**Special Relativity**

Galilean Transformation Equations

Special Relativity

In relativistic mechanics there is no such thing as absolute length or absolute time. Further, events at different locations that are observed to occur simultaneously in one frame are not observed to be simultaneous in another frame moving uniformly past the first.

An INTRODUCTION

When an observer "sees" something, what we really mean is that a particular event is recorded in the reference frame associated with the observer

A reference frame is a way of labeling each event with its location in space and the time

Galilean relativity

The laws of mechanics must be the same in all inertial frames of reference. Inertial frames of reference are those reference frames in which Newton’s laws are valid.

**An overview of Special relativity**

**2 Postulates**

**Time Dilation**

**Lorentz Transformation**

**Length Contraction**

Length Contraction

Used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics.

If a passenger in the airplane throws a ball straight up into the air, the passenger observes that the ball moves in a vertical path.

Consider an airplane in flight, moving with a constant velocity.

Describe how measurements of space and time by two observers are related.

They reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events. Is a linear transformation. It may include a rotation of space.

Lorentz Transformations

The motion of the ball is precisely of the same as it would be if the ball was thrown while at rest on Earth.

The law of gravity and the equations of motion under constant acceleration are obeyed whether the ball is thrown when the airplane was at rest or in uniform motion.

Now consider the same experiment when viewed by another observer at rest on Earth.

This stationary observer views the path of the ball to be a parabola

Further, according to this observer, the ball has a velocity to the right equals to the velocity of the plane.

Although the two observers disagree on the shape of the ball’s path, both agree that the motion of the ball obeys the law of gravity and Newton’s laws of motion, and they even agree on how long the ball is in the air

There is no preferred frame of reference for describing the laws of mechanics

Untenable at speeds that approach the speed of light.

Does the concept of Galilean relativity in mechanics also applies to experiments in electricity, magnetism, optics and other areas

NO

If we assume the laws of electricity and magnetism are the same in all inertial frames, a paradox concerning the speed of light immediately arises.

According to electromagnetic theory, the speed of light always has a fixed value of 2.997 x 10^8 m/s in free space.

Hence, Galilean relativity is inconsistent with Maxwell’s well-tested theory of electromagnetism.

According to Galilean relativity, however, the speed of the pulse relative to the stationary observer S outside the reference frame should be vector c + v.

All physical laws valid in one frame of reference are equally valid in any other frame of reference moving uniformly relative to the first

Postulate 1

The speed of light is measured as constant in all frames of reference.

Consider a train, of height h, with a beam of light bouncing up and down between mirrors of the floor and ceiling

The time between bounces can be interpreted as ticks of a clock

This interval, as measured on the train, is independent of whether the train is moving

From the ground, the light appears to move diagonally, and hence traveling a longer path than the vertical path seen on the train

But since light must moves at the same speed for both observers, each "tick" take longer for the ground-based observer.

**Time dilation**

The time of the round trip can be used to measure the length of the train

If it takes The length of the train t' to make a round trip, length of the train is x'

Compared to an observer at ground level

A beam of light bounces back and forth between mirrors at the front and back of a moving train

The distance traveled in one direction is different from that in another. Light starting from the back of the train must chase that in the other. If it takes time t1 to catch up, then distance traveled is x + v t1.

Length of train as seen from the ground

Distance the front of the train traveled while the light was under way

Postulate 2

it is the concept that distant simultaneity is not absolute, but depends on the observer's reference frame.

Relativity of Simultaneity

Observer on platform

T=1

T=2

A flash of light is given off at the center of a train just as two observers pass each other. The observer on board the train sees the front and back of the train at fixed distances from the source of light.

The observer standing on the platform sees the rear of the train moving (catching up) toward the point at which the flash was given off and the front of the train moving away from it. By Postulate 2, the light headed for the back of the train will have less distance to cover than the light headed for the front.

T=1

T=2

Observer on train

according to this observer, the light will reach the front and back of the train at the same time.

the flashes of light will strike the ends of the train at different times.