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Modern Physics

Luis A. Anchordoqui

Department of Physics and Astronomy Lehman College, City University of New York

Lesson III September 17, 2015

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 1 / 26 Table of Contents

1 Foundations of Special Relativity Einstein postulates Relativity of simultaneity Lorentz transformations

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 2 / 26 Foundations of Special Relativity

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 3 / 26 Foundations of Special Relativity

Historical Overview (1892 -1909) To explain nature’s apparent conspiracy to hide æther drift Lorentz developed theory based on two ad hoc hypotheses: Longitudinal contraction of rigid bodies slowing down of clocks (time dilation) when moving through æther at speed v both by (1 v2/c2)1/2 − This would so affect every aparatus designed to measure the æther drift as to neutralize all expected effects (1898) Poincare arugued that æther might be unobservable and suggested concept would be thrown aside as useless BUT he continued to use concept in later papers of 1908 (1905) Einstein advanced principle of relativity

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 4 / 26 Foundations of Special Relativity Einstein postulates

1 The laws of physics are identical in all inertial frames or equivalently the outcome of any experiment is the same when performed with identical initial conditions relative to any inertial frame 2 There exists an inertial frame in which light signals in vacuum always travel rectilinearly at constant speed c (in all directions) independently of the motion of the source

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 5 / 26 Foundations of Special Relativity Relativity of simultaneity

How does observer in inertial reference frame describe event? Event an occurrence characterized by: three space coordinates and one time coordinate

Events are described by observers who do belong to particular inertial frames of reference

Different observers in different inertial frames would describe same event with different spacetime coordinates

Observer’s rest frame is also known as proper frame

Two ordinary methods to construct coordinate system use of confederates at each place single observer method

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 6 / 26 Foundations of Special Relativity Relativity of simultaneity 14 Chapter 1 RelativityConfederate I scheme for coordinatizing any event § ¤ Figure 1-13 Inertial reference frame formed y from a lattice of measuring rods with a clock at each intersection. The clocks are¦ all ¥ synchronized using a reference clock. In this diagram the measuring rods are shown to be 1 m long, but they could all be 1 cm, 1 ␮m, or 1 km as required by the scale and precision of the measurements being considered. The three space dimensions are the clock positions. The fourth spacetime dimension, time, is shown by indicator readings on the clocks.

Reference clock x

z Observer establishes lattice of confederates When an event occurs, its location and time are recorded instantly by the nearest clock. Suppose that an atom located at x ϭ 2 m, withy ϭ 3 m, z identicalϭ 4 m in Figure 1-13 synchronized emits clocks a tiny flash of light at t ϭ 21 s on the clock at that location. That event is recorded in label ofspace any and in eventtime or, as we in will spacetime henceforth refer to it, in the spacetime coordinate system with the numbers (2,3,4,21). The observer may read out and analyze these data at is readinghis leisure, of within clock the limits and set by the location information transmission of nearest time (i.e., the light confederate travel to event time) from distant clocks. For example, the path of a particle moving through the lattice L. A. Anchordoquiis revealed (CUNY) by analysis of the records showingModern the Physics particle’s time of passage at each 9-17-2015 7 / 26 clock’s location. Distances between successive locations and the corresponding time dif- ferences make possible the determination of the particle’s velocity. Similar records of the spacetime coordinates of the particle’s path can, of course, also be made in any inertial frame moving relative to ours, but to compare the distances and time intervals measured in the two frames requires that we consider carefully the relativity of simultaneity.

Relativity of Simultaneity Einstein’s postulates lead to a number of predictions about measurements made by observers in inertial frames moving relative to one another that initially seem very strange, including some that appear paradoxical. Even so, these predictions have been experimentally verified; and nearly without exception, every paradox is resolved by an understanding of the relativity of simultaneity,which states that

Two spatially separated events simultaneous in one reference frame are not, in general, simultaneous in another inertial frame moving relative to the first. Foundations of Special Relativity Relativity of simultaneity

Single clock at spatial origin § ¤ ¦ ¥

Observer continuously sends out light rays in all directions keeping track of emission time

At any event incoming light ray is reﬂected back to observer

Observer has 2 times + a direction associated with any event

To yield spatial coordinatizing consistent with confederate scheme:

~r = c (τ τ ) (1) | | 2 − 1 τ + τ t = 1 2 (2) 2

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 8 / 26 Foundations of Special Relativity Relativity of simultaneity

Einstein’s thought experiments Idealized clock light wave is bouncing back and forth between two mirrors Clock “ticks” when light wave makes a round trip from mirror A to mirror B and back Assume mirrors A and B are separated by distance d in rest frame Llight wave will take ∆t = 2d/c for round trip A B A 0 → → B B B B

S0 S0 d

A A A A

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 9 / 26 1.5 CONSEQUENCES OF SPECIAL RELATIVITY 15

At this point, you might wonder which observer is right concerning the two events. The answer is that both are correct, because the principle of relativity states that there is no preferred inertial frame of reference. Although the two observers reach different conclusions, both are correct in their own reference frame because the concept of simultaneity is not absolute. This, in fact, is the central point of relativity—any uniformly moving frame of reference can be used to describe events and do physics. However, observers in different inertial frames will always measure different time intervals with their clocks and different distances with their meter sticks. Nevertheless, they will both agree on the forms of the laws of physics in their respective frames, because these laws must be the same for all observers in uniform motion. It is the alteration of time and space that allows the laws of physics (including Maxwell’s equations) to be the same for all observers in uniform motion.

Time Dilation The fact that observers in different inertial frames always measure different time intervals between a pair of events can be illustrated in another way by consider- ing a vehicle moving to the right with a speed v, as in Figure 1.10a. A mirror is fixed to the ceiling of the vehicle, and observer OЈ, at rest in this system, holds a laser a distance d below the mirror. At some instant the laser emits a pulse of light directed toward the mirror (event 1), and at some later time, after reflecting from the mirror, the pulse arrives back at the laser (event 2). Observer OЈ carries a clock, CЈ, which she uses to measure the time interval ⌬tЈ between these two events. Because the light pulse has the speed c, the time it takes to travel from OЈ to the mirror and back can be found from the definition of speed: distance traveled 2d ⌬tЈϭ ϭ (1.6) speed of light c This time interval ⌬tЈ—measured by OЈ, who, remember, is at rest in the moving vehicle—requires only a single clock, CЈ, in this reference frame. v v Mirror y′ y′ d

Foundations of Special Relativity Relativity of simultaneity O′ O′ O′ O′ c∆t 2 O d x′ x

v∆t v∆t 2 Time dilation (a) (b) (c) Since light has velocity c in all directions Figure 1.10 (a) A mirror is ﬁxed to a moving vehicle, and a light pulse leaves OЈ at ∆t 2 c∆t 2 rest in the vehicle. (b) Relative to a stationary observer on Earth, the mirror and OЈ d2 + v = (3) move with a speed v. Note that the distance the pulse travels measured by the station- 2 2 ary observer on Earth is greater than 2d. (c) The right triangle for calculating the rela- 2d ∆t0 ∆t = = (4) tionship between ⌬t and ⌬tЈ. √c2 v2 √1 v2/c2 − − Ticking of clock in Vinnie’s frame which moves @ v in direction to separation of mirrors ⊥ is slower by γ = (1 v2/c2) 1/2 − − L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 10 / 26 Copyright 2005 Thomson Learning, Inc. All Rights Reserved. Foundations of Special Relativity Relativity of simultaneity

Twin Paradox Consider two synchronized standard clocks A and B at rest at point P of intertial frame S

Let A remain @ P while B is brieﬂy accelerated to some velocity v with which it travels to distant point Q

There it is decelerated and made to return with velocity v to P

If one of two twins travels with B while other remains with A B twin will be younger than A twin when meet again

Can’t B claim with equal right it was her who remained where she was while A went on round-trip A should be younger when meet again?

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 11 / 26 Foundations of Special Relativity Relativity of simultaneity Twin Paradox

The answer is NO and this solves the paradox Answer is NO this solves paradox A remained at rest in single inertial frame A has remained at rest in a single inertial frame while accelerated out of his rest frame: while B wasB accelerated out of his rest frame at P, at Q, @andP,@ onceQ, andagain once at againP @P Accelerations recorded on B’s accelerometer Theseshe accelerations can be under are no recorded illusion that on itB’s was her accelerometerwho remain and at resthe can therefore be under no illusion that it was he who remain at rest Two accelerations at P are not essential (age comparison could be made in passing) Of coursebut acceleration the two accelerations in Q is vital at P are not essential (the age comparison could be made in passing) but the acceleration in Q is vital

Monday, August 2, 2010 L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 12 / 26 Foundations of Special Relativity Relativity of simultaneity Twin Paradox

The answer is NO and this solves the paradox Answer is NO this solves paradox

A hasEperimet remained involves at rest 3 in inertial a single frames: inertial frame while B1 wasearth-bound accelerated frame outS of his rest frame at P, at Q, and2 S once0 of outbound again at rocket P 3 S00 of returning rocket These accelerations are recorded on B’s accelerometerExperiment and not he symmetrical can therefore between be under twins: no illusion thatA stays it was at resthe inwho single remain inertial at framerest S but B occupies at least two different frames

Of courseThis allowsthe two result accelerations to be unsymmetrical at P are not essential (the age comparison could be made in passing) but the acceleration in Q is vital

Monday, August 2, 2010 L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 13 / 26 1-1 The Experimental Basis of Relativity 7 techniques available at the time had an experimental accuracy of only about 1 part in 104,woefully insufficient to detect the predicted small effect. That single exception was the experiment of Michelson and Morley.5

Questions 1. What would the relative velocity of the inertial systems in Figure 1-4 need to be 1-1 The Experimental Basis of Relativity 7 in order for the SЈ observer to measure no net electromagnetic force on the charge q? techniques available at the time had an experimental accuracy of only about 1 part in 2. Discuss why the very large value for the10 speed4,woefully of the electromagneticinsufficient to detect waves the predicted small effect. That single exception was would imply that the ether be rigid, i.e.,the have experiment a large bulk of modulus.Michelson and Morley.5

Questions The Michelson-Morley Experiment 1. What would the relative velocity of the inertial systems in Figure 1-4 need1-1 Theto be Experimental Basis of Relativity 7 All waves that were known to nineteenth-centuryin order scientists for the required SЈ observer a medium to measure in no net electromagnetic force on the order to propagate. Surface waves moving acrosscharge the q ocean? obviously require the techniques available at the time had an experimental accuracy of only about 1 part in water. Similarly, waves move along a plucked guitar string, across the surface of a 2. Discuss104,woefully why the very insufficient large value to detect for the the speed predicted of the small electromagnetic effect. That single waves exception was struck drumhead, through Earth after an earthquake, and, indeed, in all materials acted wouldthe imply experiment that the of ether Michelson be rigid, and i.e., Morley. have 5a large bulk modulus. upon by suitable forces. The speed of the waves depends on the properties of the medium and is derived relative to the medium. For example, the speed of sound waves in air, i.e., their absolute motion relative to still air, canQuestions be measured. The Doppler effect for sound in air depends not only on the relativeThe Michelson-Morley motion of the source and Experiment listener, but also on the motion of each relative to still air. Thus,1. it Whatwas natural would for the scientists relative velocity of of the inertial systems in Figure 1-4 need to be All waves that were known to nineteenth-century scientists required a medium in that time to expect the existence of some material like thein ether order to for support the S Јtheobserver propa- to measure no net electromagnetic force on the order to propagate. Surface waves moving across the ocean obviously require the gation of light and other electromagnetic waves and to expectcharge that q? the absolute mo- water. FoundationsSimilarly, waves move of Specialalong a plucked Relativity guitar string,Relativity across the surface of simultaneity of a tion of Earth through the ether should be detectable, despite the fact that the ether had struck drumhead,2. Discuss through why Earththe very after large an earthquake, value for the and, speed indeed, of the in electromagnetic all materials acted waves not been observed previously. upon by suitablewould forces. imply The that speedthe ether of be the rigid, waves i.e., depends have a large on the bulk properties modulus. of the Michelson realized that although the effect of Earth’s motion on the results of any Albert A. Michelson, here medium and is derived relative to the medium. For example, the speed of sound waves “out-and–back” speed of light measurement, such as shown generically in Figure 1-5, playing pool in his later in air, i.e., their absolute motion relative to still air, can be measured. The Doppler ef- would be too small to measure directly, it should be possible to measure v2 c2 by a dif- years, made the first 34 Chapter 2 The Special Theory of Relativity fect for sound in air depends not only on the accuraterelative measurementmotion of the of source and listener, | ference measurement, using the interference property Theof the Michelson-Morley light waves as a sensitive Experiment butRotate also on the motion clock of each relative by to 90stillthe air. speed Thus,before of light it was while natural an setting for scientists of it in motion “clock.” The apparatus that he designed to make the measurement is> called the ◦ that timeAll to expect waves the that existence were known of some to nineteenth-centurymaterialinstructor like at thethe U.S.ether scientists Naval to support required the propa- a medium in Michelson interferometer. The purpose of the Michelson-Morley experiment was to gation oforder light toand propagate. other electromagnetic Surface waves wavesAcademy, moving and whereto across expect he the had that ocean the obviouslyabsolute mo- require the measure the speed of light relative to the interferometer (i.e., relative to Earth), thereby tion of Earthwater. through Similarly, the ether waves should move be along detectable,earlier a plucked been despite a cadet.guitar the [ AIP string,fact that across the ether the hadsurface of a detecting Earth’s motion through the ether and thus verifying the latter’s existence. To not been struckobserved drumhead, previously. through Earth afterEmilio an earthquake, Segrè Visual and,Archives. indeed,] in all materials acted illustrate how the interferometer works and the reasoning behind the experiment, let us Michelsonupon byrealized suitable that forces. although The the speed effect of of theEarth’s waves motion depends on the on results the properties of any ofAlbert the A. Michelson, here first describe an analogous situation set in more familiar surroundings. “out-and–back”dL medium speed and is of derived light measurement, relative to the such medium as shown. For example, generically the in speed Figure of sound 1-5, wavesplaying pool in his later would bein too air, small i.e., to their measure absolute directly, motion it shouldrelative be to possiblestill air, canto measure be measured. v2 c2 by The a dif- Doppleryears, ef- made the first A-B accurate measurement of ference measurement,fect for sound usingin air thedepends interference not only property on the relative of the light motion waves of the as sourcea sensitive and listener, but also on the motion of each relative to still air. Thus, it was natural for scientiststhe speedof of light while an Light source“clock.” Mirror The apparatus that he designedvu ∆ t tot11 make the measurement is> called the that time to expect the existence of some material like the ether to support the propa-instructor at the U.S. Naval Michelsonc – v interferometerA v. The purpose of the Michelson-Morley experiment was to Academy, where he had O O measure gationthe speed of lightof light and relative other toelectromagnetic the interferometer waves (i.e., and relativeto expect to Earth), that the thereby absolute mo- d + v t = c t (5) S S′0 c + v earlier been a cadet. [AIP∆ 1 ∆ 1 detectingtion Earth’s of Earth motion through through the theether ether should and bethus detectable, verifying despitethe latter’s the existence.fact that the To etherEmilio had Segrè Visual Archives.] Observer A B illustrate nothow been the interferometerobserved previously. works and the reasoning behind the experiment, let us L Albert A. Michelson, here first describe Michelsonan analogous realized situation that setalthough in more the familiar effect ofsurroundings. Earth’s motion on the results of any playing pool in his later Figure 1-5 Light source, mirror, and observer are moving “out-and–back”with speed vLrelative u speed tot the of ether. light measurement, such as shown generically in Figure 1-5, d would bed too+ +smallv∆ to1t measure1 directly, it should be possible to measure v2 c2 by a dif- years, made the first According to classical theory, the speed of light c, relative to the ether, would be c Ϫ v relative accurate∆ measurementt = of (6) ference measurement, using the interference property of the light waves as a sensitive 1 to the observer for light moving from the source toward the mirror and c ϩ v for light the speed of light while an Light source Mirror > c v reflecting from the mirror back toward the source. “clock.” The apparatus that he designed to make the measurement is called the instructor at the U.S. Naval c – v Michelson interferometer. The purpose of the Michelson-MorleyB v experiment was to Academy, where he had − S0 SO measure the speedO of′ light relative toc the+ v interferometer (i.e., relative to Earth), thereby earlier been a cadet. [AIP detectingObserver Earth’s motionA through the ether and thusB verifying the latter’svu ∆t texistence.2 To Emilio Segrè Visual Archives.] illustrate how the interferometer worksL and the reasoning behind the experiment,2 let us first describe an analogous situation set in more familiar surroundings. B-C Figure 1-5 Light source, mirror, and observer are moving with speed v relative to the ether. d L −v u ∆tt2 According to classical theory, the speed of light c, relative to the ether, would be c Ϫ v relative to the observer for light moving from the source toward the mirror and c ϩ v for light reflecting from the mirror back toward theLight source. source Mirror C c – v v ~v d v∆t2 = c∆t2 (7) O OS′0 u S c + v − Observer A B ⃗ L d Figure 1-5 Light source, mirror, and observer are moving with speed v relative to the ether. ∆t2 = (8) FIGURE 2.10 Here the clockAccording carried to classical by theory,O′ emits the speed its of light c, relative ﬂash to in thethe ether, direction would be c Ϫ ofv relative to the observer for light moving from the source toward the mirror and c ϩ v for light c + v motion. reflecting from the mirror back toward the source.

c !t1,equaltothelengthL of the clock plus the additional distance u !t1 that the mirror moves forward in this interval. That is, L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 14 / 26 c !t L u !t (2.9) 1 = + 1

The ﬂash of light travels from the mirror to the detector in a time !t2 and covers adistanceofc !t2,equaltothelengthL of the clock less the distance u !t2 that the clock moves forward in this interval:

c !t L u !t (2.10) 2 = − 2

Solving Eqs. 2.9 and 2.10 for !t1 and !t2,andaddingtoﬁndthetotaltime interval, we obtain L L 2L 1 !t !t !t (2.11) = 1 + 2 = c u + c u = c 1 u2/c2 − + −

MODERN PHYSICS

L0 L0

MODERN PHYSICS

L0 L

FIGURE 2.11 Some length-contracted objects. Notice that the shortening occurs only in the direction of motion. Foundations of Special Relativity Relativity of simultaneity

Length contraction

1 Interval between two consecutive ticks in the moving frame is

2d ∆t = ∆t + ∆t = 1 2 c(1 v2/c2) − d ∆t = 0 , (9) d 1 v2/c2 0 − 2 Because of time dilation

p 2 2 ∆t0 = ∆t 1 v /c (10) − we get 1/2 v2 d = 1 d0 (11) − c2

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 15 / 26 34 Chapter 2 The Special Theory of Relativity |

u ∆t1 A O O′

L + u ∆t1

B O O′

u ∆t2

L − u ∆t2

C O O′ u !

FIGURE 2.10 Here the clock carried by O" emits its light ﬂash in the direction of motion.

c !t1,equaltothelengthL of the clock plus the additional distance u !t1 that the mirror moves forward in this interval. That is,

c !t L u !t (2.9) 1 = + 1

c !t L u !t (2.10) 2 = − 2

Solving Eqs. 2.9 and 2.10 for !t1 and !t2,andaddingtoﬁndthetotaltime interval, we obtain L L 2L 1 !t !t !t (2.11) = 1 + 2 = c u + c u = c 1 u2/c2 Foundations of Special Relativity Relativity of simultaneity − + − Example

MODERN PHYSICS

L0 L0

MODERN PHYSICS

L0 L

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 16 / 26 FIGURE 2.11 Some length-contracted objects. Notice that the shortening occurs only in the direction of motion. 34 Chapter 2 The Special Theory of Relativity |

u ∆t1 A O O′

L + u ∆t1

B O O′

u ∆t2

L − u ∆t2

C O O′ u !

FIGURE 2.10 Here the clock carried by O" emits its light ﬂash in the direction of motion. c !t1,equaltothelengthL of the clock plus the additional distance u !t1 that the mirror moves forward in this interval. That is,

c !t L u !t (2.9) 1 = + 1

c !t L u !t (2.10) 2 = − 2

Solving Eqs. 2.9 and 2.10 for !t1 and !t2,andaddingtoﬁndthetotaltime interval, we obtain L L 2L 1 !t !t !t (2.11) = 1 + 2 = c u + c u = c 1 u2/c2 Foundations of Special Relativity− Relativity+ of simultaneity −

MODERN PHYSICS

L0 L0

MODERN PHYSICS

L0 L

FIGURE 2.11 Some length-contracted objects. Notice thatL. A. Anchordoqui the shortening (CUNY) occursModern only Physics in the direction of motion.9-17-2015 17 / 26 Foundations of Special Relativity Relativity of simultaneity

Einstein postulates retain Galilean invariance there is no experiment that can detect a uniform state of motion

However Galilean transformation rule must be changed

Since light travels driven by Maxwell’s equations transformation law between inertial observers must preserve Maxwell’s equations

Actually it is even more general than that: we will have a set of transformations that leave c invariant

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 18 / 26 Foundations of Special Relativity Lorentz transformations

Recall that... When relatively moving observers label event all observers must use same two light rays

Any event is characterized uniquely by two light rays that pass through it

All observers ﬁnding labels of particular event MUST use same transmitted and received rays

This apparent coincidence implies that all observers agree: 1 on speed of light

2 that intersection of two light rays deﬁnes event

3 indeed unique label of event

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 19 / 26 Foundations of Special Relativity Lorentz transformations Example Vinnie and Brittany share same origin to coordinatize same event Transverse coordinates are same used to construct clocks

Red event is coordinatized by: Brittany as (xB, tB) Vinnie as (xV, tV)

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 20 / 26 Foundations of Special Relativity Lorentz transformations Lorentz transformations are relationship between (x , t ) and (x , t ) § B B V V ¤ ¦ ¥

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 21 / 26 Foundations of Special Relativity Lorentz transformations

Start by ﬁnding coordinates of Vinnie’s proper times τ10 and τ20 in terms of coordinates of red event in Brittany’s reference frame

Event τ10 has form (vt1, t1) in Brittany’s coordinates since it is on Vinnie’s time axis and he is moving at a speed v with respect to her

This event is also on light ray with red event

The equation of that light ray is

x x = c(t t ) (12) − B − B

Putting in coordinates of event τ10 vt x = c(t t ) (13) 1 − B 1 − B

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 22 / 26 Foundations of Special Relativity Lorentz transformations

Solving for t1 ct x t = B − B (14) 1 c v − Because of time dilation

p 2 2 τ0 = t 1 v /c (15) 1 1 − Combining these r 2 ctB xB v τ0 = − 1 (16) 1 c v − c2 − Similarly for event τ0 2 r v2 τ0 = t 1 (17) 2 2 − c2 and r 2 ctB + xB v τ0 = 1 (18) 2 c v − c2 −

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 23 / 26 Foundations of Special Relativity Lorentz transformations Substituting (16) and (18) into the deﬁnitions (1) and (2)

xB vtB xV = − = γ(xB vtB) √1 v2/c2 − yV = yB −

zV = zB 2 tB vxB/c 2 tV = − = γ(tB vxB/c ) (19) √1 v2/c2 − − Lorentz transformations if transverse directions are unaffected by velocity transformation

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 24 / 26 Foundations of Special Relativity Lorentz transformations Homework hint: exercise 5.3

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 25 / 26 Foundations of Special Relativity Lorentz transformations Homework hint: exercise 5.5

L. A. Anchordoqui (CUNY) Modern Physics 9-17-2015 26 / 26