Gravitational time dilation is another aspect of Einstein’s theory of general relativity, and it states that time passes at different rates in regions of different gravitational potential. Essentially, the stronger the gravitational field, the slower time progresses.

The formula for gravitational time dilation is given by:

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

where: - \(\Delta t'\) is the dilated time in the stronger gravitational field. - \(\Delta t\) is the proper time in a weaker gravitational field. - \(G\) is the gravitational constant. - \(M\) is the mass causing the gravitational field. - \(r\) is the distance from the center of the mass. - \(c\) is the speed of light.

Let’s consider an example to illustrate gravitational time dilation:

Example:

Suppose there are two observers, Alice and Bob. Alice is on the surface of a massive planet, and Bob is far away in space where the gravitational field is weaker.

  1. Before Alice Descends:
    • Alice and Bob synchronise their watches. They are the same age, say 30 years old.
  2. Alice Descends to a Massive Planet:
    • Alice descends to the surface of a massive planet, experiencing a stronger gravitational field.
  3. Time on the Planet:
    • Alice ages more slowly than Bob due to the stronger gravitational field on the planet.
  4. When Alice Returns:
    • Alice returns to Bob’s location in space.
  5. Age Difference:
    • Despite the fact that their watches were initially synchronised, Alice’s clock shows less time has passed compared to Bob’s clock. The actual age difference depends on the strength of the gravitational field experienced by Alice.

In this case, the gravitational field near the massive planet causes time to dilate for Alice compared to Bob.

Gravitational time dilation has been experimentally confirmed with highly precise atomic clocks placed at different altitudes on Earth and through observations of clocks on satellites. It is a significant concept in understanding the effects of gravity on time and has implications for cosmology and astrophysics.