Gravitational Time Dilation Calculator

Gravitational Time Dilation Formula:

\[ \Delta t' = \Delta t \sqrt{1 - \frac{2 G M}{r c^2}} \]

Dilated Time (Δt'):

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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. Clocks closer to a massive object (where gravity is stronger) run slower than clocks further away.

2. How Does the Calculator Work?

The calculator uses the gravitational time dilation equation:

\[ \Delta t' = \Delta t \sqrt{1 - \frac{2 G M}{r c^2}} \]

Where:

\( \Delta t' \) — Dilated time (seconds)
\( \Delta t \) — Proper time (seconds)
\( G \) — Gravitational constant (6.67430 × 10⁻¹¹ m³/kg·s²)
\( M \) — Mass of the gravitating body (kg)
\( r \) — Radial coordinate (distance from center, meters)
\( c \) — Speed of light (3 × 10⁸ m/s)

Explanation: The equation shows how time slows down near massive objects. The effect becomes significant only in extremely strong gravitational fields, like those near black holes or neutron stars.

3. Importance of Time Dilation

Details: Gravitational time dilation has practical implications for GPS satellite systems, where clocks in orbit must be regularly adjusted to account for both gravitational and velocity time dilation effects.

4. Using the Calculator

Tips: Enter proper time in seconds, mass in kilograms, and radial coordinate in meters. All values must be positive. The radial coordinate must be greater than the Schwarzschild radius (2GM/c²) to avoid imaginary results.

5. Frequently Asked Questions (FAQ)

Q1: How significant is time dilation on Earth?
A: On Earth's surface, the effect is minimal (~0.02 seconds per year compared to space), but crucial for GPS accuracy.

Q2: What happens at the Schwarzschild radius?
A: At r = 2GM/c² (the event horizon of a black hole), time dilation becomes infinite - time appears to stop from an outside observer's perspective.

Q3: Does this formula work for any mass?
A: Yes, but the effect is only noticeable near extremely massive objects like neutron stars or black holes.

Q4: How does this relate to velocity time dilation?
A: Both are relativistic effects, but gravitational dilation depends on gravitational potential while velocity dilation depends on relative speed.

Q5: Has gravitational time dilation been proven?
A: Yes, through numerous experiments including the Pound-Rebka experiment and GPS system corrections.