After understanding that time is an dimension and space and time is merged into single space-time and time slows down for the things that are in motion by the factor of Gamma.
Now we will see how it happens and in what ways we can imagine and understand time dilation clearly. Here i will give some of the situations that will clearly make our mind to think what we actually mean by time. So the situation goes like this imagine a spaceship in which your friend is in and his heart is beating at the rate of 60 beats per minute. There are mechanical and digital watches which are ticking exactly at the same speed. Now there is also one flash bulb inside the spaceship and one plane mirror kept parallely to the flash bulb. Let us consider the distance between flash bulb and the mirror is "D" then if one is interested in finding the time that it takes for a beam of light to go from bulb to mirror and from mirror to flashlight due to reflection then it is simply calculated by equation that
T=2D/c
Where c is the speed of light approximately equal to 300000km/sec.
Now we say that spaceship along with with all the clocks is passing by us with velocity "v". And here on earth we also have a similar set of clocks all ticking at the same rate as one another. This follows from Einstein's first postulate that no inertial frame is fundamentally Better than any other. However if we observe the SpaceShip and find the difference in ticking rates of clock then only it will be cleared to all of us why velocity is important factor for the change in rate of time with respect to different observers. Now to find the time it takes for a beam of light emitted from flashlight to travel distance of "2D". In order to find the time we assume that light clock is oriented perpendicular to the direction of motion of spaceship moving with velocity "v". If we imagine this situation from our point of view we know that spaceship is in motion so the beam of light which goes from bulb to mirror and vice-versa is not going to be straight instead it is going to make some angle because as the ship is in motion the trajectory of light is going to appear different from our point of view. Its like someone is throwing ball from moving train vertically. Its going to be vertical from the ground frame but for the person who is in moving train its not going to be straight instead it is going to follow a parabolic trajectory. It means light has to travel greater distance in our frame just because its not straight. Speed of light is same in both frames of reference but the greater distance means it is going to take more time for clock to tick once from our point of view than it took for observers in spaceship. now we have to use simple mathematics to come to the conclusion. We know spaceship is moving with velocity v and speed of light is c and light is not going in straight direction from our point of view, using Pythagoras theorem the vertical component of light must be (√c²-v²) so the time it takes for light to go from our point of view is
T`= 2D/(√c²-v²) = 2D/c(√1-v²/c²)
= Time between ticks of our own light clock/√c²-v².
And this is how time changes for different observers. Time dilation means that it would take less time foe the people who are in moving space ship. Let us discuss one more example on this. Suppose a person wants to visit Alpha Centauri one of the nearest stars and their ship moves at the speed (4/5)c relative to Earth. The star is nearly at a distance of 4 light years that is it takes 4 years for light to cover the distance so the time it is going to take is from Earth's point of view is
T=4c.yrs/(4/5)c=5yrs
To arrive. Spaceship runs slow by the factor √1-(4/5)²=3/5, however so the trip takes only (3/5)*5yrs=3yrs according to spaceship clocks. All shipboard clocks runs slow do the person who is in spaceship is just only 3 years older when they arrive.
Now we will see how it happens and in what ways we can imagine and understand time dilation clearly. Here i will give some of the situations that will clearly make our mind to think what we actually mean by time. So the situation goes like this imagine a spaceship in which your friend is in and his heart is beating at the rate of 60 beats per minute. There are mechanical and digital watches which are ticking exactly at the same speed. Now there is also one flash bulb inside the spaceship and one plane mirror kept parallely to the flash bulb. Let us consider the distance between flash bulb and the mirror is "D" then if one is interested in finding the time that it takes for a beam of light to go from bulb to mirror and from mirror to flashlight due to reflection then it is simply calculated by equation that
T=2D/c
Where c is the speed of light approximately equal to 300000km/sec.
Now we say that spaceship along with with all the clocks is passing by us with velocity "v". And here on earth we also have a similar set of clocks all ticking at the same rate as one another. This follows from Einstein's first postulate that no inertial frame is fundamentally Better than any other. However if we observe the SpaceShip and find the difference in ticking rates of clock then only it will be cleared to all of us why velocity is important factor for the change in rate of time with respect to different observers. Now to find the time it takes for a beam of light emitted from flashlight to travel distance of "2D". In order to find the time we assume that light clock is oriented perpendicular to the direction of motion of spaceship moving with velocity "v". If we imagine this situation from our point of view we know that spaceship is in motion so the beam of light which goes from bulb to mirror and vice-versa is not going to be straight instead it is going to make some angle because as the ship is in motion the trajectory of light is going to appear different from our point of view. Its like someone is throwing ball from moving train vertically. Its going to be vertical from the ground frame but for the person who is in moving train its not going to be straight instead it is going to follow a parabolic trajectory. It means light has to travel greater distance in our frame just because its not straight. Speed of light is same in both frames of reference but the greater distance means it is going to take more time for clock to tick once from our point of view than it took for observers in spaceship. now we have to use simple mathematics to come to the conclusion. We know spaceship is moving with velocity v and speed of light is c and light is not going in straight direction from our point of view, using Pythagoras theorem the vertical component of light must be (√c²-v²) so the time it takes for light to go from our point of view is
T`= 2D/(√c²-v²) = 2D/c(√1-v²/c²)
= Time between ticks of our own light clock/√c²-v².
And this is how time changes for different observers. Time dilation means that it would take less time foe the people who are in moving space ship. Let us discuss one more example on this. Suppose a person wants to visit Alpha Centauri one of the nearest stars and their ship moves at the speed (4/5)c relative to Earth. The star is nearly at a distance of 4 light years that is it takes 4 years for light to cover the distance so the time it is going to take is from Earth's point of view is
T=4c.yrs/(4/5)c=5yrs
To arrive. Spaceship runs slow by the factor √1-(4/5)²=3/5, however so the trip takes only (3/5)*5yrs=3yrs according to spaceship clocks. All shipboard clocks runs slow do the person who is in spaceship is just only 3 years older when they arrive.
The faster the spaceship moves time will run more slowly because the person is as much possible going in space dimension so time has to contract itself. This all means that we don't understand what is time and we just know how to calculate it.
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