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Notes on Relativity

Disclaimer: this document is not to be taken as a full depository of concepts,

methods and explanations on the subject, but merely a summary. Lectures will be

based on those notes, but consequent complementary materials and explanations

that will be covered in class do not appear in this document. To state it plainly, if

you only study from those notes without reading the book and taking

complementary notes in class, you can most likely kiss goodbye to any passing

grade.

(thats a heck of a diplomatic statement isnt it?)

I like it to...:)

Herve Collinherve@hawaii.edu

Kapiolani Community CollegeMath and Sciences Department

Created by LATEX

August 24, 2006

2

Contents

1 Special Relativity 51.1 Galileo Transformation . . . . . . . . . . . . . . . . . . . . . . 51.2 Maxwells Equations . . . . . . . . . . . . . . . . . . . . . . . 71.3 Michelson-Morley experiment . . . . . . . . . . . . . . . . . . 91.4 Einsteins postulate . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . 111.6 Length contraction . . . . . . . . . . . . . . . . . . . . . . . . 121.7 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8 Consequence of time dilation: Doppler effect . . . . . . . . . . 161.9 Transformation of Velocities . . . . . . . . . . . . . . . . . . . 181.10 Relativistic momentum . . . . . . . . . . . . . . . . . . . . . . 191.11 Mass and Energy . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 General Relativity 252.1 Principle of Equivalence . . . . . . . . . . . . . . . . . . . . . 252.2 Gravitational Red Shift . . . . . . . . . . . . . . . . . . . . . . 262.3 Schwarzschild radius and Black Holes . . . . . . . . . . . . . . 282.4 Gravitational lensing . . . . . . . . . . . . . . . . . . . . . . . 282.5 Space-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Appendix 313.1 ... divergence and curl . . . . . . . . . . . . . . . . . . . . . . 31

3

4 CONTENTS

Chapter 1

Special Relativity

1.1 Galileo Transformation

The principle of relativity was not new when Einstein published his specialrelativity in 1905. In fact, this principle was thought to be well understood.Indeed, one can experience the concept of relativity in everyday life. For in-stance, when being stationary, a cars velocity passing by can be measured atsay, 30 miles an hour. If the observer is now moving in the same direction ofthe car at a speed of 20 miles an hour, the velocity measured by the movingobserver would then be 30 20 or 10 miles an hour. This simple example isbased on the Galileo transformation.

Galileo Transformation in the x direction:

x = x vt (1.1)y = y (1.2)

z = z (1.3)

Inverse Galileo Transformation in the x direction:

x = x + vt (1.4)

y = y (1.5)

z = z (1.6)

In this transformation, t = t as Newton considered time to be absolute.The concepts of space and time are entirely separable, and time is consideredan absolute quantity in Galilean transformation. To illustrate the above

5

6 CHAPTER 1. SPECIAL RELATIVITY

transformation, imagine two inertial reference frames S and S which movealong their x axis with uniform relative velocity v with respect to each other(Figure 1.1). It is indeed clear that if the position of an object measured byan observer in the system S is x, the position of the same object measuredby an observer in the system S is x + vt.

Figure 1.1: System S moving with a uniform velocity v with respect to system S

Newtons laws are in fact invariant under this transformation; that is,they have the same form in both systems S and S when those systems movewith a constant velocity with respect to each other. Using the equation (1.4),the first and second derivate of position with respect to time becomes:

x = x + vt (1.7)

x = x + v (1.8)

x = x (1.9)

so Newtons law becomes:

F = mx (1.10)

F = mx (1.11)

F = F (1.12)

When applying the Galilean transformation to an electromagnetic wave,it is also clear that its speed depends on the reference frame, and shouldhave different measurement values depending on which reference frame theobserver is in. Experimentaly however, this has never been verified.

1.2. MAXWELLS EQUATIONS 7

1.2 Maxwells Equations

Motivation for a new transformation: E&M theory (Maxwells equations) isnot invariant under Galileo transformation. Recall that Maxwells equationsare composed of Gausss law for the electric and magnetic field, Ampereslaw and Faradays law:

S

~EdA =1

0Qinside (1.13)

S

~BdA = 0 (1.14)C

~E.d~l = ddt

S

~BdA (1.15)C

~B.d~l = 0I + 00d

dt

S

~EdA (1.16)

In fact, Maxwells equations can be rewritten in a much elegant mannerusing the divergence and curl notation. In free space, Maxwells equationsbecome:

. ~E = 0 (1.17). ~B = 0 (1.18)

~E = d~B

dt(1.19)

~B = 00d ~E

dt(1.20)

Please note that there are 8 Maxwells equations: the last two equationsare in three dimensions (curl). One of the powers of Maxwells equation isthe prediction of a constant speed for the velocity of light which was thoughtto be a wave at that time. Indeed, what can follow from the Maxwells equa-tions is the wave equation (either in term of the electric or the magneticfield), which involves a constant speed value of 1

00. Lets derive it to ease

our mind for the electric field in the x direction. Lets take the curl of bothsides of equation (1.19), using: ~A = (. ~A)2 ~A (see Appendix A).

we get:

(. ~E)2 ~E = (d~B

dt)

8 CHAPTER 1. SPECIAL RELATIVITY

The curl operator acts on the position coordinates only, and is thereforeindependent of time. Hence, when the order in which the curl and the timederivative act upon a quantity does not matter one can switch their order.The derivation can be finalized by using the first Maxwells equation (1.17)to drop one term, and the fourth Maxwells equation to expand another; oneobtains:

2 ~E = t

(00 ~E

t)

where 0 and 0 are called respectively the permittivity and permeabilityof free space and are constant quantities. Hence, they can be taken out ofthe time derivative. After rearranging the terms:

2

x2~E 00

2

t2~E = 0

This is the fonctional form of the wave equation for the electric field infree space in the x direction, which travels with velocity 00, defined as:00 =

1c2

, or:

c =1

00

(1.21)

A common form of the wave equation, traveling in the x direction in freespace for the electric field found in literature is:

2

x2~E 1

c22

t2~E = 0 (1.22)

Obviously, if Maxwells equation predicts a constant speed for light in allreference frames, either Galileo transformation is not adequate or Maxwellsequations need to be revised. However before Einstein was about to pub-lish his special relativity, both theories had so many experimental evidencesto support them that a puzzle was growing annoyingly large in physicistsminds. In the attempt to fit Galileo transformation to this paradigm, thequest for ether had started. The same way as sound waves need a medium topropagate, it was thought at that time that light needed a similar mediumto propagate: the ether. The Michelson-Morley experiment was one of suchattempt: reveal the existence of ether. All experiments performed were how-ever unsuccessful.

1.3. MICHELSON-MORLEY EXPERIMENT 9

1.3 Michelson-Morley experiment

If light needs a medium to propagate we should be able to measure it. Thefirst attempt was made by Michelson and Morley with a very clever setupwhich was expected to reveal fringes due to the interference of two incominglight beams delayed by a phase shift produced by the different paths thebeam took with respect to the ethers velocity.

Figure 1.2: Michelson Morley experiment

Homework :Derive the path difference between the two incoming beams on thedetector in the Michelson-Morley setup. The path difference isdefined as: d = ct. So really, the purpose of the game is toobtain t.Proceed as follow:Lets define distance OA and OB as L (only consider Path I andPath II; that is, the upper right quadrant of the above diagram)and lets assume that the speed of the photon is c.

10 CHAPTER 1. SPECIAL RELATIVITY

Path I: calculate the time needed for a photon to travel From O toA (tU) and back from A to O (tV ), considering the ether wind v(use Pythagores theorem). Add those two times and define it ast1. Express t1 in terms of (1 v

2

c2)1/2.

Path II: calculate the time needed for a photon to travel From Oto B (tR) and back from B to O (tL), considering the ether windv. Add those two times and define it as t2. Express t2 in terms of(1 v2

c2)1.

Compute t as t = t2 t1 and approximate your quantity tusing the Binomial Theorem for Non-integral Exponents, stated asfollow:

(1 + x) = 1 + x +( 1)

2!x2 +

( 1)( 2)3!

x3 + ... (1.23)

and apply it to the two terms in t that involves 11 v2

c2

and 11 v2

c2

assigning v2c2

to the x variable in the Binomial expansion. Keeponly up to the second order of the expansion (two terms). Youshould obtain: d = Lv

2

c2.

This path difference d has never been found experimentally.

1.4 Einsteins postulate

1. The principle of relativity: The laws of physics are the same in allinertial systems. There is no way to detect absolute motion, and nopreferred inertial system exists.

2. The speed of light is constant: Observers in all i

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