Lecture 39: Special Relativity PDF

Lecture_39 Let's dig into the significant changes that have shaped our world. When Alexander the Great, made his mark on the inhabited regions of the world, it brought about tremendous changes. History has seen the rise and fall of countries, sultanates, and capitals over thousands of years. Howeve...

Lecture_39 Let's dig into the significant changes that have shaped our world. When Alexander the Great, made his mark on the inhabited regions of the world, it brought about tremendous changes. History has seen the rise and fall of countries, sultanates, and capitals over thousands of years. However, in my opinion, the revolution that occurred in the year 1905 stands out distinctly. The Imaginary Revolution This revolutionary thought troubled many minds for a long time, presenting a strange yet fascinating picture. It was certainly an imaginary revolution, but its implications were profound. This revolution was about breaking the atom and harnessing infinite amounts of energy from it. This discovery revealed how energy is released from the sun and stars, leading to the development of atomic power plants and even the atomic bomb. This breakthrough allowed us to uncover the secrets hidden inside the atom. On one hand, the atom is the smallest unit, while on the other, it is as if the universe itself started from an atom. Einstein contributed significantly to this understanding. From the Theory of Relativity, we learned that before Newton's time, people had a specific understanding of time and space. The Concept of Space-Time Previously, time was perceived as a flowing river, moving in one direction with a constant flow. Space was considered something that could only be measured, and it was viewed as a fixed entity. However, Einstein changed this picture, introducing the concept of space-time. He explained that space and time are not separate but rather a single, interconnected entity known as space-time. The Nature of Time Physicists are concerned with how we can measure time and space. Understanding time involves recognizing periodic processes. For example, a pendulum swings from one side to the other, and this periodic motion can be used to measure time. However, pendulums are not perfect clocks. Factors like temperature changes can alter the length of the pendulum, affecting its time period. A more accurate clock is the Earth's rotation, which completes one revolution in 24 hours. The Earth's orbit around the sun, completing one revolution in a year, also serves as a reliable timekeeper. Even more precise are atomic clocks, which measure time with such accuracy that we can observe the slowing rotation of the Earth. Measuring Space To measure space, we can use various methods. One way is to use standardized lengths, although creating identical units of measure can be challenging due to slight differences. Modern methods use atomic properties to determine length accurately. The distance between atoms can be used to measure total length. Inertial Frames To grasp Einstein's Theory of Special Relativity, we must first understand the concept of an inertial frame. Imagine sitting in a train that moves smoothly without any vibrations or noise. In such a situation, you cannot tell if the train is stationary or moving. This describes an inertial frame—a frame of reference where there is no acceleration. Newton's First Law of Motion states that an object will remain at rest or continue in uniform motion unless acted upon by an external force. This law holds true in an inertial frame. For example, if a body is at rest or moving with constant velocity, it will continue to do so in the absence of external forces. Non-Inertial Frames If the train you are in starts to accelerate or decelerate, you will feel a force acting on you, indicating a non-inertial frame. In such frames, external forces cause changes in motion. Newton defined an inertial frame as one where his First Law is valid. In non-inertial frames, forces such as acceleration can affect objects' motion. The Earth as an Inertial Frame While we often consider the ground as a reference frame, it is not truly inertial. The Earth rotates on its axis and orbits the sun, causing slight acceleration. Newton approximated the Earth as an inertial frame for practical purposes, noting that objects on Earth experience minimal acceleration. The Universal Frame and Time keeping Newton also proposed a universal frame, fixed with respect to distant stars, providing an exact inertial frame. He imagined a clock in the universe that could tell the exact time, with innumerable reference frames and inertial frames. Observations and Events in Different Frames In any frame of reference, we can observe events. For example, a person traveling in a car or on a ship experiences different reference frame. Observations made in these frames are considered events. For instance, striking a match could be an event, and its occurrence can be noted in different frames of reference. Figure 1.1: An event at time t = t  in S frame of reference. In one frame, as shown in figure 1.1, we describe an event by specifying its coordinates, such as (x, y, z, t). The same event in another frame will have different spatial coordinates but the same time coordinate, according to Newton's view of absolute time. If we denote the second frame's coordinates with primes, we get transformations like: x = x − vt , y = y, z = z, t = t These transformations are known as Galilean transformations, which give common sense results. Here is an example of a rod, as shown in figure 1.2, that is at rest, lies on x-axis. Figure 1.2: A rode AB is placed along x-axis. length in S-frame = xB − x A length in S-frame = xB − xA Galilean transformation gives: xB = xB − t B and xA = x A − t A Thus, lenght in S-frame xB − xA = xB − x A − v ( t B − t A ) Since the two end points, A and B are measured at the same instant tB = t A , xB − xA = xB − xA length in S-frame = length in S-frame This shows that the length of the object remains the same in both frames. Similarly, the velocities and accelerations also transform predictably. Velocity transformation Galilean transformation of position along x-axis is x = x − vt differentiate wrt time: dx dx dt = −v dt dt dt dx dx = − v eq(1) dt dt we know t = t dx dx  = dt dt  eq(1) becomes dx dx = −v dt  dt similerly, y and z components are dy dy dz  dz = , = dt  dt dt  dt dx = u x = the x-component of the velocity dt  measured in S-frame dx = u x = the x-component of the velocity dt measured in S-frame Therefore, the velocity transformation is u y = u y uz = uz ux = ux − v In general, the vector form is u = u − v Acceleration transformation  Galilean transformation of velocity along x-axis is ux = u x − v differentiate wrt time: dux d = ( ux − v ) dt  dt dux du x dv = − dt  dt dt dux du x dv = = 0 because v = constant dt  dt dt similerly, y and z components are duy du y duz du z = , = dt  dt dt  dt Therefore, the acceleration transformation is ax = ax , ay = a y , az = az In general, the vector form is a = a , Thus, the acceleration in the S′ frame is the same as in the S frame, and Newton's laws apply equally in all inertial frames. Newton's second law: F = ma in S-frame But in S-frame: F  = ma But: m = m a = a  F  = ma = ma = F This is known as the principle of Newtonian relativity, which states that the laws of physics are the same in all inertial frames. The Speed of Light and Ether Theory Now, let's consider the speed of light. Suppose you measure the speed of light in a laboratory and find it to be 3×1010 centi-meters per second or 3×108 meters per second. If you place a light source on a moving car, according to Newtonian mechanics, the speed of light should be affected by the car's speed. Scientists once believed that light, like sound, needed a medium to travel through, called ether. However, the Michelson-Morley experiment, which aimed to detect the presence of ether, failed to find any evidence of it. This experiment showed that the speed of light is constant, regardless of the motion of the source or the observer. This finding puzzled scientists because it contradicted the notion of ether. The Dawn of New Physics Einstein’s journey into the realm of relativity began with deep questions about the nature of electromagnetism and light. He wondered why Maxwell's equations, which describe electromagnetic waves, would hold true in one frame and not another. This curiosity led to the development of his two fundamental postulates of Special Relativity, which have been tested and validated countless times. Let's delve into the concepts that revolutionized our understanding of space and time. The Two Pillars of Special Relativity Einstein proposed two main principles: 1. The Principle of Relativity: The laws of physics are the same in all inertial frames of reference. This means that no matter how fast you are moving, if you are not accelerating, the same physical laws apply to you as to someone who is stationary. 2. The Constancy of the Speed of Light: The speed of light in a vacuum is constant and independent of the motion of the light source or the observer. This means that no matter how fast you are moving, you will always measure the speed of light as 3×1010 centi-meters per second or 3×108 meters per second. These principles seem straightforward, but they lead to some surprising and counterintuitive results. These two postulates led to a profound shift in our understanding of space and time. They implied that measurements of space and time are relative to the observer's frame of reference. This realization paved the way for the development of Special Relativity, fundamentally changing our perception of the universe. Synchronizing Clocks in Different Frames Figure 1.3: Two clocks in a single frame of reference which are far apart. To understand how time and space are related, let's consider synchronizing clocks, in a single frame of reference. Suppose we have two clocks, one at point A and another at point B, 90 million kilo-meters apart, as shown in figure 1.3. If a light signal is sent from B to A, it will take 5 minutes to travel this distance (since light travels at 3×108 meters per second). d Time taken by in reaching from B to A = t = c  90 106 km t= 3 108 m / sec or 9  1010 m t= = 300sec = 5 min 3 108 m / sec When the light signal reaches clock A, it shows 3:00 PM. Since it took 5 minutes for the light to travel, clock B must have shown 2:55 PM when the light left. Thus, we can synchronize the clocks by accounting for the time it takes for light to travel between them. This method ensures that all clocks in the frame show the same time, considering the finite speed of light. Measuring Time in Moving Frames Figure 1.4: A moving train with a flashlight fixed at ceiling pointing to the floor. Now, imagine you are on a train moving smoothly and rapidly, as shown in figure 1.4. You have a flashlight pointing from the ceiling to the floor. If the distance from the ceiling to the floor is h, and the speed of light is c, the time t it takes for the light to travel from the ceiling to the floor, according to your clock, is: h Time it take the light to reach at floor = t  = eq(i) c However, an observer standing on the ground sees the train moving. For this observer, the light also has to cover the horizontal distance that the train travels while the light is moving from the ceiling to the floor. If the train's speed is v and the light takes time Δt, the horizontal distance covered is vΔt. The total distance the light travels, according to the ground observer, forms the hypotenuse of a right triangle with sides h and vΔt. Using the Pythagorean theorem: d 2 = h 2 + ( vt ) 2 d = h 2 + ( vt ) 2 as we know d = ct ct = h 2 + ( vt ) 2 h 2 + ( vt ) 2 t = c Solving for t , taking square on both sides (ct ) 2 = h 2 + ( vt ) 2 (ct ) 2 − ( vt ) = h 2 2 (t ) 2 (c 2 − v 2 ) = h 2 h2 (t ) 2 = (c 2 − v 2 ) taking squareroot on both sides h t = c − v2 2 taking c 2 common h t = c 2 {1 − (v 2 / c 2 )} h t = c 1 − (v 2 / c 2 ) or h 1 t = c 1 − ( v / c )2 using eq(i) 1 t = t  1− (v / c) 2 Therefore t ' = t 1 − ( v / c ) 2 t 1 or t ' = where  =  1− ( v / c) 2 for v  c,   1  t '  t This time Δt is longer than the time t measured by the observer on the train. The difference is due to the fact that the light has to cover more distance when the train is moving. Gamma (γ) is always greater than or equal to 1. When the speed v is much less than the speed of light c, γ is very close to 1, and we do not notice time dilation in our everyday experiences. However, as v approaches c, γ increases significantly, and time dilation becomes noticeable. For example, if you travel in a fast rocket, the time measured on your clock (moving clock) would appear to run slower compared to a clock on Earth (stationary clock). This effect is more pronounced at speeds close to the speed of light. Example: A muon is traveling through the laboratory at three-fifths the speed of the light. How long does it last? Solution: 3 Speed of muon = v = c 5 t 1 t ' = where  =  1− (v / c) 2 lets calculate  1 = 1− ( v / c) 2 1 = 2 3c 1−   5 c 1 5 = = 1 − ( 3 / 5) 4 2 So, it lives longer (than at rest) by a factor of 5/4 The Twin Paradox: An Intriguing Comparison To grasp the nuances of Special Relativity, let's delve into the Twin Paradox, a fascinating thought experiment that challenges our understanding of time and motion. The Setup Imagine two identical twins, A and B, born at the same time. Twin A remains on Earth while Twin B embarks on a journey into space, traveling at speeds close to that of light. After many years, Twin B returns to Earth and finds that Twin A has aged significantly, while Twin B has aged much less. The Paradox At first glance, this seems paradoxical. According to the Theory of Relativity, all inertial frames are equivalent. Thus, Twin A could argue that Twin B should age faster, and vice versa. However, in practice, the traveling twin (B) is younger upon return. Breaking the Symmetry The resolution lies in understanding that the traveling twin (B) was not in a purely inertial frame. Twin B experienced acceleration and deceleration phases during the journey, breaking the symmetry of inertial frames. Thus, Twin B's frame was not equivalent to Twin A's purely inertial frame on Earth. Lorentz Contraction: Objects Shrink in Motion Now, let's explore another fundamental aspect of Special Relativity: Lorentz Contraction. This phenomenon states that objects moving at high speeds appear shorter in the direction of motion to an outside observer. The Thought Experiment Figure 1.5: A high-speed train having a flashlight and mirror at its ends. Imagine you're on a high-speed train with a flashlight, as shown in figure 1.5. You stand at one end of the train car, and a mirror is placed at the other end. You shine the light towards the mirror. According to your frame of reference, the light travels a distance 2Δx′ (to the mirror and back) in a time 2Δx′/c, as shown in figure 1.6. x = length of the car with respect to car observer 2x t  = eq(I) c  Figure 1.6: In fast-speed train light travels an extra distance 2Δx′ x = length of the car according to ground observer t1 = time for the signal to reach the front end t2 = return time However, for an observer on the ground, the train is moving. Thus, they perceive the light traveling a longer distance because the mirror is moving away from the light when it is emitted and moving towards the light on its return journey. Mathematical Explanation For the ground observer, the total distance covered by light includes the forward distance (increased by the train's movement) and the backward distance (decreased by the train's movement). Let Δt1 be the time for the light to reach the mirror and Δt2 be the time for the light to return. 1. For the forward journey, the distance covered by light is x + vt1 : ct1 = x + vt1 solving for t1 ct1 − vt1 = x t1 (c − v) = x x t1 = (c − v ) 2. For the return journey, the distance covered by light is x − vt2 :  ct2 = x − vt2 solving for t2 ct2 + vt2 = x t2 (c + v) = x x t2 = (c + v ) The total time Δt for the round trip is: x x t = t1 + t2 = + ( c − v ) (c + v )  1 1  t = x  +   c−v c+v   c+v+c−v  t = x    c −v  2 2  c+c  t = x  2 2   c −v  2c t = x 2 c (1 − v 2 / c 2 ) 2x / c t = eq(II) (w.r.t ground observer) (1 − v 2 / c 2 ) From time dilation formula: t  = 1 − v 2 / c 2  t using eq(II)  2 x / c  t  = 1 − v 2 / c 2   (  1 − v2 / c2   )    2  2x / c  t  = 1 − v / c  2 2  (  1 − v2 / c2   )    2  2x / c  t  = 1 − v / c 2    ( 1 − v2 / c2 ) ( 1 − v2 / c2 )  2 x / c t  = 1 − v2 / c2  2x using eq(I) t  = c 2x 2x / c = c 1 − v2 / c2 x x = 1 − v2 / c2 Conclusion of Lorentz Contraction The length of the moving object (train) appears contracted by a factor of 1 − v2 / c2. For example, a one-meter rod moving at high speed will appear shorter to a stationary observer. The faster the object moves, the greater the contraction. Practical Implications and Observations In our daily lives, the effects of Special Relativity are negligible because the speeds involved are much less than the speed of light. However, in high-energy physics and astronomy, these effects are significant. For instance, particles accelerated to near-light speeds in colliders exhibit both time dilation and length contraction, validating Einstein's predictions. The Evolution of Transformations: Galilean vs. Lorentz In classical mechanics, we often use Galilean transformations to relate the coordinates of events between different reference frames. However, these transformations fall short when dealing with speeds close to the speed of light. In Einstein's theory of relativity, Galilean transformations are replaced by Lorentz transformations, which accurately describe the relationships between coordinates in different frames at high velocities. Galilean Transformations Galilean transformations work well for everyday speeds. They assume that time is absolute and the same for all observers, while positions and velocities transform simply with the relative motion between frames. The Need for Lorentz Transformations When dealing with velocities approaching the speed of light, the Galilean transformations are no longer valid. Instead, we use Lorentz transformations, which account for the relativistic effects predicted by Einstein's theory of relativity. These transformations show how time and space coordinates of an event change between two frames moving at a constant velocity relative to each other. Lorentz Transformations Consider two frames of reference, as shown in figure 1.7, one is with coordinates (x, y, z, t) and other has coordinates (x`, y`, z`, t`), moving with relative speed of v, as given below. Now, an event E occurs. Let’s calculate Lorentz transformation from moving frame to stationery and vice versa. where: x and t are the coordinates of the event in the original frame. x′ and t′ are the coordinates in the moving frame. v is the relative velocity between the frames. c is the speed of light. 1  (gamma) is the Lorentz factor, defined as  =. 1 − v2 / c 2 Figure 1.7: For an event E, frame of reference with prime coordinates has a relative speed v. x = d + vt But x d=   x =  ( x − vt ) because x is moving with respect to O − frame Figure 1.8: For an event E, frame of reference with coordinates (x, y, z, t) has a relative speed v.  x = d  − vt  But x d =   x =  ( x + vt  )  x =  ( x + vt  ) x =  (  ( x − vt ) + vt  ) x =  2 ( x − vt ) +  vt  x =  2 x −  2 vt +  vt  dividing with  2 on both sides, we get x vt  = x − vt +  2  vt  1 x(1 − v 2 / c 2 ) = x − vt +  2 =  1 − v2 / c2 vt  x − xv 2 / c 2 = x − vt +  vt  − xv 2 / c 2 = − vt +  vt  vt − xv 2 / c 2 =  vt  v(t − xv / c 2 ) =  t (t − xv / c 2 ) =  t  =  (t − xv / c 2 )  v   t =   t − 2 x   c  So, collectively, all Lorentzian equations are given below: x =  ( x − vt ) x =  ( x + vt ) y = y y = y z = z z = z  v   v  t =   t − 2 x t =   t  + 2 x   c   c  Simultaneity: Let’s test this Lorentz transformation for a specific situation. Consider we have clocks in S- frame (stationary) and in S’-frame (moving), as shown in figure 1.9. Now at t=0 check the value of t’. Figure 1.9: For a fixed position, clocks for two frames of references are presented. As we know  v  t =   t − 2 x  c  put t=0  v  t =   0 − 2 x  c  v t  = − 2 x c Clearly, t is not equal to t`. Time Dilation and Length Contraction The Lorentz transformations reveal that time and space are intertwined, and their measurements depend on the observer's frame of reference. This intertwining leads to two significant effects: time dilation and length contraction. Time Dilation Time dilation refers to the phenomenon where time appears to pass slower for an object moving relative to an observer's frame. From the Lorentz transformation, if we consider an event occurring at the same location in the moving frame ( x = 0 ), the relationship between time intervals in the two frames can be calculated.  We know  v  t =   t  + 2 x   c   v   t  =   t − 2 x   c  putting x = 0, we get t  = t This equation shows that the time interval t′ in the moving frame is longer than the time interval t in the stationary frame, indicating that moving clocks run slower. Length Contraction Length contraction states that objects moving relative to an observer will appear shortened along the direction of motion. If we measure the length of an object moving with velocity v, the contracted length x can be calculated as: We know x =  ( x − vt )  x =  ( x − vt ) putting t = 0, we get x = x Here, x is the length of the object in its rest frame. As the speed increases, γ increases, making x shorter. Einstein velocity addition rule: The Lorentz transformation also helps us relate the velocities of objects in different frames. If a particle moves with velocity u in frame S, its velocity u′ in frame S′ moving with velocity v relative to S is derived below. Suppose a particle moves a distance dx in atime dt. Its velocity u is then dx u= (in S-frame) dt In S-frame, meanwhile, it has moved a distance dx =  ( dx − vdt ) in time   v  dt  =   dt − 2 dx   c  The velocity in S-frame is therefore dx u = dt  putting values of dx and dt , we get  ( dx − vdt ) u =  v    dt − dx   c2  dx − vdt u = v dt − 2 dx c dt (dx / dt − v) u =  v  dt 1 − 2 dx / dt   c  dx / dt − v u = v 1 − 2 dx / dt c u−v u = eq(a)  u = dx / dt uv 1− 2 c lets calcualte for u, we have u−v u = uv 1− 2 c  uv  u 1 − 2  = u − v  c  uv u − u 2 = u − v c  uv u + v = u + u c2  u v  u + v = u 1 + 2   c  u + v =u  u v  1 + 2   c  or u + v u= eq(b) u v 1+ 2 c Eq(a) and eq(b) are Einstein velocity addition rule. Example: Suppose, instead of particle with speed u we have light itself then u becomes c. What will be its speed in prime coordinates? Solution: If u = c then u  = ? Using the Einstein velocity addition rule,we get u−v u = uv 1− 2 c putting u = c c−v u = cv 1− 2 c c−v u = v 1− c c−v u = c =c c−v  u = c Speed of light same in all reference frames Example: Suppose, an ordinary particle of speed u and relative speed of v have very low speed as compared to speed of light c. What will be its speed in prime coordinates? Solution: If u, v c then u  = ? Using the Einstein velocity addition rule,we get u−v uv u = → u − v  uv c 2 so 1 − 2  1 uv c 1− 2 c u = u − v For ordinary speeds, Einstein velocity addition rule reduces to Galilean velocity addition rule The Famous Equation: E=mc2 Einstein's most famous equation, E=mc2, illustrates the equivalence of mass and energy. It implies that mass can be converted into energy and vice versa. This principle is fundamental to nuclear reactions and particle physics, where tiny amounts of mass convert into vast amounts of energy.