Relativistic Parameters (Traveler)

Ship Velocity (v/c)

≈ 149900 km/s · Slider up to 0.999; manual input < 1

00.999

Ship Time (Δt₀)

Time measured by the observer at rest relative to the event

0.0011000

Ship Length (L₀)

Length of the object measured in its rest frame

0.00110000

Emitted Wavelength (nm)

Wavelength of light emitted by the source at rest

101000

Frame: Observer (Rest)

Results

Lorentz Factor (γ)

1.155

Dilated Time (Δt)

1.155 s

Elapsed Time (On Earth)

Contracted Length (L)

86.6 m

Observed Length (From Earth)

Simultaneity Offset Magnitude

1.926e-7 s

Observed λ (nm)

921.5 nm

Wavelength (On Earth)

Doppler Effect

Emitted

Observed

Fundamentals & Explanation

Special Relativity and Frames of Reference

In 1905, Albert Einstein revolutionized physics with his paper “On the Electrodynamics of Moving Bodies”, founded on two postulates:

Principle of relativity: The laws of physics are invariant (identical) in all non-accelerating inertial frames of reference.
Invariance of c: The speed of light in a vacuum is a universal constant (c = 299,792,458 m/s) for all observers, regardless of the motion of the light source or observer.

Analytical Rigor (Observer vs. Traveler): In this simulator, proper quantities (Δt₀, 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.

Lorentz Factor (γ)

Developed by Hendrik Lorentz, this scalar factor appears in the Lorentz transformations, which relate spatial and temporal coordinates between different inertial frames:

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

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.

Time Dilation and Length Contraction

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).

\Delta t = \gamma \, \Delta t_0

L = \frac{L_0}{\gamma}

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). Length contraction is a frame- and simultaneity-dependent measurement, not a mechanical compression of atoms.

Relativity of Simultaneity

The concept of “now” is relative. Two events separated by proper length L₀ and simultaneous on the ship are not simultaneous from Earth’s perspective. The simulator shows the offset magnitude; its sign depends on the spatial orientation and endpoint ordering:

\left|\Delta t_{\text{Earth}}\right| = \gamma\frac{v L_0}{c^2} = \gamma\frac{\beta L_0}{c}

Relativistic Doppler Effect

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)

\lambda_{\text{obs}} = \lambda_0 \sqrt{\frac{1 + v/c}{1 - v/c}}

Approach (Moving towards)

\lambda_{\text{obs}} = \lambda_0 \sqrt{\frac{1 - v/c}{1 + v/c}}

⚡ Speeds (v)

ISS: 7.7 km/s

Voyager: 17 km/s

Muons: 0.998c

LHC: 0.9999999c

📐 γ Factor

0.1c → 1.005

0.5c → 1.155

0.866c → 2.000

0.99c → 7.089

Academic References

[1] Einstein, A. (1905). Zur Elektrodynamik bewegter Körper. Annalen der Physik, 322(10), 891-921.
[2] Resnick, R., & Halliday, D. (1992). Basic Concepts in Relativity and Early Quantum Theory. Macmillan.
[3] Hafele, J. C., & Keating, R. E. (1972). Around-the-world atomic clocks: Predicted relativistic time gains. Science, 177(4044), 166-168.

Relativistic Kinematics

Relativistic Parameters (Traveler)

Results

Doppler Effect

Fundamentals & Explanation

Special Relativity and Frames of Reference

Lorentz Factor (γ)

Time Dilation and Length Contraction

Relativity of Simultaneity

Relativistic Doppler Effect

⚡ Speeds (v)

📐 γ Factor

Academic References

Special Relativity and Frames of Reference

Lorentz Factor (γ)

Time Dilation and Length Contraction

Relativity of Simultaneity

Relativistic Doppler Effect

⚡ Speeds (v)

📐 γ Factor

Academic References