a hi i

rotating systems in general relativity

by

RUNE VALLE

THESIS

for the degree of

MASTER OF SCIENCE

Theoretial Physis Division, Department of Physis

Faulty of Mathematis and Natural Sienes

University of Oslo

June 2009

This thesis might be regarded as a review over a subjet that has a history

of ative researh of more than 100 years. However, it diers from many

otherreviews inertainrespets. I have madeeorts tonot onlypresentthe

results, but alsohow they were found. The hopeis that this willbe enough

to get a deeper understanding of the results, and that it mightexpose ways

to extend them. I alsohave made a serious eortto keep the mathematial

levelassimpleaspossiblewithoutthelossofpreisionthatoftenisassoiated

with suh popularisations. My own ontributionhas mainlybeen toprovide

my own interpretations,examples and some suggestions where appropriate.

There are three setions I want to mention espeially: The rst two are

those that over two very reent results. One of those is the improved data

analysis of the gravity probe B experiment detailed in 3.3.3. The other is

Shmid's result on linear perturbations on FRW-universes that is presented

in4.1. FinallyIwouldliketomentionthesetionalulatingdraggingeets

inasimplegalaxymodel2.3. WhileIameverpresentthroughoutthis thesis

in seleting, rening and ommenting on works of others, this isthe setion

where I trulyfeel that I ampresenting work that isentirely my own.

Thistextisprobablybestusedasanintrodutiontotheeldinquestion,

or as a reading ompanion to the main artiles presented in this thesis. It

mayalsobereadmorelightlyasasimpleoverviewof thehistoryofthe more

reent researh on an engagingphilosophial problem, or as a seond point

of viewfor those already familiarwith the eld.

This thesis is arranged partially historiallyand partiallybased on om-

plexity. The rst hapterisasimple introdutionnarrowing thefous ofthe

restofthethesiswhileprovidingsomehorizonsforfurtherstudy. Theseond

hapteronly examinesthe simplestdeviationsfromspeial relativitytheory.

The third hapter extends on this, going to more ompliated systems, but

stillkeeping theMinkowskiboundary. Finally inthe fourthhapter thease

of entire universes are treated. The lasthapter is justa short wrapping up

of the previous hapters.

Most of this text should be possible to enjoy for anyone having lower

grade ourses in basi mehanis and vetor eld theory. I also assume su-

perial familiarity with the main onepts of the general relativity theory

like the metri tensor and the eld equations. Full understanding will how-

ever demand some more advaned lassial mehanis and familiarity with

ertain analyti methods. The exeption is the setion on galaxy rotation

2.3. Here some numerial methods and programming is used. This setion

ishowever not neessary forenjoyingthe rest of the thesis.

Inorder tobeasuseful aspossibleasa readingompanion Ihavemostly

preserved the notation of the soures formulas are based from. Exeptions

are notedinthe text. This willbeexplainedinthe relevantsetions. I use a

fewommononventions I would liketomention here: I use Einstein'ssum-

mation onvention.

g µν

^{is the metri tensor.}

T ^µν

^{is the} energy-momentum tensor. The time like omponent is the

0

^{-omponent of tensors. Greek in-}

dies represent all 4 dimensions, while Latin indies mark only the spatial

omponents.

Of partiular note is it that there are dierent onventions onthe gravi-

tationalonstant. Some use Newton's, while others use that of Einstein. In

addition,itisquiteommontouse theonvention that setthe speed oflight

and the gravitational onstant (Newton's) tounity.

I would like to thank my supervisor professor Øyvind Grøn for all his

help, and my familyfor support and feedbak. Also a bigthank toallthose

books,artilesand webpagesthat haveservedasinspirationand shaped my

view of this amazing subjet. Not nearly all of them did nd their way to

the bibliography, asthey did not diretly relate toany of the ontent.

1 Introdution 1

1.1 Mah's priniple. . . 1

1.1.1 What Mah said . . . 1

1.1.2 Interpretations of Mah . . . 2

1.1.3 First usageof the term . . . 4

1.1.4 Present formulations . . . 4

1.2 Alternatives toRotation . . . 5

1.2.1 Boundary onditions . . . 5

1.2.2 Requirementfor determinability . . . 6

1.2.3 Absoluteelements. . . 7

1.3 Alternatives togeneral relativity . . . 8

1.3.1 Restritions of solutionsto eld equations . . . 8

1.3.2 Einstein-Cartan theory . . . 9

1.3.3 Siama . . . 9

1.3.4 Brans-Diketheory . . . 10

2 Gravitomagnetism 11 2.1 The fundamentalformulas . . . 11

2.1.1 Simplemotivation. . . 11

2.1.2 Linearized generalrelativity . . . 13

2.1.3 Gravitomagneti equations . . . 14

2.2 Examples . . . 17

2.2.1 Classiallaws . . . 17

2.2.2 Fore strengths . . . 19

2.2.3 Gyrosopes . . . 20

2.2.4 Inside ring . . . 21

2.2.5 Hollowinnite ylinder . . . 24

2.3 Rotating galaxy . . . 25

2.3.1 Method . . . 26

2.3.2 Results. . . 29

2.3.3 Conlusions . . . 32

3 Asymptotially Minkowski spaes 37 3.1 Minkowski universe . . . 37

3.1.1 Rotatingobserver . . . 38

3.2 Inside ahollowshell . . . 41

3.2.1 Thirring . . . 41

3.2.2 Brill-Cohen . . . 44

3.2.3 Psterand Braun . . . 47

3.2.4 Revisitingthe rotatingylinder . . . 50

3.3 Outside rotatingbodies. . . 51

3.3.1 Approximate solutions . . . 51

3.3.2 The Kerr metri. . . 55

3.3.3 Gravity probeB. . . 57

4 Universe models 61 4.1 FRW/Shmid . . . 61

4.1.1 FRW universes . . . 62

4.1.2 Linearperturbationon FRW. . . 63

4.1.3 Eigenelds of Laplaian . . . 65

4.1.4 Perfet draggingin perturbed FRW . . . 67

4.1.5 Summaryand onlusions . . . 69

4.2 Rotating universes . . . 71

4.2.1 Goedel Universe . . . 72

4.2.2 Ozsváth and Shüking . . . 74

4.2.3 Gravitationalwaves solution . . . 75

4.2.4 Spinning partilessolution . . . 78

5 Conluding remarks 81

A Soure ode for galaxy model 83

Introdution

I will in this hapter give an introdution to the topi of this thesis, both

historially and oneptually. This I will do by starting at the parts of

the title and desribing those in more detail, in addition to other possible

approahes to the problemat hand.

1.1 Mah's priniple

I will in this setion give a short historial and philosophial introdution

to how the term "Mah's priniple" ame to be, and give a short overview

of possible meanings. In the later setions I will narrow down the sope of

the rest of this thesis. This isneessary as Mah's prinipleitself isa far to

broad onept for me to serve it justie to in the limited time and spae of

a master thesis. The historial treatment is primarily based on Norton [37℄

and Hoefer[23℄

1.1.1 What Mah said

Mah's prinipleis the name given to a very looselydened onept that is

attributedtothe physiistErnst Mah. Oneofthe keyquotes fromhimthat

has lead to this onept being attributed to him is a ritique of Newton's

buketexperiment. Inthis experimentNewtononsiders abuketlled with

water, initially held at rest. He observes that the water has a at surfae.

He then starts to rotatethe buket around itshorizontalaxis. After a little

while the water is moving toward the edges, so that it is shallower in the

middle than toward the sides. This he explains by referring to a entrifugal

eet that arises when the water in the buket start to rotate with respet

toabsolute spae. Mah's answerto this is [33℄:

Newton's experiment with the rotating vessel of water simply informs

us that the relative rotation of the water with respet to the sides of

thevesselproduesnonotieableentrifugalfores,butthatsuhfores

are produed by its relative rotation with respet to the mass of the

Earth and the other elestial bodies. No one is ompetent to say how

the experiment would turn out if the sides of the vessel inreased in

thikness and mass tillthey were ultimatelyseveral leaguesthik. The

one experiment onlyliesbefore us, and our business is,to bringitinto

aord with the other fats known to us, and not with the arbitrary

tions of our imagination.

Thisquote shouldbe seeninthe ontextthat Mahinhistextadvoates

the viewthat allobservations isof howdierentbodies relatetoeah other.

Heneitisproblematieventotrytodeneaoneptsuhasabsolutespae.

1.1.2 Interpretations of Mah

Exatly what Mah wanted to say with this quote has been up to some

speulation. One possibility seems to be that itis an emphasis of the point

that wean't know anything about situationswe an't observe. In this ase

themainmessageofMahseemstobeaallforaredesriptionofthephysis

so that it only was desribed as how bodies move in relation to eah other

with noreferene toabsolute spae. This may atually be done even within

the framework of Newtonian physis under the simple assumption that the

universeitselfisnot rotatingwithrespettosuharealabsolutespae. This

isfor instane shown by Donald Lynden-Bell in [32℄.

A seondway toreaditisthat he isproposingthat thereouldbesome-

thing other than absolute spae that determines the outome of Newton's

buket experiment. The problemis that if this isthe ase, he isgivinglittle

suggestions astowhat and how, exept that itshould have something todo

with howmatter movesin relationto eah other. One striking thing isthat

if this interpretation is right, then he is very vague about it ompared with

someof hisontemporaries. Forinstane thebrothers ImanuelandBenedit

Friedlaenderpresented a paperin1896 desribing anexperimentthat would

attempt to determine if the rotation of the Earth had any modifying eet

onthe lawof inertia. They were however unabletond any deviationsfrom

Newton'smehanis, onsideringtheir error margin.

But why should there be any reason to searh for fators that might

hange the outome of Newton's buket experiment? There are two im-

portant somewhat distint lines of reasoning that lassially seem to reah

the same onlusion, but in later times have turned out to give quite dif-

ferent ways to approah the problem. The rst is an argument onerning

the aesthetis of ausality: Aording to Newton's mehanis - If you know

the relative distanes and veloities of all bodies in the universe at some

time, you know almost enough to determine how the system will evolve at

all times. What is required to make the system ompletely determinable

seems ridiulouslylittleomparedwith the huge amount of informationyou

have onthe universe by then. One way is to put these bodies into aframe-

work like that of Newton. Another way is simply stating that the universe

isnot rotating,ormore generalgivingan axisandmagnitude of rotation. It

should be possible to determine this axis by observation by observing a few

of the double-dierentials of the relative positions of the matter. But even

when this extra information is available,a theory where this it wouldn't be

neessary would seem leanerthan Newton's.

The seond line of reasoning is similar to that above, but stops before

observing the double-derivatives. One should rather note that this extra

needed information seems arbitrary. Why should it be so that a single axis

of rotation should be so important for being able to ompletely desribe

nature? Could this rotation axis really be totallyarbitrary, or is it possible

that it is atually determined by the relative distanes and veloities of the

bodies inthe universe?

There is one important observational fat that has been used to argue

that it is unlikely that what has been alled absolute spae is independent

of the masses of the universe: That suh an absolute spae seems to be

unaelerated with respet to the "xed stars". Consider Newton's buket

experiment. When we are standing on the Earth, nearly at rest relative to

thexedstars, weobservethewaterlimbingtheedgeswhilewearerotating

the buket. Weare proneto arguethat the reasonfor this is that the water

inthe buket is rotating,and hene itexperienes a entrifugaleet. If we

ontheotherhandsitinsidethebuket, westillseethe waterbeingshallower

inthe middlethan farther out. But the water and the buketis not moving

relatively to us in this ase. It is simple to laim that we are experiening

this beausewe are rotatingourselves, but howan we say? If youlookup,

maybe youan see the stars raing aroundthe sky athigh speed. Wouldn't

itthen be plausiblefromyour point of viewtolaim that the reason forthe

water moving away from the entre atually is that the stars in the sky is

rotatingaround it?

1.1.3 First usage of the term

Regardless of motivation, it is the last interpretation that has beome the

mainidea ofwhatistoday alledMah's priniple. WhenMahwas solittle

lear about this himselfone might wonder how this priniple ame to bear

his name? This is mostly attributed to Albert Einstein. He rst used the

term inhis paper ongeneral relativityfrom 1918 [18℄:

Das G-Feld ist restlos durh die Massen der Körper bestemmt. Da

MasseundEnergienahdenErgebnissenderspeziellenRelativitëtsthe-

orie das Gleihe sind und die Energie formaldurh den symmetrihen

Energie-tensor(

T µ ν

^{) beshrieben wird,sobesagtdies, dass das G-Feld}

durhden Energietensor der Materie bedingt und bestimmtsei.

Thisdenitionishowevernotstandingverystrong. ItseemslikeEinstein

during the period 1912-1918 had some idea he attributed to Mah that he

really wanted the theory he was working on to satisfy. But his atual for-

mulation of this idea was hanging over time. This denition doesn't stand

muhstrongerwhen oneonsidersthat Einsteinhimselfmoreorlessgaveup

theentire idea thesummer1918. Thebakgroundfor thiswas thendingof

the de Sitter spae that was an empty-spae solution with the osmologial

onstant. As it is hard to argue that the G-eld is then aused by some

matterdistribution the generaltheory ofrelativity doesn't seem tofullthe

abovegiven denition.

1.1.4 Present formulations

Even though Einstein's formulation of 1918 isn't very popular, the term

"Mah'spriniple"hasbeen muhusedintheliteraturewithothermeanings

sinethen. But therehas been noommononsensus astowhat the preise

meaning of the term should be, and thus it has been used with quite a

fewdierentmeaningsdependingonthewriter. Commonisthatitsomehow

triestograsptheideasgivenbytheseondinterpretationoftheMahquote.

Several attempts have been made to ollet the dierent uses of the term,

for instane in[21℄, the index of [25℄ and in [7℄.

As several of these denitions fall outside the sope of this text I will

here only listthose formulationsof Mah's prinipleI'll work with, for easy

referene. Common for all of them is that it tells us something about how

things faraway have loaleets.

•

^{Formulation1: The universe isspatially losed.}

•

^{Formulation2: There is nothingthat ats that is not ated upon.}

•

^{Formulation 3: In the rest frame of any body the total} gravitational eld on the body arising from all the other matter in the universe is

zero.

•

^{Formulation4: Massesshouldsomehowdeterminetheinertialsystems.}

•

^{Formulation 5/6: The inertial systems should be} partially/ompletely determinedby the masses of the universe.

•

^{Formulation7: Theaxesofinertialframesareperfetlydraggedaround}

by a weighted average of the motion of partilesinthe universe.

Finally I will add a formulation that I have not enountered anywhere,

but that will be onsidered briey later by me as it seems to be a possible

interpretation.

Formulation x1: Mah's priniplesays that the boundary onditions are

tobedetermined by loalbehaviour.

1.2 Alternatives to Rotation

Inthe previoussetionIonsideredMah's prinipleingeneral. Mostofthis

textwillasthetitlesuggests fousonrotationalaspetofthepriniple,butI

willdevote thissetiontoashortoverview ofsomeotherpossibleapproahes

toMah's priniplethat doesn't diretly involverotation.

1.2.1 Boundary onditions

When examining how things far away may aet loal physis it may be

interesting toexaminethe asewhere"faraway" goestothelimitofinnity.

Inatheory governedby eldsanddierentialeld equationslikethegeneral

theory of relativity this translates to boundary-onditions of the equations.

Aording to[23℄ even Einstein himselftried this approah forsome time in

1916-1917.

I an see major ways that the boundary-ondition problem may be at-

temptedrelatedtoMah's priniple. TherstistodeneMah'sprinipleas

the boundary-onditions that give us the loalbehaviour we observe inthis

universe. TheotheristobeginwithsomeotherformulationofMah'sprini-

pleandseeifthatposesanylimitationsonwhatkindofboundary-onditions

an be allowed. Neither of theseapproahes has proven very fruitful. I have

found no examples of the suggested denition in the literature. I an see

several possiblereasons for that:

•

^{It doesn't inorporate any relevane to things loser than innity to}

Mah's priniple, whih breaks with the ommon idea attempted to

put intoMah's priniple.

•

^{It has little or no physial signiane as more than a} self-fullling requirement to the boundary onditions.

•

^{It is hard to do the alulations involved with it, and it may ome in}

onitwith the desire of having ontinuity/onvergene.

To nd boundary onditionsthat t anidea of Mah's priniplehas also

proven most diult oreven impossible. A goodillustrationof howdiult

this seems is that one of the main formulations of Mah's priniple is that

the spaeis spatiallylosed. This formulationdates bak to Albert Einstein

in1917 [23℄. In this ase the need for boundary-onditions disappears. One

majorargumentforthisdenitionisthisproperty. Andinertainframeworks

(mostnotablygeneralrelativity)thisdenitionalsoturnsouttodiretlylead

toseveral eets thatare onsidered Mahian. Andeven inotherdenitions

ofMah's prinipleitistempting tohavespatiallosureas arequirementto

avoidthe boundaryproblems.

1.2.2 Requirement for determinability

In 1.1.2 it was argued that inNewton's theory we need to know all relative

positions,veloities andsomethingelse atagiven timeinordertodetermine

how the system evolves indenitely. I also provided a sketh of why this

somethingelse was undesirable. Toonvert this notiontothe generaltheory

ofrelativityprovesdiultasitoperateswithelds,not partiles,and there

areissuestrying todene"agiventime". Itistherebyofinteresttoexamine

whatinformationyouneedinordertobeabletodeterminetheonguration

of the entire spae-time.

One suh formulation that an be onsidered important in relation to

Mah's priniple is the thin sandwih onjeture proposed in [3℄. This on-

siders the intrinsi geometries of two spae-like surfaes lose to eah other

(nearlyalike). Inthisase the dierenebetween thesespaesbehaveslikea

derivative. Inthe generaltheory ofrelativityitturnsout thatthis shouldbe

enoughtodetermine the geometryofthe entire 4-spae. Thisis very similar

to the lassially formulated wish that the physis shouldbedetermined by

relativepositionsand their rstdierentialsalone,withoutany extrafator.

JulianBarbourandBrunoBertonidevelopsthisideafurtherin[4℄. Thisis

nielyexplainedin[5℄. Hereitisnotposedanyompat denitionofMah's

priniple. The main dierenefromthe above argument ishowever thatthe

terminology is sharpened and generalized. The required knowledge should

onlybeapointinaphase-spaeof geometries,andadiretion. Appealingto

thethinsandwihonjetureitislaimedthatgeneralrelativityisompletely

Mahian. One interesting idea that is proposed is that we only require the

thin sandwih onjeture to be applied loally, at every point, not globally.

This way it seems like one may avoid the problems related to boundary-

onditions even inuniverses that isn't spatially losed.

1.2.3 Absolute elements

Anotherapproahistosetthefousatthe"absoluteness"ofabsolutespaeof

Newtonian theory thatMahseems toprotestagainst. This is donein some

formality by Jürgen Ehlers in [17℄. Here he attempts a denition of Mah's

priniplegoing something along the line"There is nothing that ats that is

not ated upon". Newton's absolute spae is suh a thing that determines

howthings move,while nothing may hange that spae.

He thenompares dierent theorieswithregard towhatgeometrial and

physialpropertiesofasystem ittakesintoaountandgoverns. Heshows a

generaltendenythatthegeneralrelativitytheoryhasfewer"Absoluteelds"

than the speial relativity theory, and that the speial relativity theory in

turnhasfewerthanNewton'stheory. Thoseeldsthatarenolongerabsolute

inthe moregeneraltheoriesarefound asdynamialeldsthatareintimately

onneted with the other elds of the theory. In partiular this involvesthe

metriand onnetion-elds,in additiontoa oneivable "Ether eld".

The denition of what may be onsidered a eld in a theory, and how

todetermine/deneabsolutenessishowevernot verywellexplainedhere. In

the disussion found in the proeedings after the paper [17℄, Karel Kuhar

points out a possible absolute element in the underlying geometry of the

generalrelativitytheory. Ehlers aknowledgesthis, but says he feelsthere is

afundamentaldierenebetween this and the elementshe has onsideredin

his paper. He was however unable to formulate this dierene. I have not

found any more reent treatment of this approah.

One extension of this idea is also to look at the onstants of a theory.

Shouldthese be onsideredelds of the theory? In this ase, shouldthey by

Mah's priniplenot be true onstants, but somehow be determined by the

physialstate? This and similar onsiderations have been raised and led to

several theories that laim to t better with Mah's priniple than general

relativity. I willgive these some treatmentin the next setion.

1.3 Alternatives to general relativity

Thereare lotsoftheoriesofgravitationthatsomehowaddressesMah's prin-

iple, and even the spei questionof rotationrelated toit. Many of these

are intimately related to the general relativity theory as an extension, gen-

eralizationor restrition ofit. I willinthe remaininghapters onlyonsider

basi general relativity (and its standard lower order approximations). In

ordertonarrowdown andspeify thesopeof whatI willhereonsider, and

asI feelitdeserves mentioningin areview regardingMah's priniple,I will

here say abit about some of the more proledtheories that I amnot going

toover inthe later hapters.

1.3.1 Restritions of solutions to eld equations

Einstein's eld equations do have solutions that by some have been hara-

terized as "un-Mahian". I will get into some of these in later hapters. A

way to deal with this ould be to nd some onditions that have to be ap-

pliedinadditiontothe usual eld equationsthat ruleout suhsolutions. In

partiularthis ouldbe relatedtosetting boundary-onditionsasmentioned

inthe previous setion.

Onlyallowingloseduniversesisalsoanexampleofthis. AsfarasIknow

only the restrition to losed universes has been somewhat suessful, and

this has the major problemthat itis anopen question whether the universe

atuallyislosed. Someoftheproblemsarediretlyrelatedtothelakofany

stritdenitionof"Mah'spriniple"andheneitishardtoagreeonexatly

what solutions should be ruled out. Formulatingboundary-onditions faes

similarproblems, but is alsomade diultby the mathematialomplexity

involved.

I will in the remaining hapters use the full general relativity without

restritions. This way I willalsobe able tostudy some of the more dubious

solutionsseen from a Mahian perspetive and examine rotationaleets in

them.

1.3.2 Einstein-Cartan theory

Einstein-Cartan theoryis thenaturalextensionof generalrelativitytoallow

for spinningmasses. The basis are given in areview artile from 1976 [22℄.

The theory owesits name in part toÉlie Cartanwho in the rst halfof the

1920smade somebasiwork ondierentialgeometryrelatedtotorsion. But

as afull theory itwas only developed later.

As atheorythat allowsfor spinthis theoryouldbehighlyinterestingin

the ontextof investigatingrotationalphenomena. Thefat thatthere isan

extension to general relativity allowing spinning masses shows that general

relativity operates with non-spinning masses. This I will use to pose some

qualitative suggestions on physial interpretation on some systems in 4.2.4.

Togiveaproperanalysisofspin-eetswouldhoweverrequirethisframework

and hene fall outsidethe sope of this thesis.

1.3.3 Siama

Inhis1953artile[53℄Siamaoutlinesasimpliedtheory thatisbasedupon

the quite ommon view that Mah's priniple tells that inertia should be

determined by matter. This is made more aurate in this quote:

In the rest frame of any body the total gravitationaleld atthe body

arising fromall the other matterin the universe iszero.

He then sets out todemonstrate atoy-theory that shows howthis might

get implemented. He assumes for simpliity that gravitation is governed by

a vetor eld in a Minkowski spae. He points out that the gravitational

potentialatually has to be a seond rank tensor, and that this modelthus

onlyis illustrative.

The result is a model with some similarities with eletromagnetism. A

omparison between this and the gravitomagnetism desribed in the next

hapter ould be interesting, but falls outside the sope of this text. There

is however one importantresult here, namely the relation:

Gρτ ² ≈ 1

^(1.1)

Where

G

^{is the} gravitational onstant,

ρ

^{is the density of the universe, and}

τ

^{is the age of the universe. The} approximation should be onsidered very

"oarse" only meaning"inthe order of".

In his paper he ontinuously refers to a "subsequent paper" where he is

supposed to develop this theory in a muh more realisti manner. I have

however been unable to nd this referene, or anyone referring to suh an

artile. In 1964 Siama seems to be working in the framework of general

relativity, with possible extensions and restritions [54℄. The equation 1.1

stillseemedto be entral inhis idea of Mahianitythen, however.

1.3.4 Brans-Dike theory

TheBrans-Dike theory wasrst presented ina paperbyBrans and Dikein

1961 [11℄. This theory is based on the idea that the gravitational onstant

ould indeed be dierent at dierent plaes determined by the mass distri-

bution. They give two importantmotivations for the gravitational onstant

tobe non-onstant.

The rst isthe relation1.1 somewhat rewritten:

GM/Rc ² ≈ 1

^where

M

isthe visiblemass ofthe universe,

R

^{isthe radiusof the visibleuniverse and}

c

^{isthe speedof light. This relationifsolved withrespet to}

G

^givesanidea

of howthis quantity ouldbedetermined by the mass inthe universe.

The seond is the dimensionlessnumber

m e (G/¯ hc)

^where

m e

^{isthe ele-}

tron rest-mass. This has asize that ismathematially simplyrelatedtotwo

seeminglyunrelated observed and varying numbers: The age of the universe

in atomi time units and the mass of the visible universe in proton masses.

Wanting to keep

m e ¯ h

^and

c

^{onstant the remaining fator that an be ad-}

justed totake this intoonsideration isG.

Theytherebyonstrutedatheoryformulatedinsimilartermsasthegen-

eraltheoryof relativity,butwith asalareldnotpresent inthe other. This

theory is alsodetermined by a parameter that has tobeset by observation.

This makesithard tofalsify,but there has been set ratherstrit onstraints

onthe free parameter of the theory by the Casini-Hugens experiment [6℄.

Gravitomagnetism

As said in the introdution, Mah's prinipleonerns how objets far away

mayaetertainexperimentsloally. OnesuhexampleisNewton'sbuket.

In Newton's theory, if you have a situation where the stars are rotating in

the universe around a buket that stands still (relative to absolute spae),

then the water in it stays at. There are no entrifugal, or "inertial"fores

thatgivetheresultthatthewater movesuptoward thewall. Onemayargue

that this situation should be equivalent to the situation where you have an

observer sitting inside a rotating buket observing the universe. Hene we

should look for some eet that makes the water in the buket urve in all

possiblesenarios wherethe universeisrotatingrelativetoit. Suh aneet

may atually be found in general relativity and is gravitomagnetism. This

hapter willoverthis phenomenon in simpleloalsystems.

2.1 The fundamental formulas

Iwillinthissetiondeduetheequationsofgravitomagnetismfromlinearized

general relativity. I will start by giving a simple argument from speial

relativity that shouldmotivate that there issuh aneet. After that I will

go through the more detailed and aurate alulation of the equations for

gravitomagnetismin linearized generalrelativity.

2.1.1 Simple motivation

Iwillherepresent anargumentthatmaymotivatethe existeneofagravita-

tionaleetwithsimilaritytoeletromagnetisminarelativistitheory. This

is inspiredby adesription of eletromagnetismattributed to E. M. Purell

as desribed in [52℄. In the given referene one onsiders a partile moving

alongawirearryinganeletrialurrent,andarguesthatdependingonthe

frameof referene the fores ating on the partile may be seen upon as an

eletrioramagneti eld. I willhere simplifythis toaless realistisystem,

but one that is simplerto relate tothe gravitational ase.

Consider a negatively harged partile initiallyat rest beside an innite

positively harged wire. In this ase we know from lassial eletrostatis

that there is an attrative fore between the partile and the harged wire.

If we however hanges refereneframe to one moving at aonstant veloity

relative to the rest frame of the partile, parallel to the wire, the partile

is moving as an eletrial urrent in the wire in the same diretion as the

initialveloityof the partile. Aording tolassialeletromagnetism there

is then a magneti fore that pushes the partile away from the wire. As

the partile has to behave similarly in both frames of referene one needs

an eet that makes up for the eet of the magneti fore. Suh an eet

an be found in the speial relativity theory. The length ontration of the

wire inthe moving referenesystem relative tothe initialrest system of the

partile makes the harge density higher. Thus we get a stronger eletri

forethat anels the eet of the magneti fore.

One an argue that this argument laks several fators that may modify

the relation between the magneti and the eletri fores like relativisti

time dilation and mass inrease. The key point that the length ontration

makes a net inrease in eletri fore is better founded in Purell's original

treatmentasitis theredemonstrated howone may gofroma framewith no

eletrial, onlymagneti fores, to aframe withno magneti, onlyeletrial

fores bya simpleveloitytransition. I would alsoliketomentionthe paper

[16℄ where an attempt is made to develop the entire eletromagnetism in a

similarwayfromonlyspeialrelativityandeletrostatis,even thoughIhave

been unable toverify whether this paper istrustworthy.

So, keeping in mind that Lorentzontration may give frame dependent

foresIturntheattentiontoasimilargravitationalmodelastheeletromag-

neti ase examined above. Wenow have anunharged partileand a wire.

Inthe rest-frameweknow thatthere isaertaingravitationalforebetween

these. In a moving frame one may expet a stronger gravitational fore as

the mass-density of the wire inreases due to length ontration. Opposite

tothe above ase we then seek aneet that opposes this inreased forein

theframe, and onemightbetempted tosuggestthatthere isagravitational

ounterpart tothe magneti eld.

Tomakeanyformalalulationsonthisishoweveroflittleinterest. There

areseveralothereetsthat playintothis piture. Mostimportantisproba-

blythespeialrelativistinotionof inreasedinertialmassunderhighvelo-

itiesthat I suspet may be enough togive aomplete explanation model of

the presented ase withouthavingtorefer toany kindof "gravitomagneti"

onept at all. In addition omes the question of how to formulate gravi-

tation in a relativisti framework, whih is exatly what general relativity

does.

What I want toshowin this setion ishowever that itshouldn't bevery

surprising when it turns out that general relativity atually displays eets

very similar to eletromagnetism, and point out one idea that might give a

understanding of how this dierene fromNewtonian physis mightarise.

2.1.2 Linearized general relativity

The theoryof gravity thatwe getbylinearizingthegeneralrelativitytheory

maybetraedbaktoEinstein'spaperin1916aordingtoforinstane [21℄.

After that it has been treated in several works. I will here go through the

main points inthe derivation fromgeneral relativity following the approah

given in[36℄.

Consider the situationwhere the metri may bewritten inthe form

g µν = η µν + h µν

^(2.1)

where

η

^{is diagonal with signature}

− + ++

^{, that is the metri of the}

Minkowski spae in standard oordinates. We also assume for simpliity

that

c = 1

^.

h

^{is a small} perturbation of this metri, with small derivatives and seond derivatives. This gives us a weak-eld universe, that is without

any high densities orrelativistiveloities.

The onnetion oeients may then be written:

Γ ^µ _αβ = 1

2 g ^µν (g αν,β + g βν,α − g αβ,ν ) ≈ 1

2 η ^µν (h αν,β + h βν,α − h αβ,ν )

^(2.2)

In the approximation we have omitted produts between the perturbation

and it'sderivatives, and used that

η

^{is onstant.}

AsweareatleastlosetoaoordinateframewehavefortheRiitensor:

R µν = Γ ^α _µν,α − Γ ^α _µα,ν + Γ ^α _βα Γ ^β _µν − Γ ^α _βν Γ ^β _µα

^(2.3)

Intheapproximationthe twolasttermsare negletedasseondorderterms.

Theindiesfrom2.2israisedusing

η

^{insteadofusing the fullmetri}

g

^{. This}

isalso donewhen alulating the Riisalar

R = g ^µν R µν ≈ η ^µν R µν

^(2.4)

It turns out that the eld equations take a partiularly nie form if we

introdue

h ¯ µν = h µν − ¹ ₂ η µν h

^where

h

^{isthe ontrationof the} orresponding tensor. Then we may impose on the system the following ondition due to

freedom of hoie of oordinatesystem:

¯ h ^µα , α = 0

^(2.5)

Fixingoordinateslikethisisalledtoimposeagaugeonditionandthison-

dition is analogous to the Lorenz gauge

A ^α , α = 0

^of eletromagneti theory.

The eld equationsthen beome

− 2 ¯ h µν = 2κT µν

^(2.6)

This equation along with the gauge and the expressions for the metri

and

¯ h

^{formsthe basis for the linearized theory of} relativity.

2.1.3 Gravitomagneti equations

Aording to [35℄, Einstein suspeted a relation between his eld equations

and Maxwell's equations for eletrodynamis. It is laimed inthis referene

thatThirringdid apaperonthis in1918,but Ihaveunfortunately not been

abletogetholdofthisreferenetoseehowfarthiswasdone. Inafootnotein

therstartileinthistranslationpaper,hedoeshoweverstronglysuggestthe

orrespondenes desribed in this setion. It is worth tomention that there

are other approahes that give similar equations. In 1977 a general version

of Maxwellian relations was found in [10℄ that was based on parameterized

post-Newton formalism whih is a formalism to desribe a broad lass of

theoriesthat inludegeneralrelativity. However, this fallsoutside the sope

of this text.

TheapproahI willtaketoshowhowonemayrelatethe linearizedequa-

tion with Maxwell'sequation is inspiredby [21℄, [56℄, [34℄ and [57℄.

In eletromagnetism we know that

2A ν = µ 0 J ν

^{along with the Lorenz}

gauge, where

A ν

^{is an} eletromagneti four potential, gives us Maxwell's equationsinstandardform. Ifollowthesamereasoningasineletrodynamis

and restrit the attention to the

¯ h _0α

^{terms. This even give us diretly the}

orret

c

^{dependeny. We an dene}

E ~ G

′ = − c ∇ h ¯ ₀₀ − d ¯ h 0i

dt

^(2.7)

B ~ G

′ = ∇ × ¯ h 0i

^(2.8)

where

h ¯ 0i

^{denotes the normal 3-vetor} orresponding to the usual vetor potential.

The eld equations then take the familiarMaxwell-equation form:

∇ · B ~ G

′ = 0

^(2.9)

∇ · E ~ G

′ = − c ² κT 00

^(2.10)

∇ × B ~ G

′ = − 2κT _0i + 1

c ² d ~ E G

′

dt

^(2.11)

∇ × E ~ G

′ = − d ~ B G

′

dt

^(2.12)

Here

µ ₀ J ν

^{is replaed by}

− 2κT _0ν

^{fromthe standard} expression.

Wehaveherefoundsomequantitiesrelatedtogeneralrelativitythatobey

anequivalentofMaxwell'sequations. However, apartfromtheirounterparts

ineletrodynamis,

B _G ^′

^and

E _G ^′

^don't immediatelyhaveany simplephysial interpretation. They are here simply dened so that they behave in the

desiredway. They arethus oflittlephysialinterestyet. Theresult aboveis

thus onlyto be seen as a step in a alulation that willeventually lead to a

physiallyinteresting result.

We leave

B _G ^′

^and

E _G ^′

^{for now and rather turn our attention to a simple}

physial system. Consider the ase where

¯ h ij = 0

^{, that is all non-zero el-}

ements of

¯ h

^{an be found as}

¯ h 0α

^{. In this ase we have from 2.6 that also}

T ij = 0

^{. This may be a reasonable modelof a perfet uid with nopressure}

and lowveloities. In this ase

T µν = ρu µ u ν

^{. With}

u ₀ ≈ c

^wehave

T ₀₀ ≈ c ² ρ

and

T 0i ≈ ρcu i = cj i

^where

~j

^is orrespondingto lassial matter ow. The produts

u i u j

^{are onsidered vanishingasboth terms are small.}

We now onsider the movement by a partile having low veloity inthis

system. It willfollowageodeti urve given by

d ² x ^µ

dt ² = − Γ ^µ _αβ dx ^α dt

dx ^β

dt

^(2.13)

Ignoringseondorderspatialveloityterms,and using

dx ⁰

dt = 1

^{and thesym-}

metry in the lower indies of the onnetion oeients allowus tosimplify

2.13 to

d ² x ⁱ

dt ² = − Γ ⁱ ₀₀ − 2Γ ⁱ _0j dx ^j

dt

^(2.14)

We are thus interested in ndingthese onnetion-oeients.

In order to keep the equations simple I again introdue

c = 1

^{. By on-}

trating the equation

h µν = ¯ h µν + ¹ ₂ η µν h

^{we get in this ase}

h = h ^α _α = ¯ h 00

whihinturn gives

h αα = ¹ ₂ ¯ h ₀₀

^otherwise

h αβ = ¯ h αβ

^{. Thenweanuse 2.2to}

alulatethe onnetion oeients interms of

¯ h Γ ⁱ ₀₀ = 1

2 (2¯ h 0i,0 − 1 2 ¯ h 00,i ) Γ ⁱ _0j = 1

2 (¯ h 0i,j − ¯ h 0j,i + δ ij

2 ¯ h 00,0 )

We now denethe vetor elds

B ~ G

^and

E ~ G

^by

E ~ G = ( ∇ h ¯ 00

4 − d ¯ h 0i

dt )

^(2.15)

B ~ G = ∇ × ¯ h 0i

^(2.16)

This gives usa movement equation of the form:

~a = E ~ G + ~v × B ~ G + a~v

^(2.17)

where

~a ⁱ = d~v ⁱ

dt = d ² x ⁱ dt ² a = 1

2 ¯ h 00,0

We see that in this ase

E G

^and

B G

^{are the elds that play exatly}

the same role in the equations of motion in the ase of gravitation as their

eletromagneti ounterparts. In addition, the denition of these elds are

very similar to those of

E _G ^′

^and

B _G ^′

^{. The equivalent of the magneti eld is}

the same. However the

E G

^{term is not quite so nie. We see that the time}

variation of the vetor-potentialplays a smaller role ompared to the salar

potentialindeterminingthe pathof the partilethaninthe eletromagneti

ase. I will here restrit attention to the stationary ase, that is

¯ h µν,0 = 0

^.

In this ase, we get preisely:

E _G ^′ = 4E G

B _G ^′ = B G

InsertingthisintotheMaxwellequations2.9-2.12whileignoringtimederiva-

tives give us afterinsertionof to makethe units right:

∇ · B ~ G = 0

^(2.18)

∇ · E ~ G = − 4πGρ

^(2.19)

∇ × B ~ G = − 16πG

c ~j

^(2.20)

∇ × E ~ G = 0

^(2.21)

We see that the main dierenes from the stationary eletromagneti ase

is that the fores behave oppositely relative to the urrents, and that the

gravitomagnetiforethatouples tomovement is4timesstronger thanthe

gravitoeletri ompared tothe orresponding eletromagneti ase.

In summary, I have here ompared two approahes at ombining lin-

earized theory with lassial eletrodynamis. The rst nds quantities in

generalrelativitythat behave aordingtoMaxwell'sequations. The seond

examinesthemovementof partilesand trytomakeitinaformomparable

to eletrodynamis. There are some referenes where this inequivalene is

poorly stated. This inlude [31℄, [56℄ and [1℄. The rst two do state that

their Lorentz forelawonlyholds in the stationaryase, and the Wikipedia

artile seems tobe based upon the rst of these due to the referene list. I

added this lariationto the Wikipediaartile at the stated retrievaldate.

2.2 Examples

In this setion, I will give some examples of simple systems where we may

use the above theory. I willalsorelate this to anidea of Mah's priniple.

2.2.1 Classial laws

From the Maxwellequations, wemay immediatelydedue two laws thatare

important in stationary eletrodynamial systems: Ampere's Law, and the

law of Biotand Savart.

The equivalent of Ampere's law is gotten by using Stokes' theorem on

2.20. It beomes:

I B ~ G · d~l = − 16πG

c I

^(2.22)

where the integral is around alosed path and

I

^{is the matter owthrough}

any surfae having the path as edge.

The equivalent of the law of Biot and Savart is trikier to dedue. It is

done in[57℄ so I willsimplyset up the main result here:

B ~ G (~r) = 4G

Z ~r − ~ r ^′

~r − r ~ ^′

3 × ~j( ~ r ^′ )

c dV ^′

^(2.23)

Here it is usual in eletromagnetism to make the substitution

~jdV ^′ = Idl

where

I

^{is the urrent through a line element of a wire}

dl

^{. However, it}

is worth noting that suh a one-dimensional redution of the gravitational

system is not without problems. The reason for this is the assumption of a

weakeld in the linearizingof the gravitationaltheory. This meansthat we

need to have a limited mass-density, and urrent veloity. In this situation

the mass urrent

I M

^{through the wire has to vanish in the limit of a one-}

dimensionalwire.

As the wire-formof the lawof Biot and Savartis very useful, I willshow

thatitis areasonableapproximationifwe are alulatingthe magnetield

far from the "wire". Consider a 3-dimensional wire divided into surfaes

S

that is normal to

~j

^{. Assume further that}

~j

^{is onstant on the surfaes and}

parallelto the wire. In this ase 2.23 beomes:

B ~ G (~r) = 4G

Z I ~r − ~ r ^′

~r − r ~ ^′

3 × ~j( r ~ ^′ )

c dSdl

^(2.24)

Ifthe surfaes

S

^{arerelativelysmallandfarfromthepointweareevaluating}

the magneti eld for we may assume

~r − r ~ ^′

^{to be onstant through the}

integration. If we then set

I M d~l = dl ^H ~jdS

^{, we get the familiar form of the}

lawof Biot and Savart:

B ~ G (~r) = 4G c

Z (~r − r ~ ^′ ) × I M d~l

~r − ~ r ^{′ 3}

(2.25)

I will add that the above argumentation may be used to alulate the

elds farfroma smallonentrationof mass with veloity

~v

^{, andtotal mass}

M

^:

B ~ G (~r) = 4G c

(~r − ~ r ^′ ) × M~v/c

~r − r ~ ^{′ 3}

(2.26)

It is alsoworth noting that 2.19 is the same as the formulafor the grav-

itational eld in Newton's theory of gravity, and hene we may use all the

results weknow from there.

2.2.2 Fore strengths

I will here set up a model in order to try to ompare the strength of the

gravitomagnetieet omparedtothatof thefamiliargravitoeletri. Con-

sider a smallspherialgravitationalsoure with mass

M

^{and speed}

v M

^{. We}

then examinethe behaviourofatest partileso farfromthis sourethat we

may onsider the distane a onstant

r

^{. From 2.26 we an see that we get}

the strongest magnetield ifwe assumethatthe test partilethenisinthe

planenormaltotheveloitydiretionofthemass-onentration. Inthisase

the magnitude of the magneti eld beome

B G = 4GMv M

r ² c

^(2.27)

From 2.17 wesee thatthe aeleration eet fromthe gravitomagneti term

beomes greatest if the test partile has veloity normal tothe eld. So we

make this assumption, and set the speed to be

v

^{. Keep in mind that in}

2.17 we have assumed

c = 1

^{so that in general units we have to divide the}

veloity by

c

^{in order to get the right units. Hene, the magnitude of the}

gravitomagneti eet to the aelerationof the test partileis atmost

a B = 4GMv M v

r ² c ²

^(2.28)

WegettheaelerationfromthegravitoeletritermdiretlyfromNewtonian

mehanis:

a E = GM

r ² = c ²

4v M v a B

^(2.29)

From these equations alone, it might seem like there is a possibility for

theaelerationfromthegravitomagnetieettobeomeaslargeas4times

thatofthegravitoeletri. However, fromthe weakeldapproximationdone

inthe linearizingwe have that

v M << c

^{, and fromthe dedution of 2.17 we}

also used

v << c

^{. So indeed the} gravitomagneti aeleration is smaller thantraditionalgravityintheseondorderof smallveloities. Thusinmost

appliations it seem like this eet is too small to be worth any attention.

However, it leads to eets that is not found in Newtonian gravitational

theories, and itmay turn out to be important ata universal sale. Just like

ordinary gravitation, it is a

r ⁻²

^{law not} "bloked" by anything and thus is long-range.

2.2.3 Gyrosopes

Intheprevioussetion,wesawthatthegravitomagnetieetofaeleration

seems to be hard to detet. In this setion, I will examine the behaviourof

a gyrosope in a gravitomagneti eld. This is of partiular interest, as we

knowthatNewtoniangravitationdoesnotaetthediretionofagyrosope.

Itturnsoutthat thegravitomagnetieetdoes. Thismaybeusedasaway

todetetthe eetwithouthavingtoworrysomuh thatthe muhstronger

gravitoeletrieet willdisturbthe experiment.

Consider a right-handed Cartesian oordinatesystem with agravitoele-

tri eld in the positive z diretion. At the origin, there is a gyrosope

with angularmomentum along the x-axis. We then see that in slightly pos-

itive z-diretion it has a veloity in the negative y-diretion. From 2.17 we

an onlude that it thus experienes an aeleration/fore in the negative

x-diretion. Similarly, in the slightly negative z-diretion it experienes an

aeleration/fore in the positive x-diretion. This adds up to a torque in

thenegativey-diretion,andmakesthe angularmomentumofthegyrosope

turn toward the negative y-diretion. A similar argument holds whenever

the angular momentum is in the x-y plane, and we an onlude that the

gyrosope is preessing around the z-axis. This is equivalent to the Larmor

preession of eletrodynamis.

The strength of the eet may be dedued from only 2.17 and lassial

rotationalmehanisaspresentedinforinstane[58℄. UsingNewton'sseond

law, the torque-formula, and the relation

~v = ~ω × ~r

^{we get that the total}

torque onthe system beomes:

~τ =

Z

~r × ρ((~ω × ~r) × B ~ G )dV

^(2.30)

where the integral is over any volume ontaining the entire rotating body.

Using the Cartesian oordinates with

~ω = (ω, 0, 0)

^,

B ~ G = (0, 0, B G )

^and

~r = (x, y, z)

^{this evaluates to:}

~τ =

Z Z Z

ωρB G (0, − z ² , zy )dxdydz

^(2.31)

Wenowapply theassumption thatthe gyrosope hasitsrotation-axisas

asymmetryaxis. Asitisthensymmetriuponhangingsigns of

z

^and

y

^we

anonludethat thez-termofthetorque anelsout undertheintegration.

Going to ylindrial oordinates so that

r ² = y ² + z ²

^and

cos θ = ^z _r

^{we get}

for the magnitude of the torque:

τ = ωB G

Z R 0

Z 2π 0

Z

ρr ² cos ² θdx r dθ dr

^(2.32)

Usingthat

ρ

^and

r

^are independentof

θ

^{due torotationalsymmetry, and}

that

cos ² θ

^is independent of

x

^and

r

^{we may separate this integral into}

τ = ωB G

I 2π

Z 2π

0 cos ² θdθ

^(2.33)

where

I

^{isthe ordinarymomentof inertia aroundthe x-axis given by}

I =

Z R 0

Z 2π 0

Z

ρr ² dx r dθ dr

^(2.34)

The remaining integral in 2.33 is well known, and may be found in for

instane[46℄. Itevaluatesto

π

^{. Asthesystem isrotatingaroundasymmetry}

axis we have

~τ = I ^d~ _dt ^ω

^{. Further, I will assume a perfet gyrosope. As we}

are working in a framework that depends on low veloities, the best way

to implement this would be to use a spherially symmetri distribution. In

this ase the above argumentation holds atalltimes. The timederivative of

the angular veloity vetor is always of magnitude

B G

2 ω

^{, orthogonal to the}

angular veloity itself and the z-axis. This means that the angular veloity

vetor isitself rotatingaround the z-axiswith an angularveloity:

Ω G = B G

2c

^(2.35)

where the

c

^{term is inserted to make the units right, and appears as is}

assumed to be 1 in 2.17. One may note that this agrees with the result

given in for instane [34℄ (up to a 2-fator due to dierent saling of the

gravitomagnetield). Heretheresultisalsogeneralizedtothesituationthat

thegyrosopehavingnon-ortogonalangularmomentum,with theresultthat

it is stillpreessing around the axis of the magneti eld. It is of partiular

interest that this result is independent of

ω

^{and the} mass-distribution, as long as the symmetry restritions are satised.

2.2.4 Inside ring

Iwillhereturnmyattentiontothesituationattheentreofarotatingringof

radius

R

^{and with a onstant angular veloity}

ω

^{relative to the bakground}

metri. We may hoose ylindrial oordinates with z-axis orthogonal to

the plane spanned by the ring, and origin at the entre of the ring. Due

to symmetry we an onlude that there is no lassial gravitational fore

at the entre of the ring;

E ~ G = 0

^{. If we further assume that the ross-}

setionof the ring vanishesompared to Rwe may use 2.25 toalulatethe

gravitomagnetield:

B ~ G = 4G c

Z R ~ × I M d~l

R ³

^(2.36)

Weseethat weonlyhavenon-zeroz-omponentsinthisintegral. Weassume

that

I M

^{is onstant, where}

I M = ωRAρ

^{. Here}

A

^{is the area of onstant}

θ

ross-setion of the ring and

ρ

^{is the mass density, both assumed onstant.}

Aswe are onlyworking with orthogonalvetors,it issimpletoalulatethe

magnitude of the magneti eld:

B ~ G

= 8πGωAρ

c

^(2.37)

It is interesting tonote that this expression isindependent of the radius

of the ring. This may seem like a deviation from the standard result in

eletromagnetism

B = ^µ _2R ^{0 I}

^{[58℄. However, in the standard} eletromagneti ase itispratialtouse the expressions for onstanturrent

I

^{, whileI here}

hold the angularveloity

ω

^{onstant. This aounts for this dierene.}

If wenow use 2.35 wesee that inthis ase:

Ω = 4πGAρ

c ² ω

^(2.38)

It isinteresting tonote that we get

Ω G = ω

^when

Aρ = c ²

4πG = 10 ²⁶ kg/m ≈ ρ U R _U ²

^(2.39)

where

ρ U

^and

R ² _U

^{are the measured} mass-density and radius of the observa- tional universe. As there are huge unertainties on these two quantities the

approximationisatbest an"inthe orderof". (Onemayuse forinstane the

ritialmass density ofthe order of

10 ⁻²⁹ g/cm ³

^{and a radius of the orderof}

10thousand million lightyears. These are in aordwith [13℄)

Testing the diretion of the preession, we nd that it has the same sign

as the angular veloity of the ring. Hene we have that if the ondition

2.39 is satised a gyrosope at the entre of the ring will onstantly point

at the same point on the ring. For other values of

Aρ

^{we still get that the}

gyrosope is preessing in the same diretionas the ring rotates relative to

the bakground. Thus, we say that the gyrosope is dragged by the ring.

2.39 issaid tobe a ondition forthis draggingtobeperfet.

Wemaynowturnour attentiontofreelymovingpartiles. Asmentioned

abovethereisnogravitoeletrieet,sothat weonlyhavetopay attention

to any gravitomagneti eets. Partiles moving parallel to the magneti

eldwillhenebeunaelerated,andloallymoveinastraightline. Partiles

movingintheplaneoftheringwithveloity

~v

^{willexperieneanaeleration}

in the ring-plane orthogonal to the veloity with magnitude

B G v

^{, where}

v = ^{|~ v|} _c

^{is normalized to be} dimensionless. Comparing with the argument in 2.2.3 we see that this means that if the partile had moved through a

onstant eld it'sveloity vetor would rotate with a angularspeed of

2Ω G

^.

This atually gives a nie onnetion between the movement of a gyrosope

and the movement ofthe free partile. Consider agyrosopepointing inthe

same diretionas the initialveloity ofthe free partile. Theinitialposition

of the gyrosope is the same as that of the partile, but the gyrosope is at

rest. During a short time

t

^{we may assume the aeleration of the partile}

to be onstant. In this ase we nd that the partile after a short time is

at a distane

r = vct

^{, and has a deviation from the original gyrosope axis}

xed to the bakground metri of

1

2 B G vt ²

^{. The gyrosope axis has however}

hangedby anangle

θ = Ω G t

^{. Thismeans thatthe pointthat the gyrosope}

now points at, and that is a distane

r = vct

^{from the gyrosope, has to be}

at adistane of

vct sin Ω G t ≈ vct ² Ω G = ¹ ₂ B G vt ²

^{from the original axis. This}

is the same point as we found the free partile to be at. We an onlude

that the gyrosope is stillpointing at the free partile.

Fromtheaboveargument,weanonludethatinaloalrefereneframe

at the origin with axes xed by gyrosopes free partiles are moving along

a straight line. This is the dening property of an inertial frame. It is here

we get the onnetion with Mah's priniple. Imagine a sientist living in

a box at the entre of this rotating ring. Using gyrosopes and wathing

the motions of free partiles lose to him he nds that there is a ertain

framein whih the gyrosopes keep axed diretionthat ishard to hange,

and in whih the partiles move along a straight line. As he is unable to

determine any ause for this, he is prone just to take it as a fat of nature

thatthereisa"preferred"framethathappenstobeasitis,andthusmaybe

explainedby means of anabsolutespae. Assume further that the equation

for perfet dragging2.39 issatised. If the wallsof the boxsuddenly should

beome transparent so that the sientist ould see the ring of dust around

hislaboratory,it shouldbeeasyto envisionhimwonder why this ring turns

out tobeatrest relative tohisinertial frame. Above wehavereasoned that

this is no oinidene at all. No matter how the ring rotates (as long as it

iswithinthe weakeld approximation),the sientist'sframewould turnout

tonot rotaterelative toit.

This raises the question, ould we be in a similar situation? From the

approah in this setion, it would be natural to say that the result of the

experiments the boxed sientist used to determine his inertial frame was,

at least in part, aused by the properties of the surrounding ring. Mah's

priniplemaybeinterpretedasastatementthatitisthiskindofexplanation

that is preferred, and even neessary. I am thus ready to formulate the

main denition of Mah's priniple I willonentrate most of the remaining

treatmentaround:

The inertialsystemsshould bepartially/ompletelydeterminedby the

masses of the universe.

2.2.5 Hollow innite ylinder

Iwillheregiveashortpresentation ofarotatinghollowinnitelylongylin-

der. Itmightbeaninteresting systemfromagravitomagnetipointofview,

butI havefound littleuse foritregarding Mah's priniple. Itwillalsolater

be used to demonstrate the limitationsof the simpliations used to arrive

atthese equations forthe gravitodynamis.

Thissituationmayfromagravitomagnetiviewbetreatedthe sameway

as the magneti eld of a solenoid as desribed in [58℄. In this ase, we use

Ampere's lawonaretangle with one side insidethe ylinder paralleltothe

sides and the opposite side outside. The remaining sides are orthogonal to

the sides of the ylinder. The simplied idea is that due to symmetry the

magneti eld must be normal to the lines that pass through the ylinder.

Thelineoutsidetheylinderexperienesnomagnetield. Onewaytoargue

for this is that it may be as far away as we want showing that it at least

an be set to zero. Personally, I am more fond of an argument regarding

the magneti eld to be divergene less, hene its density must be the same

inside and outside the ylinder; but outside is innitely bigger. Anyway, we

ndthat the onlyontributiontothe path integralof Ampere's lawis along

the line inside the ylinder, and that the eld is parallel to this. If we say

that the length of this lineis

L

^{we get that 2.22 goesto}

B G L = − 16πG

c LDρRω

^(2.40)

where

D

^{isthethiknessoftheylinder,}

ρ

^isthemass-density,

R

^istheradius

of it, and

ω

^{is its angular veloity.}

L

^{may be anelled at both sides. We}

nd that we have a onstant gravitomagnetield inside the ylinder.

I have notfound any treatmentofthe lassialgravitation insideaylin-

der, and in the eletrodynami ase, the solenoid is usually onsidered neu-

tral. The following argument should however show that there is indeed no

gravitoeletri eld inside the ylinder: Consider a losed nite ylinder in-

sidethe inniteylinder. Itssidesare paralleltothatof theinniteone,and