Time Dilation

We all know that time is one of the fundamental units of measurement. Time is measured in seconds, minutes, hours, and days. This usual measurement of time is known as linear time. Time is the most essential part of our living. To define any event, we use time as the frame of reference. We use the time to define predefined events. The time elapsed is dependent on the motion of an observer with respect to the process being measured. In this session, let us learn in detail about a concept known as time dilation, along with the time dilation formula and equation. Time dilation is observed when the clock is in motion with respect to the other clock.

What is Time Dilation?

Einstein developed the theory of general relativity to deal with accelerated frames and with gravity, a prime source of acceleration. Einstein’s special relativity explained the concept of slowing of a clock which is experienced by an observer who is in motion with respect to that clock. Einstein’s special relativity theory states that time can pass at different rates in different reference frames. It is the experience of time passing slower for an observer who is in relative motion to another observer. The difference in the passed time measured by two clocks (different frames of reference) gives time dilation.

Before we know what time dilation exactly is, and how it is measured, let us understand what is meant by proper time.

As we know, time dilation is due to the effect of the gravitational potential of their locations or due to the effect of velocity. It is measured mostly due to the effect of velocity. Proper time is measured by a clock that has the same motion as the observer, i.e, the time measured by a clock that has the same motion as the observer. Proper time is also referred to as one-position time. In the observer’s reference frame, the time between events is called observer time or two-position time.

Note: The observer time and proper time are measured in seconds. Observer time is always greater than the proper time. This effect is termed time dilation.

Time Dilation Formula

Consider the proper time or one-position time is represented by \Delta t_{0}. The observer time or two-position time is represented by \Delta t. Since we know the observer time will be more than the proper time. The formula to find the time dilation is given by:

Where,

Δt and Δt0 are measured in seconds

v = velocity (m/s)

c = speed of light (3.0 × 10⁸ m/s)

Note: As per the Lorentz transformation, in a moving frame the clock will be considered to be running slow. In the rest frame, time will always be the shortest.

Read more: Time dilation formula along with solved example

Solved Example

1. Gopi boards a spaceship and flies past the Earth at a speed of 0.800 times the speed of light. While his twin brother Goutham stays on the Earth. At an instant, Gopi’s ship passes the Earth and they both start timers. Gopi watches his timer, and after he sees 60.0 seconds have passed, he stops it. At that instant, how much time would Goutham’s timer say has passed?

Solution:

Given: In Gopi’s reference time frame, two events take place at the same time in the same position, and are considered as the proper time Δt0.

Gopi’s timer happens in two positions.

In Goutham’s reference frame, the time difference on the Earth is the observer time (Δt).

The amount of time that passes in Goutham’s reference frame is found using the formula:

When Gopi is in a reference frame moving at 0.800c relative to Goutham’s reference frame, Gopi observes that 60 seconds pass, his brother will observe that 100 seconds have passed.

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