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!!!!!!!!Solutions Manual For Special Relativity A Heuristic Approach Sadri Hassani

Contents

List of Symbols 1

1 Qualitative Relativity 5

2 Relativity of Time and Space 9

3 Lorentz Transformation 19

4 Spacetime Geometry 49

5 Spacetime Momentum 75

6 Relativity in Four Dimensions 85

7 Relativistic Photography 99

8 Relativistic Interactions 127

9 Interstellar Travel 139

10 A Painless Introduction to Tensors 151

11 Relativistic Electrodynamics 157

12 Early Universe 167

A Maxwell's Equations 177

B Derivation of 4D Lorentz transformation 181

C Relativistic Photography Formulas 185

List of Symbols, Phrases, and Acronyms

^aunit vector in the direction of~a anti-particlea particle whose mass and spin are exactly the same as its corresponding particle, but the sign of all its \charges" are opposite. If a particle is represented by the letterp, then it is customary to denote its anti-particle by p. If a particle is represented by the letterq(orq+), then it is customary to denote its anti-particle byq+(orq). asarcsecond; an arcsecond is an angle 1=3600 of a degree. baryona hadron whose spin is an odd multiple of~=2. Baryons are composed of three quarks. Examples of baryons are protons and neutrons. fractional velocity of one observer relative to another,~=~v=c bosona particle whose spin is an integer multiple of~. All gauge particles are bosons as are all mesons, as well as the Higgs particle. causally connectedreferring to two two events. If an observer or a light signal can be present at two events, those events are said to be causally connected. causally disconnectedreferring to two two events. If an observer or a light signal cannot be present at two events, those events are said to be causally disconnected.

CBRCosmic Background Radiation

CMcenter of mass

CScoordinate system

ex;^ey;^ezunit vectors along the three Cartesian axes. EMelectromagnetic or electromagnetism, one of the four fundamental forces of nature. equilibrium temperaturetemperature of the universe at which matter and radiation densities are equal eVelectron volt, unit of energy equal to 1:61019J

2 List of Symbols

fermiona particle whose spin is an odd multiple of~=2. Fermions obey Pauli's exclusion principle: no two identical fermions can occupy a single quantum state. Electrons, protons, and neutrons are fermions, so are all leptons and quarks, as well as all baryons. the Lorentz factor, = 1=p12= 1=p1(v=c)2 gauge bosonsAccording to the modern theory of forces, fundamental particles interact via the exchange of gauge bosons. Excluding gravity, whose microscopic behavior is not well understood, there are 12 gauge bosons whose exchange explains all the interactions:Z0,Wand (photon) are responsible for electroweak interaction, while 8 gluons are responsible for strong interaction. gluonsthe particles responsible for strong interactions: two or more quarks participate in strong interaction by exchanging gluons. There are four gluons, which with their antiparticles comprise the eight gluons whose exchange binds quarks together. GTRgeneral theory of relativity; the relativistic theory of gravity.

Gyrgigayear, equal to 109years

hadrona particle capable of participating in strong nuclear interactions. Examples of hadrons are protons, neutrons and pions. All hadrons are made up of quarks and/or anti-quarks. half lifethe time interval in which one half of the initial decaying particles survive.

LAVLaw of Addition of Velocities

leptona particle that participates only in electromagnetic and weak nuclear interactions, but not in strong nuclear interactions. Leptons are elementary particles in the sense that they are not made up of anything more elementary. There are three electri- cally charged leptons: electron, muon, and tauon. Each charged lepton has its own neutrino. So, altogether there are six leptons.

LHCLarge Hadron Collider

light cone(at an eventE) The set of all events that are causally connected toE. light hourthedistancethat light travels in one hour,1:081012m light minutethedistancethat light travels in one minute,1:81010m light secondthedistancethat light travels in one second,3108m lightlikereferring to two events, whenct= xor (s)2= 0. luminally connectedreferring to two two events. If a light signal can be present at two events, those events are said to be luminally connected. lylight year; one light year is 9:4671015m. mean timethe time interval in which 1=eof the initial decaying particles survive.

List of Symbols 3

mesona hadron whose spin is an integer multiple of~. Mesons are composed of one quark and one anti-quark. Examples of mesons are pions. MeVmillion electron volt, unit of energy equal to 1:61013J mmicrometer = 106m Minkowskian distancealso called \spacetime distance," (s)2= (ct)2(x)2 is an expression involving the coordinates of two events which is independent of the coordinates used to describe those events. MM clocksometimes called \light clock" is described on page 13.

MpcMegaparsec

muonan elementary particle belonging to the group of particles named \leptons," to which electron belongs as well. Muon is called a \fat electron" because it behaves very much like an electron except that it is heavier. neutrinoa neutral lepton with very small mass. Neutrinos participate only in weak nuclear force. That's why they are very weakly interacting. nsnanosecond or 109s ParsecA distance of about 3.26 light years. One parsec corresponds to the distance at which the mean radius of the Earth's orbit subtends an angle of one second of arc. positronthe anti-particle of the electron quarkselementary particles which make up all hadrons. There are six quarks: up, down, strange, charm, bottom, top. Quarks participate in all interactions, in particular, the strong interaction.

RFreference frame

spacelikereferring to two events, whenct STRspecial theory of relativity tauonan elementary particle belonging to the group of particles named \leptons," to which electron belongs as well. It is the heaviest lepton discovered so far. timelikereferring to two events, whenct >xor (s)2>0.

CHAPTER1Qualitative Relativity

Problems With Solutions

1.1.A rod of lengthLemits light from all of its points simultaneously (in its rest frame)

when a remote switch is turned on. Its center is on thex-axis and is moving on the axis in a plane parallel to a very large photographic plate and innitesimally close to it. When it reaches the middle of the plate, the switch is turned on. (a) Compare th elength Ljjof the image on the photographic plate withLwhen the rod is along thex-axis:Ljj> L,Ljj=L, orLjj< L? Give a reason for your answer. (b) Compare the length L?of the image on the photographic plate withLwhen the rod is perpendicular to thex-axis:L?> L,L?=L, orL?< L? Give a reason for your answer.

Solution:

(a) Length parallel to the direction of motion shrinks regardless of who sees the ev ents of light emission simultaneously. See the discussion in Section 1.3 for the reason (as well as how to capture the length of a moving object). (b) Length p erpendicularto the dir ectionof motion do esnot c hange. Note the importance of the fact that the distance between the rod and the photographic plate is zero.

1.2.A rod is placed along thex-axis with its center at the origin. A pinhole cameraC1is

located on thez-axis and takes a picture of the stationary rod. Now the rod starts moving along thex-axis parallel to itself from1. CameraC1is removed and another pinhole cameraC2replaces it on thez-axis. As soon as the center of the rod reaches the origin (call itt= 0),C2takes a picture. (a) Is the pinhole of C2collecting the light rays from the two ends of the rod that were emitted att= 0?

6 Qualitative Relativity

(b) Is the pinhole collecting the ligh tra ysfrom the t woends of the ro dthat w ereemitted simultaneously, but not att= 0? (c) If the answ erto (b) is no, whic hend emitted its ligh trst, the trailing end or th e leading end? (d) Is it p ossiblefor the image of the ro di nC2to belongerthan its image inC1? Hint: Consider the location of each end as it emits the light ray captured byC2.

Solution:

(a) No. It tak estime for the ligh tto reac hthe camera once it lea vesits source. (b) No. (c) The trailing end is farther a wayfrom the camera, so i tm ustemit the ligh tso oner than the leading end. (d) The trailing end emits its ligh t,the ro dmo vesa little, then the leading end emits its light. So, the distance between the source of the light from the trailing end and that of the leading end is indeed larger than the length of the rod. Note that the image in cameraC2, which is longer than the image inC1, has nothing to do with the actual length of the rod!

1.3.A rod is placed along they-axis with its center at the origin. A pinhole cameraC1is

located on thez-axis and takes a picture of the stationary rod. Now the rod starts moving along thex-axis parallel to itself from1. CameraC1is removed and another pinhole cameraC2replaces it on thez-axis. As soon as the center of the rod reaches the origin (call itt= 0),C2takes a picture. (a) Is the pinhole of C2collecting the light rays from the two ends of the rod that were emitted att= 0? (b) Is the pinhole collecting the ligh tra ysfrom the t woends of the ro dthat w ereemitted simultaneously, but not att= 0? (c) If the answ erto (b) is no, whic hend emitted its ligh trst, the top or the b ottom? (d) Is it p ossiblefor the image of the ro din C2to be longer than its image inC1? Hint: Consider the locations of the ends as they emit their light rays captured byC2, the distance between those locations and the pinhole, and the angle they subtend at the pinhole.

Solution:

(a) No. It tak estime for the ligh tto reac hthe camera once it lea vesits source. (b) Y es.The p erpendiculardistance do esnot c hange.So, the top and b ottomof the ro d are equidistant from the pinhole, and to reach it att= 0, the rays must have been emitted at the same time in the past. (c)

The answ erto (b) is y es!

7 (d) No. Since the lo cationsof the sources of the ra ysare farther from C2(they have negativex-coordinates) than the locations inC1, they must have asmallerimage. Chapter 7 discusses this in gory mathematical detail!

1.4.A circular ring emits light from all of its points simultaneously (in its rest frame) when

a remote switch is turned on. It is moving in a plane parallel to a photographic plate and innitesimally close to it. When it reaches the plate, the switch is turned on. What is the shape of the image of the photograph? Hint: See Problem 1.1. Solution:The diameter along the direction of motion shrinks; the diameter perpendicular to the direction of motion stays the same. So, the shape is an ellipse attened in the direction of motion.

1.5.A conveyor belt moving at relativistic speed carries cookie dough. A circular stamp

cuts out cookies as the dough rushes by beneath it. What is the shape of these cookies?

Are they

attened in the direction of the belt, stretched in that direction, or circular? Solution:The answer is identical to that of the previous problem. So, the cookies are attened in the direction of the belt.

1.6.A conveyor belt moving at relativistic speed carries cookie dough. A laser gun one

meter above the belt emits a beam in the shape of the surface of a circular cone that cuts the dough perpendicularly. Are these cookies attened in the direction of the belt, stretched in that direction, or circular? Hint: Concentrate on the two ends of the diameter of the beam along the dough, and note that their light beams arrive simultaneously at the stationary bed on which the dough is moving. Now consider how the two events appear in the RF of the moving dough and what implication it has on the length of the diameter. The discussion surrounding Figure 1.3 may be helpful. Solution:The image of the laser beam on the stationary bed is circular and the two events of the arrival of the beams from the two ends of the horizontal diameter occur simultaneously. Consider two experiments. In the rst experiment, there is no dough and the laser imprints a circle on the stationary bed. In the second experiment, an observer riding with the dough records the coincidence of the location of the event in front of her with the front end of the imprintbeforethe coincidence of the event in the back. She concludes that for her, the distance between the two events islargerthan the two marks on the stationary bed. So, the image is an ellipseelongatedalong the direction of the belt. Thus, the cookies are stretched in the direction of the belt.

1.7.A circular ring is centered at the origin in thexy-plane. A pinhole cameraC1is located

on thez-axis and takes a picture of the stationary ring. Now the ring starts moving along thex-axis from1. CameraC1is removed and another pinhole cameraC2replaces it on thez-axis. As soon as the center of the rod reaches the origin (att= 0),C2takes a picture. (a) Is the pinhole of C2collecting the light rays from the two ends of the horizontal diameter (along thex-axis) of the ring that were emitted att= 0? Hint: Look at

Problem 1.2.

(b) Is the pinhole of C2collecting the light rays from the two ends of the horizontal diameter of the ring that were emitted simultaneously, but not att= 0?

8 Qualitative Relativity

(c) If the answ erto (b) is no, whic hend emitted its ligh trst, the trailing end or th e leading end? (d) Is it p ossiblefor the image of the horizon taldiameter in C2to belongerthan its image inC1? (e) Is the image of the v erticaldiameter (along the y-axis) inC2equal to, longer than, or shorter than its image inC1? Hint: Look at Problem 1.3. (f) Can y ougues swhat the shap eof the image of the ring i sin C2?

Solution:

(a) No. (b) No. (c)

The trailing end.

(d)

Y es,it alw aysis.

(e)

It is shorter than its image in C1.

(f) It is an ellipse elongated along the direction of m otion. See Example 2.2.6 for a quantitative analysis of this problem.

CHAPTER2Relativity of Time and Space

Problems With Solutions

2.1.Take the most rigid rod you can nd, and hit one end of it with the hammer. The

rodas a wholestarts to move because it is rigid.1Actually not! It takes time for the information that one end of the rod was hit to reach the other end, because of Note 2.1.7. Now go to the rest frame of the rod which is now moving relative to the hammer. Assume that the hammer hits the rod in such a way as to cause (the front end of) it to stop. But the other end knows nothing about the hammer yet. So, it keeps moving! What does this say about the concept of \rigidity" in relativity? Solution:Rigidity is not a well dened concept in relativity. The other end of the rod gets compressed because of its motion.

2.2.In this problem you'll learn more about superluminal transverse speeds.

(a) Sho wthat the angle that maximizes Equation (2.6) is giv enb ycos =. (b)

Substitute this in (2.6) to obtain ( vtr)max=c

(c)

Sho wth at( vtr)maxis larger thancfor any >1=p2.

(d) What sp eedmak es( vtr)maxten times faster than light? What is the angle corre- sponding to this speed?

Solution:

(a)

Set the deriv ativeof vtr

v

0tr() =c(cos)(cos1)2

equal to zero to get cos=. The second derivative test shows thatv00tr()<0 when cos=. Therefore,vtris indeed maximum at cos=.1 If you hit one end of a slinky, the other end does not move, at least not immediately.

10 Relativity of Time and Space

(b) (vtr)max=csin1cos=cp1212=cp12=c (c) c > c()2

2>1()2>12()2>1=2:

(d) = 10()2

2= 100()2= 1001002()2= 100=101;

or if= 0:995. The angle corresponding to thisis= cos10:995 = 0:0997 or = 5:71.

2.3.Consider an MM clock moving horizontally with speedrelative to observerO. Denote

its length in motion byLand at rest byL0. Let t1be the time it takes light to go from the emitter to the mirror according toO. Let t2be the time it takes light to go from the mirror to the emitter according toO. (a)

Sho wthat

ct1=L+ct1; ct2=Lct2: (b)

Sho wthat a \tic k"according to Ois

t= t1+ t2=2L=c12: (c) No wuse the time d ilationform ulawith = 2L0=cto derive the length contraction formula.

Solution:

(a) By the time the ligh tthat is emitted at the emitter reac hesthe mirror, the MM clo ck has moved. So, the distance that the light covers isLplus the distance that the MM clock moves in the same time interval. So,ct1=L+ct1, and ct1(1) =L()t1=Lc(1): On re ection, the light and the MM clock move in opposite directions. Therefore, ct2=Lct2, and ct2(1 +) =L()t2=Lc(1 +): (b)

A tic kaccording to Ois therefore,

t= t1+ t2=Lc(1)+Lc(1 +)=2L=c12:

But t=

(2L0=c). Thus, (2L0=c) =2L=c12=

2(2L=c)()L=L0=

=L0p12: 11

2.4.The spaceship Enterprise goes to a planet in a star system far away with a speed of

0:9c, spends 6 months on the planet, and comes back with a speed of 0:95c. The entire trip

takes 5 years for the crew. (a) Ho wfar is the p lanetaccording to Earth observ ers? (b) Ho wl ongdid it tak ethe crew to get to the planet? (c) Ho wlong did the en tiretrip tak efor the Earth observ ers?

Solution:

(a) On the outb oundpart, the distance for th ecrew is L0p10:92and the time it takes them to get there isL0p10:92=(0:9c), whereL0is the distance according to Earth observers. Similarly, the time it takes them to come back isL0p10:952=(0:95c).

So, the entire round trip time is

L

0p10:920:9c+L0p10:9520:95c= 4:5 years:

This givesL0= 5:54 light years.

(b) outbound=L0p10:920:9c= 0:484 years: (c) The distance according to Earth is 5.54 ligh ty ears.So, for outb oundtrip toutbound=5:54 light years0:9c= 6:15 years:

Similarly,

tinbound=5:54 light years0:95c= 5:83 years: Therefore the entire trip takes 6:15 + 0:5 + 5:83 = 12:48 years.

2.5.A rocket ship leaves the Earth at a speed of 0:8c. When a clock on the rocket says 1

hour has elapsed, the rocket ship sends a light signal back to Earth. (a) According to Earth clo cks,when w asthe s ignalsen t? (b) According to Earth clo cks,ho wlong after the ro cketleft did the signal arriv ebac k on Earth? (c) According to the ro cketclo ck,ho wlong after the ro cketleft did the sign alarriv ebac k on Earth?

Solution:

(a) The ro cketis measuring the prop ertime of 1 hour. So, f orEarth t= =1 hourp10:82= 1:67 hours:

12 Relativity of Time and Space

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