Twin Paradox Calculator

Consider two twins, Alice (traveler) and Bob (stays on Earth). Alice boards a spacecraft and travels at constant velocity $$v$$ to a distant star, then returns at the same speed. From Bob's perspective on Earth:

• The total round-trip time is $$t_{\text{Earth}}$$ years.
• The one-way distance to the star is $$d = v \cdot t_{\text{Earth}} / 2$$.

From Alice's perspective, time dilation means her clocks run slower by the Lorentz factor:

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

So Alice experiences only:

$$t_{\text{Alice}} = \frac{t_{\text{Earth}}}{\gamma} = t_{\text{Earth}}\sqrt{1 - \frac{v^2}{c^2}}$$

Additionally, Alice observes length contraction: the distance to the star appears shortened to $$d' = d/\gamma$$, which is consistent with her shorter travel time at velocity $$v$$.

Resolving the "paradox": One might ask: from Alice's frame, isn't Bob moving, so shouldn't Bob age less? The resolution is that the situation is not symmetric. Alice must undergo acceleration — at departure, at the turnaround point, and at arrival — changing reference frames. Bob remains inertial throughout. In general relativity terms, Alice follows a longer worldline through spacetime, and the principle of maximal aging states that the inertial (unaccelerated) path between two events accumulates the most proper time.

At $$v = 0.9c$$, $$\gamma \approx 2.294$$, so a 20-year Earth trip means Alice ages only ~8.7 years. At $$v = 0.99c$$, $$\gamma \approx 7.089$$, and the age difference becomes even more dramatic. At $$v = 0.9999c$$, Alice would age only ~100 days for every 20 Earth years.

The Traveler's Elapsed Time is how much the traveling twin actually ages during the journey — always less than the Earth time. The Age Difference shows how much younger the traveler is upon return. The Contracted Distance is the distance as measured by the traveler, shorter than the Earth-frame distance by the factor $$\gamma$$. As speed approaches $$c$$, the traveler's time approaches zero and the age difference approaches the full Earth time — in principle, one could travel to distant galaxies within a human lifetime at sufficient speed, though the energy requirements would be astronomical.