相対 Special Relativity · Simulation

Addition of Relativistic Velocity & Spacetime Invariance

Aim: to investigate Einstein's relativistic velocity addition theorem in Special Relativity, demonstrate why no physical object can exceed $c$, observe the invariance of the speed of light under Lorentz boosts, and visualize worldlines on a Minkowski spacetime diagram.

Twin Reference Frames & Relativistic Flight

Frame S (Lab) vs Frame S' (Cruiser)

Watch the scout probe fire from Cruiser A. Even when both travel at 0.90c, their combined speed from Earth remains strictly < c.

Spacetime Minkowski Diagram ($ct$ vs $x$)

Einstein vs. Galilean Velocity Addition

Relativistic Transformation Catalog

Derivation from the Lorentz Transformation

Consider two inertial frames: $S$ (Earth base station) and $S'$ (Cruiser spacecraft) moving along the common $x$-axis at velocity $v$. The coordinates transform via the Lorentz transformation:

$$ x' = \gamma (x - v t), \qquad t' = \gamma \left(t - \frac{v x}{c^2}\right) $$

Taking differentials $dx'$ and $dt'$:

$$ dx' = \gamma (dx - v dt), \qquad dt' = \gamma \left(dt - \frac{v dx}{c^2}\right) $$

Dividing $dx'$ by $dt'$ gives the velocity in $S'$:

$$ u' = \frac{dx'}{dt'} = \frac{\gamma(dx - v dt)}{\gamma(dt - v dx/c^2)} = \frac{\frac{dx}{dt} - v}{1 - \frac{v}{c^2}\frac{dx}{dt}} = \frac{u - v}{1 - \frac{u v}{c^2}} $$

Inverting to solve for the velocity $u$ as observed from frame $S$:

$$ u = \frac{u' + v}{1 + \frac{u' v}{c^2}} $$

In the Newtonian limit ($u', v \ll c$), the denominator $1 + \frac{u'v}{c^2} \to 1$, recovering Galileo's addition formula $u = u' + v$.

Invariance of the Speed of Light & Light Cone

If the probe is a light pulse or photon ($u' = c$):

$$ u = \frac{c + v}{1 + \frac{c v}{c^2}} = \frac{c(1 + v/c)}{1 + v/c} = c $$

The speed of light is identical in every inertial frame, fulfilling Einstein's Second Postulate. Furthermore, if $u' < c$ and $v < c$, their composition is strictly bounded:

$$ c - u = \frac{c(1 - u'/c)(1 - v/c)}{1 + \frac{u' v}{c^2}} > 0 \implies u < c $$

No combination of sub-luminal velocities can ever reach or breach the light speed barrier.

Viva Questions

What is the Lorentz factor $\gamma$ and how does it behave as $u \to c$?

$\gamma = 1/\sqrt{1 - u^2/c^2}$. At rest ($u=0$), $\gamma = 1$. As $u \to c$, $\gamma \to \infty$, demanding infinite energy to accelerate a massive particle to the speed of light.

What causes the tilt in the simultaneity axis $x'$ on the Minkowski diagram?

The time transformation $t' = \gamma(t - vx/c^2)$. Events simultaneous in $S'$ ($t' = 0$) satisfy $t = (v/c^2)x$, forming a tilted line of slope $v/c$ in the $(x, ct)$ plane.