Special Relativity

1. Special Relativity An Introduction To ‘High Speed’ Physics

2. 18th Century 19th Century 20th Century Newton must be right!? diffraction/interference wave-particle duality What Is Light? There were two contradicting theories as to the nature of light: Newton – light is corpuscular Huygens – light is a wave If light is a wave, through what does it propagate?

3. The Michelson-Morley Experiment The Aether Space is permeated by an invisible lumineferous aether (light-bearing medium) Medium through which light can propagate The Earth must be moving relative to the aether So light will travel faster or slower, depending on the orientation Differences can be determined by experiment Test for the existence of the aether

4. B speed c – v C A v ms-1 relative to the aether speed c + v O d The Michelson-Morley Experiment Test for the presence of an aether using an interferometer A is a half-silvered mirror B/C are mirrors O is a detector incoming light There is a phase difference between the two beams Light at O should be phase-shifted, but no phase shift was observed

5. changing E-Field E-Field B-Field changing B-Field Maxwell’s Predictions Electric and magnetic fields interact Accelerating charges produce EM waves Maxwell’s equations predict that these waves propagate through a vacuum at a constant speed

6. v ms-1 u ms-1 Galilean velocity transformation What Has Gone Wrong? Consider a train: The resultant velocity of the person is u + v The resultant velocity of the light is c not c + v Is Maxwell wrong? Are Michelson and Morley’s results wrong? Or is Galileo wrong?

7. Einstein’s Postulates We can now state two postulates: • The laws of physics are the same in all inertial frames of • reference • The speed of light in a vacuum is the same in all inertial • frames of reference But what is an inertial frame of reference?

8. y y y v ms-1 a ms-2 x x x z z z inertial frames indistinguishable from a gravitational field Frames Of Reference A frame of reference is the coordinate system of an observer stationary frame frame moving at constant velocity v accelerating frame

9. y y’ S S’ v ms-1 x x’ z z’ object in frame S’ The Galilean Transformation Consider two inertial frames, S and S’ S’ is moving at velocity v away from S, and vy= vz= 0 ms-1 The object has the same y- and z-coordinates in both frames The time measured at any instant is the same in both frames The x-coordinate is constant in S’, but changes in S

10. The transformation from S’to S The transformation from S to S’ The Galilean Transformation At a time t, the x-axis of S’is a distance vt from the x-axis of S So the x-coordinate in S is the x-coordinate in S’+ vt All other coordinates are unchanged This transformation between frames can be written as: Galilean Transformation

11. The speed of light must be a constant It must tend tothe Galilean transformation for low velocities A New Transformation… The Galilean transformation contradicts Einstein’s second postulate We need to derive a new transformation, with the properties: We are not assuming that time is absolute So we need to be careful when referring to time An event has both space and time coordinates This can be written as a four-vector(x, y, z, t)

12. d A Thought Experiment Alice is on a long train journey, and is rather bored She decides to build a clock using her mirror and a torch What is the time interval between a pulse leaving and returning to the torch?

13. l l d v ms-1 vt’ 9 ¾ A Thought Experiment Bob is standing on platform 9¾ and watches Alice in the train What is the time interval Dt’ in Bob’s frame of reference?

14.  1 0 c Speed / ms-1 The Lorentz Factor The factor g is the Lorentz Factor For v << c, g(v) = 1 As v tends to c, g(v) tends to infinity

15. Time Dilation We have this relationship, but what does it mean? If a body is travelling slowly w.r.t. an observer (g = 1) time intervals are the same for the body and the observer If a body is travelling fast w.r.t. an observer (g >> 1) time intervals appear longer to the observer If you are in a spacecraft travelling close to c, time will pass normally for you, but will speed up around you Notice, therefore, that photons do not age

16. Bill Ben The Twin Paradox Once upon a time there were two twins… Ben goes on a journey into space, but Bill stays on Earth When Ben returns, which of the twins is oldest? Bill thinks he will be older, as Ben travelled very fast away from him Ben thinks he will be older, as Bill travelled very fast away from him Who is right? Bill is older, because he stayed in the same inertial frame, but Ben had to accelerate in the rocket The viewpoints are not identical

17. Length Contraction How do lengths appear in a different frame? Similar derivation as for time dilation There is no change in the directions perpendicular to travel In the direction of travel, we can show that: So, for a body travelling with v close to c, relative to an observer, the body will appear shorter to the observer, in the direction of v

18. How do you measure your velocity? It is meaningless to have an absolute velocity Velocity can only be measured relative to another body But what about length contraction and time dilation? If lengths are shortened, then can you measure the change? No – the ruler is length-contracted as well Similarly, clocks slow down, so changes in time can’t be measured But doesn’t relativity define c as a maximum absolute speed? Not quite – this is a maximum relative speed, as light has the same speed (c) relative to any frame

19. y y’ S S’ v ms-1 x x’ this length is contracted in the frame S z z’ The Lorentz Transformation We can now derive a relativistic transformation The y- and z-coordinates will be the same in both frames, as before From the Galilean transformation, x = x’+ vt But in frame S, x’is length-contracted to g-1x’

20. Lorentz time transformation The Lorentz Transformation To get the time transformation is a little trickier Notice that the transformations are linear

21. v << c The Lorentz Transformation We can now state the full Lorentz Transformation: This satisfies the conditions for the transformation

22. relativistic momentum Can We Go Faster Than Light? From Newton’s Second Law, F = ma(constant mass) So if we provide a continuous force, we can achieve v > c But momentum must be conserved m0 is the rest mass The g factor in effect increases the mass, as v increases A greater force is needed to provide the same acceleration To reach the speed of light, an infinite force would be required

23. Acceleration and gravity are equivalent Geometrical interpretation of gravitation Much more difficult theory! Conclusions We have looked at the theory of Special Relativity This resolved the conflict between Newtonian mechanics, and Maxwell’s equations It is a ‘special’ theory, as it doesn’t consider accelerations This is dealt with in General Relativity: Derivation left as an exercise to the student 