B.Tech sem I Engineering Physics U-III Chapter 1-THE SPECIAL THEORY OF RELATIVITY

• 1. Special Theory of Relativity

2. The dependence of various physical phenomena on relative motion of the observer and the observed objects, especially regarding the nature and behaviour of light, space, time, and gravity is called relativity. When we have two things and if we want to find out the relation between their physical property i.e.velocity,accleration then we need relation between them that which is higher and which is lower.In general way we reffered it to as a relativity. The famous scientist Einstein has firstly found out the theory of relativity and he has given very useful theories in relativity. Introduction to Relativity 3. What is Special Relativity? In 1905, Albert Einstein determined that the laws of physics are the same for all non-accelerating observers, and that the speed of light in a vacuum was independent of the motion of all observers. This was the theory of special relativity. 4. FRAMES OF REFERENCE A Reference Frame is the point of View, from which we Observe an Object. A Reference Frame is the Observer it self, as the Velocity and acceleration are common in Both. Co-ordinate system is known as FRAMES OF REFERENCE Two types: 1. Inertial Frames Of Reference. 2. non-inertial frame of reference. 5. FRAMES OF REFERENCE We have already come across idea of frames of reference that move with constant velocity. In such frames, Newtons laws (esp. N1) hold. These are called inertial frames of reference. Suppose you are in an accelerating car looking at a freely moving object (I.e., one with no forces acting on it). You will see its velocity changing because you are accelerating! In accelerating frames of reference, N1 doesnt hold this is a non-inertial frame of reference. 6. Conditions of the Galilean Transformation Parallel axes (for convenience) K has a constant relative velocity in the x-direction with respect to K Time (t) for all observers is a Fundamental invariant, i.e., the same for all inertial observers speed of frame NOT speed of object x' x v t y' y z' z Galilean Transform 7. Galilean Transformation Inverse Relations Step 1. Replace with . Step 2. Replace primed quantities with unprimed and unprimed with primed. speed of frame NOT speed of object x x vt y y z z t t 8. General Galilean Transformations ' ' ' tt yy vtxx 11 ' ' ' ' ' ' dt dt dt dt vv dt dy dt dy vvvv dt dx dt dx samethearetandt yy xx yy yy xx xx aa dt dv dt dv aa dt dv dt dv samethearetandt ' ' '0 ' ' inertial reference frame FamFam ' 11 ' ' ' ' ' ' dt dt dt dt tt dt dy dt dy vuuv dt dx dt dx samethearetandt yy xx frame K frame K Newtons Eqn of Motion is same at face-value in both reference frames PositionVelocityAcceleration 9. Einsteins postulates of special theory of relativity The First Postulate of Special Relativity The first postulate of special relativity states that all the laws of nature are the same in all uniformly moving frames of reference. 10. Einstein reasoned all motion is relative and all frames of reference are arbitrary. A spaceship, for example, cannot measure its speed relative to empty space, but only relative to other objects. Spaceman A considers himself at rest and sees spacewoman B pass by, while spacewoman B considers herself at rest and sees spaceman A pass by. Spaceman A and spacewoman B will both observe only the relative motion. The First Postulate of Special Relativity 11. A person playing pool on a smooth and fast- moving ship does not have to compensate for the ships speed. The laws of physics are the same whether the ship is moving uniformly or at rest. The First Postulate of Special Relativity 12. Einsteins first postulate of special relativity assumes our inability to detect a state of uniform motion. Many experiments can detect accelerated motion, but none can, according to Einstein, detect the state of uniform motion. The First Postulate of Special Relativity 13. The second postulate of special relativity states that the speed of light in empty space will always have the same value regardless of the motion of the source or the motion of the observer. The Second Postulate of Special Relativity 14. Einstein concluded that if an observer could travel close to the speed of light, he would measure the light as moving away at 300,000 km/s. Einsteins second postulate of special relativity assumes that the speed of light is constant. The Second Postulate of Special Relativity 15. The speed of light is constant regardless of the speed of the flashlight or observer. The Second Postulate of Special Relativity The speed of light in all reference frames is always the same. Consider, for example, a spaceship departing from the space station. A flash of light is emitted from the station at 300,000 km/sa speed well call c. 16. The speed of a light flash emitted by either the spaceship or the space station is measured as c by observers on the ship or the space station. Everyone who measures the speed of light will get the same value, c. The Second Postulate of Special Relativity 17. 18 The Ether: Historical Perspective Light is a wave. Waves require a medium through which to propagate. Medium as called the ether. (from the Greek aither, meaning upper air) Maxwells equations assume that light obeys the Newtonian-Galilean transformation. 18. The Ether: Since mechanical waves require a medium to propagate, it was generally accepted that light also require a medium. This medium, called the ether, was assumed to pervade all mater and space in the universe. 19. 20 The Michelson-Morley Experiment Experiment designed to measure small changes in the speed of light was performed by Albert A. Michelson (1852 1931, Nobel ) and Edward W. Morley (1838 1923). Used an optical instrument called an interferometer that Michelson invented. Device was to detect the presence of the ether. Outcome of the experiment was negative, thus contradicting the ether hypothesis. 20. Michelson-Morley Experiment(1887) Michelson developed a device called an inferometer. Device sensitive enough to detect the ether. 21. Michelson-Morley Experiment(1887) Apparatus at rest wrt the ether. 22. Michelson-Morley Experiment(1887) Light from a source is split by a half silvered mirror (M) The two rays move in mutually perpendicular directions 23. Michelson-Morley Experiment(1887) The rays are reflected by two mirrors (M1 and M2) back to M where they recombine. The combined rays are observed at T. 24. Michelson-Morley Experiment(1887) The path distance for each ray is the same (l1=l2). Therefore no interference will be observed 25. Michelson-Morley Experiment(1887) Apparatus at moving through the ether. u ut 26. Michelson-Morley Experiment(1887) First consider the time required for the parallel ray Distance moved during the first part of the path is || || || ct L ut L t (c u) (distance moved by light to meet the mirror) u ut 27. Michelson-Morley Experiment(1887) (distance moved by light to meet the mirror))( || uc L t |||| utLct Similarly the time for the return trip is )( || uc L t The total time )()( || uc L uc L t u ut 28. Michelson-Morley Experiment(1887) The total time || 2 2 2 2 ( ) ( ) 2 ( ) 2 / 1 L L t c u c u Lc c u L c u c u ut 29. Michelson-Morley Experiment(1887) For the perpendicular ray we can write, ct vt 2 2 2 2 2 2 2 2 2 2 2 2 2 ( ) ( ) ct L ut L c t u t c u t L t c u (initial leg of the path) The return path is the same as the initial leg therefore the total time is 22 2 uc L t u ut 30. Michelson-Morley Experiment(1887) ct vt 2 2 2 2 2 2 / 1 L t c u L c t u c The time difference between t two rays is, 1 212 2 || 2 2 2 2 2 3 2 1 1 2 2 L u u t t t c c c After a binomial expansi L u Lu t c c c on u ut 31. Michelson-Morley Experiment(1887) The expected time difference is too small to be measured directly! Instead of measuring time, Michelson and Morley looked for a fringe change. as the mirror (M) was rotated there should be a shift in the interference fringes. Results of the Experiment A NULL RESULT No time difference was found! Hence no shift in the interference patterns Conclusion from Michelson-Morley Experiment the ether didnt exist. 32. The Lorentz Transformation We are now ready to derive the correct transformation equations between two inertial frames in Special Relativity, which modify the Galilean Transformation. We consider two inertial frames S and S, which have a relative velocity v between them along the x-axis. x y z S x' y' z' S' v 33. Now suppose that there is a single flash at the origin of S and S at time , when the two inertial frames happen to coincide. The outgoing light wave will be spherical in shape moving outward with a velocity c in both S and S by Einsteins Second Postulate. We expect that the orthogonal coordinates will not be affected by the horizontal velocity: But the x coordinates will be affected. We assume it will be a linear transformation: But in Relativity the transformation equations should have the same form (the laws of physics must be the same). Only the relative velocity matters. So x y z c t x y z c t 2 2 2 2 2 2 2 2 2 2 y y z z x k x vt x k x vt a f a f k k 34. Consider the outgoing light wave along the x-axis (y = z = 0). Now plug these into the transformation equations: Plug these two equations into the light wave equation: x ct x ct in frame S' in frame S 1 / & 1 / x k x vt k ct vt kct v c x k x vt k ct vt kct v c ct x kct v c ct x kct v c t kt v c t kt v c 1 1 1 1 / / / / a f a f a f a f 35. Plug t into the equation for t: So the modified transformation equations for the spatial coordinates are: Now what about time? t k t v c v c k v c k v c 2 2 2 2 2 2 1 1 1 1 1 1 / / / / a fa f c h x x vt y y z z a f x x vt x x vt x x vt vt a f a f a f inverse transformation Plug one into the other: 36. Solve for t: So the correct transformation (and inverse transformation) equations are: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 / 1 1 / / 1 / / x x vt vt x vt vt v c x vt vt v c xv c vt vt t xv c vt v t t vx c x x vt x x vt y y y y z z z z t t vx c t t vx c a f a f c h c h/ /2 2 The Lorentz Transformation 37. Application of Lorentz Transformation Time Dilation We explore the rate of time in different inertial frames by considering a special kind of clock a light clock which is just one arm of an interferometer. Consider a light pulse bouncing vertically between two mirrors. We analyze the time it takes for the light pulse to complete a round trip both in the rest frame of the clock (labeled S), and in an inertial frame where the clock is observed to move horizontally at a velocity v (labeled S). In the rest frame S t L c t L c t t L c 1 2 1 2 2 = time up = time down = mirror mirror L 38. Now put the light clock on a spaceship, but measure the roundtrip time of the light pulse from the Earth frame S:t t t t c L v t c t L c v t t L c v t L c v c v c 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 2 1 1 1 time up time down The speed of light is still in this frame, so / / / / / c h L c t/ 2 v t / 2 39. So the time it takes the light pulse to make a roundtrip in the clock when it is moving by us is appears longer than when it is at rest. We say that time is dilated. It also doesnt matter which frame is the Earth and which is the clock. Any object that moves by with a significant velocity appears to have a clock running slow. We summarize this effect in the following relation: 2 2 1 2 , 1, 1 / L t cv c 40. Length Contraction Now consider using a light clock to measure the length of an interferometer arm. In particular, lets measure the length along the direction of motion. In the rest frame S: Now put the light clock on a spaceship, but measure the roundtrip time of the light pulse from the Earth frame S: L c 0 2 1 2 1 2 1 1 1 2 2 2 time out, time backt t t t t L L vt ct t c v L L vt ct t c v A A C C vt1L 41. In other words, the length of the interferometer arm appears contracted when it moves by us. This is known as the Lorentz-Fitzgerald contraction. It is closely related to time dilation. In fact, one implies the other, since we used time dilation to derive length contraction. 1 2 2 2 2 2 2 2 2 2 0 2 2 2 2 1 1 / 1 / 2 But, from time dilation 1 / 1 1 1 / Lc L t t t c v c v c ct L v c t v c L L v c 42. Engineering physics By Dr. M N Avadhnulu, S Chand publication ENGINEERING PHYSICS ABHIJIT NAYAK http://www.maths.tcd.ie/~cblair/notes/specrel.pdf http://www.newagepublishers.com/samplechapter/0 00485.pdf