PHIR11: UNIT-III SPECIAL THEORY OF RELATIVITY Dr. AWNISH KUMAR TRIPATHI 1 Theory of Relativity General Theory of Relativity (1915) : massive objects cause a distortion in space-time, which is felt as gravity. Gravitational lensing Changes in the orbit of Mercury Frame-dragging of space-time around rotating bodies Gravitational redshift Gravitational waves 2 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Special Theory of Relativity (1905) 3 Galilean-Newtonian Mechanics 4 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY GTR STR G-N 5 6 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Time evolution in the Theory of Mechanics 7 UNIT-III Special Theory of Relativity Topics to be covered: Impact of Theory of Relativity in daily Life The Michelson-Morley experiment Relativistic transformations Length contraction It was found that there was no displacement of the interference Time dilation fringes, so that the result of the experiment was negative and Variation of mass with velocity would, therefore, show that there is still a difficulty in the theory itself… mass-energy equivalence - Albert Michelson, 1907 8 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Impact of Einstein's Theory of Relativity in daily Life Electromagnetism Global Positioning System Gold's yellow color Gold doesn't corrode easily Mercury is a liquid Old (Cathode ray based) TV Light Nuclear Power Sunlight Our existence https://interestingengineering.com/10-ways-can-see-einsteins-theory-relativity-real-life-keyword-theoryrelativity https://www.livescience.com/58245-theory-of-relativity-in-real-life.html 9 Newtonian (Classical) Relativity Assumption: It is assumed that Newton’s laws of motion must be measured with respect to (relative to) some reference frame. A reference frame is basically a coordinate system where the origin is either fixed with the system or sometimes with the observer depending on the convenience. 10 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Inertial Reference Frame A reference frame is called an inertial frame if Newton laws are valid in that frame. Such a frame is established when a body, not subjected to net external forces, is observed to move in rectilinear motion at constant velocity. 11 Newtonian Principle of Relativity If Newton’s laws are valid in one reference frame, then they are also valid in another reference frame moving at a uniform velocity relative to the first system. This is referred to as the Newtonian principle of relativity or Galilean invariance/relativity. The laws of mechanics are independent on the state of movement in a straight line at constant velocity. 12 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Inertial Frames K and K’ K is at rest and K’ is moving with constant velocity along x direction. Axes are parallel. K and K’ are said to be INERTIAL COORDINATE SYSTEMS. 13 The Galilean Transformation For a point P In system K: P = (x, y, z, t) In system K’: P = (x’, y’, z’, t’) P x’ x 14 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Conditions of the Galilean-Newtonian Transformation Parallel axes, NO change in y and z coordinates. K’ has a constant relative velocity in the x-direction with respect to K ux’ = ux – v ax’ = ax uy’ = uy ay’ = ay u z’ = u z a z’ = a z Time (t) for all observers is a Fundamental invariant, i.e., it remain the same for all inertial observers. 15 The Inverse Galilean-Newtonian Transformation Step 1. Replace with . Step 2. Replace “primed” quantities with “unprimed” and “unprimed” with “primed.” ux= ux’ + v ax= ax’ uy= uy ’ ay= ay ’ u z= u z ’ a z= a z’ Different observers (in O and O’) will assign different velocities because of their relative motion, but will measure the same acceleration ! 16 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Consequences: Length intervals and time intervals are the same for all inertial observers of the same events. Two meter sticks (same length) would be of the same length when compared in relative motion to one another; the distance between two points, A and B, measured at a given instant, is the same for each observer: xB’ – xA’ = xB - xA if clocks are calibrated and synchronized when at rest, their readings and rates will agree even if they are in relative motion to one another. It follows from these transformations that the time interval between two events, P and Q, is the same for each observer: tP’ – tQ’ = tP – tQ 17 Consequences: Galilean-Newtonian Relativity The acceleration of a particle is the same in all reference frames which move relative to one another with constant velocity. Since in classical physics, the mass is unaffected by the motion of the reference frame, the product of mass and acceleration (ma) will be the same for all inertial observers. Since F = ma is the definition of force, this will also be the same for each inertial observer. Newton’s laws of motion and the equations of motion of a particle are exactly the same in all inertial frames the laws of mechanics are the same in all inertial frames. 18 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY The Transition to Modern Relativity Although Newton’s laws of motion had the same form under the Galilean transformation, Maxwell’s equations (ELECTROMAGTIC THEORY) did not. In 1905, Albert Einstein proposed a fundamental connection between space and time and that Newton’s laws are only an approximation. 19 The Need for Ether It was assumed that like sound wave, even Electromagnetic waves require a medium to travel through. The wave nature of light suggested that there existed a propagation medium called the luminiferous ether or just ether having below listed properties. Ether had to have such a low density that the planets could move through it without loss of energy. It also had to have an elasticity to support the high velocity of light waves. Let’s call the “ether” frame O, and assume it to be an inertial one in which the observer measures the speed of light to be exactly c. In a frame O’ moving at a constant speed v with respect to the ether frame, an observer would measure a different speed for the light pulse, ranging from c + v to c - v, depending on the direction of relative motion, according to the Galilean velocity transformation. Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY 20 Whereas: Maxwell’s Equations In Maxwell’s theory the speed of light, in terms of the permeability and permittivity of free space, was given by Thus the velocity of light between even in moving systems must be a constant. 21 ETHER: An Absolute Reference System Ether was proposed as an absolute reference system in which the speed of light was the constant c and from which other measurements could be made. The Michelson-Morley experiment was an attempt to show the existence of ether and to figure out Earth’s relatives movement through (with respect to) the ether so that Maxwell’s equations could be corrected for this effect. 22 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY 23 The details of Michelson-Morley experiment at the end as Annexure. 24 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Michelson’s Conclusion He concluded that the hypothesis of existence of the stationary ether must be incorrect. After several repeats and refinements with assistance from Edward Morley (1893-1923), both Michelson and Morley concluded again a null result. Finally it was established that, ether does not seem to exist! 25 Einstein’s Postulates Albert Einstein (1879–1955) was only two years old when Michelson reported his first null measurement for the existence of the ether. At the age of 16, Einstein began thinking about the form of Maxwell’s equations in moving inertial systems. In 1905, at the age of 26, he published his startling proposal about the principle of relativity, which he believed to be fundamental. 26 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Einstein’s Two Postulates With the belief that Maxwell’s equations must be valid in all inertial frames, Einstein proposed the following postulates: 1) The principle of relativity: The laws of physics are the same in all inertial systems. There is no way to detect absolute motion, and no preferred inertial system exists. 2) The constancy of the speed of light: Observers in all inertial systems measure the same value for the speed of light in a vacuum. 27 Re-evaluation of Time In Newtonian physics we previously assumed that t = t’ Thus with “synchronized” clocks, events in K and K’ can be considered simultaneous Einstein realized that each system must have its own observers with their own clocks and meter sticks Thus events, happening at the same instant of time in both K and K’, may not be simultaneous in both K and K’. 28 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY The Problem of Simultaneity Frank at rest is equidistant from events A and B: A −1 m B +1 m 0 Frank “sees” both flashbulbs go off simultaneously. 29 The Problem of Simultaneity Frank, moving to the right with speed v, observes events A and B in different order: −1 m A c-v 0 c+v +1 m B Frank “sees” event B, then A. 30 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY We thus observe… 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. This suggests that each coordinate system has its own observers with “clocks” that are synchronized… 31 Synchronization of Clocks Step 1: Place observers with clocks throughout a given system. Step 2: In that system bring all the clocks together at one location. Step 3: Compare the clock readings. If all of the clocks agree, then the clocks are said to be synchronized. 32 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY A method to synchronize… One way is to have one clock at the origin set to t = 0 and advance each clock by a time (d/c) with d being the distance of the clock from the origin. Allow each of these clocks to begin timing when a light signal arrives from the origin. t=0 t = d/c d t = d/c d 33 The Lorentz Transformations The special set of linear transformations that: 1) preserve the constancy of the speed of light (c) between inertial observers; and, 2) account for the problem of simultaneity between these observers known as the Lorentz transformation equations 34 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Lorentz Transformation Equations Hendrik Antoon Lorentz 35 Lorentz Transformation Equations A more symmetric form: 36 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Properties of γ Recall β = v/c < 1 for all observers. equals 1 only when v = 0. 1) 2) Graph of β vs γ : (note v ≠ c) 37 Derivation of Lorentz Transformations equations Use the fixed system K and the moving system K’ At t = 0 the origins and axes of both systems are coincident with system K’ moving to the right along the x axis. A flashbulb goes off at the origins when t = 0. According to postulate 2, the speed of light will be c in both systems and the wavefronts observed in both systems must be spherical. K K’ 38 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Derivation Spherical wavefronts in K: Spherical wavefronts in K’: Note: These are not preserved in the classical transformations with 39 Derivation 1) Let x’ = (x – vt) so that x = (x’ + vt’) 2) By Einstein’s first postulate: 3) The wavefront along the x, x’- axis must satisfy: x = ct and x’ = ct’ 4) Thus ct’ = (ct – vt) and ct = (ct’ + vt’) 5) Solving the first one above for t’ and substituting into the second... 40 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Derivation Gives the following result: from which we derive: 41 Finding a Transformation for t ’ Recalling x’ = (x – vt) substitute into x = (x’ + vt) and solving for t ’ we obtain: which may be written in terms of β (= v/c): 42 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Thus the complete set of Transformation equations Galilean-Newtonian Transformation Lorentz Transformation Inverse-Lorentz Transformation Inverse = v/c 43 Remarks 1) If v << c, i.e., β ≈ 0 and ≈ 1, we see these equations reduce to the familiar Galilean transformation. 2) Space and time are now not separated. 3) For non-imaginary transformations, the frame velocity cannot exceed c. 44 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Time Dilation and Length Contraction Consequences of the Lorentz Transformation: Time Dilation: Clocks in K’ run slow with respect to stationary clocks in K. Length Contraction: Lengths in K’ are contracted with respect to the same lengths stationary in K. 45 Time Dilation To understand time dilation the idea of proper time must be understood: The term proper time,T0, is the time difference between two events occurring at the same position in a system as measured by a clock at that position. Same location 46 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Time Dilation Improper Time Beginning and ending of the event occur at different positions 47 Time Dilation Frank’s clock is at the same position in system K when the sparkler is lit in (a) and when it goes out in (b). Mary, in the moving system K’, is beside the sparkler at (a). Melinda then moves into the position where and when the sparkler extinguishes at (b). Thus, Melinda, at the new position, measures the time in system K’ when the sparkler goes out in (b). 48 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY According to Mary and Melinda… Mary and Melinda measure the two times for the sparkler to be lit and to go out in system K’ as times t’1 and t’2 so that by the Lorentz transformation: Note here that Frank records x2 – x1 = 0 in K with a proper time: T0 = t2 – t1 or with T ’ = t ’2 – t ’1 49 Time Dilation 1. T ’ > T0 or the time measured between two events at different positions is greater than the time between the same events at one position: time dilation. 2. The events do not occur at the same space and time coordinates in the two system 3. System K requires 1 clock and K’ requires 2 clocks. 4. A moving clock slows down. 50 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Length Contraction To understand length contraction the idea of proper length must be understood: Let an observer in each system K and K’ have a meter stick at rest in their own system such that each measure the same length at rest. The length as measured at rest is called the proper length. 51 What Frank and Mary see… Each observer lays the stick down along his or her respective x axis, putting the left end at xℓ (or x’ℓ) and the right end at xr (or x’r). Thus, in system K, Frank measures his stick to be: L0 = xr – xℓ Similarly, in system K’, Mary measures her stick at rest to be: L’0 = x’r – x’ℓ 52 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY What Frank and Mary measure Frank in his rest frame measures the moving length in Mary’s frame moving with velocity. Thus using the Lorentz transformations Frank measures the length of the stick in K’ as: Where both ends of the stick must be measured simultaneously, i.e, tr = tℓ Here Mary’s proper length is L’0 = x’r – x’ℓ and Frank’s measured length is L = xr – xℓ 53 Frank’s measurement So Frank measures the moving length as L given by but since both Mary and Frank in their respective frames measure L’0 = L0 and L0 > L, i.e. the moving stick shrinks. 54 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Addition of Velocities Taking differentials of the Lorentz transformation, relative velocities may be calculated: 55 So that… defining velocities as: ux = dx/dt, uy = dy/dt, u’x = dx’/dt’, etc. it is easily shown that: With similar relations for uy and uz: 56 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY The Lorentz Velocity Transformations In addition to the previous relations, the Lorentz velocity transformations for u’x, u’y , and u’z can be obtained by switching primed and unprimed and changing v to –v: Lorentz Velocity Transformation 57 Relativistic Mass and Momentum The rest mass of an object is its inertial mass : m0 However, the mass of an object as defined by: m F a Etotal c2 will be larger if the object is moving with respect to a given frame of reference since the object will also have kinetic energy. Einstein suggested that the relativistic energy of an object would be given by: 2 Etotal m0 c v2 1 2 c Etotal m0 c 2 58 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Relativistic mass Etotal m 2 c m0 1 2 v c2 m0 Relativistic Momentum The momentum of an object is also relativistic: p mv m0 v 1 2 v c2 m0 v 59 Relativistic Energy Due to the new idea of relativistic mass, we must now redefine the concepts of work and energy. Therefore, we modify Newton’s second law to include our new definition of linear momentum, and force becomes: 60 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Relativistic Kinetic Energy To derive an expression for the relativistic energy we start with the work-energy theorem W K Fdx (Work done) W dP dx dt Using the chain rule repeatedly this can rewritten as W dP vdv dv m0 v d vdv 2 2 dv 1 v c 0 v (Work done accelerating an object from rest some velocity) 61 v 0 m0 v d vdv W dv 1 v 2 c 2 m0 1 v W m0 v 2 c 2 3 dv v 1 u c 3 2 W m0 c 1 u c 2 Dr. AWNISH KUMAR TRIPATHI m0 v 1 v 0 2 c 2 3 dv 2 Integrating by substitution: 2 m0 1 u c 2 1 2 2c 2 v du m0 2v 2 2 du 1 u c 2 3 2 u 1 2 m0 c 2 1 u c 2 0 2 1 2 m0 c 2 SPECIAL THEORY OF RELATIVITY 1 u 2 0 62 W m0 c 2 1 u c 2 Therefore W 1 2 m0 c 2 m0 c 2 1 v 2 c2 1 2 m0 c 2 However this equal to K therefore the Relativistic Kinetic energy is K m0c 2 1 v 2 c2 1 2 m0c 2 The velocity independent term ( m0 c 2 ) is the rest energy – the energy an object contains when it is at rest. At low speeds v << c we can write the kinetic energy as K m0 c 2 1 v 2 c 2 K m c 1 12 Using a Taylor expansion we get, 2 0 1 2 v 2 1 c v c 2 3 8 2 2 2 63 5 16 v 2 c 2 3 ... 1 Taking the first 2 terms of the expansion K m0 c 2 1 12 v 2 c 2 1 So that K 12 m0 v 2 (classical expression for the Kinetic energy) The expression for the relativistic kinetic energy m0 c 2 1 v2 c2 K m0 c or mc 2 K m0 c 2 2 where E mc is the total relativistic energy If the object also has potential energy it can be shown that mc 2 K V m0 c 2 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY 64 Relativistic and Classical Kinetic Energies 65 Relativistic Energy The relationship E mc is just Einstein’s mass-energy equivalence equation, which shows that mass is a form of energy. An important fact of this relationship is that the relativistic mass is a direct measure of the total energy content of an object. It shows that even a small mass corresponds to an enormous amount of energy. This is the foundation of nuclear physics. Anything that increases the energy in an object will increase its relativistic mass. In certain cases the momentum or energy is known instead of the speed. 2 66 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY m02 c 4 Note that E m c 1 v2 c2 2 2 4 m 2 c 4 1 v 2 c 2 m02 c 4 m 2 c 4 E 2 m 2 c 2 v 2 m02 c 4 Since P mv E 2 P 2 c 2 m02 c 4 When the object is at rest, p 0 so that E mc2 67 0 so that E mc2 When the object is at rest, p For particles having zero mass (photons) we see that E pc the expected relationship relating energy and momentum for photons and neutrinos which travel at the speed of light. When the object is at rest, p For particles having zero mass (photons) we see that E pc the expected relationship relating energy and momentum for photons and neutrinos which travel at the speed of light. 0 so that E mc2 68 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY ANNEXURE MICHELSON-MORLEY EXPERIMENT Dr. AWNISH KUMAR TRIPATHI 69 The Need for Ether In early 19 century, It was assumed that like sound wave, even Electromagnetic waves require a medium to travel through. The wave nature of light suggested that there existed a propagation medium called the luminiferous ether or just ether. Ether had to have such a low density that the planets could move through it without loss of energy. It also had to have an elasticity to support the high velocity of light waves. Let’s call the “ether” frame O, and assume it to be an inertial one in which the observer measures the speed of light to be exactly c. In a frame O’ moving at a constant speed v with respect to the ether frame, an observer O’ would measure a different speed for the light pulse, ranging from c + v to c - v, depending on the direction of relative motion, according to the Galilean velocity transformation. 70 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Maxwell’s Equations In Maxwell’s theory the speed of light, in terms of the permeability and permittivity of free space, was given by Thus the velocity of light between moving systems must be a constant. 71 ETHER: An Absolute Reference System Ether was proposed as an absolute reference system in which the speed of light was the constant c and from which other measurements could be made. The Michelson-Morley experiment was an attempt to show the existence of ether and to figure out Earth’s relatives movement through (with respect to) the ether so that Maxwell’s equations could be corrected for this effect. . 1. Before understanding Michelson-Morley experiment, we have to understand Michelson interferometer. 2. Before understanding Michelson interferometer we have to understand INTERFERENCE because of Division of Amplitude. 72 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Change of Phase in Reflection String Analogy Point to remember: An EM wave undergoes a phase change of 180° upon reflection from a medium that has a higher refractive index than the one in which it is traveling. 73 n1 nf n2 C t t D C for the first two reflected beams A D i B t A B Optical path difference Path difference n f [AB BC] n1 (AD) AB BC d /cos t AD AC sin i 2d tan t d nf sin t n1 2n f dcost For air n1=1 Condition for maxima dn f cos t (2m 1) f 4 m 0, 1, 2,... Condition for minima dn f cos t 2m f 4 m 0, 1, 2,... 74 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Interference in Thin Films • A wave traveling from a medium of index of refraction of n1 towards a medium with index of refraction of n2 undergoes a 180° phase change upon reflection if n2 > n1 and no phase change if n2 < n1. • The wavelength of light n in a medium with index of refraction n is given by, n = / n. For constructive interference For destructive interference 1 2t m n 2 2nt m 1 2nt m 2 m = 0,1,2,… m = 0,1,2,… 75 What is a typical Michelson’s Interferometer ? 76 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Michelson’s Interferometer: Working principle S Monochromatic light source S1’ Beam splitter partially silvered at the back side S2’ Beam compensator M1’ M2 M1 Fixed mirror and can be tilted M2 Movable mirror S’ Image of S because of Beam splitter M1’ Image of M1 because of Beam splitter S2’ Image of S’ because of mirror M 2 S S1’ Image of S’ because of mirror M 1’ M1 Interference fringe pattern at constant angles S’ Screen 77 Non-localized Interference pattern 78 S 1' SG＝S'G S'M 2 =M 2S'2 2d S 2' ΔL (L 2d)2 R 2 L2 R 2 L 2 Ld / L R 2d cos 2 2 The condition of bright fringe is M 1' M2 d Since L>>d, expression becomes G L S S' 2dcos =k k=0,1,2, R E Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY P M1 Interferogram demonstration 79 2d cos k Maxwell’s Equations In Maxwell’s theory the speed of light, in terms of the permeability and permittivity of free space, was given by Thus the velocity of light between moving systems must be a constant. 80 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY The Michelson-Morley Experiment 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 (Michelson interferometer) to measure the minute phase difference between two light waves traveling in mutually orthogonal directions. 81 The Michelson Interferometer 1. AC is parallel to the motion of the Earth inducing an “ether wind” 2. Light from source S is split by mirror A and travels to mirrors C and D in mutually perpendicular directions 3. After reflection the beams recombine at A slightly out of phase due to the “ether wind” as viewed by telescope E. 82 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY 83 Typical interferometer fringe pattern expected when the system is rotated by 90° NEVER OBSERVED !!!! 84 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY The Analysis Assuming the Galilean Transformation Time t1 from A to C and back: Time t2 from A to D and back: So that the change in time is: 85 The Analysis (continued) Upon rotating the apparatus 90o, the optical path lengths ℓ1 and ℓ2 are interchanged producing a different change in time: (note the change in denominators) 86 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY The Analysis (continued) Thus a time difference between rotations is given by: and upon a binomial expansion, assuming v/c << 1, this reduces to 87 Results Using the Earth’s orbital speed as: V = 3 × 104 m/s together with ℓ1 ≈ ℓ2 = 1.2 m So that the time difference becomes Δt’ − Δt ≈ v2(ℓ1 + ℓ2)/c3 = 8 × 10−17 s The (Δt’ − Δt ) will introduce a path difference of c(Δt’ − Δt ) which will lead to a phase difference 2 c(Δt’ − Δt)/ between ray AC and AB. Although a very small number, it was within the experimental range of measurement for light waves. 88 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY Michelson’s 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! 89 Possible Explanations Many explanations were proposed but the most popular was the ether drag hypothesis. This hypothesis suggested that the Earth somehow “dragged” the ether along as it rotates on its axis and revolves about the sun. This was contradicted by stellar abberation wherein telescopes had to be tilted to observe starlight due to the Earth’s motion. If ether was dragged along, this tilting would not exist. 90 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY The Lorentz-FitzGerald Contraction Another hypothesis proposed independently by both H. A. Lorentz and G. F. FitzGerald suggested that the length ℓ1, in the direction of the motion should be contracted by a factor of …thus making the path lengths equal to account for the zero phase shift. This, however, was an ad hoc assumption that could not be experimentally tested (at that time but later proved). 91 Length contracted for the moving muon, it’s own life time just 2.2 micro seconds Life time of the muon delayed for observer on Earth so that it can travel the whole distance as observed from Earth Great thing about special relativity is that one can always take two viewpoints, moving with the experiment, watching the experiment move past, the observations need to be consistent in both cases They move with about 98 % the speed of light Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY 92 END 93 Dr. AWNISH KUMAR TRIPATHI SPECIAL THEORY OF RELATIVITY