To calculate the time dilation between two clocks at different altitudes, you can use the formula for gravitational time dilation derived from general relativity. The time dilation factor depends on the gravitational potential difference between the two locations. Here's how you can calculate it:
Determine the gravitational potential difference: The gravitational potential at a given altitude is given by the formula V = gh, where g is the acceleration due to gravity (approximately 9.8 m/s² near the Earth's surface) and h is the height difference between the two locations.
Calculate the time dilation factor: The time dilation factor, often denoted as γ (gamma), is given by the equation:
γ = √(1 - (2GM) / (rc²))
Here, G is the gravitational constant (approximately 6.67430 × 10^(-11) m³/(kg·s²)), M is the mass of the Earth (approximately 5.972 × 10^24 kg), and r is the distance from the center of the Earth to the clocks' locations. For simplicity, you can use the average radius of the Earth, which is approximately 6,371 km (or 6,371,000 meters).
Calculate the time dilation: The time dilation factor γ represents the ratio of the time measured by the higher clock (at altitude) to the time measured by the lower clock (at sea level). Therefore, the time dilation between the two clocks is given by:
Δt_altitude = γ * Δt_sea_level
Here, Δt_altitude is the time experienced by the clock at altitude, and Δt_sea_level is the time experienced by the clock at sea level.
Note that the time dilation calculated in this manner accounts for the gravitational effect on time due to the difference in altitude. It does not account for other factors that may influence timekeeping, such as velocity or local gravitational anomalies.