Compute time dilation, length contraction and the relativistic Doppler effect from velocity as a fraction of c. Based on Special Relativity.
In 1905, Albert Einstein revolutionized physics with his paper "On the Electrodynamics of Moving Bodies", founded on two postulates:
Analytical Rigor (Observer vs. Traveler): In this simulator, "Proper" quantities (t₀, L₀) represent measurements made by the traveler at rest relative to the ship. The calculated results represent the measurements of a static inertial observer (on Earth) watching the ship pass by.
Developed by Hendrik Lorentz, this factor is the mathematical operator that transforms spatial and temporal coordinates between different inertial frames:
Because a massive object's velocity (v) can never equal or exceed c, the quantity under the radical is always positive. As v approaches c, γ tends to infinity, demonstrating that accelerating mass to the speed of light requires infinite energy.
To preserve the constancy of the speed of light, space and time are no longer absolute. Time ticks more slowly for a moving object (Dilation) and its length shortens in the direction of motion (Lorentz-FitzGerald Contraction).
Empirical Evidence: Time dilation was proven by the Rossi-Hall experiment (1941) measuring anomalous cosmic muon survival, and verified by atomic clocks on commercial flights (Hafele-Keating, 1971). Spatial contraction is a spacetime deformation, not a mechanical compression of atoms.
The concept of "now" is relative. Two spatially separated events that are simultaneous on the ship will occur with a time offset from Earth's perspective:
Unlike classical sound waves, the frequency shift of light emitted by sources at relativistic speeds incorporates transverse time dilation (verified by the Ives-Stilwell experiment, 1938).
Recession (Moving away)
Approach (Moving towards)
ISS: 7.7 km/s
Voyager: 17 km/s
Muons: 0.998c
LHC: 0.9999999c
0.1c → 1.005
0.5c → 1.155
0.866c → 2.000
0.99c → 7.089