THEORY OF RELATIVITY : THEORY OF RELATIVITY AKASH SHUKLA
Topics: Topics Introduction Frame of reference Galilean transformation Michelson’s Morlay Experiment Postulates of Special Theory of Relativity Lorentz Transformation Length contraction Time dilation Variation of Mass with velocity Addition of Velocity Mass Energy relation
CLASSICAL RELATIVITY: CLASSICAL RELATIVITY 1,000,000 ms -1 1,000,000 ms -1 How fast is Spaceship A approaching Spaceship B? Both Spaceships see the other approaching at 2,000,000ms -1 . This is Classical Relativity.
Modern/ Einstein relativity: Modern/ Einstein relativity 1,000,000 ms -1 0 ms -1 300,000,000 ms -1 Both spacemen measure the speed of the approaching ray of light
Galilean postulates : Galilean postulates Galileo argued that there is no such thing as "Absolute Rest". In his view: The mechanical laws of physics are the same for every observer moving with a constant speed along a straight line (this is called "inertial observer" for short). Galileo Galilei 1564 -1642
Galilean transformation: Galilean transformation x y z ( x,y,z ) ( x',y',z ') v x' y' z' We have two frames of reference, K and K', and K' is moving along axis x with some constant speed v. Something happened at point A. According to Galileo, there is no one special reference frame -- if we know where A happened in one frame, we are done! That's because: Galileo transformations: know what happened in one frame, can tell what happened in another
Newton’s laws : Newton’s laws We know how positions of an object transform when we go from one inertial frame of reference to another What about velocities? What about accelerations? velocity of an object in K is equal to its velocity in K', plus the velocity of K’ with respect to K = 0 as v = const Accelerations are the same in both K and K’ frames! So Newtonian forces will be the same in both frames
Michelson-morley experiment: Michelson- morley experiment Principle:- Earth is moving around the sun with a speed of 10 -4 C. Ether through which the light is propagated with a fixed velocity C. Earth is passes through this ether with a speed10 -4 C. The purpose of this experiment was to measure this time difference from which the velocity of earth relative to the ether. Ether: It was proposed as an absolute reference system in which the speed of light was constant and from which other measurements could be made. The Michelson-Morley experiment was an attempt to show the existence of ether. Albert Michelson (1852–1931) was the first U.S. citizen to receive the Nobel Prize for Physics (1907), and built an extremely precise device called an interferometer to measure the minute phase difference between two light waves traveling in mutually orthogonal directions.
PowerPoint Presentation: M’ 1 M’ 2 G’ 1 M 1 M 2 G 1 d d P A A’ P’ B B’ c Velocity of Earth (v) Michelson- morley experiment
Cont….: Cont…. Let C be the velocity of Light through the ether and v is the velocity of Earth (i.e. v is the velocity of apparatus) The ray reflected from P and moving transversally will strike the mirror not at A but A’ due to motion of Earth If t be the time taken by the ray starting from P to reach M’ 1 Total path of the ray until it come back to the plate path Now Time for the lateral or Transverse Journey
Cont….: Cont…. If t 1be the total time taken by the ray to travel the whole path PA’P’ Applying the binomial theorem Time for the Longitudinal Journey The ray transmitted through P and moving longitudinally towards M 2 has the velocity (C-v) relative to the apparatus from P to B and ( C+v ) on the return journey from B’ to P’, If t 2 be the total time taken by this ray to return to the plate
Cont….: Cont…. The difference between the time of travel of the longitudinal and transverse ray Now the optical path difference will be The no of Fringes The interference pattern would be shifted through
Cont….: Cont…. If whole the apparatus is rotated through 90 0 then mirror M 1 and M 2 change their position so that time interval will be change from Longitudinal to Transverse of or Opposite Total time for the longitudinal journey =(t 2+ t 1 ’) Total time for the Transverse Journey = (t 1+ t 2 ’) Total time difference between the two journey The optical path difference between the interfering wave
PowerPoint Presentation: In this experiment the fringe shift will be if T his calculated fringe shift due to experimental error so no fringe shift was observed
conclusion: conclusion Michelson noted that he should be able to detect a phase shift of light due to the time difference between path lengths but found none. He thus concluded that the hypothesis of the stationary ether must be incorrect. After several repeats and refinements with assistance from Edward Morley (1893-1923), again a null result. Thus, ether does not seem to exist!
Postulates of Special Theory of Relativity: Postulates of Special Theory of Relativity Einstein (1905) The laws of physics are the same in all inertial reference frames Light travels through empty space at c, independent of speed of source or observer There is no absolute reference frame of time and space
Lorentz Transformation: Lorentz Transformation S and S’ be the two reference frame S’ is moving relative to S is positive X direction. If a event is happened at some point and wave front of light reaching at point P. So the coordinates of point P with respect to S and S’ are ( x,y,z,t ) and ( x’,y’,z’,t ’) x ,y,z,t P x’ ,y’,z’,t ’ S S’ Principle of Simultaneity of event: Two events that are simultaneous in one reference frame ( K ) are not necessarily simultaneous in another reference frame ( K ´) moving with respect to the first frame.
Cont….: Cont…. When the pulse of light is observed from S then we have When the pulse of light is observed from S’ then we have Now the transformation relation between ( x,y,z,t ) and ( x’,y ’, z’,t ’) Because the motion is along x axis so y = y ’ and z = z’ Now transformation between x and x’ can be represented by the relation Now transformation between x’ and x can be represented by the relation ---------(1) ---------(2) ---------(3)
Cont….: Cont…. Put the value of x’ and t’ in the transformation equation (1) Equating the coefficient of t 2 equal to zero ---------(4)
Cont….: Cont…. Now equating the coefficient of xt equal to zero On simplification we find ---------(4) ---------(5)
Cont….: Cont…. Put the value of in the x’ equation (2) Put the value of in the t’ equation (5) ---------(6) ---------(8) ---------(7) Now equation (6), (7) & (8) are the L orentz Transformation Equation Now Inverse L orentz Transformation Equation
Length contraction : Length contraction S S’ x 1 x 2 If the coordinates of rod with respect to S is x 1 and x 2 and S’ is x’ 1' and x’ 2. S’ is moving with respect to S in X axis Length of the rod with S is L= (x 2 -x 1 ) and with S’ is L’= (x’ 2 -x’ 1 ) Form the Lorentz transformation equation x’ 1 x’ 2
Conclusion : Conclusion Moving objects appear shorter in the dimension parallel to their velocity. Here L is the proper length and L’ is the imagery length. Case :-1 If v<<C Then v 2 /C 2 is negligible Case :-2 Then v 2 /C 2 is less then unity Case :-3 Then v 2 /C 2 equal to 1
Time dilation : Time dilation S S’ S’ is moving with respect to S in X axis Form the Lorentz transformation equation Signals from the clock have the interval Δ t =t 2 -t 1 with respect to S or Δ t’ =t’ 2 -t’ 1 with respect to S’
Conclusion : Conclusion Case :-1 If v<<C Then v 2 /C 2 is negligible Case :-2 Case :-3
Addition of velocities: Addition of velocities S’ is moving with respect to S in X axis. If a body is moving dx distance in dt time in system S and moving dx ’ distance in dt ’ time in system S’. Form the Lorentz transformation equation Taking the derivatives Dividing the equation
Conclusion : Conclusion This is the relativistic law of addition of velocities Case :-1 If u’ and v both are smaller then C Case :-2 If one object moves with velocity c with respect to other then their relative velocity will be c, it does not depends upon the velocity of other
Variation of mass with velocity : Variation of mass with velocity v u’ -u’ m 1 m 2 v Using the law of addition of velocity According to law of principal of conservation momentum
Cont….: Cont….
CONT….: CONT…. Let body of mass m 2 is at rest in system S before collision i.e. u 2 =0 m 1 =m, m 2 =m 0 u 1= v This is the formula for variation of mass with velocity
Conclusion : Conclusion Case :-1 If v<<C Then v 2 /C 2 is negligible Case :-2 Case :-3
Mass energy equivalence relation : Mass energy equivalence relation The energy is defined in term of work (force x distance) and force is defined as According to relativity mass and velocity are variable If particle is moved through a distance dx by application of a Force then its Kinetic Energy is increase Variation of mass with velocity ----(1)
PowerPoint Presentation: Taking the derivatives Comparing (1) and (2) ----(2) If body is initially at rest and after applying a Force it acquire a velocity so mass of the body increase bym o to m then total kinetic energy defines as
PowerPoint Presentation: Now Total Energy (E) is Total Energy (E) = Kinetic Energy (K)-Potential Energy (P) This is the universal equivalence between Mass and Energy