Introduction

This calculator demonstrates Gravitational Time Dilation, a phenomenon predicted by Albert Einstein's theory of General Relativity. It shows how time passes at different rates for observers located in regions of differing gravitational potential.

Specifically, time passes slower in stronger gravitational fields (closer to a massive object) compared to weaker gravitational fields (further away from the mass).

How to use:

1. Enter the mass of the celestial body (e.g., Earth, Sun).
2. Enter the distance from the center of that mass at which an observer is located.
3. Enter the "proper time" – a duration of time experienced by this observer near the mass.
4. Click "Calculate Time Dilation".
5. The results will show the equivalent time passed for a distant observer (where gravity's effect is negligible) and the step-by-step calculation.

Theory & Formulas

Gravitational time dilation arises because gravity is the curvature of spacetime caused by mass and energy. The stronger the gravity (the more spacetime is curved), the slower time flows.

Key Formula:

The relationship between time for an observer in a gravitational field (t_proper) and an observer far away (t_{observer_far}) is given by:

Where:

• t_{observer_far}: Time elapsed for the distant observer (in a weaker gravitational field, often considered at "infinity"). This is the time being calculated. (seconds)
• t_proper: Proper time elapsed for the observer near the massive object (in the stronger gravitational field). This is your input. (seconds)
• G: Gravitational constant (6.67430 × 10^-11 m³ kg^-1 s^-2).
• M: Mass of the celestial body creating the gravitational field. Your input. (kg)
• r: Distance from the center of the mass to the observer. Your input. (meters)
• c: Speed of light in a vacuum (299,792,458 m/s).

Schwarzschild Radius (R_s):

The term 2GM/c² is known as the Schwarzschild Radius (R_s). It represents the radius of the event horizon of a non-rotating, uncharged black hole of mass M.

If r ≤ R_s, the observer is within or at the event horizon, and the standard time dilation formula (as observed from outside) is not applicable or leads to singularities. This calculator requires r > R_s.

The time dilation formula can also be written using R_s:

Assumptions:

• The massive body is spherically symmetric, non-rotating, and uncharged.
• The "distant observer" is sufficiently far away that the gravitational potential they experience is negligible.
• The observer near the mass is stationary relative to the mass (i.e., no relativistic velocity effects are considered here, only gravitational).

Calculator

Calculation Results

Time for Distant Observer (t_{observer_far}): seconds

Time Difference (t_{observer_far} - t_proper): seconds

Time Dilation Factor (t_{observer_far} / t_proper):

Step-by-Step Solution