Relativistic rocketry and the requirements for instellar flight

von

KARMAN INSTITUTE

1-3

z

0'\

FOR

P

LUID DYNAMICS

\11

TECHNICAL NOTE 65

, i.

RELATIVISTIC ROCKETRY AND v •

THE REQUIREMENTS FOR INTERSTELLAR FLIGHT \

by

K.R. ENKENHUS

RHODE-SAINT-GENESE, BELGIUM

von KAR MAN INSTITUTE FOR FLUID DYNAMICS' TECHNICAL NOTE 65

RELATIVISTIC ROCKETRY AND

THE REQUIREMENTS FOR INTERSTELLAR FLIGHT.

by

K.R. ENKENHUS

MARCH 1970

,

\

The author wishes to thank Miss I. Wuilbaut for

t yping this manuscript. The encouragement provided by

SUMMARY

The kinematic requirements for interstellar travel are examined, and the theoretical performance of nuclear and photon rockets is evaluated. The first part of the paper contains a detailed discussion of relativistic rocketry, including the problems of time-dilatation in an accelerating system, and the twin paradox. In the latter part of the paper, formulas and charts are presented which permit a quantitative evaluation of the requirements and possibilities for

inter-stellar flight. It is concluded that, while nuclear rockets are marginal in performance even for flight to the nearest stars, the photon rocket, employing matter-antimatter anni-hilation reactions, would permit voyages to stars several hundred light years away in the lifetime of the astronaut. Although such flights are theoretically possible, the

engineering problems which must be solved before such flights become a reality are of such a mag~itude that it is impossible to predict when interstellar travel will be achieved.

CONTENTS

SUMMARY

LIST OF FIGURES NOTATION

I. INTRODUCTION

11. RELATIVISTIC MECHANICS AND THE TIME DILATATION

Page

i ii 1

EFFECT 7

2.1 Newtonian Mechanics and the Galilean

Trans-formation 7

2.2 Consequences of the Invariance of the Speed

of Light 11

2.3 The Special Theory of Relativity and the

Lorentz Transformation 16

2.4 Representation of the Lorentz Transformation

by the Brehme Diagram 19

2.5 The Laws of Mechanics in Special Relativity 22 2.6 Space-Time Transformatioos in Accelerated

Frames 25

2.7 The Twin Paradox 28

2.8 Discussion of Observations made by an

Accelerating Observer 33

2.9 Proof that an Inertial Observer Ages more

than a Non-Inertial Observer 39

111. REQUIREMENTS FOR INTERSTELLAR FLIGHT 42

3.1 Kinematic Requirements 42

3.2 Performance Analysis of Nuc1ear and Photon

Rockets 47

3.3 Equations for Ideal Photon Rockets 52

IV . RESULTS AND DISCUSSION

4.1 Performance of Nuc1ear Rockets 4.2 Performance of Photon Rockets

V. CONCLUSIONS REFERENCES FIGURES 53 53 55 58 59

-

~

-LIST OF FIGURES

1. The Brehme Diagram

2. Significance of the Rotational Angle

e

3. Simultaneity of Events

4.

The Lorentz Contract ion

5.

The Time Dilatation Effect

6.

Brehme Diagram of the Twin Paradox

1

.

Concepts of Time of Various Observers ~n the Twin Paradox

8

.

The Space Ship Flotilla Problem

9. Brehme Diagram of the Space Ship Flotilla Problem 10. Relative Aging of Non-inertial vs Inertial Observer 11. Kinematics of an Interstellar Journey

12. Distance Flown and Time Elapsed on Earth for Nuclear Rockets (Astronaut' s Flight Time

=

20 years)

13. Ratio of Astronaut ' s Flight Time at Minimum Acceleration to that at Infinite Acceleration, for Nuclear Rockets (Astronaut's Flight Time

=

20 years)

14. Accelerati on of Nuclear Rockets

15. Specific Exhaust Power of Nuclear Rockets

16. Optimi zation of the Performance of Photon Rockets by Inert Mass Addition to Exhaust Stream

11

.

Optimum Fraction of Total Propellant Load Converted to Energy in Photon Rockets

18. Di stance Flown and Time Elapsed on Earth for Photan Rockets 19. Ratio of Astranaut's Flight Time at Minimum Acceleration to

that at Infinite Acceleration, for Photon Rockets (Astronaut's Flight Time

=

20 years)

20. Acceleration af Photon Rockets

r

I A a c E F g m ~ p R T t

u,v,w

u

w

X,Y.Z x Cl E y NOTATION

an inertial Cartesian co-ordinate system (X.Y,Z); unprimed quantities are those measured by an observer in this frame

Cartesian co-ordinate system (X'.Y',Z'). initially coincident with

r,

and translating in the positive X-direction. This frame may be either inertial or accelerating. Primed quantities are those measured by an observer in this frame

acceleration

acceleration in light-years/year/year - Aa/c

(-0.96874S)

velocity of l i g h t .

2.998

x 108 meters/sec

kinetic energy (also used to denote an event) force in X or X' direct ion

acceleration due to gravity

=

9.80665

meters/sec 2 mass

momentum

mass ratio

=

(m + m_f + m. )/m

s 1 s

time in seconds time 1n years

=

T/a

~ Cartesian velocity components of a body of velocity U dimensionless X-component of velocity : U/c

velocity of frame

r'

relative to

r

work

Cartesian co-ordinates

distance in light-years

=

X/ca

(1

light-year

=

9.40691

x lOl~ meters)

fraction of rocket propellant load converted into energy by nuclear or mass annihilation reactions fraction of nuclear fuel converted into energy potential energy

n

n

e a

x.'

e

SUbscri&ts a ce cr c d e f ~ p r s - i i i -NOTATION (Continued)

fraction of energy produced by power plant converted to directed kinetic energy in exhaust nozzle

specific exhaust power (Eqo 3.40a)

number of seconds in a year

=

3.1557 x 107

distance and time tranformations defined by Eqs. (2.3) angle of rotation in Brehme space-time diagram

acceleration

classical value calculated by earth observer classical value calculated by rocket observer

value during coasting period af ter acceleration of rocket deceleration

exhaust fuel

inert propellant added to fuel

total propellant load (fuel + inert propellant)

rest value

structure + payload

X,X'.T,T' partial derivatives with respect to indicated variables

value measured by an observer in the co-ordinate frame E'

• first derivative with respect to independent variable • • second derivative with respect to independent variable

- 1

-1 0 INTRODUCTION

0 0 0 for my purpose holds

To sail beyond the sunset~ and the baths of all the western stars o

"Ulysses" - Tennyson

Af ter the exploration of the solar systemi it is inevitable that manis thoughts will turn to interstellar voyages. At first glancej it might seem that the enthusiasm

for longer journeys would be dampened by the fact that the possibility of finding life in our own solar system appears increasingly remoteo The contrary is trueo Astronomers not only

observe that the primordial gas clouds have condensed into clusters of hundreds of stars like the Hyades or Pleiades~ but that

75%

of the stars have one or more orbiting companions -sextupIets like Castor~ triplets like Alpha Centaurii or

doublets like the Dog Stari Siriuso There are numerous stars

which wobbIe due to the gravitational pull of unseen companions which may be pl anetso Indeedi modern theories of stellar

evolution find no reason why most st ars should not have planetsj

born with them out of the same gas clouds by the same multiple-birth processeso And since most stars have habitable zones

around them where water can exist and gases can be hel d as an atmosphere. astronomers have little doubt that life may have evolved around several million of the 100 billion stars in our galaxy, the Milky Wayo{l ) It therefore seems highly probable that we could find habitable worlds much like our owng and event

intelligent beings with whom to discuss and share our experiences~

if only an adequate means of interstellar travel were avail ableo We might even go so far in these speculations as to question why some super- race from the stars has not already contacted us~ The answer~ as we shal l seei would seem to lie in t he incredible

vastness of space; in the requirement for a phenomenally advanced technology to make the longer journeysj and in the great deal of searching which would have to be done before a habitable

worlde rare on a statistical basis, would be foundo

Interstellar distances are so immense that the only convenient 'yardstick for measuring the gulf between us and

other suns is the light-year - the distance travelled by light, at 1869000 miles per second - in one year. A light year is

5.87 x 10 12 mileso Even if a space ship could attain optic

velocity, the time required to travel across our galaxy and return would be 200,000 yearso However, within 30 1ight-years

from 'our sun there are some 170 stars, of which Alpha Centauri, located in the constellation Centaurus near the Southern Cross, is the closest, it is 403 light-years awayo But at the speed of an earth-orbiting satellite (18,350 miles per hour) it would s t i l l take 3650 years to reach Alpha CentauriQ

Clearly, even the most modest interstellar journeys would be impractical unless a space ship can be constructed which will reach a significant fraction of the speed of light. As we approach nearer to the speed of light, the laws of physics must be modified to allow for relativistic effects, and it is precisely here that the astronaut finds a "100phole" which speeds him to his goalo This loophole is the time dilatation

effect, which indicates that when travelling at a speed U. time will slow down by a factor

1

1 - U2 /cZ , where c is the speed of lighto Thus, at a speed of

9905%

that of light,

Il_U2/c2~0.l,

and a journey to a star 100 light-years away could be made in an elapsed time of 10 years for the astronauti although a little over 100 years will have passed by on earthQ In principle, a trip around the entire universe could be made during the

life-~ime of the astronaut providing his ship is accelerated to a

speed sufficiently close to that of lighto In the billions of

years which would have elapsed on earth the sun would long since have grown cold, and even the familiar constellations would have changed beyond recognitiono

A fundamental question is whether it is possible, with known energy sources i to construct a ship capable of interste11a.r

- 3

-flight. The answer is yes. The use of nuclear energy. generated, for example. by the hydrogen fusion reaction which stokes the

"fires of our sun, would allow u~ to reach the nearest stars in one or two lifetimes. even though the maximum speed reached

would bet at best, a few tenths of the speed of lighto

The performance obtainable using any nuclear reaction

is limited'by the fact that only a small fraction of one pe~­

cent of the 'massis converted into energye The key to longer

journeys "is the wholesale conversion of mass to energy using

matter-antimatter annihilation reactions. In this cases the

exhaust of the rocket would consist of pure energy in the form

of photons, i.e. , a beam of light. Such a vehicle has been called

a photon rocket.

A practical issue is whether it is possible with current technology to harness these energy sources. The answer must be a qualified yes for the nuclear rocket, and no for the

photon rocketo

Although a nuclear fission rocket suitable for t ravel

within "thesolar system has already been constructed (the NERVA

engine). the fraction of the mass converted to energy is an

order of magnitude less than for the deuterium fusion reaction

in the sune Even i f we "tame the H-bomb", a problem arises from

the fact that the rate of power generation required to produce

areasonabIe thrust in any nuclear reaction i s extremely hi gh.

The energy released must therefore be converted into a jet

of hi gh speed exhaust gases issuing from the nozzle with nearly

100% efficiency. or else the wast ed energy~ in the form of heat~

would melt the working parts. With current technology, this

problem can be controlled only by l imit ing the rate of power

generation to the point where the thrust produced is less t han

1/ 100 the acceleration of gravitYi and perhaps as l ittle as

We find that the higher value of acceleration quoted would make a trip to Alpha Centauri feasibleo At an acceleration

of oOlg~ the ship would reach approximately 1/100 the speed of

light in one year. or 1/5 c in 20 years Q The average speed during

th1S acce erat10n per10d woul ' l ' , d lbe

Tö

c. and hence the d1stance ,

travelled would be 20 x

Ïo

or 2 light-years. or about half the

distance to the nearest staro If the ship now began to

decele-rate at the same decele-rate, it would reach Alpha Centauri af ter an additional 20 years. The travel time would thus be 40 years one-waYi or 80 years for a round tripo For a maximum speed of

fa

c, the time-dilatation effect would be negligible.

At an acceleration of .001g, the round trip would take

800 yearso It has been suggested that such a journey might still

be made if we accept the fact that many generations wou1d have to be born and die on board the star ship. In the 800 year journey a11uded to. it wou1d be as though Lief Erickson had departed for America and his distant descendants had on1y

arrived today. The physica1 and psychologica1 prob1ems invo1ved in maintaining the hea1th and sen se of purpose of the sma11

community setting out on such a pi1grimmage wou1d be profoundo

The likeliest solution wou1d seem to 1ie in perfecting a

method of maintaining the crew in a state of suspended animation

unti1 the destination were approached.

It wi11 be asked whether. if a means were found to control the dissipation of energy so that a nuc1ear Ship cou1d acce1erate at. say. 19. and thus approach the speed of light in one year. wou1d not this make long interstellar voyages possib1e? The answer is. unfortunate1y. no. because the mass

ratio of a nuc1ear powered ship (i.e. , the ratio of fue1+structure

to empty weight) wou1d have to be prohibitive1y large in order to re ach a speed where the time-di1atation effect becomes

significanto For examp1e_s to reach U/c = 0.995. where the time

di1atation is 1/10. a ship powered by the deuterium fusion

reaction wou1d require a mass ratio of

6

x 1011 ! To re ach U/c=Oo2i

5

-The photon rocket, in contrast~ is particularly promising because it can re ach speeds approaching that of

light with reasonable mass ratios; for example, U/c

=

00

995

is obtainable with a mass ratio of 20, and U/c

=

00

9998

with a mass

ratio of 100, as we shall see from our analysis. In the latter case, time is slowed down by a factor of ~~ fer the astronaut, so that a voyage of 1000 light years could be made in 20 yearso However, the engineering problems are incredibly greater than those of the nuclear rocketo The specific rate of power

gene-ration required to produce a given accelegene-ration is at least 100 times as high for a photon rocket as for a nuclear rocketo At one g acceleration, it is, in fact i about 3 million

mega-watts per ton of weight of the ship. A 500 ton ship would

there-fore have to develop a power which exceeds the present rate of electrical production of the entire world by a factor of 4000

times! The energy in the exhaust beam could sear an entire continent; what then of the reflector which directs this beam? The answer is that entirely new means of shielding against intense radiation, which can only be faintly guessed at the present time, would have to be employed.

Another problem which will have to be solved before the photon rocket can be created is the product i on and storage of anti-mattero Such matteri in the form of anti-protons or positrons, for example_t has already been created in minute quantities in atom-smashers o Howeverj there is no escape from the fact that an enormous amount of energy - at least as much energy as is released again in the annihilation process - would have to be employed in the product ion of anti-matter. Furthermore, the antimatter so produced would have to be confined and stored

by force fields-electrical or magnetic in nature - to prevent its contact with ordinary matter until ready for useo Some experiments with such confinement- methods have already been done. Should the necessary apparatus fail during a long journey~

the ship would be instantly vaporized in an explosion dwarfing a million H-bombsa

Having briefly indicated some of the concepts for interstellar flight and their associated difficulties, it is clear that a full solution will not be achieved until perhaps af ter several centurieso This does not prevent us, however, from undertaking the task now of establishing quantitatively the basic requirements for interstellar vehicles, and carrying out performance calculationso

Although nuclear-p~wered rockets can hardly re ach the speeds where relativistic effects are important, the theory of relativity plays a central role when considering the per-formance of photon rockets, which will ultimately make the longer interstellar journeys possibleo In Section 11 we shall therefore first turn OUT atten~ion to a review of the salient features of the special Theory of Relativityo We shall consider carefully the time-dilatation effect, about which there has been a surprising amount of controversy even in recent times, and discuss the famous Twin Paradoxo In this Section. the analysis will also be extended to include accelerated mot ion

of a relativistic rocketi and the space-time transformations

obtainable from the General Theory of Relativityo

In Section lIlt the kinematics of interstellar

voyages are first established, ioeo, the velocities and acce-le rations necessary to make trips of a given length in a

specified timeo An analysis of the performance of both nuclear and photon rockets is then presented, together with a

consi-deration of how it may be optimized.

The results of the analysis are given in Section IVo

From the graphs obtained, a number of interesting conclusions about the possibilities fort and limitations on interstellar flight can be drawno

7

-II. RELATIVISTIC MECHANICS AND THE TIME DILATATION EFFECT

2.1 Newtonian Mechanics and the Ga1i1ean Transformation

The 1aws of physics are genera1izations of experiencei

the va1idity of which rest on the accuracy of experiment al measurements. Thus Pto1emy's astronomy of cyc1es and epicyc1es agreed wel1 with observations of p1anetary motions at a time when

the te1escope was unknowni and the passage of time cou1d on1y

be reckoned by sun dia1s or water c10cks. The idea that the heaven1y bodies were inbedded in ce1estia1 spheres set into motion by the Prime Mover was 10gica1 because the concept of gravitationa1 interactions had not been imagined o This picture

was profound1y a1tered by Newton9_{s mechanics and the 1aw of}

universal gravitationi which received their most start1ing

verification in the prediction of Kep1eri _{s 1aws of p1anetary}

motion o However e at the turn of this centurYe Einstein showed

that the Newtonian theories are on1y the limiting case for 10w

ve10cities of more general 1aws of mechanics and gravitation

which we knowas the Special and General Theories of Re1ativityo (2-5 )

From a phi10sophica1 point of viewi the most

revo1u-tionary aspect of EinsteinOs theories is that the fundament al

physical quantities of mass 0 1ength and time can no 10nger be

considered invariants but depend on the motion of the o'bserver

re1ative to the phenomenon being studiedo On1y the Eroper va1ues

of these quantities, i Oeoi the va1ues measured when at rest

re1ative to the observer~ are invarianto It is worthwhi1e to

consider precisely what we mean by the latter before proceeding

furthero

-Unti1 recent1y, the me~er was defined as the distance

between the centers of two 1ines traced on a p1atinum-irridium

bar kept at OoC in a vau1t at the International Bureau of

Weights and Measures at Sèvres, Franceo Today, the standard

meter has been redefined in terms of the wave1ength of a parti= cu1ar spectra1 1ine, not on1y because very accurate comparisons

of length can be made by interferometric measurements, but because the wavelength of the light emitted is readily made

very insensitive to external influences~ The kilogram of mass is

defined as the mass of a certain block of platinum which is also

preserved at the International Bureau of Weights and Standardso

Although the measures adopted are arbitrary, underlying them is the assumption that the length of a bar, or the mass of a body are invariant properties if the physical conditions are main-tained constant, independently of the location or time at which

we examine them~

We are able to conveniently define intervals of time because of the cyclic nature of many natural phenomena. The mean solar second used in physics is a submultiple of the mean

solar day, which is the average interval between the times the sun crosses a meridian on successive days during the year. The mean solar second changes very gradually due to dissipative forces in the solar system, so that the most accurate measu-rement of time is made possible by using the period of

vibra-tion of atoms o We conceive of time as a general property of

matter. described by the law that all isolated material changes

occur (or would occur) in invariant ratios to each othero

(6)

Thus all clocks, based on any mechanism whatsoeveri but free

from external changes or dissipative influences, will, af ter

initi al synchronizationi indicate the same elapsed time

regard-less of when or where they are examined.

The classical Newtonian mechanics can now be

formula-ted in the following way :

(a) The quantities length

(x),

mass (m) and time (T)

are assumed to be understood, together with the notions of

Euclidean geometry.

(b) We

~

acceleration A

=

~ d!tine the vector quantities velocity U

dU ~ ~

dT

tand momentum P

=

mUg

~

dX

=

dT~

(c) When several bodies interact, the sum of their

- 9

-and the mass of each body remains constant (the law of conser

-vation of mass ).

(d) The momentum of a body does not change unless there

is some outside interference. We call this interference force~ and

d do • ~ dP . .

de fine its units~ magnitude

~

NewtonWs second lawo If F

=

an ~rect~on by F

=

!T.

Th~s ~s

~

O~ P

=

constant. This is Newtonis

first lawo When a body causes a force to act upon a second body~

the second simultaneously exerts an equal and opposite force

upon the firsto This is NewtonOs third law, which follows from

thus

1

₂

=

-Fl· ~

(e) We de fine the work W done by a force

F

as the

scalar

r~ ~

Since ~ d 2X W = F. dX. F = m

-

, we have dT 2 rl d2X dX ~ T2 1 ~2 ~2 W =

(2

m

- -

dT =

_J

d(~

mU2 ) =

_'2

m(U₂ - UI) dT2 Tl The work do ne 1 energy E

=

'2

i. eo i one for ~ • such that F dT Tl

on the body thus equals the change in kinetic

_--

_..

mU2 • If the work is done by a conservative force~

!

~ ~

which

j

FodX

=

0, then there exists a potentia!

=

-grad . , and r2

W = -

J

grad~odX

= -(+2-+1).

r l

Combining this result with the previous equation for Wi we obtain the familiar form of the law of conservation of energy

1 ~2

There is a fundamental point in the above exposition vhich we have glossed over. It is~ is there any restriction on hov we must choose our co-ordinate system in order that Newtonian mechanics be valid? The answer is that we must make our measure-ments using an inertial or Galilean reference frame which is not undergoing linear or angular accelerations with respect to the matter in the universe as a wholeo In considering the motions of bodies orbiting each other under mutual gravita-tional attraction in the solar system, for example, we obtain a Galilean frame by choosing a set of co-ordinates which are non-rotating relati e to the stars, the origin o~ which is fixed at the center of mass of the solar systemo The center of

mass is selected because the laws of dynamics show that it is not accelerated unless some external force acts on the bodies_a and the net attraction of the stars on the solar system can be reckoned as negligible.

Whether a given set of co-ordinates can be considered

inertial depends. of course, on the kind of phenomenon which

is being studied and the accuracy with which results are to be predicted o Thus i the surface of the earth is a good

approxima-tion to an inertial frame when we study the ballistics of a

cannon balI. but not when ve wish to calculate the trajectory of a long-range missile~ nor when we study the Foucault pendulumo

If we insist on employing a non-inertial frame of

reference, then artificial forces must be introduced to allow for the linear and angular accelerations of the co-ordinate system o If the frame has a linear acceleration

A,

then Newton~s lavs may be applied providing we assume that all masses are subjected to a fictitious gravitational force producing an acceleration

-Ag

Artificial centrifugal and Coriolis forces must be introduced to make Newtonis laws valid for a rotating

- 11

-Let us now examine what the form of NewtonUs law would be for an observer in a frame EU moving at a constant vel ocity

-+

Ua vith respect to an inertial frame Eo If the frames are

coin-cident at T

=

Ti

=

0. then according to classical concepts of space and timet the co-ordinates are related by the Galilean transformation

:t -+ -+

.x9 _•

_X

_-

_U

_o

_T

}

Tt

=

T

The observer in Et measures an acceleration

d (-+ -+) dT

dT

U - Ua -dTi -+ dU

=ëiT

-+ 0

due to t he force F act1ng on a mass m; ioeo~ Newton gs l aw is

-+

:t: dU

.l"=md'T=m

and is invariant under the Galilean tranformation o Since NewtonQs law holds in EO~ any frame moving with a constant velocity

relative to an inertial frame is also an inertial frameo It may be concluded that there is no way~ through the l ava of classical

mechanics~ to teIl whether we are "reall y at rest"~ or in

uniform motiono

202 Consequences of the Invariance of the Speed of Li5ht

Of course i we may immediately aski in motion vith

respect to what? I f all the laws of physics were invariant under

the Galilean transformation~ then, clearly~ the concept of

absolute motion would become meaningless. Physici sts of the

19th cent ury found this was not the caseo The laws of light propagation~ expressed by Maxwellgs electromagnetic equations~

were found

-

not to be invariant under the Galil ean tranformationo

o 0 0 + 0

a different and more complicated formo An attempt was made to

explain this by the ether hypothesis - that all of space is

uniformly pervaded by a substance, the etheri relative to which

electromagnetic waves propagate at the velocity of light~ like

waves on a pond of water o It was argued that Maxwell's equations

w~re in the correct form for an observer at rest with respect

to the ether o A number of experiments were devised to check the

ether hypothesis 5 and it didi in fact, sail through one test

af ter another until it was finally shipwrecked by the

Michelson-Morley experimento

(7)

Michelson and Morley reasoned that~ since the earth

undergoes a complicated motion, spinning on its axis and

rota-ting around the suni it should have some net velocity relative

to the ether most of the time~ which should be detectable from

variations in the measured speed of light o Despite many

carefully-repeated experimentsi no such effect was ever found o The

velo-city of light was also shown to be independent of the motion

of the source relative to the observer o Physicists were

there-fore forced to the conclusion that the velocity of light is th~

same for all observerso

Einstein, in 1905_i

(8)

rescued science from the

dilemma caused by the collapse of the ether hypothesis and

simultaneously revolutionized our concepts of space and time,

by proposing the Special Theory of Relativityo The first hypo=

thesis in this theory is that the velocity of light is constant

for all observers. The second hypothesis is the so-called

principle of relativity~ which states that the laws of physics

are the same in all inertial frames, ioeo~ there is no way an

observer can establish whether he is "really at rest" or in

uniform motiono Einstein later considered accelerated frames of reference and revised Newtonis law of gravitation in the General

Theorl of Relativitlo Since the postulate of the invariance of

the speed of light is common to both the special and General

Theories, it is appropriate to see its significance before

- 13

-We shall consider a frame of co-ordinates E{XiY~Z~T } and a second frame E~ {XQtyoiZ9~T~} which is initially

coin-cident with t but moving in an arbitrary manner along the X-axiso

Then the transformation between the frames will be

Y • y' • Z . Z9

1

X· x(X~.T')

_•

where X and T are functions to be determined in such a way that they satisfy the first hypothesiso Then

Division of Eq. (2 0

4)

by (205)~ followed by division of the numerator and denominator on the right hand side by dT9

yields the velocity transformation

where

u

=

!!!

dT

are the X-components of the velocity of a body measured in E

and tQ~ respectivelyo

Application of the hypothesis of the constancy of t he speed of light ~ i oeot that when U

=

±cp UQ

=

±C leads to the

Addition and substraction of these equations then gives

t

(2.8)

Differentiation of the first re1ation with respect

to T~ and the second with respect to XO and equating the

resulting expressions for 'T'X'

=

'X8T9 leads to the resu1t

while differentiation of the first with respect to XO and the

second with respect to T' and equating the resu1ting expressions

for XX'T'

=

XT'X' yie1ds

It is thus a consequence of the invariance of the speed of light that the position and time transformations X and • obey the

one-dimensional wave equation with propagation velocity co

The general solution of Eqo (20 9) is that g~ven by

diAlembert' s formula :

where Xl and X2 are arbitrary functions of the arguments XO+cTo

and X t_cTo~ respective1y. A simi1ar solution for T can be written

from Eq. (2.10) . We see that the transformations are governed

by expressions anal ogous to those describing waves which pro

-pagate in the positive and negative X (or Xi) directions at

the speed of 1i ght o

The relative velocity of the co-ordinate systems E

and EO may now be foundo The velocity of a point at rest in

E'

(U

'

=O

)

as measured in

E

is

(2.,12)

~

15

-at rest in E (U=O) as measured in E is

Uo'

= -

_{XT, XXQ} /

= -

U 0

The observers in the two frames agree that their relative veloeities are equal and oppositeo

We are now in a position to derive a more speeific form of the to eliminate we find that Ui _{+ Uo} U

-UQUO 1 + e 2 velocity ~Xo and

transformation~ Eqo (2 06)0 Using Eqso (2 08)

TTo from (2 06)i and then applying Eqo (20l2)~

This is the Einstein velocity addition formula o It shows that veloeities do not add linearlYi and that their sum is always less than the speed of lighto For example, if the frame t U

is moving wi th a velocity Uo = OoBe relative to E~ and a body has a velocity UU = 008e as measured in EQ~ then the velocity measured in t will be (OoBe + 008e)/(1 + OoB2e2)

~

Oo9756e < Co

e2

We now proeeed to obtain relations between the length and time intervals measured in the two frameso A length dXi in

E~ is measured by an observer in t at some instant T (dT ~ 0)0

dX is then given by Eqo (204 )~ with dT~ =

The use of Eqs. (20B) and (20l3) then leads to the re sult

ConverselYi a distanee dX in E is measured by ~n

observer in EO at some instant T' (dTv=O )o Eqo (2 04) immediately

yields

Each observer finds that lengths measured in the other system are j in general, altered o Eqso (2015) and (2.16) are generalized expressions for the Lore~z contractiono

A clock at rest in tQ(dX~=O) reads a time interval dT'. An observer in t measures that the time which has elapsed is, by Eqs. (20 5) and (2.8)i

A clock at rest in E(dX=O) reads a time interval dTo

An observer in t ' measures an elapsed time which is given by Eq. (20 5)i with dX' = -xT,/Xxoo With the aid of Eqso (208)

and (2013)~ we find that

dT

Each observer finds that the rate of a clock in the other frame is~ in general, alteredo Eqso (2017) and (2018)

are general expressions for the time dilatation effect o

To obtain explicit results, it is now necessary to

solve the wave equation (Eq. (20

9»

for specified boundary

conditions. This we now proceed to do.

203 The Special Theory of Relativity and the Lorent z

Transformation

We consider the ~ase where the frames E and EO have

a constant relative velocity UO o Then, by Eqso (206)~

(2 08)

and (2011), • Uo Xl - X2 = constant,

-

_c

=

Xl + X2

- 17

-argument X'+cT' of Xl. and the argument Xi-cT' of X2. It then

follows thati since Xl and X2 are independent,

ua

Xl (1

-) =

_c

where k is a constant. The solution is UA

Xl =

~

(x'

+ cT')/(l -

c-)

so that the transformationi Eq. (2011) becomes

x

=

k

(xt

+ UOT')

1 -

U~

/

c2

An additive constant has not been included in the integrations

because we wish X to be zero when X'=T'=O. T=T can now be found

by integrating Eqso (20 8). We obtain

T

=

k

(T' +

uoxt/c

2 )

2

1 - Uo/c 2

If we solve Eqso (2.19) and (2.20) explicitly for X·

and T', we obtain the inverse transformations

x'

=

x

-

UOT

k T' =

T - UOX/C2

k

In order to evaluate the constant k, it is necessary to invoke Einstein's second postulate, the principle of

rela-tivityo Let t be an inertial frame o Then E' is also an inertial

frame i since it is moving at a constant velocity Ua relative to

Eo From E~ 's point of viewi E is moving in the negative Xi

direction at a velocity -UOo Therefore, in E', the co-ordinates

(x - UOT) (T - UOX/C 2 )

x'

=

k

,

₂

1 - UO/C 2

It is seen that Eqs. (2.21) and (2.22) are equivalent if, and only if

We thus obtainfrom Eq~. (20 19), (2020) and (2.23) the Lorentz transformation

X· + UOT'

x

=

T

=

(2.24a)

Eqs. (2021) and (2. 23) Y ield the inverse Lorentz trans format ion

x -

UOT

x'

=

,

T' = T - UOX/c 2

/1 -

U~/c2

The Y and Z co-ordinates remain unchanged, i.eo , Y

=

Y' t Z

=

Z'.

Eqs . (2.15) and (20 16) show (sinee XX'

that a length dX' in _{1: '} is therefore measured in

dX =

11

-

U~/c2

dX'

and conversely, a length dX ~n E is measured in

= 1/II-U%/C2 ) E to be

(2.25a) E ' to be

(2.25b) The proper length measured by either observer in his own frame is found to be contracted by a factor /1 -

U~/c2

by an observer

~n the other frame. This is the Lorent z contraction. which takes place in the direction of motion only~ since dY'

=

dY, dZ9

=

_dZ

19

-Furthermore~ Eqso (2017) and (2018) show that a time

interval dT~ recorded by a clock at rest in tt is recorded

by an observer in t as

dT

=

1 dT9

h-

U~/c2

and converselyp a time interval dT recorded by a clock at rest

in t is read in t 0 _as

dT~

=

1 dT

)1 - U~ /c2

Each observer finds that the other's Eroper time interval is

increased (dilated) Dy a factor 1//1 -

U~/c20

This is the time

dilatation effect o

We note that when UO/c ~ 0i eqs. (2024) reduce to the

Galilean transformation i Eqo {2 01}j and the Lorenz contract ion

and time dilatation effects disappearo

that

2&4 Representation of the Lorentz Transformation by

the Brehme Diagram

From the Lorentz transformation (Eqo (20 24» we find

This equation forms the basis of a simple graphical

represen-tation of the Lorentz transformation due to Brehmeo (10)

Other graphical representations include the complex rotationa

Minkowskii and Loedel diagrams (see Ref~ 3i ppo 12-27). Various

applications of the Brehme diagram have been discussed by Sears.(ll)

This diagram, shown in Figo li is constructed by first drawing

an orthogonal set ofaxes (Xii cT) and then a second set (Xi cTij)

co-ordi-nates of a point E are obtained by dropping perpendiculars on the axes. Then. by the theorem of Pythagoras.

which is Eq.

(2027).

It is easy to show from the geometry of

the diagram (see Fig. 2) that

Xi = OA

=

OB - AB

=

Xcos8 - cT'sine

cT'

=

DE

=

CE - CD

=

cT co·se'.- .X;--'sine

50 that

x'

+ cT9 _sine

X = cos e ~ T

=

T' + (X'/c)sine _cose

Comparison of these expressions with Eq. (2.24a) shows that

the Lorenz transformation is represented by the diagram if e

=

sin-1(Uo/c)D The Brehme diagram forms a convenient means of illustrating the important features of the Special Theory

of RelativitYQ

(a) Events and World Lines

An event is something which occurs at a definite place and timeo It is represented by a point such as E on the

Brehme diagramo The local and time (X,T) of the event as seen

by an ooserver in t are found by dropping perpendiculars from E to the X and cT axes. respectivelyo An observer in t ' finds that the same event occurs at {X',T')j obtained by dropping perpendiculars on the X' ,cT' axes. respectively. Specification

of any pair of the four co-ordinates X,T.X' ,T' defines the

other two. Note that from a mathematical point of view, the

inverse Lorenz transformation, Eq. (2.24b), adds nothing to

the direct transformation, Eq. (2.24a); it is simply more

convenient for certain problems.

A world line is the path traced out in space-time by

- 21

-line is normal to the X or X9 _{axis~ respectivelyo If the body}

is in motion~ its wor1d line is the curve X(T) in t~ and X9 (TO)

• ~g • • • 1 dX U

1n ~ 0 The veloc1ty of the body 1S g1ven by the s ope

CdT

=

-dXo UO c

in t~ and by

= --

_c in t Oo

(b) Simultaneity of Events

The Brehme diagram in Figo 3 shows very simply how observers in E and E~ both find that a light wave propagates in all directions with the same speed fr om their respective originsi even though the observers have a relative velocity UO o

Consider the world lines OP and OQ of optic wave fronts propa-gating in the positive and negative X-directions~ which have the equations X

=

i cTo We see from the diagram that XU

=

±CT9 as well o Thus Eqo (2027) is obviously satisfied~ with the left

and right hand members both equal to zero for a light ra~o At T

=

Tli ~ an observer in tobserves the simultaneous events El

and E2~ and finds that the light rays have propagated equal distances OA~ OB in the positive and negative X-directionso For an observer in tU ~ however~ events El and E₂ are not simultaneous;

...

El and E2u _{are simultaneous o The observer in tO thus also finds}

that the light rays have propagated equal distances OA9 ~ OBi in the positive and negative XU-directionso Both observers agree that the wave front propagates in all directions with the same speed c because they do not agree on the simultaneity of eventso

(c) Graphical Illustration of the Lorenz Contraction and Time Dilatation Effects

The Lorent z contract ion is illust rated in the Brehme diagram of Figo

4

0

The world lines representing the ends of a rod of length L

=

X

B ~ X

A in I are "gormal to the X-axiso The position of the rod AB is shown at a time Tl o To an observer in tt _~ however~ at a time TU g when one end of the rod is at AI the

other end is _,... not at Bp but at BUo The length of the rod as measured in t O i s found by projecting ABO onto the Xi axisi

giving Li

= X~

-

Xl

=

Lcose

=

Lil -

U

~/

c

2

o

The reciprocal effect is found by considering a rod at rest in t O_{, the ends of which} have world l ines perpendicular to the XO-axiso

The time dilatation effect is shown in Fig. 5. In Fig. (5a)~ the time dilatation of events Ei and Ei taking place at XO

=

constant in tO~ as seen in ti is illustrated. Fig. 5b

shows the converse situation for two events El and E2. occurring

at X

=

constant in t, as seen by an observer in tt.

205 The Laws of Mechanics in Special Relativi\r

H. A" Lorentz had discovered that Maxwell

°

s electro-magnetic equations were invariant under the Lorenz transfor-mation before Einstein derived the trans format ion on the basis

of the two postulates discussed above. Howeveri NewtonQs laws

are

E.2i

invariant under the Lorentz transformation. Einstein

took the bold step of stating in his Principle of Relativity

that all physical laws are invariant in form under the Lorenz

transformation. He then sought and found revised laws of

mecha-nics which would satisfy this postulate. A general discussion

of relativistic vs Newtonian mechanics which is quite illumi-nating has been given by Park. (11)

To begin the treatment of relativistic mechanics,

consider a body which has an instantaneous velocity U in the inertial frame t. We choose a second inertial frame t ' which has a constant velocity Uo relative to E which matches the

instantaneous velocity of the body. Then the body is

instanta-neously at rest relative to an observer in tO, i.e., ut

=

00

By differentiating the Einstein velocity transformation, Eq. (2.14)9

with respect to Ti we obtain

dU

TT=

uou~ 2 (1 + _ ._-) c2 dut dT' dTt

dT

Setting Uo

=

U. ut

=

0 in this expressions and using Eq. (2.26a)

dT 9 •

dU

where A =

dT

'

_A'

= -

dU'

dT9

- 23

-(2.28)

are the accelerations of the body measured by observers in E

and

E',

respectively. Since the body is instantaneously at

rest in the frame Et, its mass is the proper, or rest mass m •

r

and Newton9_{s law}

FO

=

m A'

r

is assumed to be valid in the frame

E'.

To derive the form of

Newton's law which holds in E, we first have to establish the

relativistic transformation of force. This is readily done by

means of the following thought experiment. If a bar of ultimate

tensile strength a and cross-sectional area ~y~Z is aligned in

the X-direct ion anu subjected to an axial tension F. it will be

seen to snap in two by observers in both E and E' when F=~y~z. a.

Now a is a property of the material~ and ~y~z

=

~y'~z' since

the i and Z co-ordinates are unchanged by the Lorenz

transfor-mationo Therefore F9

=

~Y'~Z' a

=

~Y~Za = F. From Eqso (2.28)

and (2 029) we th us obtain F'

=

m AQ

=

F

=

r m A r m A r +

=

(2.30) m A r

Hence we may write the relativistic equation of motion as

F

=

h-

(mU)

where m = m

//1

-

U2/c2

Newton's 1aw must be modified in re1ativistic mechanics br taking

into account the increase in mass with ve10citr re1ative to the

ojserver. The relativistic mass m exceeds the rest mass m

r ~

-factor

11/1

-

u

2

/c

2 • For brevity here, we have not given the

transformations of velocity, acce1eration and force in the

y- and Z-directions, which may be found in any of the books

on re1ativityo However, the resu1t is that the equations of

motion in the Y- and Z-directions a1so take the form of Eqa (2031),

i.eo, the forces in these directions are, respective1y, equa1

d d

to

dT

(mV).

dT

(mW) where V and Ware the ve10cities in the

Y- and Z-directions, and m

=

m

1/1

-

u2/c 2 •

*

r

We now proceed to find an expression for the energy

of the body. When the force F acts through a distance dX, the

work dW which is done equa1s the increase in kinetic energy,

dE~ Thus, by Eqo (2.31),

FdX

=

dE =

~

(mU)dX = d(mU)

~

= Ud(mU)

=

U2 dm + mUdU •

where, from m

=

mr

/

~

dm = mUdU

Thus~ dE = mUdU

Comparison of the expressions for dm and dE shows that dE=c2dm~

(2.32)

The additive constant of integration has been dropped. because

when m = Oi E

=

O. Eqo (2032) expresses Einstein's famous pri~­

cip1e of the equiva1ence of mass and energy. It shows that a

*

These expressions are va1id in a frame with the X-axis pointing in the direct ion of the instantaneous velocity, so that

+

- 25

-body at rest has an energy m c 2• This is an enormous amount

r

of energYi equa1 to 9 x 1016 watts per kilogram. The partia1

or tota1 release of this energy is accomp1ished in nuc1ear and matter-antimatter annihi1ation reactions, respective1y.

20

6

Space-Time Transformations in Acce1erated Frames

The transformation equations for the case where the

frame I ' is acce1erating ~way from I wi11 now be found by

assu-ming that the re1ative velocity is a function of time. Uo=Uo(~lo

For a slight1y different ana1ysis of this prob1em. see Lass. (12)

UO(T)

Beginning as in section 203. we have

c

Differentiation with respect to X' leads to

.

.

Xl

-

_•

=

- =

X2 _• Af/C2 Xl X2 • Xl X2

=

_{• •} Xl + X2 (2.33)

where A' is a constant which has the units of acce1eration.

The symbo1s (0) and (0' ) denote the fi rst and second derivatives

of Xl and X2 with respect to their arguments X' + cT' and

Xi - cT'o Solving Eqso (20 33) for Xl and X2, we find that

c2

X

=

Xl + X2

=

where klj k2 and k~ are the constants of integrationo To

deter-mine them we assume as a boundary condition that the frames

I and I t _{are init i a11y coincident, i oe. , X}

=

°

_{at X'}

=

_T'

=

₀

0

This yie1ds

kl + k2 + k3

=

°

We sha11 a1so specify that the frames are initia11y at rest

with respect to each other. Then, when At ~ 0, X ~ X' for any

Since eY

=

1 + Y as Y + 0, we obtain

lim X

=

(kl + k2)X' + (kl - k2)cT~

A9+0

which gives X + X' for any T' if kl + k2

=

1, kl - k2

=

0.

1

Thus kl

=

k2

=

2 '

k3

=

-(kl + k2)

=

-1, and the Bo1ution is

X

=

.s..:

[eA9X'/C2 COSh(A'T')

AI c (2.34)

The corresponding time transformation, obtained by integrating Eqso (2.8), is c T = -A' A'X'c 2 A'T' e sinh{----) c (2.35)

The re1ative ve10city between the frames is, by Eq. (2.12).

=

c

tanh(~)

Uo c

The Lorentz contract ion and time-di1atation re1ations are

obtained from Eqso (20 34), (2.35) and (2.15)-{2. 18). They are_g

,

dX' dT

=

A'X'/c2 e dT'

/1

'

-

U~/c2

,

dT' = -A'X'/c2 e dT

/1 -

U~/c2

'

(2.38)

27

-The acceleration of the origin of E' as seen in Eis, by

Eqs. (2.35) and (2.36),

dUO

A = ~ = A'sech 2 (---)(---A'T' aT' )

c aT

x'

=0

Comparison of this result with Eq. (2.28) shows that, if we

take E as an inertial frame, then the origin of E' has a

cons-tant proper acceleration A'. The transformation equations (2.34)

and (2.35) thus do not represent a general solution to the

problem, but correspond to the case where, say, a rocket ship

located at X'

=

0 accelerates away from the earth (X

=

0) at

a constant proper acceleration A'o

From Eqs. (2.37) and (2038) it is seen that, at the

origin of the accelerating frame (Xi

=

0), the Lorenz

contrac-tion and time dilatacontrac-tion factors are the same as given by the

Lorenz transformation for an inertial frame which

instanta-neously has the same velocity Uo as the accelerating frame Et.

Thus, at the origin of the accelerating frame, the Lorenz

contract ion and time dilatation factors are given by the Special

Theory of RelativitYi even though the frame is accelerating.

The dependenee of the time dilatation factors in

Eq. (2.38) on the location Xi of a clock in E' is that indicated

by the General Theory of Relativity for the red-shift in a

uniform gravitational field of strength -A'.

The transformations we have given can easily be

generalized to the case where the frame Et has an initial

velocity Uo . relative to E at T

=

Tt

=

O. Consider an inertial

1.

frame E" which i s initially coincident with E and Et, but moving

at a velocity-UO - relative to E. Di stances and times in E" are

1.

related to those in E by the Lorentztransformation

X"

=

,

Til

=

T + Ue _X/c2

1.

If we substitute for X and T from Eqso (2034) and

(2035).

we

obtain X"(Xt~T9)~ T"(X'.Tg)~ where now E' initially has the

desired velocity

Uo

.

relative to E"o If E" is now considered

1

as the initial inertial frame~ we have, dropping the (")

sUbscriptse the desired transformations

Uo

₁

·

-

C

A' T' )

coSh(C)

-These equations reduce to the Lo~ntz transformation when Ai+O~

The solution for a spaceship with an arbitrary acceleration

A'(TS) could be built up by approximating the actual motion by

a series of bursts of constant acceleration, each of a slightly

different value o But to compare the time elapsed on the rocket

ship with that elapsed on earth we merely need to use the

time-dilatation relations of Special Relativity, as has been noted

by numerous authors (see, for example~ the discussion by Scott (13»

207 The Twin Paradox

The famous twin paradox may be stated as follows ~

suppose an astronaut leaves his twin on earth and travels to

a distant star and returns with a speed approaching that of

lighto Due to the fact that the time dilatation effect is

symmetrical_e both the astronaut and his twin on earth find the

other is aging more slowly. Therefores when they meet again

on earth each will be younger than the other - an impossible

29

-ad absurdum? Even recently, there have been scientists who we re willing to embrace the accepted conclusion that the twin who does the travelling comes back younger, but have claimed that, since accelerations are involved. the problem can only be treated by the General Theory. This is not in fact necessary. for we have seenthat the time-dilatation relations for an observer at the origin of the accelerating frame reduce to those of the Special Theory, Eqs. (2.26a) and (2.26b). These equations show that each observer finds the other's time interval dilated by a

factor 1//1 -

U~/c2.

We conclude that the time-dilatation effect is symmetrical, even when one observer is in an inertial frame and the other in an accelerating frame. Thus, the idea that "taking into account that the astronaut is accelerating" i.e., applying the General rather than the Special Theory, will re-solve the twin paradox, is erroneous. As we shall see, the paradox is resolved when we take into account the fact that the astronaut's concept of simultaneity changes when his

velo-city changes i and differs fr om that of the inertial observer

on earth.

In our calculations, we will find if more convenient

to first express the Lorentz transformation in terms of the

commonly-used astronomical units. Let us measure velocities

in multiples of the speed of light, distances in light-years,

and time in years. Then the new variables are

u

=

U/c. t

=

T/a, x

=

X/co

(2.42)

where a is the number of seconds in a year. Unprimed quantities will denote those measured in the earth-frame

r,

primed quanti-ties those i n the astronaut's frame

r'.

Then the LoreLz trans-formation and its inverse become

x = (x' + uQt,}/ll

_Uo

2

Ua t )/11

x' = (x - Ua 2

ua x

)/1

1

t ' = (t

-

-

Ua 2

A Brehme diagram may now be constructed in which the axes Xi Xi

and t, t ' are marked direct1y in 1ight-years and years,

res-pective1y.

The specific examp1e of the twin paradox is as fo11ows

(see Fig.

6)0

Observers E and 8 are located on earth and on a

p1anet of a star 15 1ight-years aways respective1yo An

astro-naut A 1eaves the earth, and acce1erates very rapid1y to 0 06 of

the speed of lights trave1s t o the star, dece1erates equa11y

quick1y, and lands on the planet. A short time 1ateri he

rever-ses the journey and returns to earth. The assumption that the

periods of acce1eration and dece1eration require neg1igib1e

time is on1y made for convenience. We wi11 show later how the

re1ative aging can be found when the astronaut has an arbitrary

velocity historyo

In the Brehme diagram of Fig. 6a~ x and t are the

distance and time measured in the frame E which is common to

the earth observer E and star observer 8. with x

=

0 at E and

x

=

15 at 80 x

Q

and t ' are the co-ordinates in an in~rtia1

frame Et, initia11y coincident with E, and moving in the

posi-tive x-direction at a speed ua

=

0.60 The wor1d 1ines of E and

8 are the straight 1ines xE

=

0, Xs

=

15 i respective1yo During

the outward journeYi the wor1d 1ine of the astronaut is the 1ine xÀ

=

0; on the return trip it is the 1ine E_{2 -} E40

(a) Time for the journey to the star ca1cu1ated by

A p E and 8

Immediate1y af ter takeoff and acce1eration to the

speed ua

=

006~ the astronaut measures t he distance to the star~

whicht at tÀ

=

t~

=

Os is 10cated at Xs

=

15 (Event El )o From

- 31

-obtainab1e from Eq. (2.43a)0 The distance to the star is reduced

to 12 1ight-years by the Lore~ contraction. Hence the time t '

at which A reaches the star at a velocity Uo

=

0.6 is, according

to him, 12/006

=

20 yearso The observers E and S, on the other

hand, calculate the time required as t • 15/0.6

=

25 years.

The Brehme diagram confirms that the wor1d lines of A and S

meet at t

=

25, t '

=

200 Or mathematical1y, Eqs. (2.43c) and

(2.43d) show that when x

=

15, t

=

25, x'

=

0 (the astronaut

s s

is at this location) and t '

=

200

b} Aging of the astronaut ca1cu1ated by E and S

If the c10ck of the astronaut (at x'

=

0) sends out

signals at intervals 6t'. then, by Eg. (2.43b). the time

inter-val recorded by E or S is 6t

=

6t'/ll -

u~

(the time dilatation

effect}o Thus when 6t

=

25 years have elapsed for E and S, both

agree that the astronaut has aged by 6t'

=

0 08 x 25

=

20 years,

i.e. that A is now 5 years younger than they areo

c) Aging of S ca1culated by A

Just af ter acce1eration to Uo

=

006, A measures that at

t '

=

Oi Xi

=

12_i t

=

+9 (Eqo 2043b} o A concludes that the

s s

c10ck on the star-planet is 9 years fast, or. to put it another

way, that 9 years have suddenly gone by on the star during

his brief accelerationo The Brehme di agram confirms this va1ue

of t

s as the co-ordinate of the event E10 This effect is due

to the change in A' s concept of simu1taneity resu1ting from

the rotation of his space-time frame through the ang1e

e

=

sin-1uo o During the journey, Eqo (2043d) shows that if a

c10ck at x

=

x

=

15 emits signals at intervals 6t, then the

astronaut

reco~ds

the interval as 6t'

=

6t//l -

u~

(the inverse

time dilatation effect}o Thus, af ter 6t'

=

20 years have e1apsed

for the astronaut, he finds that 6t

=

0.8 x 20

=

16 additiona1

years have elapsed on the star. The tota1 time e1apsed for St

according to the astronaut, is therefore 9 + 16

=

25 years.

Thus, when A and Smeet, both agree that A has aged 20 years

d) Aging of E ca1cu1ated by A

Now let us examine the astronaut 9_{s impression of the}

time e1apsed on earth o Just af ter takeofii at t ' = 0, a c10ck

on earth (at xE = 0) reads tE = O~ according to Eq. {2.43d} •

. During the voyage to the starp the e1apsed time At at xE = 0 is

re1ated to the

At

~

of the astronaut by At = /1 -

u~At·.

according to Eq. {2.43d}. Hence when A reaches the vicinity of the stari

he finds that E has aged by At = 0.8 x 20 = 16 years. This resu1t can a1so be obtained direct1y from the trans format ion equations; at ti = 20~ xE = Ot Eq. (2043d) yie1ds tE = 16i and Eq. (2.43c) gives x~ = -12, the Lorentz-contracted distance back to earth. These numbers correspond to the co-ordinates of the event E~ in the Brehme diagram of Fig. 6b (conditions on earth (x=O) at t'=20). During dece1eration. AiS frame EV

rotates back through the ang1e e~ and the time tE = 16 on earth which previous1y appeared to be simultaneous with t l = 20 is

changed to tE = 25. Thus

9

additiona1 years slip by on earth due to A9_{s changing concept of simu1taneity. When A lands on}

the star9_{s p1anet}

p he returns to the frame Ei and finds the

time on earth i s tE = tE = 25 i just as S does. A conc1udes that

E has aged 16 +

9

= 25 years i whi1e he has aged 20 yearso This

agrees with Egs conc1usion and there is no paradox.

e) The return journey

The return journey to earth (see Fig. 6c ) is just the reverse of the one to the star. E and Sage an additiona1 25 years, A ages 20 years. A returns home af ter a tota1 of 40 years of his time, E and S have aged 50 years~ and everyone agrees that A is 10 years younger than E and SQ A reaches the

earth again at event E4t which has the co-ordinates x = O~ t = 50~ Xl = -3705~ t~ = 620 50 The latter two numbers may

require some exp1anationo They are the co-ordinates of A when he again reaches the earth~ measured in the frame Eg, which has continuous1y moved at ua = 0.6 in the positive x-direction.

According to the Einstein velocity transformation. Eqo (2.14)~ when A has a velocity

Uo

= -0.6 in E during the return journeYi