Chapter 5. Relativity

A special and important theory that is not only for objects moving at a high speed but also for revealing ( 揭示 ) the sources of energies. This theory changes our conception to space and time.

New terms related to this chapter

1. Electromagnetic (theory) field ( 电磁（理论）场）

2. Special (general) relativity ( 狭义（广义）相对论 )

3. Transformation ( 变换 ), radiation ( 辐射 ) , collapse ( 倒塌 )

4. Postulates , hypothesis, assumption, ( 假设 )

5. Spaceship ( 飞船 ), missile ( 导弹 ) ， patch

6. Simultaneous ( 同时的 ), synchronized （同步的） relativistic ( 相对论的 ), absolute, relative, classical, modern

7. Dilation (expansion) ( 时间膨胀 ) ， contraction (shrinkage) ( 收缩 ), conception, outlook ( 观点 ), reception, receipt

8. Thermodynamics ， electrodynamics, quantum mechanics, statistical physics, optics,

Introduction to relativity

1. Physics theory at the end of 19th century (Newtonian mechanics, thermodynamics, statistical physics, theory of electro-magnetic field).

2. Speed of light predicted in Maxwell’s theory.

3. Problem: which inertia frame of reference is the right one for the speed of light.

4. two patches of “clouds” in the physics sky.

5. Special relativity and general relativity

Generally speaking, relativity contains two parts, one is called special relativity and the other is called general relativity. The former describes the phenomena of objects moving at a very high velocity and the latter explains the behavior of objects moving in a strong gravitational field.

5.1 Galileo transformation • The absolute outlook of space-time of classical

mechanics and the classical relativity principle• Galileo transformation.

tt

zz

yy

utxx

x x´

y y´

• P

y y´

O O´

utx´

x

A frame of reference S moves with velocity u relative to a frame S.

Speed and acceleration of the particles can be obtained by the differentiation with respect to the equation of motion.

zz

yy

xx

zz

yy

xx

aa

aa

aa

adt

dv

vv

vv

uvv

This explains that Newton’s mechanical laws are identical in all inertia reference frames. Problem EM

amamF

5.2 Lorentz transformation

5.2.1 Tow postulates:

1. The relativity principle: All the laws of physics have the same form in all inertial frames of reference (no absolute reference frame exists)

2. Constancy of the speed of light: Light propagates through empty space with a definite speed c, independent of the speed of the source or observer.

5.2.2 Lorentz transformation

)(

1

)(1

2

2

2

2

22

xc

ut

cu

xcu

tt

zz

yy

utxcu

utxx

Time and space are no longer independent.

2

2

1

1

cu

)(

)(

2x

c

utt

zz

yy

tuxx

The reversal ( 逆转 ) relations between the two frames of reference are given below:

The speed of an object after Lorentz transformation can be found by differentiation with respect to time.

Note that

)(),(

2dx

c

udttdudtdxxdor

td

dt

dt

dx

td

dx

dt

dxv

td

xdv xx

The following relations are obtained:

2

2

2

2

2

22

11

,11

,1

c

u

vcuv

v

c

u

vcu

vv

vcuuv

v

x

zz

x

yy

x

xx

2

2

2

2

2

22

11

,11

,1

c

u

vcuv

v

c

u

vcu

vv

vcuuv

v

x

zz

x

yy

x

xx

The reversal velocities are:

Example 5-1. A spaceship moving away from earth with a speed 0.9c fires a missile in the same direction as its motion, with a speed of 0.9c relative to the spaceship. What is the missile’s speed relative to earth?

Solution: Let the earth’s frame of reference be S, the spaceship’s S´. Then vx´=0.9c and u = 0.9c

The non-relativistic velocity addition formula would give a velocity relative to the earth of 1.8c. The correct relativistic result can be obtained from:

cccc

cc

vcuuv

vv

cuuv

v

x

xx

x

xx 994.0

/)9.0)(9.0(1

9.09.0

112

22

When u is less than c, a body moving with a speed less than c in one frame of reference also has a speed less than c in every other frame of reference. This is one way of thinking that no material body may travel with speed greater than that of light, relative to any inertial frame of reference.

5.3 Special relativistic conception of space-time

5.3.1 Time dilation ( 时间延迟 , 时间膨胀 )

Moving clocks run slower than clocks at rest, an effect of relativity known as Time dilation. This indicates that intervals of time are not absolute but are relative to the motion of the observers. If two identical clocks are synchronized ( 同步的 ) and placed side by side in an inertial frame of reference they will read the same time as long as they both remain side by side. However, if one of the clocks has a velocity relative to the other, which remains beside a

stationary observer, The traveling clock will show, to that observer, that less time has elapsed than the stationary clock. If two events ((x1, t1 ), (x2, t2 )) happen in S in the same place but at different time. What is the difference between the time intervals in the two relative moving frames S and S? It is easy to find that

Where 0 is called proper time ( 固 有时 ) which is less than the time interval observed by the stationary observer. This is so called “the moving clock runs slower than it is at rest”.

0012

1221212

tt

xxc

utttt

Example 5-2 A spaceship passes the earth at a speed of 108 m/s. The time interval between two events that take place at the same point on the spaceship is 100 seconds. What is the time interval between the events according to the observers recorded on the earth. Solution: Begin by solving for v/c = 1/3, so

ss

cv106

9/11

100

/1 22

0

5.3.2 Lorentz contraction ( 洛伦兹收缩）

Moving objects measures shorter than when it is at rest, an effect of relativity known as Lorentz (or length) contraction. Note that Length contraction occurs only in that dimension of an object parallel to its direction of motion.

As before, a frame of reference S moves with velocity u relative to a frame S.

x x´

y y´

O O´

ut x´1 x´2

L0

120 xxL

utxx

1 02

2

00

12120

Lc

uL

LL

LxxxxL

Moving ruler becomes shorter!

P1 P2

The ends of the ruler passes P1 and P2 at the same time (t1=t2)

Example 5-3 A spaceship moves past the earth at a relative speed of 1.0 108 m/s. A 4.0 m length of pipe on board the spaceship is parallel to the spaceship’s direction of velocity. Calculate the length of pipe as recorded on the earth.

Solution: Directly from the above equation,

mcvLL 77.39/110.41 220

Examplem 5-4 A rigid object on a moving spaceship is parallel to the direction of motion. Its length, as measured on the spaceship, is 3.00 m. When measured by observers on the earth, its length is 2.598 m. Calculate the relative speed of the spaceship.

Solution: This problem belongs to the Lorents contraction. So we have

20

2

02

2

11 LLcvL

L

c

v

5.3.3 The relativity of simultaneity ( 同时的相对性 )

There two events ((x1, t1 ), (x2, t2 )) happen in S at the same time but in different places. Using the formula

)(2x

c

utt

0 122

1221212

xxc

u

xxc

utttt

Therefore,

The two events happened simultaneously ( 同时的 ) in S but not in S frame.

x

c

utt

2

Something happens in S’ at the same time, but to the observer in S, they will not happen simultaneously. While 0t

xc

ut

2

5.3.4 Causality ( 因果关系 ) and signal speed

In S frame, two events P(xP, tP) and Q(xQ, tQ) has causality, one happened after the other.

0 PQ tttThe propagating speed in S is

t

x

tt

xxv

PQ

PQs

In S´ frame,

svc

ut

t

x

c

utx

c

utt

2

22

1

1

As vs < c, t´> 0. Therefore, causality is unchanged.

5.4 Relativistic Mechanics

In relativity, the concepts of mechanics are facing redefinition. However, this redefinition has to be satisfied with the correspondence principle, that is, when v << c, the redefined physical quantities have to be their classical corresponding physical quantities. On the other hand, the conservational laws are kept valid as much as possible.

5.4.1 Momentum and mass-speed relation

In order to satisfy the conservation of momentum and to hold the correctness of Lorentz transformation, the momentum can still have the save form:

vmp

But mass m must be a function of velocity and isgiven by

22

0

/1 cu

mm

This is a very important mass-velocity equation in relativity. It reveals the relation between the mass and velocity for an object in motion. When the velocity of the object approaches to the light speed, its mass will approach infinity and for an object with infinity mass, it is impossible to accelerate it. This also explains that the light speed is the upper speed limit for all objects.

vmp

0When u<<c, the relativistic momentum is very closely equal to the classical result.

Example 5-5 An electron with a rest mass of 9.11 x 10-31 kg moves at a very high speed in a linear accelerator. The relativistic mass of the moving electron is 12.22 x 10-31kg. Calculate the speed of the electron.

Solution: Using the relativistic mass equation:

2

022022 1 1

m

mcv

m

mcv

smmmcv /1000.2)(1 820

5.4.2 Force and kinetic energy

1. Force Newtonian mechanics:

amdt

vdmvm

dt

d

dt

pdF

Relativistic mechanics:

vmdt

d

amdt

vdmvm

dt

d

dt

pdF

0

2. Kinetic energy

using relativistic force and classical idea of mechanics, the kinetic energy could be the work by external force from stationary to the state in motion. Therefore, we have

vv

vss

k

cv

cmd

cv

vmdv

vvmdrdvmdt

drdFE

022

20

022

0

000

11

20

2

2022

20

1

cmmc

cmcu

cmEk

This is the relativistic point mass kinetic energy. This is very important formula in relativity.

3. Mass-energy relation

02

022

202

1Ecm

cu

cmmcE

This relation not only reveals the relation between mass and energy but also unifies the mutually independent two conservational laws of mass and energy.

420

222 cmcpE

4. Energy-momentum relation:

22

2242022

2222222

22222422

/1

)/1(

)(

)(

cv

cvcmcp

vccmvcm

vvccmcmE

This relation predicts the possibility of the existence of zero-mass particles and it can be shown that these zero- mass particles are moving with the speed of light.

mcp So the total energy could be written in terms of the momentum p or in terms of the kinetic energy.

Example 5-6 An particle with a rest mass of 2.5 x 10-28 kg moves at a speed v = 0.8c. What is its kinetic energy?

Solution: Using the relativistic mass equation:

kgcv

mm 28

22

0 101.4/1

Thus its KE is

)(/104.1)( 221120 JsmkgcmmKE

5.5 Introduction to General relativity

1.  In 1915, Einstein proposed the equivalent principle based on the assumption that the inertia mass and gravitational mass are equivalent and founded the general theory of relativity. Einstein extended his earlier work to include accelerated system, which led to his analysis of gravitation. He interpreted the universe in terms of a four-dimensional space-time continuum in which the presence of mass curves space in such a way that the gravitational field is created. This explains that the mass curves space or the presence of gravitational field will also curve space.

5.5.1 The two hypotheses of general relativity

1. Equivalent principle:

For all physical processes, the reference frame with uniform acceleration is equivalent to the local region of gravitational field and the inertial force is equivalent to the local region of gravitation.

2. General relativity principle:

Physics law has the same form in all reference frames, no matter inertial or non-inertial.

5.5.2 The characteristics of space-time in gravitational field

1. Light is curved in gravitational field.

2. Space bend

The three-dimensional space used in Newton’s mechanics and the four-dimensional space used in special relativity are Euclidian space ( 欧几里得空间 ). Light traveling linearly in such a space can be considered as the result of the characteristics of the level-straight space ( 平直空间 ).

c

ABt 0

c

ABt

0tt is called time dilation caused by gravitational filed.

3. Time dilation effect in gravitational field

A B

Curved space

Straight space

In gravitational field, light rays are curved. This is determined by the characteristics of space-time in gravitational field which curves the four-dimensional space.

4. Gravitational collapse ( 坍缩 ) and black hole

Gravitational collapse is the phenomenon in the process of star evolution ( 演化 ) while black hole ( 黑洞 ) is a strange star with infinite density.

High density stars can be divided into three kinds: white dwarf star ( 白矮星 108kg/m3), Neutron star ( 中子星 107 kg/m3) and black hole.

5. Gravitational wave

An accelerating electrical charges will emit electromagnetic wave. Einstein proposed that the accelerating body in gravitational field will excite gravitational wave.