3. E2=p12c2−μ2c4=9.00×10−20−8.10×10−21=(9.00−0.81)×10−20=8.19×10−20 J2.E^2 = p_1^2 c^2 - \mu^2 c^4 = 9.00\times10^{-20} - 8.10\times10^{-21} = (9.00 - 0.81)\times10^{-20} = 8.19\times10^{-20}\ \mathrm{J^2}.

  4. E=8.19×10−20=8.19×10−10≈2.861×10−10 J.E = \sqrt{8.19\times10^{-20}} = \sqrt{8.19}\times10^{-10} \approx 2.861\times10^{-10}\ \mathrm{J}.

    • (Because 8.19≈2.861\sqrt{8.19}\approx2.861 and 10−20=10−10\sqrt{10^{-20}}=10^{-10}.)

  5. Numerator for vv: p1c2=1.00×10−18×9.00×1016=9.00×10−2=0.0900.p_1 c^2 = 1.00\times10^{-18}\times9.00\times10^{16} = 9.00\times10^{-2} = 0.0900.

  6. v1=0.09002.861×10−10=9.00×10−22.861×10−10.v_1 = \dfrac{0.0900}{2.861\times10^{-10}} = \dfrac{9.00\times10^{-2}}{2.861\times10^{-10}}.

Compute ratio:

9.00×10−22.861×10−10=9.002.861×108≈3.146×108 m/s.\frac{9.00\times10^{-2}}{2.861\times10^{-10}} = \frac{9.00}{2.861}\times10^{8} \approx 3.146\times10^{8}\ \mathrm{m/s}.

Compare to c=3.00×108 m/sc = 3.00\times10^{8}\ \mathrm{m/s}: v1≈3.146×108>cv_1 \approx 3.146\times10^{8} > c.

Case 2 — p2=1.00×10−17p_2 = 1.00\times10^{-17}

  1. p22=1.00×10−34.p_2^2 = 1.00\times10^{-34}.

  2. p22c2=1.00×10−34×9.00×1016=9.00×10−18 J2.p_2^2 c^2 = 1.00\times10^{-34}\times9.00\times10^{16} = 9.00\times10^{-18}\ \mathrm{J^2}.

  3. E2=9.00×10−18−8.10×10−21.E^2 = 9.00\times10^{-18} - 8.10\times10^{-21}.
    Convert to same exponent: 9.00×10−18=9000×10−21.9.00\times10^{-18} = 9000\times10^{-21}.
    So E2=(9000−8.10)×10−21=8991.90×10−21=8.99190×10−18.E^2 = (9000 - 8.10)\times10^{-21} = 8991.90\times10^{-21} = 8.99190\times10^{-18}.

  4. E=8.99190×10−18=8.99190×10−9≈2.999×10−9 J.E = \sqrt{8.99190\times10^{-18}} = \sqrt{8.99190}\times10^{-9} \approx 2.999\times10^{-9}\ \mathrm{J}.
    (8.99190≈2.999\sqrt{8.99190}\approx2.999.)

  5. Numerator: p2c2=1.00×10−17×9.00×1016=9.00×10−1=0.900.p_2 c^2 = 1.00\times10^{-17}\times9.00\times10^{16} = 9.00\times10^{-1} = 0.900.

  6. v2=0.9002.999×10−9=9.00×10−12.999×10−9.v_2 = \dfrac{0.900}{2.999\times10^{-9}} = \dfrac{9.00\times10^{-1}}{2.999\times10^{-9}}.

Compute ratio:

9.00×10−12.999×10−9=9.002.999×108≈3.001×108 m/s.\frac{9.00\times10^{-1}}{2.999\times10^{-9}} = \frac{9.00}{2.999}\times10^{8} \approx 3.001\times10^{8}\ \mathrm{m/s}.

So v2≈3.001×108>cv_2 \approx 3.001\times10^{8} > c (slightly).

Case 3 — p3=1.00×10−16p_3 = 1.00\times10^{-16}

  1. p32=1.00×10−32.p_3^2 = 1.00\times10^{-32}.

  2. p32c2=1.00×10−32×9.00×1016=9.00×10−16 J2.p_3^2 c^2 = 1.00\times10^{-32}\times9.00\times10^{16} = 9.00\times10^{-16}\ \mathrm{J^2}.

  3. E2=9.00×10−16−8.10×10−21.E^2 = 9.00\times10^{-16} - 8.10\times10^{-21}.
    Convert to same exponent: 9.00×10−16=900000×10−21.9.00\times10^{-16} = 900000\times10^{-21}.
    So E2=(900000−8.10)×10−21=899991.90×10−21=8.9999190×10−16.E^2 = (900000 - 8.10)\times10^{-21} = 899991.90\times10^{-21} = 8.9999190\times10^{-16}.

  4. E=8.9999190×10−16=8.9999190×10−8≈2.9999865×10−8 J.E = \sqrt{8.9999190\times10^{-16}} = \sqrt{8.9999190}\times10^{-8} \approx 2.9999865\times10^{-8}\ \mathrm{J}.

  5. Numerator: p3c2=1.00×10−16×9.00×1016=9.00×100=9.00.p_3 c^2 = 1.00\times10^{-16}\times9.00\times10^{16} = 9.00\times10^{0} = 9.00.

  6. v3=9.002.9999865×10−8.v_3 = \dfrac{9.00}{2.9999865\times10^{-8}}.

Compute ratio:

9.002.9999865×10−8=9.002.9999865×108≈3.0000135×108 m/s.\frac{9.00}{2.9999865\times10^{-8}} = \frac{9.00}{2.9999865}\times10^{8} \approx 3.0000135\times10^{8}\ \mathrm{m/s}.

So v3≈3.0000135×108>cv_3 \approx 3.0000135\times10^{8} > c (very slightly).

Summary table

pp (kg·m/s)EE (J)vv (m/s)v/cv/c
1.00×10−181.00\times10^{-18}2.861×10−102.861\times10^{-10}3.146×1083.146\times10^{8}1.0491.049
1.00×10−171.00\times10^{-17}2.999×10−92.999\times10^{-9}3.001×1083.001\times10^{8}1.00031.0003
1.00×10−161.00\times10^{-16}2.9999865×10−82.9999865\times10^{-8}3.0000135×1083.0000135\times10^{8}1.00000451.0000045

(Shown to a few significant figures; c=3.00×108 m/sc=3.00\times10^{8}\ \mathrm{m/s}.)

Notes & caveats (important)

  • This is a toy calculation using a speculative dispersion relation E2=p2c2−μ2c4E^2 = p^2 c^2 - \mu^2 c^4. It does not mean tachyons exist or that this model is physically consistent with causality, quantum field theory, or experiment. Tachyons bring severe conceptual problems (instabilities, causality violations) in standard physics.

  • The numerical choices (particularly μ\mu) were chosen to make the algebra clear and to produce visible superluminal vv. Realistic particle masses and units would make the effects either impossible or require reinterpreting the model.

  • In this model, as momentum grows much larger than μc\mu c, E≈pcE\approx p c and vv approaches cc from above — i.e., superluminal but asymptotically close to cc. For very small pp near μc\mu c the formulas break down (or become imaginary) if p≤μcp\le \mu c.

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