General relativity interprets gravity as the warping of spacetime.

When we view a picture of the gravitational field around a massive object,

it's usually represented as a depression in space.

However, we need to understand that gravity also warps the passage of time.

It's strangely difficult for the human mind to grasp the concept of warping spacetime.

We understand what it means to bend or warp a material like plastic,

but what does it mean when the actual space and

time that we live in are bent and twisted?

In a sense, warped spacetime means that

the paths we choose to cross space and time will be shorter or

longer in distance between two points and in the duration it

takes to travel between them

depending on what the gravitational fields are along the path.

Let's focus specifically on how gravitational fields warp

the time component in an effect called gravitational time dilation.

Let's start with an example by considering

two astronauts exploring an unstudied planet around a distant star,

perhaps planet e in the nearby Trappist-1 System,

which we'll shorten to trappy.

One astronaut needs to stay with the ship in order to orbit around

the parent star while the astronaut descends to trappy surface.

Since we are talking about time,

both astronauts will need to carry clocks,

which they synchronize before they separate.

Far from the surface of the planet,

both clocks tick in perfect synchronicity.

One astronaut now descends to the surface of trappy.

On the surface, he is deep in the planet's gravitational well and therefore,

experiences a greater gravitational force.

The spacetime in the vicinity of the planet will also be warped.

The effect that the warping has on

the astronauts' clocks causes it to tick more slowly than the one in orbit.

For every tick of the clock on the surface,

the orbital clock ticks more rapidly.

On the surface of trappy,

the astronaut doesn't experience the change in the passage of time

because all biological processes are likewise slowed down by the warping of gravity.

Just like the ticks of the clock,

a distant observer would see the heartbeat

of an astronaut on the surface to beat more slowly.

Once the surface mission is complete,

the two astronauts rejoin one another in orbit around trappy.

The astronaut who stayed in orbit will be dismayed.

She experienced a longer time than the astronaut who was on the surface.

Depending on the duration of the stay and the strength of the gravitational field,

the astronaut who went down to the planet's surface will

experience fewer ticks of the clock and therefore,

be several seconds younger than the one who stayed in orbit.

To calculate how time has worked in a strong gravitational field,

the following equation is employed.

Delta t planet, the elapsed time on the surface of the planet,

is equal to Delta t orbit,

the elapsed time on the orbiting spaceship,

times the square root of

one minus two times G times mass divided by radius times c squared.

In this formula, the mass and radius refer to the mass and radius of the planet.

But if instead of a planet,

you were a distance R from a star or a black hole with mass M,

you could use the same formula.

The important thing in this formula is that the quantity

inside the square root sign is smaller than one.

So the amount of time that passes when you're in

a gravitational well is smaller than

if you're out in space far from the gravitating object.

Note that this formula doesn't make sense if

the ratio of the mass to radius gets too large.

This formula only makes sense if R is

larger than two times G times mass divided by c squared.

You might think that your everyday life is not much

affected by time dilation due to special or general relativity.

However, you may be surprised to learn that almost everyone carries

a piece of technology that would be useless without both theories, GPS.

The Global Positioning System that you use every time you navigate with a map on

your smartphone depends on Einstein's theory of relativity to function correctly.

Handheld GPS works because the device inside your smartphone is

capable of measuring and comparing the signals

from multiple satellites in orbit around the Earth.

These satellites are placed in well-known orbits and carry very precise clocks.

By broadcasting a timing signal that can be picked up on a GPS receiver,

the difference in timing signals from

different satellites can be used to triangulate your position.

Since GPS satellite travel at about 14,000 kilometers per hour,

they experience a very slight time dilation due to special relativity.

Each day, a satellite's clock would appear to slow down by about seven microseconds.

That doesn't sound like much,

but if you neglected this drift,

your GPS would accumulate an error of about two kilometers every day.

General relativity predicts that the clocks aboard a GPS satellite traveling at

an altitude of 20,000 kilometers would appear to tick faster than clocks on Earth.

Every day, a satellite clock would appear to speed up by

45 microseconds compared to clocks on Earth's surface.

If this error wasn't corrected,

the GPS would accumulate an error of over 13 kilometers a day.

Since special relativity works to slow down

the apparent rates of the clock on a GPS satellite,

and general relativity speeds up their apparent rates,

the combined effects add up to a 38 microseconds per day error.

Without relativity, our GPS devices would drift by over 10 kilometers every day,

roughly the same as 12 centimeters a second.

Luckily, we know about the effects of relativity.

So we can correct for this drift.

GPS devices are some of

the most robust tests we have for Einstein's theories of relativity.

In the movie "Interstellar",

the main character Cooper is sent to retrieve

a fellow explorer Mann from the surface of

Miller's planet orbiting the nearby black hole Gargantua.

On the surface of Miller's planet,

an hour of time is equivalent to seven years on Earth.

This is an example of the correct use of an effect called gravitational time dilation.