Time Dilation Calculator Gravity

Gravitational Time Dilation Formula:

\[ t' = t \times \sqrt{1 - \frac{2GM}{rc^2}} \]

Dilated Time (t'):

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1. What is Gravitational Time Dilation?

Gravitational time dilation is a phenomenon predicted by Einstein's theory of general relativity where time passes at different rates in regions of different gravitational potential. The closer to a massive object, the slower time passes.

2. How Does the Calculator Work?

The calculator uses the gravitational time dilation formula:

\[ t' = t \times \sqrt{1 - \frac{2GM}{rc^2}} \]

Where:

\( t' \) — Dilated time (seconds)
\( t \) — Proper time (seconds)
\( G \) — Gravitational constant (6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²)
\( M \) — Mass of the gravitational source (kg)
\( r \) — Radial coordinate (distance from center, meters)
\( c \) — Speed of light (299,792,458 m/s)

Explanation: This formula describes how time is affected by gravity, with time passing slower in stronger gravitational fields.

3. Importance of Time Dilation Calculation

Details: Understanding gravitational time dilation is crucial for GPS satellite systems, astrophysics research, and testing fundamental principles of general relativity. Even small time differences become significant over large distances or long periods.

4. Using the Calculator

Tips: Enter proper time in seconds, mass in kilograms, and radius in meters. All values must be positive numbers. The calculator uses standard physical constants for gravitational constant and speed of light.

5. Frequently Asked Questions (FAQ)

Q1: What is the significance of the term under the square root?
A: The term \( \frac{2GM}{rc^2} \) represents the gravitational potential. When this approaches 1 (at the Schwarzschild radius), time dilation becomes infinite, indicating an event horizon.

Q2: How significant is time dilation in everyday life?
A: For Earth's gravity, the effect is very small (about 1 second difference every 30 years between sea level and space), but it's crucial for precision systems like GPS.

Q3: Does this formula work for all gravitational fields?
A: This is the exact solution for a non-rotating, uncharged spherical mass. For other configurations, more complex formulas are needed.

Q4: What happens when 2GM/rc² equals or exceeds 1?
A: This represents the Schwarzschild radius or event horizon of a black hole. Beyond this point, the formula gives imaginary results, indicating no information can escape.

Q5: How is this different from velocity time dilation?
A: Gravitational time dilation is caused by differences in gravitational potential, while velocity time dilation (from special relativity) is caused by relative motion between reference frames.