General Relativity: Time Dilation

Do you feel dizzy looking at the clock above? Equally fascinating and perplexing as the picture above, is general relativity's consequence of gravitational time dilation. While this concept was introduced in special relativity as a result of travelling at relativistic speeds, its application in general relativity's gravitation has more prominent, observable examples. This concept, now confirmed experimentally, describes how time runs slower in a stronger gravitational field.

HOW DO WE CALCULATE GRAVITATIONAL TIME DILATION?

Einstein's derived the following formula for time dilation:

where     𝑡f is the relativistic time measured outside the gravitational field

       𝑡0 is the proper time measured inside the gravitational field  

       𝐺 is the gravitational constant

       𝑀 is the mass of the massive body

        𝑟 is the distance from the center of the massive body

        𝑐 is the speed of light in a vacuum

      𝑣𝑒 is the escape velocity at that point in the field: 𝑣𝑒=√(2𝐺𝑀/𝑟)

      𝑟𝑠 is the Schwarzschild radius, where 𝑣𝑒=𝑐 : 𝑟𝑠=2𝐺𝑀/𝑐^2 

While the formula itself is elegant, it gives rise to exciting conclusions about the relation of spacetime:

• Due to time dilation, the time elapsed in a gravitational field is always lesser than the time elapsed outside the gravitational field.
• However, using the formula, in order to achieve a 1% lag in time, a radius 𝑟 of 6000 km and a mass 𝑀 of approximately 8 × 10^31 kg [80 suns] is required.
• For a lag of factor 60, with the same radius 𝑟 of 6000 km, a mass 𝑀 of approximately 4 × 10^34 kg [40,000 suns] is required 

Gravitational time dilation is a far more common form of time dilation, given that we don't have the equipment for motion at high relativistic speeds required for time dilation using special relativity. An appearance of this phenomenon is seen in Interstellar, when proximity to a black hole causes extreme time dilation (1 hour near black hole = 7 years earth time).

Overall, time dilation helps to correct anomalies where precision and accuracy is extremely important, such as satellites for GPS systems. Even a tiny anomaly, can cause errors equivalent to miles on earth. Learn more about gravitational time dilation, and about general relativity in its entirety, through this oversimplified presentation.