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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**Theory of Special Relativity**Correspondence Principle**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.**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.**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.**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.**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.**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.**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.**Consequences of Special Relativity**• We will be looking at three particular phenomena: time dilation, length contraction and simultaneity.**Consequences of Special Relativity**Time Dilation**V**S′ S 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.**D**Time Dilation • An observer in the car flashes a light, the path of which is shown below. • The diagram shows the path as seen by an observer in the car.**D**Time Dilation • Time taken for the ray to be reflected to observer is**D**Time Dilation • 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.**Do**A B Time Dilation • An observer outside the car will see a moving mirror. • As the car travels a distance , the light moves a distance of .**Do**A B Time Dilation • The time taken for the light to be reflected back is • Two clocks measured the time, one at A and the other at B**Do**A B Time Dilation • The relationship between and must be found. • First using Pythagoras we determine an expression for containing :**Time Dilation**Simplifying we get However Therefore (Time dilation)**Time Dilation**This can be written as (Lorentz Factor) the quantity is often represented by Therefore**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.**Experimental Evidence of Time Dilation**Decaying Muons**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 .**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 .**Decaying Muons**• How can the muons travel ?**Decaying Muons**• How can the muons travel ? Answer • Time Dilation**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!!**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**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,**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**The Muon Frame**• In the moving frame the muons see a shorter length to be travelled . • Achievable in their lifetime.**Consequences of Special Relativity**Length Contraction**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.**v**E Length Contraction 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.**Earth frame**Length Contraction • Since at rest wrt the stars he measures a proper length . • From his frame the time for the trip is**v**Ship frame Length Contraction • For him the star moving towards him a velocity . • Therefore measures a shorter distance. • He measures a time and distance**v**Ship frame Length Contraction • Substituting for t0 we get (Length contraction)**Features of Length Contraction**• A moving observer measures a contracted length in the direction of motion. • There is no contraction perpendicular to the motion.**Consequences of Special Relativity**Simultaneity**Simultaneity**• Unlike our everyday experiences, two events which are simultaneous in one frame are in general not simultaneous in another.**v**O′ O A B Simultaneity The following thought experiment highlights this. • 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.**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.**v**O′ O A B Train Frame • By the time the light reaches , the observer has moved (and hence his point of reference) tA tB t′B < t′A**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.**Twin Paradox**• Famous thought experiment in special relativity.**Twin Paradox**• Famous thought experiment in special relativity. • Experiment involves two identical twins**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.**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.