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**Theory of Special Relativity**

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**Theory of Special Relativity**

Einstein is recognised for: encompassing all of the known results restating the laws of relativity and the required modifications for the laws of Physics to be consistent

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**Theory of Special Relativity**

Correspondence Principle

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**Correspondence Principle**

Any new theory proposed must agree with an older theory in terms of correct predictions which the theory gave. In this case the theory of special relativity must agree with classical mechanics at low speeds.

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**Correspondence Principle**

This requirement guided Einstein in developing the transformations which are the foundation of Special Relativity. It is a good idea to verify that these equations reduce to those from classical physics.

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**Einstein’s Postulates (1905)**

The postulates of special relativity are: The laws of Physics are the same in all inertial reference frames. The speed of light in a vacuum is c in all inertial reference frames.

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**Einstein’s Postulates (1905)**

Thus there is no ether (or rather no need for an absolute frame) as there is no preferred reference frame. The Galilean transform was replaced by a Lorentz transformation.

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**Consequences of Special Relativity**

Several phenomena are predicted as a result of these formulations which conflict with our everyday experiences. Specifically an effect on length and time.

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**Consequences of Special Relativity**

For example: there is no absolute length or absolute time events at different locations occurring simultaneously in one frame are not simultaneous in another frame.

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Definitions Proper frame: the reference frame in which the observed body is at rest. Proper length: the length in the frame where the object is at rest wrt the observer. Proper time: the time interval recorded by a clock attached to the observed body. Although results in both frames are still correct we use the following definitions.

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**Consequences of Special Relativity**

We will be looking at three particular phenomena: time dilation, length contraction and simultaneity.

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**Consequences of Special Relativity**

Time Dilation

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Time Dilation To illustrate how time is altered consider the case where there is a mirror attached to the ceiling of a car moving with a velocity v. V S′ S

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Time Dilation An observer in the car flashes a light, the path of which is shown below. D The diagram shows the path as seen by an observer in the car.

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Time Dilation D Time taken for the ray to be reflected to observer is

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Time Dilation D The time measured by this observer is the proper time since only one clock is needed to measure the time. The length is the proper length since the observer is at rest wrt the mirror.

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**Time Dilation An observer outside the car will see a moving mirror.**

Do A B As the car travels a distance , the light moves a distance of

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**Time Dilation The time taken for the light to be reflected back is**

Do A B The time taken for the light to be reflected back is Two clocks measured the time, one at A and the other at B

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**Time Dilation The relationship between and must be found.**

Do A B The relationship between and must be found. First using Pythagoras we determine an expression for containing :

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Time Dilation Simplifying we get However Therefore (Time dilation)

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**Time Dilation This can be written as (Lorentz Factor) the quantity**

is often represented by Therefore

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**Features of Time Dilation**

The time measured in the stationary frame is longer than that measured in the moving frame. Moving clocks run slower!! This effect is called time dilation.

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**Experimental Evidence of Time Dilation**

Decaying Muons

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**Decaying Muons Muons are particles produced by cosmic radiation**

have a life of If they move at a speed of the max. distance a muon can travel is They are measure moving at 0.99c wrt to the lab.

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**Decaying Muons Muons are particles produced by cosmic radiation**

have a life of If they move at a speed of the max. distance a muon can travel is But muons traveled They are measure moving at 0.99c wrt to the lab.

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Decaying Muons How can the muons travel ?

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Decaying Muons How can the muons travel ? Answer Time Dilation

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Decaying Muons Consider the two reference frames – the earth frame and the muon reference frame. The time measured by an observer is the proper time since only one clock (attached to the muons) is required. However an observer in the earth frames needs to clocks!!

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The Earth Frame Because of time dilation the lifetime of a muon is longer as measured by an observer in the earth frame. by a factor

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The Earth Frame Because of time dilation the lifetime of a muon is longer as measured by an observer in the earth frame. by a factor decay time in the earth frame,

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The Earth Frame Because of time dilation the lifetime of a muon is longer as measured by an observer in the earth frame. by a factor decay time in the earth frame, Hence available time

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The Muon Frame In the moving frame the muons see a shorter length to be travelled Achievable in their lifetime.

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**Consequences of Special Relativity**

Length Contraction

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Length Contraction length in a moving frame of reference will appear contracted in the direction of the motion. NB: The proper length is the length measured in which the object is at rest wrt the frame of reference.

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Length Contraction v E Consider a spaceship moving with a velocity between two stars. Two observers, one on earth at rest with respect to the stars and the other on the ship measure the distance.

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Length Contraction Earth frame Since at rest wrt the stars he measures a proper length . From his frame the time for the trip is

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**Length Contraction For him the star moving towards him a velocity .**

Ship frame For him the star moving towards him a velocity . Therefore measures a shorter distance. He measures a time and distance

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**Length Contraction Substituting for t0 we get Ship frame**

v Ship frame Substituting for t0 we get (Length contraction)

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**Features of Length Contraction**

A moving observer measures a contracted length in the direction of motion. There is no contraction perpendicular to the motion.

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**Consequences of Special Relativity**

Simultaneity

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Simultaneity Unlike our everyday experiences, two events which are simultaneous in one frame are in general not simultaneous in another.

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**Simultaneity The following thought experiment highlights this.**

v O′ O A B A train is moving with a velocity to the right when its ends are struck by lightning. Two observers at and both points midway between the two ends record the occurrence via light reaching them.

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Stationary Frame The light from the strikes at A and B reach the observer at the same time. Since the distance travelled by the light is the same, he concludes that the events were simultaneous.

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Train Frame By the time the light reaches , the observer has moved (and hence his point of reference) tA tB O A B v O′ t′B < t′A

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Train Frame Since the speed of light is constant, light from B′ reaches the observer first. Hence he concludes that the events are not simultaneous. The lightning strikes the right side first.

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Twin Paradox

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Twin Paradox Famous thought experiment in special relativity.

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**Twin Paradox Famous thought experiment in special relativity.**

Experiment involves two identical twins

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**Twin Paradox Famous thought experiment in special relativity.**

Experiment involves two identical twins – one stays on earth and the other journeys in a spaceship traveling near the speed of light to a star 30lys away.

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**Twin Paradox Famous thought experiment in special relativity.**

Experiment involves two identical twins – one stays on earth and the other journeys in a spaceship traveling near the speed of light to a star 30lys away. On reaching the star he immediately returns to earth at the same speed.

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Twin Paradox Experiment involves two identical twins – one stays on earth and the other journeys in a spaceship traveling near the speed of light to a star 30lys away. On reaching the star he immediately returns to earth at the same speed. He returns to find his brother 80 yrs while he only aged 10 yrs.

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Twin Paradox Experiment involves two identical twins – one stays on earth and the other journeys in a spaceship traveling near the speed of light to a star 30lys away. On reaching the star he immediately returns to earth at the same speed. He returns to find his brother 80 yrs while he only aged 10 yrs. Thus the traveler appears younger than the brother who was at rest.

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Twin Paradox From the frame of the brother on earth, he was at rest while his brother travelled at high speed. While according to his brother he was at rest in his frame and it was brother who sped away from him and then returned.

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Twin Paradox The “apparent” paradox is who was travelling at high speed and therefore should have aged? The view by both brother appears symmetric and hence the paradox.

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Twin Paradox When this hypothetical situation was first stated it appeared that there was a conflict.

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**Twin Paradox (Solution)**

The problem is that trip is not symmetrical as stated.

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**Twin Paradox (Solution)**

The problem is that trip is not symmetrical as stated. The brother on spaceship experiences a series of accelerations and decelerations as he leaves and returns to earth.

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**Twin Paradox (Solution)**

The problem is that trip is not symmetrical as stated. The brother on spaceship experiences a series of accelerations and decelerations as he leaves and returns to earth. Therefore he is not always in an inertial frame!!

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**Twin Paradox (Solution)**

The problem is that trip is not symmetrical as stated. The brother on spaceship experiences a series of accelerations and decelerations as he leaves and returns to earth. Therefore he is not always in an inertial frame!! Hence the laws of special relativity are not valid in his frame.

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**Twin Paradox (Solution)**

The other brother has remained in an inertial frame. Therefore his measurements are correct.

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**Twin Paradox (Solution)**

Therefore the brother on the spaceship is indeed younger.

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Time Dilation Example The period of a pendulum is measured to be 3s in the inertial frame of the pendulum. What is the period when measured by an observer moving at a speed of 0.95c wrt the pendulum?

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Time Dilation Example The period of a pendulum is measured to be 3s in the inertial frame of the pendulum. What is the period when measured by an observer moving at a speed of 0.95c wrt the pendulum? Solution:

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Time Dilation Example The period of a pendulum is measured to be 3s in the inertial frame of the pendulum. What is the period when measured by an observer moving at a speed of 0.95c wrt the pendulum? Solution:

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Time Dilation Example The period of a pendulum is measured to be 3s in the inertial frame of the pendulum. What is the period when measured by an observer moving at a speed of 0.95c wrt the pendulum? Solution:

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**Length Contraction Example**

An observer on earth sees a spaceship at an altitude of 435m moving downward towards the earth with a speed of 0.97c. What is the altitude of the spaceship as measured by an observer on the spaceship?

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**Length Contraction Example**

An observer on earth sees a spaceship at an altitude of 435m moving downward towards the earth with a speed of 0.97c. What is the altitude of the spaceship as measured by an observer on the spaceship? Solution:

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**Length Contraction Example**

An observer on earth sees a spaceship at an altitude of 435m moving downward towards the earth with a speed of 0.97c. What is the altitude of the spaceship as measured by an observer on the spaceship? Solution:

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Simultaneity Example A relativistic snake of proper length 100cm is moves with speed v=0.6c to the right across a table. A boy holding two hatchets 100cm apart plans to bounce them simultaneously so that the left hatchet lands immediately behind the snake’s tail. The boy sees the length of the snake contracted so that the hatchets miss the snake. The snake sees a contracted distance between the hatchets so it is cut. Who is correct?

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**Simultaneity Example Solution: A problem on simultaneity! Boy’s Frame**

The length of the snake = 80cm

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**Simultaneity Example Boy’s Frame**

Let us looks at the time which the two events occur. Since xL=tL=0

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Simultaneity Example In S′ the two hatchets do not fall simultaneously.

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