Gravitational Time Dilation Calculator With Velocity

Time Dilation Equation:

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

Proper Time Interval (Δt):

seconds

Gravitational Constant (G):

m³/kg·s²

Mass (M):

Radial Coordinate (r):

meters

Speed of Light (c):

m/s

Velocity (v):

m/s

Dilated Time Interval (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 more slowly in stronger gravitational fields. When combined with velocity, the effect becomes more pronounced.

2. How Does the Calculator Work?

The calculator uses the combined gravitational and velocity time dilation equation:

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

Where:

\( \Delta t' \) — Dilated time interval (seconds)
\( \Delta t \) — Proper time interval (seconds)
\( G \) — Gravitational constant (6.67430 × 10⁻¹¹ m³/kg·s²)
\( M \) — Mass creating the gravitational field (kg)
\( r \) — Radial coordinate from the center of mass (m)
\( c \) — Speed of light (299,792,458 m/s)
\( v \) — Velocity relative to the gravitational field (m/s)

Explanation: The equation combines both gravitational and velocity effects on time dilation, with the square root term representing the time dilation factor.

3. Importance of Time Dilation Calculation

Details: Understanding time dilation is crucial for GPS satellite systems, astrophysics, and testing general relativity. Precise calculations are needed for satellite navigation systems to maintain accuracy.

4. Using the Calculator

Tips: Enter all values in SI units. For Earth's mass use 5.972 × 10²⁴ kg, for Earth's radius use 6.371 × 10⁶ m. Velocity should be relative to the gravitational field.

5. Frequently Asked Questions (FAQ)

Q1: How significant is time dilation on Earth?
A: For Earth's gravity at sea level, the gravitational effect is about 0.7 nanoseconds per second. Combined with orbital velocity, GPS satellites experience about 38 microseconds per day difference.

Q2: What happens at the Schwarzschild radius?
A: When r = 2GM/c² (the Schwarzschild radius), time dilation becomes infinite, representing the event horizon of a black hole.

Q3: How does velocity affect time dilation?
A: Higher velocities increase time dilation, with the effect becoming significant as v approaches c (speed of light).

Q4: Can time dilation be observed in daily life?
A: The effects are too small to notice without precise instruments, but they're accounted for in GPS systems and atomic clocks.

Q5: What's the difference between gravitational and velocity time dilation?
A: Gravitational dilation depends on gravitational potential, while velocity dilation depends on relative speed. Both effects combine multiplicatively.