Physics

UNIT 12:RELATIVITY CONCEPTS AND POSTULATES OF SPECIAL RELATIVITY

Topic Area: Relativity and Particle Physics
Sub-Topic Area: Concepts and Postulates of Special Relativity

Key unit competence: By the end of the unit, I be able to analyse Relativity Concepts and postulates of special relativity.

Unit Objectives:
By the end of this unit learners will be able to;

◊ Explain the concept of general and special relativity.

◊ Explain the concept of the frames of reference and apply it in other theories.

12.0 INTRODUCTION

The general theory of relativity developed in the early 20th century, originally attempted to account for certain anomalies in the concept of relative motion. But it has developed into one of the most important basic concepts in physical science. The theory of relativity, developed primarily by German American physicist Albert Einstein, is the basis for later demonstration by physicists of the essential unity of matter and energy of space and time of gravity and acceleration.

12.1 DEFINITION OF RELATIVITY

This is a theory developed by Albert Einstein which says that anything except light moving with respect to the time and space depends on the position and movement of the observer. Einstein’s special theory of relativity (special relativity) is all about what’s relative and what’s absolute about time, space and motion.
The theory states that the laws of motion are the same for all inertial (non‑accelerating) frames of reference and that the speed of light (in a vacuum) is the same for all inertial reference frames. This leads to the equivalence of mass and energy, time dilation, and length contraction.
Special relativity requires us to think of space and time as inextricably linked. All our measurements of distance and time depend on the motion of the observer. The effects of time dilation and length contraction are only observed at very high speeds (close to the speed of light).

12.2 CONCEPT OF SPACE, TIME AND MASS

Time Dilation
Time dilation is the phenomenon where two objects, moving with respect to each other (or even just a different intensity of gravitational field from each other) experience different rates of time flow.
Time dilation becomes most apparent when one of the objects is moving at nearly the speed of light, but it manifests at even slower speeds. Here are just a few ways we know time dilation actually takes place:
•
Clocks in airplanes click at different rates from clocks on the ground.
•
Putting a clock on a mountain (thus elevating it, but keeping it stationary relative to the ground-based clock) results in slightly different rates.
•
The Global Positioning System (GPS) has to adjust for time dilation. Ground-based devices have to communicate with satellites. To work, they have to be programmed to compensate for the time differences based on their speeds and gravitational influences.
Let’s construct a light beam clock. It consists of two mirrors, one at a distance D above the other. At t = 0, we launch a photon of light upwards from the bottom of the mirror. It reflects from the top mirror and returns to its starting position, use c as the speed of the photon;

First, the photon is released. When the photon hits the top mirror, the whole clock has moved a distance L to the right. Thus, the photon travelled a longer distance    as seen by this other observer. When it returns to the bottom mirror, it has travelled a distance still at speed c (recall all observers measure this same speed). Thus, Dt (the time for 1 tick according to the new observer)

Since nothing can travel faster than light, therefore, γ ≥ 1 and it appears to the observer who watches the clock go by at velocity v that it takes longer to tick
(Dt > Dt0) or runs slowly compared to his own clock. This is called time dilation and is a property of time, not just our unusual clock.

EXAMPLE 12.1

An astronaut travels to a distant planet with a speed of 0.5c. According to his clock, the trip takes one year.

(a) How long does the trip appear to take to an observer on the earth?

(b) How fast should the astronaut travel so that the travel time appears two years to the observer on the earth?

Solutions

(a) The time measured in the spacecraft is the proper time since the clocks in the spacecraft are at rest with respect to the astronaut. So,

Length Contraction

If we turn our light beam clock to face in the direction of motion, time dilation implies length contraction. If the observer at rest with respect to the clock (now a ruler) says it has proper length L0, then an observer on the earth watching him and his clock/ruler by velocity v sees the ruler having length L. Objects look shorter (they are contracted) in the direction of motion.

A metre stick zips by you with a speed of 0.9c. The length of the stick is along its direction of motion. How long does it appear to be?

Solution:

Momentum, Mass and Energy
Einstein found that momentum;

EXAMPLE 12.3

A proton travels at a speed of 0.9c. Compare its relativistic and classical momenta.

Solutions:

As v → c, m → ∞. Thus, infinite energy would be needed to accelerate an object to the speed of light. Einstein showed the total energy of a free body;

where m0c² is the mass energy of the body.
As space and time are united in the theory, so are momentum and energy. We see here m0 is rest mass, p is momentum and c is speed of light.

EXAMPLE 12.4

An electron has a speed of 0.8c. What is its kinetic energy?

EXERCISE 12.1

1. An electron has a speed of 0.1c. What is its kinetic energy?

2. Calculate the rest mass energy of the electron in electron volts.

3. Find the kinetic energy released in the fusion reaction given below:

12.3 CONCEPT OF FRAME OF REFERENCE

Imagine you threw and caught a ball while you were on a train moving at a constant velocity past a station. To you, the ball appears to simply travel vertically up and then down under the influence of gravity. However, to an observer stood on the station platform, the ball would appear to travel in a parabola, with a constant horizontal component of velocity equal to the velocity of the train. This is illustrated in Fig.12-4 below.

The different observations occur because the two observers are in different frames of reference.

A frame of reference is a set of coordinates that can be used to determine positions and velocities of objects in that frame; different frames of reference move with respect to one another.

This means that when you are standing on the ground, that is your frame of reference. Anything that you see, watch or measure will be compared to the reference point of the ground. If a person is standing in the back of a moving truck, the truck is now the frame of reference and everything will be measured compared to it.

Types of Frame of Reference

There are two types of frames of reference.

Inertial Frame of Reference: It is a frame of reference in which a body remains at rest or moves with constant linear velocity unless acted upon by forces. Any frame of reference that moves with constant velocity with respect to an inertial system is itself an inertial system. In other words, it is the frame of reference in which Newton’s first law of motion holds good.

Non-inertial Frame of Reference: This is a frame of reference that is undergoing acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will in general detect a non-zero acceleration. In this frame of reference, Newton’s first law of motion does not hold good.

12.4 GALILEAN EQUATION OF TRANSFORMATION

Galilean transformations, also called Newtonian transformations, are set of equations in classical physics that relate the space and time coordinates of two systems moving at a constant velocity with respect to each other. Galilean transformations formally express the ideas that space and time are absolute; that length, time, and mass are independent of the relative motion of the observer; and that the speed of light depends upon the relative motion of the observer.

Let there be two inertial frames of references S and S′ where S is the stationary frame of reference and S′ is the moving frame of reference. At time t = t′ = 0, i.e., in the start, they are at the same position, i.e., observers O and O′ coincide. After that S′ frame starts moving with a uniform velocity v along x axis.

Let an event happen at position A in
the frame S′. The coordinate of the P will be x′ according to O′, the observer in S′ and it will be x according to O in S. The frame S′ has moved a distance vt in time t.
The Galilean transformation relates the coordinates of events as measured in both frames. Given the absolute nature of time, Newtonian physics, it is the same for both frames. So, this may look over-elaborate if we write

t = t′

It is seen that in direction y and z, displacements remain the same. So, we may summarise these displacements as:

This set of equations is known as the Galilean Transformation. They enable us to relate a measurement in one inertial reference frame to another.

EXAMPLE 12.5
If a vehicle is moving in x-direction in system S, then what would be the velocity of the vehicle in S’?

and

⇒

12.5 POSTULATES OF SPECIAL THEORY OF RELATIVITY

With two deceptively simple postulates and a careful consideration of how measurements are made, Einstein produced the theory of special relativity.

First postulate: The Principle of Relativity

This states that the laws of physics are the same in all inertial frames of reference.

This postulate relates to reference frames. It says that there is no preferred frame and, therefore, no absolute motion.
To understand the meaning of this postulate, consider the following situation.

You are sitting in a train that is stopped at a railway station. Another train is facing the opposite direction on the track directly beside you. Ten minutes before your train is due to leave, you look out through the window at the other train and see that it is slowly starting to move relative to yours. Your first reaction would probably be one of surprise: your train was leaving.

early! After passing the train from your window, you might notice that the station was still there, and you realize that it was the other train that was moving.

Second postulate: The Principle of Invariant Light Speed

The speed of light is a constant, independent of the relative motion of the source and observer.

The speed of light in vacuum (c = 3 × 108 m/s ) is so high that we do not notice a delay between the transmission and reception of electromagnetic waves under normal circumstances. The speed of light in vacuum is actually the only speed that is absolute and the same for all observers as was stated in the second postulate.

12.6 CONCEPT OF SIMULTANEITY

The concept of simultaneity says that two events that are simultaneous to one observer are not necessarily simultaneous to a second observer. Both observers are correct in their observations -- there is no best or preferred frame of reference.

If the speed of light is the same in all moving coordinate systems, this means that events that occur simultaneously in one system may not be observed as being simultaneous in another coordinate system.
An example is illustrated in the Fig. 12.7 below.

An observer O′ stands in the middle of a moving boxcar and another observer O stands at rest beside the track. When the positions of the observers coincide, a lightning bolt strikes at each end of the boxcar, leaving mass on the ground and at each end of the boxcar. The light from the lightning strikes at A and B reaches to observer O at the same time, so observer O′ concludes that the lightning strikes occurred simultaneously. But to observer O′ in the moving boxcar, the lightning strikes do not appear to occur at the same

time. The light traveling from A′ to O′ travels further than the light from B′ to O′. Because of the motion, O′ moves towards the incoming beam from B′ and away from the incoming beam from A′. So to observer O′ the strike at B′ appeared to occur before the strike at A′.

                                   END OF UNIT QUESTIONS

1. If you were on a spaceship travelling at 0.50c away from a star, when would the starlight pass you?

2. Does time dilation mean that time actually passes more slowly in moving references frames or that it only seems to pass more slowly?

3. If you were travelling away from the Earth at 0.50c, would you notice a change in your heartbeat? Would your mass, height, or waistline change? What would observers on the earth using a telescope to see you say about you?

4. What happens to the relativistic factor when objects travel at normal everyday velocities?

5. A spaceship travels at 0.99c for 3 years ship time. How much time would pass on the earth?

6. A spaceship is travelling at a speed of 0.94c. It has gone from the earth for a total of 10 years as measured by the people of the earth. How much time will pass on the spaceship during its travel?

7. A spaceship has gone from the earth for a total time of 5 years ship time. The people on the earth  have measured the time for the ship to be away to 25 years. How fast was the ship travelling?

8. A 520 m long (measured when the spaceship is stationary) spaceship passes by the earth. What length would the people on the earth say the spaceship was as it passed the earth at 0.87c?

9. A 25 m long beam is shot past a stationary space station at 0.99c. What length does the people on  board the space station measure the beam to be?

10. A 100 m long steel beam is moving past the earth. Observers on the earth actually measure the steel beam to be only 50 m long. How fast was the beam travelling?

UNIT SUMMARY

Definition of relativity

Anything except light moves with respect to time and space depends on the position and movement of someone who is watching.

Concept of space, time and mass

• Time Dilation
Time dilation is the phenomenon where two objects moving relative to each other (or even just a different intensity of gravitational field from each other) experience different rates of time flow. The total time is given by

• Length Contraction
If we turn our light beam clock to face in the direction of motion, time dilation implies length contraction.

Postulates of special theory of relativity

• First postulate
This states that the laws of physics are the same in all inertial frames of reference.
This postulate relates to reference
frames. It says that there is no preferred frame and, therefore, no absolute motion.

• Second postulate
This states
that speed of light, c is a constant, independent of the relative motion of the source and observer.