# Gravitational Time Dilation Calculator

Created by Davide Borchia
Last updated: Sep 14, 2022

Einstein's theory of general relativity gives a heap of forecasts, among them, how to calculate the time dilation in a gravitational potential. If you've seen the movie Interstellar you may remember the scenes around the black hole, where the characters were aging at a slower rate than the people on Earth. There is science behind it: learn it with our gravitational time dilation calculator:

• Discover the time dilation equation in the general relativity theory;
• Learn how to calculate time dilation;
• Some examples of the time dilation formula; and
• Other relativity calculators.

## The effect of gravity on time

Mass bends the fabric of space, creating "wells". A well is a good representation of a gravitational potential: the representation of fabric stretched by a mass (like a bowling ball on a trampoline) works — though it is way more complex in three-dimensional space! Light follows the curvature too (as any other object), but since light speed remains constant, the time has to stretch to accommodate for a longer path (you can see how this work by looking at the formula for speed, as in our speed calculator).

The stronger the curvature, the deeper we move in the potential; the longer the path, the longer the time.

## How to calculate the gravitational time dilation

The formula we use to calculate the gravitational time dilation is surprisingly straightforward. You can see it here: in it, we can find two of the universe's constants: gravity and the speed of light.

$\Delta t' = \Delta t \cdot \sqrt{1 - \frac{2 \cdot M \cdot G }{ r \cdot c^2}}$

Where:

• $\Delta t$ — The result of the time dilation formula;
• $\Delta t'$ — The time duration from the point of view of a far reference frame;
• $M$ — The mass of the body generating the gravitational potential;
• $G$ — The gravitational constant: $G=6.6743 \times 10^{-11}\ \text{m}^3/(\text{kg}\cdot\text{s}^2)$;
• $r$ — The distance from the center of mass of the objects; and
• $c$ — The speed of light in the vacuum.

As for many formulas in the relativistic framework, the square root has a value at most equal to one, which implies that a far observer outside of the influence of any acceleration/gravitational potential — the equivalence principle — experiences the longest possible time duration.

Every body under the influence of the gravitational force experiences the effects of time dilation. Check some examples of the time dilation formula in the next section.

## Examples of the gravitational time dilation equation in our Universe

The effects of time dilation are negligible in our daily lives: or at least, they were. On the surface of Earth, a day is $0.00006$ seconds shorter than in interstellar space. However, with the refinement of our technology, we started facing the issues caused by time dilation: GPS satellites orbit around the planet with a delay of $45\ \text{μs}$. This difference would affect the results of our devices by tens of meters each day.

Time dilation affects the measurements of atomic clocks around the world, too: the presence of high mountains in the proximity of those extremely sensible devices affects the flow of time.

The last example: the difference in time between Mercury and Earth due to Sun's gravity: apply the equation for time dilation in the general relativity theory in both cases, and you will find two different times. Take as a reference the duration of a day, $86,400\ \text{s}$.

For Mercury:

\begin{align*} \Delta t'_{\text{Mercury}} &= 86,\!400\ \text{s} \\ &\cdot \sqrt{1 - \frac{2 \cdot M_{\odot} \cdot G }{ 0.387\ \text{UA} \cdot c^2}}\\ &=86399.9978\ \text{s} \end{align*}

For Earth:

\begin{align*} \Delta t'_{\text{Earth}} &= 86,\!400\ \text{s} \\ &\cdot \sqrt{1 - \frac{2 \cdot M_{\odot} \cdot G }{ 1\ \text{UA} \cdot c^2}}\\ &=86399.9991\ \text{s} \end{align*}

The difference is in the milliseconds range: still not enough to be noticeable. We need more massive objects, such relativity of length as neutron stars or black holes, to experience perceivable time dilations.

If you need more tools related to the relativity theories, you can visit our electron speed calculator or relativistic kinetic energy calculator.

Davide Borchia
Time interval with no gravity
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First frame of reference
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Earths
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Time interval 1
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Second frame of reference (for comparison)
Mass
Suns
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Time interval 2
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Difference between the time intervals 1 and 2
Time difference
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