Relativistic Time Dilation

Time dilation: Δt' = γ·Δt

In special relativity a moving clock ticks slow. The relation is Δt' = γ·Δt, where the Lorentz factor is γ = 1/√(1 − v²/c²). While v stays far below c, γ sits near 1 and you can ignore the effect without losing any accuracy. Push v toward c and γ blows up. At 99% c it climbs to about 7.09, so one second aboard a starship would match 7.09 s back on Earth. The everyday proof of this comes from cosmic muons. Their proper half-life is just 2.2 µs, which in principle is far too short to survive the trip from the upper atmosphere to the ground, yet time dilation stretches that lifetime in our frame and they make it down anyway. Example: at v = 10⁸ m/s, γ works out to roughly 1.061, so 1 s of proper time reads as ≈1.061 s in the lab frame.

Applications: GPS, particle accelerators, sci-fi

Drop the relativistic corrections (special and general alike) and GPS would drift by ≈38 µs/day, which piles up roughly 11 km of positional error in a single day. At the LHC and other accelerators, measuring the dilated lifetimes of muons and pions is routine. The twin paradox belongs here too. The traveling twin returns younger because the acceleration on the turnaround breaks the symmetry, and that idea is what drives films like Interstellar and a whole shelf of time-dilation novels.

FAQ

Is time dilation real or just appearance? It's real. Atomic clocks flown on airplanes come back measurably out of sync with the ones left on the ground (Hafele–Keating, 1971).

What is proper time? Δt is the reading on a clock at rest in the moving frame, while Δt' is what an outside observer measures. Each frame sees the other's clock ticking slow.

Why does the twin paradox break symmetry? The traveling twin has to accelerate to turn around, which makes the two situations different. Only the stay-at-home twin spends the whole trip in a single inertial frame.

Can we ever reach γ = ∞? No. Only massless particles travel at c. A massive object would need infinite energy to get there, so its γ always stays finite, no matter how big it grows.