# Gravitational Time Dilation Calculator

Created by Rita Rain
Reviewed by Dominik Czernia, PhD candidate and Jack Bowater
Last updated: May 02, 2022

The gravitational time dilation calculator lets you find out how much slower time passes near massive objects, for example, a black hole. It can also determine the difference in the lapse of time between Earth and Sun and any other objects, provided you know their mass and radius.

Although we don't experience either speed time dilation or gravitational time dilation on a daily basis, Einstein's theory of relativity is crucial when dealing with cosmic magnitudes. For example, it enables to calculate time dilation for satellites orbiting Earth.

Read on to learn how to use the calculator and find more gravitational time dilation examples.

## What is gravitational time dilation?

Gravitational time dilation is a change in the lapse of time caused by a gravitational field, which, in Einstein's general theory of relativity, is described as a curving of space-time. The theory predicts that the closer an observer is to a source of gravity and the greater its mass, the slower time passes. Usually, we don't experience these effects because they are minimal in everyday life.

## How the gravitational time dilation calculator works

The gravitational time dilation calculator lets you find out how gravity of an object influences a lapse of time:

1. In the first field, you can input the time interval for which you want to calculate time dilation. When you input mass and radius in the first frame of reference, you can compare time lapse with no gravity to the time influenced by the gravity of an object with a given mass and radius.

2. If you want to compare two frames of references (for example, the lapse of time on two planets), enter the masses and radii of both objects. Then enter one of the time intervals (one with no gravity or one influenced by either of the two objects). The calculator will determine the two remaining time intervals. It'll also display the time difference between the two frames of reference.

3. If you need to calculate time dilation on the surface of a celestial object, enter its radius. If you want to find the result for a place at a certain distance from the object, in the "radius" field, enter the sum of the orb's radius and the distance. This number is the distance from the object's center.

4. You may want to play around with calculating in different directions, e.g., to see how radius changes when you set a different time interval. In these cases, it may be helpful to lock the variables you don't want to change (click the grey area to the right of a given field).

5. You can reset the numbers by clicking the arrow icon below the calculator.

6. Use scientific notation to enter big numbers, e.g. "5e8" (5*108) instead of 500,000,000.

The results are accurate to ten decimal places.

## Gravitational time dilation equation

To calculate time dilation by hand, use the following gravitational time dilation equation:

Δt' = Δt * √(1 - 2 * M * G / r * c²)

where:

• Δt' - Time interval affected by gravity;

• Δt - Time interval uninfluenced by gravity (in a reference frame in an infinite distance from any mass);

• M - Mass of an object causing the gravitational dilation;

• G - Gravitational constant ( G = 6.6743 * 10-11 N*m2/kg2);

• r - Distance from the center of an object causing the gravitational dilation; and

• c - The speed of light in a vacuum (c = 299,792,458 m/s).

## Gravitational time dilation example

If you'd like to see a simple gravitational time dilation example, compare the lapse of time on Earth and the Sun. The default units will make it easy. You should see that the difference is around 67 seconds per year.

If you want a more dramatic effect, imagine yourself in a spaceship near a supermassive black hole with the catchy name S5 0014+81. It has a mass of 4.3×109 solar masses and a radius of 0.002 light-years. You're trying to keep a safe distance of a Planck length 1,000 km from the event horizon (and wearing a seatbelt, of course). Five minutes have passed and, with Einstein's theory of time (the gravitational time dilation equation) in mind, you've started wondering how much time passed on Earth.

We can use the gravitational time dilation calculator to answer the question. To do it:

1. Let's leave the first field empty, set the values in the first frame of reference to 1 Earth-mass and 1 Earth radius.

2. In the second frame of reference, we need to change the first unit to solar mass and type in "4.3e9" (4.3×109). Then set the radius unit to light-years, enter "0.002", change the unit to kilometers and add 1,000 km. Finally, set the time interval to 5 min.

3. Check the result in the fourth field of the gravitational time dilation calculator (time interval in the first frame of reference) - almost 9 minutes passed on Earth. If you spent a year near the black hole, the time difference would grow to almost 9 months!

## FAQ

### What is space-time?

Space-time is a mathematical model in which three dimensions of space and the fourth dimension - time, are described as one continuum. Although space-time is mainly associated with Einstein's theory of relativity, it was first developed by his teacher, mathematician Hermann Minkowski.

### Is time different in space?

Yes, time is different in space. The closer you are to objects like planets and the bigger their mass, the slower time runs for you. This effect is called gravitational time dilation. Another factor is velocity time dilation, which states that the faster you're moving, the slower the lapse of time. So, whether time in space goes faster for you depends on your speed, mass, and distance from sources of gravity.

### Is time dilation real?

Time dilation is real, and numerous experiments have confirmed it. For example, in 1971 physicist Hafele and astronomer Keating placed atomic clocks on airplanes, flying around the world. They left one atomic clock at an observatory. When all clocks "reunited", they showed different times. The results were consistent with the theory of relativity.

Rita Rain
Time interval with no gravity
sec
First frame of reference
Mass
Earths
R⊕
Time interval 1
sec
Second frame of reference (for comparison)
Mass
Suns
R☉
Time interval 2
sec
Difference between the time intervals 1 and 2
Time difference
sec
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