Let us today discuss and derive an interesting concept in relativity called the time dilation. You must be astonished to know that when an object moves with a speed comparative to the speed of light, its time dilated or more appropriate term is that it is appeared to move slow.

Let there are two inertial frames of references S and S1. S is the stationary frame of reference and S’ is the moving frame of reference. At time t=t’=0 that is in the start, they are at the same position that is Observers O and O’ coincides. After that S’ frame starts moving with a uniform velocity v along x axis.

Let a clock is placed in the frame S’. The time coordinate of the initial time of the clock will be t1 according to the observer in S and the time coordinate of the final tick (time ) will be will be t2 according to same observer.

The time coordinate of the initial time of the clock will be t’1 according to the observer in S’ and the time coordinate of the final tick (time ) will be will be t’2 according to same observer.Therefore the time of the object as seen by observer O’ in S’ at the position x’ will be

t’ = t’2 – t’1                             (1)

The time t’ is called the proper time of the event. Proper time is defined as the time of an event measured by the observer which is in the same frame in which the event is occurred.

The apparent or dilated time of the same event from frame S at the same position x will be

t = t2 – t1                                 (2)

Now use Lorentz inverse transformation equations for time (that I discussed during the derivation of Lorentz transformation equations), that is

t1 = (t’1 + x’v/c²⁾/(√1 – v²/c²)               (3)

t2 = (t’2 + x’v/c²⁾/(√1 – v²/c²)   (4)

By putting equations (3) and (4) in equation (2) and solving, we get

t =  (t’2 – t’1)/ (√1 – v²/c²)

Substitute equation (1) in above equation,

t = t’/(√1 – v²/c²)

This is the relation of the time dilation.

Why it known as time dilation, if we solve (√1 – v²/c²), then the value will always be in decimal (except when v = c).

If we divide t’ with a decimal value, then t will be more than t’. This is the reason that it is called time dilation as dilation means to get increase. Thus the clock will appear to be slower now.

For example let us discuss a numerical: If an object is moving with speed 0.8c and its keeping the time 1 second, then what will be its apparent time?

Solution:

Given v = 0.8c

Proper length t’ = 1 sec

To calculate t

As t = t’/(√1 – v²/c²)

If we substitute the given values, we get

t = 1.67 sec

Thus t > t’

Special Case:

If v <<< c, then v²/c² will be negligible in t’/(√1 – v²/c²) and it can be neglected

Then t = t’

The post Time dilation in relativity derivation first appeared on Winner Science.

The term relativity comes from the word “relative that is in relation”. In this universe, everything is relative, there is nothing like absolute rest. If you think that you are sitting somewhere and you are in the state of rest, then my dear you are wrong. You are also in the state of motion, because if someone is able to see you from outside the earth (suppose from moon), then you are also moving. Therefore, everything is relative.

Here question arises with respect to which we can study the motion of another object? This problem is solved with the concept of frame of reference.

Definition: The frame of reference is defined as any coordinate system with respect to which one can study the motion of the another object. For example, a bus can be a frame of reference with respect to which one can study the motion of another object.

Types of the frame of reference:

1. Inertial frame of reference:

[This you can understand only if you know the Newton’s first law or law of inertia. So what is that? Remember of forgot! Let us revise it. Newton’s first law is that a body at rest will remain at rest; a body in uniform motion will remain in uniform motion, until and unless some external force is applied on it.

Here a body at rest will remain at rest, until and unless some external force is applied on it is also called law of inertia of rest.

A body in uniform motion will remain in uniform motion until and unless some external force is applied on it is also called law of inertia of uniform motion.]

Thus inertial frame of reference is that frame in which the law of inertia is obeyed. In other words, the non-accelerated frame of reference is called inertial frame of reference.

2. Non – inertial frame of reference:

[It should be simple for you know if you understood the inertial frame of reference].

Non-inertial frame of reference is that frame of reference in which the law of inertia is not obeyed. In other words, the accelerated frame of reference is called inertial frame of reference.

Brain Teaser: Can you give an example of inertial frame of reference that is non-accelerated one?

Answer:  The answer is nothing can be perfect inertial frame. But we consider earth as an inertial frame because acceleration of the earth is negligible.

Types of theory of relativity:

On the basis of this, theory of relativity is divided into two types:

1. Special theory of relativity: This theory deals with inertial frames of reference.

2. General theory of relativity: This theory deals with non-inertial frames of reference. 

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