Time Dilation Calculator

Reviewed and verified by: David Chen, PhD in Astrophysics

This Time Dilation Calculator uses Einstein’s Special Relativity formula to determine how time changes for an object moving relative to an observer. Enter any two variables to solve for the third, or enter all three to check for consistency.

Time Dilation Calculator

Calculated Result:

Detailed Calculation Steps

Time Dilation Calculator Formula

The time dilation formula is derived from the Lorentz transformations in Special Relativity. It relates the time measured in a moving frame ($\Delta t$) to the time measured by a stationary observer ($\Delta t’$).

$$ \Delta t’ = \frac{\Delta t}{\sqrt{1 – \frac{v^2}{c^2}}} = \Delta t \cdot \gamma $$

Formula Source: Wikipedia: Time Dilation, Physics Stack Exchange

Variables Explained

• Proper Time ($\Delta t$): The time interval measured by a clock at rest relative to the events. This is the shortest time measured. (e.g., the time experienced by an astronaut traveling at high speed).
• Dilated Time ($\Delta t’$): The time interval measured by an observer in an inertial frame of reference relative to the moving clock. This time is always longer than the Proper Time. (e.g., the time experienced by people on Earth).
• Relative Velocity ($v/c$): The relative speed between the two frames, expressed as a fraction of the speed of light ($c$). Must be less than 1.
• Lorentz Factor ($\gamma$): The factor $\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}$, which quantifies the relativistic effect.

What is Time Dilation?

Time dilation is a phenomenon predicted by Albert Einstein’s theory of Special Relativity. It states that time passes differently for observers in relative motion compared to one another. Specifically, the faster an object moves, the slower time passes for that object relative to a stationary observer. This effect is negligible at everyday speeds but becomes extremely significant as the velocity approaches the speed of light ($c$).

The concept is famously illustrated by the “Twin Paradox,” where one twin takes a fast space journey and returns to find their sibling has aged significantly more. This isn’t just a theoretical curiosity; it’s a proven reality essential for modern technology. For instance, GPS satellites, moving fast and experiencing less gravity than Earth-bound receivers, must have their internal clocks constantly adjusted for relativistic effects to maintain accuracy.

How to Calculate Time Dilation (Example)

Let’s use the formula to find the Dilated Time for an astronaut who travels for 5 years at 95% the speed of light.

1. Identify Known Variables: Proper Time ($\Delta t$) = 5 years; Velocity ($v/c$) = 0.95.
2. Calculate the Lorentz Factor ($\gamma$): $$\gamma = \frac{1}{\sqrt{1 – 0.95^2}} = \frac{1}{\sqrt{1 – 0.9025}} = \frac{1}{\sqrt{0.0975}} \approx 3.20256$$
3. Calculate Dilated Time ($\Delta t’$): Multiply Proper Time by the Lorentz Factor. $$\Delta t’ = \Delta t \cdot \gamma = 5 \cdot 3.20256 \approx 16.01 \text{ years}$$
4. Conclusion: When the astronaut returns after 5 years of their journey, 16.01 years will have passed for the stationary observer (e.g., on Earth).

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Frequently Asked Questions (FAQ)

1. Is time dilation real, or just a theory?

Time dilation is a proven physical phenomenon. It is routinely measured in particle accelerators (where particles moving near $c$ decay slower than expected) and is crucial for the correct operation of the Global Positioning System (GPS).

2. What is the difference between Proper Time and Dilated Time?

Proper Time ($\Delta t$) is the time interval measured by the observer who is at rest relative to the clock (the traveler). Dilated Time ($\Delta t’$) is the time interval measured by the observer who is moving relative to the clock (the stationary observer, like someone on Earth). The Dilated Time is always greater than or equal to the Proper Time.

3. What happens if I enter a velocity greater than $c$?

If the velocity ($v$) equals or exceeds the speed of light ($c$), the term $1 – v^2/c^2$ becomes zero or negative, making the Lorentz factor ($\gamma$) mathematically undefined or imaginary. In the calculator, this will trigger an error, as exceeding $c$ is physically impossible according to current physics laws.

4. What units should I use for time input?

You can use any consistent time unit (seconds, minutes, years), but you must use the same unit for both Proper Time and Dilated Time. This calculator uses Years for ease of understanding in common scenarios like the twin paradox.