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Special Relativity. Classical Newtonian Mechanics – its limitations. At the beginning of the 20 th century, the only theory of motion was that of Newton, based on his three laws – ‘classical’ Newtonian mechanics.
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Classical Newtonian Mechanics – its limitations • At the beginning of the 20th century, the only theory of motion was that of Newton, based on his three laws – ‘classical’ Newtonian mechanics. • Newtonian mechanics was known to work well in describing the motions of objects in the macroscopic world, whose velocities are generally slow compared to the speed of light (c=3x105 km/s). • But what about objects traveling at speeds approaching the speed of light? Such objects do exist in the microscopic world, for example an electron accelerated through a 10 MV potential difference (and hence given an energy of 10MeV), will attain a speed of 0.9988c. [And note that in Newtonian mechanics, there is no reason why a particle cannot have any speed it like, with v>c!!] • However, if we accelerate an electron through a 40MV potential difference, giving it 4x the energy, experiment shows its does not have 2x the speed (as expected from K=½mv2). So instead of having v~2c, it is found to have v=0.9999c. Experiment shows that Newtonian mechanics does not predict the correct answers when it is applied to such fast particles; it fails in the limit of v c (with c appearing to be an upper limit)!
Galilean Transformations – describing events from different reference frames First some definitions: A physical event – something that happens independent of the reference frame we might use to describe it; e.g. a collision between two particles, a flash of light. The event occurs at a point in space (x,y,z) and at an instant of time t. An inertial system – a frame of reference in which the law of inertia – Newton’s 1st Law – holds. In such a system, which may also be referred to as an unaccelerated system, a body that is acted on by zero net external force will move with constant velocity. While an unaccelerated frame on earth is not truly ‘inertial’ due to the accelerating effects of the earth’s rotation and its orbital motion around the Sun, the accelerations involved are so small that it approximates an inertial frame. The Theory of Special relativity deals only with the description of events by observers in inertial reference frames; the Theory of General Relativity concerns itself with all reference frames, including non-inertial ones.
Galilean Transformations • Consider two inertial frames S and S’, with S’ moving at a constant velocity v with respect to S along common x, x’ axis. An observer in S measures P to occur at coordinates x,y,z and at time t, while an observer in S’ measure P to occur at coordinates x’,y’,z’ and at time t’. y y’ v S S’ x ● P x’ x’ vt z z’ • Fundamental assumption in making measurements in different frames using ‘classical’ procedures: length intervals and time intervals are the same for all inertial observers of the same events • if meter sticks are of the same length when measured and compared at rest, they are of the same length when compared in relative motion to one another; • if clocks are calibrated and synchronised when at rest, their readings and rateswill agree even if they are in relative motion to one another.
If the clocks of each observer read zero at the instant that the origins of the frames S and S’ coincide, then the Galilean coordinate transformations are: • x’ = x – vt • y’ = y • z’ = z • t’ = t Galilean Transformations y y’ v S S’ x ● P x’ x’ vt z z’ • 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 • and that the distance between two points, A and B, measured at a given instant, is the same for each observer: • xB’ – xA’ = xB - xA Einstein turned these concepts completely on their head by arguing that the two observers would measure different lengths and time intervals in this situation!
Galilean Transformations and Newtonian Relativity • We can differentiate the Galilean transformations to determine the velocities and accelerations that different inertial observers measure: • starting with: x’ = x – vt • and differentiating with respect to t: dx’/dt = dx/dt – v • and because t’ = t, we can write d/dt = d/dt’ • hence dx’/dt’ = dx/dt – v, dy’/dt’ = dy/dt, dz’/dt’ = dz/dt • which we can write as ux’ = ux – v, uy’ = uy, uz’ = uz • To obtain the acceleration transformation, we differentiate the above velocity relations: • dux’/dt’ = d(ux-v)/dt, duy’/dt’ = duy/dt, duz’/dt’ = duz/dt • and since v = constant, d(ux-v)/dt = dux/dt • hence we have: ax’ = ax, ay’ = ay, az’ = az • Hence we see that the different observers (in S and S’) will assign different velocities because of their relative motion, but will measure the same acceleration!
Galilean Transformations and Newtonian Relativity • Important conclusions to be drawn from the above relations: • 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 x 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.
Electromagnetism and Newtonian Relativity • We have just shown that the laws of Newtonian mechanics are invariant under a Galilean transformation. Is this the case for the laws of electromagnetism?? • Consider a pulse of light (an electromagnetic wave) traveling to the right with respect to the medium through which it is propagated at a speed c. The “medium” of light propagation was given the name “ether”, historically, since the mechanical view of light that emerged from the physics of the 19th century was inconsistent with it traveling through empty space. • Let’s call the “ether” frame S, and assume it to be an inertial one in which the observer measures the speed of light to be exactly c. • In a frame S’ 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.
Electromagnetism and Newtonian Relativity Hence the speed of light is certainly not invariant under a Galilean transformation, and there is only one inertial frame in which the speed of light is measured to be c, and hence one frame in which Maxwell’s laws of electromagnetism apply what would be an absolute frame. Attempts to locate the Absolute Frame – The Michelson-Morley Experiment • The question as to whether the “ether” frame existed and the speed of light was measured to be different in different inertial frames was deemed to be so important that in 1887 Michelson and Morley devised an experiment to determine whether this was the case. • If an ether exists, the spinning and orbiting earth should be moving through it. An observer on earth would sense an “ether wind” whose velocity is v relative to earth, where v is ~30 km/s (v/c ~ 10-4) the earth’s orbital speed about the Sun. • It was Michelson’s invention of the optical interferometer that provided a device sensitive enough to make the measurement (1 in 10,000 effect).
The interferometer consists of a beam of light (from source S) that is split by a partially silvered mirror M into two coherent beams, beam 1 being transmitted through M and beam 2 being reflected off M. Beam 1 is reflected back to M by mirror M1 and beam 2 by mirror M2. Then the returning beam 1 is reflected and the returning beam 2 is transmitted by M back to a telescope T where they interfere. The interference is constructive or destructive depending on the phase difference of the beams. If M1 and M2 are very nearly but not quite at right angles, a fringe pattern is observed in the telescope (similar to a thin wedge of air between two glass plates). Suppose v is the velocity of the ether with respect to the interferometer. The Michelson-Morley Experiment
The Michelson-Morley Experiment • The phase difference between beams 1 and 2 can arise from two causes: (i)different path lengths travelled, (ii)different speeds of travel with respect to the instrument because of the “ether wind”. The latter is the crucial one with beam 1 being a ‘up-stream’ and ‘down-stream’ situation, and beam 2 being a ‘cross-stream’ situation. • If the instrument is rotated by 90°, thereby making beam 1 the cross-stream beam and beam 2 the up-stream/down-stream beam, it can be shown that this will cause a shift in the fringe pattern, since it changes the phase relationship between the two beams. The key to detecting this is to make the lengths of the paths the two beams travel to be as long as is practically possible; M-M were able to achieve path lengths of ~11m. • The fringe shift is given by N = (2l/)(v/c)2, where l is the path length, and the wavelength of the light. Putting in the values for M-M’s experiment, we get N = (22/5.5 x 10-7)(10-4)2 = 0.4 four-tenths of a fringe!!! • M-M mounted their interferometer on a massive stone slab for stability and floated it in mercury for smooth rotation. The fringes were observed under a continuous rotation, and observations were made day and night.
The Michelson-Morley Experiment • A precision of 0.01 of a fringe was achieved in measuring their shift! • Despite making observations during all seasons of the year (and hence with the earth at different locations about the Sun and at different orientations with respect to the “ether”, NO FRINGE SHIFT WAS OBSERVED! M-M concluded from their experiment that there was no fringe shift at all. • This null result (N = 0) was such a blow to the ether hypothesis that the experiment was repeated many times by different workers over a 50-year period. The null result was amply confirmed!
Postulates of Special Relativity • Einstein published his Theory of Special Relativity in 1905 (hence this year is being celebrated as the World Year of Physics in acknowledgement that it is 100 years since Einstein’s “Annus Mirablis”). The assumptions or postulates on which his theory was based are: • The laws of physics are the same in all inertial reference frames (The Principle of Relativity) • The speed of light in free space has the same value c in all inertial reference frames (The Principle of the Constancy of the Speed of Light) • The latter principle contradicts the Galilean velocity transformation, but is consistent with the Michelson-Morley experiment. However, it is the ability of a theory to explain experimental results upon which its success is judged, and here Special Relativity was not only able to explain existing results, but it also predicted new effects which were confirmed later by experiment. Furthermore, no experimental objection to Special Relativity has yet been found.
Simultaneous Events Suppose one observer S notes that two independent events occur at the same time. Suppose another observer S’, who is moving at constant velocity v with respect to S, also records the same two events. Q: Will S’ observe the two events to occur simultaneously? A: No!! Simultaneity is not an absolute concept but a relative one, depending on the state of the observer. It was this very fact that led Einstein to recognise that time is not absolute for all observers (as assumed in the Galilean transformations) – since timing is intimately related to timing events – and of all the features of the classical theory, it was the measurement of time that was the most suspect.
Lets look at this more closely via an example: • Consider two spaceships which provide inertial frames for their observers Sally and Sam who are located at the midpoint of each ship. Sally’s ship is moving with a velocity v relative to Sam’s ship. Two large meteorites strike the ships, and leave a permanent mark on each ship at positions R, R’ and B, B’. • Suppose the wavefronts from the two events happen to reach Sam at the same time (c). Sam also determines by measurement that he is stationed halfway between the marks B and R on his ship. He will rightly conclude that the two events occurred simultaneously. • For Sally, the wavefront from R’ reaches her (b) before the wavefront from B’ does (d) . And having determined that she is halfway between B’ and R’, she concludes the two events were not simultaneous.
Simultaneous Events • Sam and Sally do not agree on whether the events were simultaneous, but both are correct the relativity of simultaneity! • Of key importance in considering what is observed in this situation is the Second Postulate of Special Relativity in that the speed of light, c, is the same in both reference frames. The Relativity of Time • The relativity of simultaneity has an important implication for measuring time: if two observers measure the time interval between a given pair of events, they will not agree as to how long that interval is. • Consider another situation: • Sally is on a train that is moving with uniform velocity v with respect to the station. She has an electronic clock, which she uses to measure the time to between two events: • Turning on of flash bulb B • Arrival of light back at its source after reflection from a mirror on the ceiling
The Relativity of Time • For the time interval between these two events Sally finds: • to = 2D/c • where D is the distance between the source and the mirror. For Sally, these two events occur at the same place and she can time the interval between them with a single clock C located at that place. A time interval between two events at the same location, as measured by a stationary clock at that location, is called a proper time interval, identified by the subscript zero. • For Sam, who is standing on the station platform watching the train go by, these two events will appear different. As shown in the diagram, while the light beam still appears to travel at speed c, it travels a larger distance given by 2L (cf. 2D for Sally). Hence the time interval between the two events is: • t = 2L/c • where L = (½vt)2 + D2) • And we can express D in terms of the proper time: L = (½vt)2 + (½cto)2
The Relativity of Time • Now L = ½tc, so substituting into the previous equation and solving for t, gives: • t = to/1 – (v/c)2 • Hence the time interval between the two events that Sam measures is longer than the proper time that Sally measures TIME DILATION The key difference here is that Sam sees the two events occur at different places in his reference frame, and so to measure t he must use two synchronised clocks, one at each place. Thus the interval he measures is not a proper time. Hence the situations for Sam and Sally are not symmetrical. • Time dilation is very real and has nothing to do with any mechanical change that takes place in a clock because of its motion; it is simply the nature of time. Lorentz Factor: this is the dimensionless 1/1 – (v/c)2 term that appears in the time dilation equation, and which gets increasingly larger as v c. This factor appears in almost all the relativity equations and is denoted by .
The Relativity of Length Measuring length of object at rest (with respect to observer): note the position of the two ends on a long stationary scale and subtract the two readings – no worries! Measuring length of a moving object: need to measure the position of the two ends simultaneously (in observer’s reference frame). Because simultaneity is relative and it enters into length measurements, then length is also expected to be a relative quantity and differ for observers in different inertial frames. Because of this relativity, we need to define proper length – this is the length Lo of an object, that is measured in an inertial reference frame in which the object is at rest. Length Contraction Consider once more the two observers Sally and Sam, Sally seated on a train moving through a station and Sam on the station platform. Suppose they both want to measure the length of the platform.
Length Contraction First step: we need to ask the question, which of the two observers measures (or can measure) the proper length? In this case it is Sam, since the platform is at rest with respect to him. Sam also notes that a marker on the train covers this length in a time: t = Lo/v where v is the speed of the train. That is Lo = v t. The time interval t is not a proper time interval because the two events that define it – that is, when the marker passes the beginning of the platform and when the marker passes the end of the platform – occur at different places and Sam must use two synchronised clocks to measure the time interval t. For Sally, however, the platform is moving, and she sees it approach and then recede at speed v, and finds that the two events measured by Sam occur at the same place in her reference frame. So she can time them with a single stationary clock, so the time interval she measures is the proper time, to. To her, the length L of the platform is given by L = v to.
Length Contraction So we have Lo = v t (Sam), and L = v to (Sally). Dividing one by the other: L/Lo = (vto)/(vt) = 1/ (since t = to). Hence: L = Lo/ and so the length L that Sally measures is shorter than the proper length, and so this is called the length contraction equation.
To conclude this section, it is appropriate to revisit the Galilean Transformations and see what form they take under Special Relativity. The transformations that describe the space and time coordinates of events in different inertial reference frames in this case are called the Lorentz Transformation. Lorentz Transformations y y’ v S S’ x ● P x’ x’ vt z z’ • Yet again, we consider two inertial frames S and S’, with S’ moving at a constant velocity v with respect to S along common x, x’ axis. An observer in S measures P to occur at coordinates x,y,z and at time t, while an observer in S’ measure P to occur at coordinates x’,y’,z’ and at time t’. • The Galilean transformations were x’ = x – vt, y’ = y, z’ = z, and t = t’. • The Lorentz transformations are: • x’ = (x – vt) • y’ = y • z’ = z • t’ = (1 – vx/c2)
