I will discuss special relativity and the concept of length, mass and time dilation compared to classical physics.
To derive the equation of time dilation one must understand the concept of light clock. SAy a person is ina train car and the height of the trian car is say ( L) and there is a light clock and it tiks when the light trvels the distance (L) and reaches it again. That is for The person in the car if he moves say at a velocity of v, the time he observes is 2L/c. However if a person is stationary and watches the time it is not 2L/C because he will not see the light travelling as observed by the person in the car. That is, the time itself is relative and not a constant for even if the clocks are perfect. In effect, if v reaches the speed of light the time observed by the person stationery will be reduced compared to the person in the car. However if v is very less compared to the speed of light then the the times for the person in the car and for the person who is stationery is approximately same. But it is not equal. That is the time looks constant for all persons because of the sppeds we move is very less compared to the spped of light which is a constant according to Special relativity Theory.
The Lorenz Time Dilation equation
As discussed above, the distance travelled for the light as per the stationery person is the slant distance which is the Pythagorean slant distance of a right angle triangle. Applying this geometry to this distance, this distance is also equal to the time observed by the person stationery and is equal to the time say (t’)* c or ct’. That is, the following equation can be written as follows:
ct’ = square root of ( 4L squared + v squared* t squared)
That is, by squaring both sides c squared* t’ squared = 4 ls quard + vsquared8 t’ squared
That is t’ squared* ( c squared – v squared) = 4 L suquared
t’ = square root of 4 L squared/ square root of ( c squared – v squared)
t’ = square root of 4 L squared/ Square root of c squared * ( 1 – v squared/ C squared)
t’ = 2L/c *square root of ( 1- v squared/ c squared)
But t = 2L/c
That is t’ = t/ square root of ( 1- v squared/ c squared)
That is for a person travelling at close to the speed of light the time is reduced compared to a person at rest very significantly.
Lorenz Length contraction equation
This is related to time dilation equation. That is, if v approaches to the speed of liight the distance will be reduced. That is, if time increases at the same rate the length reduces, That is
L (Observed) = L ( object)* square root of (1- v squared/c squared).
Mass Increases with velocity
Say a person going forward leaving a man behind in a spaceship at constant thrust. So his spaceship accelerates. He is going faster and faster. However, he cannot go faster than the speed of light which is a constant. Only he can approach it asymptotically. That is when he goes faster and faster the acceleration decreases. However one knows if F = Mass* acceleration and if F is constant, then if acceleration decreases the mass must increase. That is in other wards when the speed of the spaceship approaches the spped of light, the mass increases as explained above. That is the Mass at rest is not equal to Mass at very high velocity according to special relativity theory.
Adding relative velocities
In Newtonian Physics the relative velocities are added. For example if a train moves at v velocity and a person moving in relative to the train at u velocity the person at rest observes the person moving at u + v velocity. However at very high speeds as the speed of light is a constant adding velocities may exceed the speed of light. It cannot be as any relative velocity cannot be more than the speed of light. In Special relaticity at very high speeds the relative velocity observed is equal to (U+ v)/ ( 1+ u*v/c squared). That is the observer will see the person in slow motion due to the fact that the speed of light is a constant. If v and u are relatively small compared to the speed of light then Newtonian Physics of the universe is applicable. However at very high speeds it breaks down.
That is, Mass ( observed) = Mass of the object/square root of ( 1- v squared/ c squared)
This is the basis of the famous equation e= m c squared.