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tnternationalJournalofTheoreticalPhysics

VoI. 12, No. 1 (1975), pp. 35-45

S o m e R e m a r k s o n r t h o n or m a l T e tr ad T r a n s p o r t t

J

K R A U S E

Instituto de F[sica Universidad Cat68ca de Chile Casilla

114-.0,

Santiago

Received:

20

April

1974

Abstract

Some general features of orth ono rma l tetra d tran sport along time-like curves are con-

sidered while emphasising the k inem atical meaning o f the 'vierbein ' trans port process. Th e

issue is further illustrated by comparing Fre net -Se rret transpo rt with Fermi-W alker

transport.

1. hztroduction

T h i s p a p e r d e a ls w i t h r e l a ti v i st i c k i n e m a t i c s . S o m e f e a t u re s o f t h e m a t h e -

m a t i c a l i m p l e m e n t s b e l o n g i n g t o t h e ' v i e r b e in ' f o r m a l i s m a re h e r e d i s cu s s ed

i n o r d e r t o p o i n t o u t s o m e u s e f u l r em a r k s c o n c e r n in g t h e t r a n s p o r t o f

o r t ho nor m a l t e t r ads a long g iven tim e- l ike curves . O ne o f the ea r l i e s t d i s-

cuss ions on the o r t ho no rm al ' v i e rbe in ' f ie lds in phys ics i s due to E ins t e in

( 1 9 2 8 , 1 9 3 0 ), w h e n h e a t t e m p t e d a u n i f i ed f o r m u l a t i o n o f g r a v i ta t i o n a n d

e l e c t r o m a g n e t i s m i n h i s ' d i s t a n t p a r a l l e l i sm ' t h e o r y . S o o n a f t e r w a rd s , t h e

m a t h e m a t i c a l p r o p e r t i e s o f te t r a d f i e ld s w e r e e l u c i d a t e d , a n d t h e i m p o r t a n c e

o f t h e t e t r a d f o r m u l a t i o n f o r a q u a n t u m f ie ld t h e o r y o f g r a v it a t i o n w a s

recognized++ by F oc k (19 29) and R osen fe ld (1930) . A f t e r a pe r iod dur ing

w hic h the in t e res t i n ' v i e rbe ins ' subs ided , M611er's w ork on the conse rva t ion

laws of gene ral re la t iv i ty (M611er, 19 58, 1964) repre sents a lucid prof i t of the

t e t r ad f i e ld t echn iq ue , s ince the i r use is , i n f ac t , i nd i spensab le to o b ta in the

r e q u i r e d t r a n s f o r m a t i o n p r o p e r t i e s o f E i n s t e i n 's t a w s o f g r a v i t a t io n ( D a v i s &

Moss , 1963 , 1965) . A l so , i n the in t e res t ing a t t em pts to in t e rp re t g rav i t a t ion

a s a c o m p e n s a t i n g Y a n g - M i l l s fi e ld , t h e u s e o f a n o r t h o n o r m a l t e t r a d f o r m a l i sm

has been r ecogn i sed as ind i spensab le (K ibb le , 1961) . N ot long ago , K aem pfe r

Work sponsored by a p ost-doctoral fellowship of the University of C aliforn ia-

University of Chile Cooperative Program (Fo rd Fou nda tion Grant).

$ See, for exam ple, Dirac (1962).

Cop yright © 19 75 Plenum Publishing Company Limite d. No part o f this pub licatio n may

be reprod uced, stored in a retrieval system, or transmitted, in any form or by any means,

electron ic, mechan ical, photo copy ing, microfilming, recording or otherwise, withou t

written permission of Plenum Publishing Company Limited.

35

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36 J. KRAUSE

(1968) showed the usefulness of a 'v ierbein ' f ie ld theo ry of gravi ta t ion for

the distinction of genuine gravitational f ields from pseudogravitational ( i .e.

inertial) f ields. In summary, we could say that the f ield theoretic role of the

vector te trad form af ism t is wel l unders tood today , a lbei t no t widely em ployed

in the current li terature.

On the o ther hand, the use of or thon orm al te trads is not l imited to f ie ld

theoret ic considerations; indeed , the formalism has also p layed a fund amen tal

role in studies of paths in differential geometry (Caftan, 1935). In the present

paper we wil l dwell on these k inem atic features of the 'v ierbein ' technique,

for its intuitive value is very" appealing. In relativity t he ory we recognise th at

the specif icat ion of a local frame of reference per ta in ing to an observer

determines a vector tetrad which is being transported along a time-like curve.

Thus, for instance, the Fermi-W alker t ranspor ted te trads , with their t ime- like

com pone nt vector chosen to be tangent to the t ranspor t ing curve, appear to

give us the adequate relativistic generalisation of the Newtonian concept of

an accelerated non-ro tat ing f rame of reference. Although the c oncept of

'obse rver ' (as an idealised creature reduced to b e a moving point) is perhaps

purely rhetorical, i t is a useful picturesque notion. Each observer has a world

line ( time-like, to be sure) , representing his history as an ordered continuous

sequence of events, and carries a clock (proper time). We obviously normalise

the t heory by requir ing al l observers to bear c locks of the same make (s tandard

clocks). However, for the p urposes o f physics, even if mere kinem atics is at

issue, these idealised point-observers have to be more r ichly equipped, other-

wise they would be l ike monad s witho ut windows The wor ld formed by the

continuous co l lection of these iso lated beings would have, a t m ost , a purely

af f ine s t ructure not corresponding with the m etr ic s t ructure of the wor ld we

live in. The refore in relativity theo ry w e have to assume, as is clone for

ins tance in cosmology, that ea ch observer car ries a fundamental equipm ent

composed of the following tools: standard clocks, goniometers, l ight bulbs,

and photocells . (This is in the same spirit as the classical ' rule and com pass '

p latonic requirement of the ancient geometers . ) This means that the m easure-

ments of time and angles have a locally direct, operationally simple, meaning,

while the concept of space distance is a secondary ( i .e. derived) construct,

which has to be def ined somehow by recourse to operat ions per formed by

means of the fundame ntal too ls only . What is of in terest for us in th is context

is the fact that these imp lem ents enable the local observer to set up a physical

frame of reference by choosing three arbitrary linearly independent space-like

directions . I t m ust be b orne in mind that in making th is choice each observer

is completely free to use whatever physical cr iter ion he wants to, and that his

own h is tory is not af fected by th is choice. In consequence, corresponding to

one an d the same time-like curve, we m ay have quite differe nt local observers

(in the sense of local frame s of reference ), each one spinning relative to th e

others. F urth erm ore , i t is clear that each of these coinciding point-observers,

"~ The field theoretic role of tetrad formalism in the framework o f general relativity

is reviewed by Davis (1966).

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SOME REMARKS ON ORTHONORMAL TETRAD TRANSPORT 37

wi th t he i r a t t ached ro t a t i ng f r ames, cor r esponds t o a conco mi t an t vec to r

t e t r ad w hich i s t r anspor t ed a long the co mm on wor ld l ine accord ing t o some

prescr ibed equat ion of mot ion. Al l these te t rads have the same t ime- l ike

com pon ent vec to r , namely , t he un i t t angen t t o t he t r anspor t i ng curve , and

thei r space- l ike components represent the three-dimensional spinning f rames

used by t h e comoving obse rve r s t o r e fe r t he m ot io n o f eve ry th ing e lse in t he

universe.

In t he fo l lowing sec t ions we show how the o r tho norm al t e t r ad fo rmali sm

is indeed the space- t ime geom etr ic rea l isa t ion of the kin emat ic n ot io n o f a

local, arbi t rari ly moving, f ram e o f reference. We develop ou r theme in the

general relat ivist ic formalism. In Sect ion 2 we discuss general t ransport

p rocesses which p rese rve t he o r thonormat i t y re l a t i ons among the component

vectors of a 'v ierbein ' , emphasis ing som e useful rema rks of general val idity .

In Sec tion 3 we in t roduce t he concep t o f F rene t -Se r r e t t r anspor t a s an

interes ting ins tance of the i ssues we are discuss ing. F inal ly , in Sect ion 4 , we

br i e fly p resent t he w e l l -known Fermi -Walker t r anspor t , and com pare i t s

behav iour wi th our F rene t -Se r r e t t r anspor t .

2. Orthonormality Preserving Transport

The f i r s t point we want to emphasise i s that there are inf ini te ly many

way s of t ranspor t ing a given te t rad a long a curve whi le preserving the o r tho-

norm al i t y r e la t ions am ong the com pon ent vec to r s o f t he t e t rad . Indeed ,

conside r a te tra dt {a~v)} such that , at som e poin t on a given t ime-l ike curve,

the relat ions

ee~*v)Oe(x) = 8 ~ (2.2 )

hold . No w le t us assume the te t rad i s t ransp or ted a long the curve whi le

preserving these re la t ions . Th en the absolute ra te of change of the vectors

~ ) can be r epresen ted i n t e rms o f t he t e t r ad i ts e lf ; namely , we have

t In this paper we let Greek indices run over the range 0, 1, 2, 3, and Latin indices

over 1 2 3. We adop t signature (- 2) for the metric For an orthonorma l tetrad we write

{ (v)}, say, where ~ is a tensor index, and (v) stands for a label denoting th e com pon ents

o f the tetra d (we designate with (0) the time-like com pon ent, and with (i) the space-like

components).

In equations (2.1) and (2.2) we have used the definitions c~ff = gtzv~fh)(P)au(O

where ~(M)(p) = ~(X)(P)= diag. (1, -1 , --1, -1 ) is Minkowski's matrix, and g v is space-

time covariant metric tensor.

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38 J. KRAUSE

wh ere 6~v d eno tes th e abs olute derivat ive of o~(v) with respect t o pro per t ime ,

and the elem ents A(v) (x) of th e

transport matrix

are sca lar funct io ns o f proper

t ime. Moreover , s ince equat ion (2 .1) holds a l l a long the curve , we obta int

Ao~)(v) +A(v)O0

= 0 (2.4 )

i .e . the t ranspo r t mat r ix i s necessar i ly sk ew-symm etr ic . This resul t i s obvious ly

r e la t ed t o t he f ac t t ha t each occur r ence o f an o r t ho norm a l t e t r ad a s i t r ides

on a curve cor respon ds to r ig id ro ta t ions in space- time; thus the te t rad evolves

f rom one o r i en t a t i on t o t he nex t by means o f an i n f in i te s i ma l L oren t z t rans -

fo rma t i on who se gene ra t o r is t he t r anspor t ma t r i x . Fur t he rmo re , equa t i ons

(2 .3) and (2 .4) together represent a suf f ic ient condi t ion for preserving equat ions

(2 .1) and (2 .2) . Th at i s, provided equat ions (2 .1) (or equat ion (2 .2) fo r tha t

mat te r ) i s g iven as a se t of in it i al con di t ions a t some point of the curve , the

l aw o f mot i o n s t a t ed i n equa t i on (2 , 3 ) t oge t he r w i t h t he skew-sym met ry o f

A(u)(v) are

enoug h t o a s su re tha t t he o r t ho norm a l i t y o f the t e t r ad ho l ds al l

a l ong t he cu rve . Hence , eve ry li nea r hom ogen eous skew-symmet r i c p roces s o f

t r anspor t p r ese rves o r t hon orma l i t y .

Nex t , we wan t t o r emark t ha t a s soc ia t ed w i t h each cons i der ed l aw o f

t r anspor t f o r o r t hon orm a l t e t r ads we have a co r re spond i ng law o f t r anspor t

for vec tors and tensors . Cons ider ing again equa t ion (2 .3) as the g iven trans-

por t l aw of a v ierbe in se t {a~v)} we can then w r i te , ins tead o f equat ion (2 .3) ,

= 2 .5)

where the t ranspo r t t en sor A ~ of the te t rad i s def ined as

A ~ ~ = A(v (a)o~(~)o~(V) (2 .6 )

Clear ly , the t ranspor t t ensor i s skew-symmetr ic :

A~v +Av~

= 0 (2.7 )

Then le t V~ be a vec tor propag ated a long the curve according to the fo l lowing

law:

i~ +A ~V~= 0 2.8 )

We say tha t vec tor Vu is

comoving

with the te t rad a long the curve . Clearly ,

because o f t he skew-sym met ry o f the t r anspor t t ensor o f t he t e tr ad , t he n orm

of a comovi ng vec t o r is a cons t an t o f mot i on a l ong t he t r anspor t i ng wo r l d

l ine , i .e . only vec tors of co ns tant no rm on a g iven curve can be eventual ly

com oving wi th or th on orm al t e t rads o n tha t curve . I t is clear tha t th e sca lar

p rod uc t o f t wo vec t o r s comovi ng w i t h t he s ame t e tr ad r ema ins cons t an t a l ong

-~ Although the labels (v) have no ordinary tensorial meaning, we can manipulate them

confid ently as if they were true Lore ntz tensorial indices. Thus, in equation (2.4) we

have used the definition A (~z)(v) =

A(v)(h) o(~.)(v),

as is usually done in this approach.

This rule f or raising and lowering the labels by means o f the Minkowski matrix prevails

thro ugh out, and is introduced to secure simplicity.

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SOME REMARKS ON ORTHONORMAL TETRAD TRANSPORT 39

the curve of t ranspor t . In par t icular , the com pon ents of a vec tor re la t ive to a

com ovin g te t rad are conserved quant i t i es ( i .e . proper t ime inde pende nt ) .

Al l these prop er t ies a re wel l kn ow n for vec tors undergoing para lle l trans-

por t or Ferm i-Walker t ransp or t a long t ime-l ike wor ld l ines. What we wa nt to

s tress here i s the genera l va l id i ty of these s ta tem ents for every kind o f

o r t honorma l t r anspor t .

I t is a l so in teres t ing to observe tha t every given vector o f cons tan t no rm

along a curve can be used to define an ad hoc t r anspor t t ensor on t he wor l d

l ine , such tha t equat io n (2 .8) holds for the g iven vector . Indeed, g iven tha t

d/dr)

V~ Vu = 0, we define

A up = (V x V 'k) -I (Vitt ~ - VV[ ~#) (2.9)

This t ranspor t t ensor i s not unique , how ever . We can always add a skew-

symmet r i c t ensor t o A uv in equat ion (2 .9) , provided th i s added tensor i s

eve rywhere o r t hogo na l t o t he vec t o r a l ong t he cu rve , Such a change o f ' gauge '

does not a l te r equat ion (2 .8) , never the less i t provides us wi th a comple te ly

di f ferent l aw of t ranspo r t for te t rads . Therefo re , associa ted wi th a vec tor of

cons tan t n orm a long a curve , we have a whole fami ly of 'pr opa gato rs ' A uv

for tha t curve , of which the s imples t me mb er is g iven in equat ion (2 .9) . Con-

verse ly , qui te d i f ferent t ranspor t processes can be re la ted wi th one and the

same cons t an t -norm vec t o r. T h i s ' gauge ' f r eedom fo r cons t ruc t i ng t r anspor t

t ensor ou t o f a cons t an t -norm vec t o r p l ays an i mpor t an t r o le i n k inema t i c s,

s ince i t cor responds to the fac t , a l luded to in the In t roduct ion, tha t the same

wor ld l ine can hos t d i f ferent ro ta t ing loca l f rames .

3. Frenet-Serret Transport

Space- t ime di f ferent ia l geom et ry te l ls us tha t associa ted wi th eac h event

on a wor ld f ine there i s a par t icular ly in teres t ing or tho no rma l t e t rad , the

te t rad of Frenet and Ser re t , which i s in t imate ly re la ted wi th the geomet ry

of the curve . We shall now br ief ly d i scuss th i s 'v ierbe in ' , and the cor respond ing

law of t ranspor t for vec tors , according to the s imple ideas presented in the

previous sec t ion.

We consid er a t ime-l ike curve. L et u u represent the 4-velo ci ty relat ive to a

sys t em of space -t i me coord i na te s . I n o rde r t o i n t roduce t he Frene t -Se r r e t

tetr ad {7~v)} at s om e eve nt xU(~ ) on the curve (z is prope r t im e) w e f i rst

def ine 7(o) = c - u (c deno tes the ve loc i ty of l ight thro ug hou t ) ; we then

define, in a progressive manner, three scalars C i) and three vectors 7~') ,

i = 1 , 2 , 3 , by m eans o f t he we l l -known Frene t -Se r r e t f o rmul as?

3'fu) (;~) (3 .1 )

C v) 7(~.)

~ The c~nstruction of the Frenet-Serret orth ono rma l set should be well kno wn to

the reader. W e only recall here that, while the time-like vector 3'~o) s the un it tangent

to the curve, the space-like unit vectors 7~') point in space-time into th e three normal

directions t o th e curve, and th e scalars

C i)-are

the corresponding curvatures of the

world line.

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40 j. KRAUSE

where the Frene t -Serre t t ranspor t matr ix (or curvature matr ix) of the w orld

line is given by

o o ]

C0z)(v) = C(D 0 -C (2 ) (3.2 )

[ o c(2~ o c(,

o o

It is useful to in troduc e the rota tion mat rix co u) v) of the curve, which is

defined as the dual of the t ransport matr ix , namely

coO~) v) = e u)

(v)0,) (o) C(x) (p) (3 .3 )

wh ere e ~)(v)(x)~°) is the per mu tati on sym bol in fo ur dim ensions. Thu s, in

terms of the ro tat ion matr ix , the Frene t-Serret equat ions become

5,}zv) = e(v)(x)(p)(cgco(P)(cg y (x) u (3.4)

One easily shows that the vec tors of the Fren et-S erre t tetrad satisfy the

orth ono rm ality relations, cf. equatio n (2.1) and (2.2), all along the curve.

We now define the corresponding

Frene t -Serre t t ranspor t t ensor

as

c au = c( , , ) e)7o )x~, c°)~ (3.5)

and we say that a vec tor V u is being Frene t-Se rret tra nsp oge d along the curve

if it obeys the law o f motion , cf. equation (2.8),

T; + cuv vv = 0 (3.6)

which, wri t ten expl ici tly , becomes

~:U = {C (1)(,,/(0 )#,,/(1)p __ ,):(O)v,),(l)#)

+ C 2) y Ou@ z)v -

3,(DvT(z)u) (3.7)

+ C(3)(@2)u7 (3)v - 3,(2)vT(3)u)}

V v

From here we obtain the F renet-Serre t tensor of the curve. In order to get a

comp act and more useful expression for this tensor, we observe that, quite

generally, the tr ansport tensor o f an orthon orm al te trad is given by

A u~ = a(~')u&~) (3 .8)

Thus

uv = 7(°)u5,~ o) + 7(0u5,~ i) (3 .9 )

But, from equation (3 .4), one readily shows

~/~.) = 6 t]))C ,)7~o) + e i)

0')(k)co(k) /(/) t (3.1 O)

) ~0) = C(1 ) /~I)

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S O ME R E M A R K S O N O R T H O N O R M A L T E T R A D T R A N S P O R T 41

wher e we have def ined the scalars

66 1) = co °)(0 = -C(3)

66(2) = 09(0) (2) = 0

) (3.1 1)

( .0 (3) = ( .0 (0)(3) = -C ( 2 )

So the fo l lowing compact expression for the Frenet-S erret t ransport tensor

obtains

C up = -C (l ) ( , ) , (° ) /~ / (1 )v - , / (° )v ' ) (1 )g t ) + 6( t ) ( j .) (k )66(k)7( i )g t'~ ( / ')v (3.12)

As for any kind of orthono rma l t ransport , it fo l lows that i f we Fre net-

Serret transp ort a 'vierbein ' along a t ime-like curve (not necessarily the

Frene t-Serre t te t rad ) the 'v ierbein ' remains o r thonorm al and i ts Fre net-

Serret com pone nts are constants along the curve. But this means that every

Frenet-S erret t ransported te t rad is re lated to the Fren et-Serre t te t rad i tself

by means of a (constant) Lore ntz t ransformation.

4 . F e r m i - W a l k e r T ra n s p o r t

The well-known geom etric process called Fermi-W alker transpor t allows

us to in troduce another k ind of in terest ing orthonorma l te t rad associated with

each event on a t ime-like world l ine. According with the general scheme

~resented in Section 2, one uses the fact th at for eve ry time-like curve the

4-velocity is a vector o f constant norm , i .e.

uuu u

= c 2, and the Ferm i-Wa lker

transport tensor is thus d efined as the simplest 'propag ator' ge nerated by the

4-velocity, cf. equation (2.9); name ly

Buv = c-2 uUg v - uPg u) (4.1)

wher e g~ is the 4-acceleration. Fro m the first two Fre net- Ser ret equations,

(3.1) and (3.2), one im med iately obtains gU = cC 1)7~1), and so the Ferm i-

Walker tensor o f the curve is given by

BUt = _ C( l ) (7 (O )# , , / ( 1 )p _ ~ / , (o )t ) ,y (l )# ) (4.2)

Accordingly, a vector

V u

is said to u ndergo Fermi-W alker transport along the

curve if i ts prop er t ime rate of change is

l) u = C(1)(7(°)u3 (a )' - 7(°)v3 (1)u) V, (4 .3)

Fr om this de finition it follows that the 4-velocity i tself autom atically evolves

along the curve unde r Fermi-Wa lker transport (trivially so, since the 4-velocity

was used to generate this type of trans port). Howe ver, the 4-velocity also

undergoes Fre net- Ser ret transp ort, as can be easily seen, so that bo th kinds

of processes differ only by a 'gauge' tensor o rthog onal to uU(cf, equa tion

(3.1 2) and (4.3)), affording thus an example of the 'gauge' freed om discussed

in Section 2.

Compariing equations (3. I2 ) and (4.2), i t is interesting to observe tha t

Fermi-Walker and Frenet-S erret becom e the same t ransport for t ime-l ike

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44 j. KRAUSE

rotat ing relat ive to an instantaneous comoving inert ial ( i .e . f ree fal l ing)

observer , the F ermi-W alker t r i ad evolves wi tho ut such a ro ta t ion . (To th i s end

we in tro du ce an auxiliax3r inertial te trad , {a~v)} say, wh ich is parallel trans-

por ted along the curve, i .e . &~v) = 0, with ini t ial condi t io ns a~ )( r) = f i~)(r) =

7~v)(~'); we th en subst i t ute for a~v) in the r ight-hand me mb ers of equat io ns

(4.6) and (4.7) , and proje ct f l~? (r + dr ) and 6~})(r + dr ) along the inert ial

t e t r ad )

To end th i s sec t ion le t us f ind the Fermi-Walker t ranspor t mat r ix . According

to equat ion (2 .6) we must have

B ~, ~ ) = B U~ l x) ,~ o) v

(4 .8)

that is

= C (1) , (o

B(~)(p) (1)3'

6~lfl ~.)u - 61°lflCo)u)

(4.9)

wh ere {/3~)} is

s o m e

Fermi-W alker t ransp or ted 'v ierbe in ' . In par ti cular , i f a t

proper t im e z we take ~) ( r ) = 7~v)(r ), we ge t

_ 0 X ) 1 ) ~ 0 ) ~ ,

B (~,) (;)(r) - C(1 ) (r )( 6 Co/~ (x) - ~ Co) ~ (~,)~ (4 .1 0)

as w e akead y kn ow f rom equa t i on (4 .5 ) .

5. Conehts ion

To sum mar i se , we observe tha t the 'prop agator s ' of the reference te t rads ,

a t t ached to a g iven t ime- l ike curve , consi s t of the Fermi-W alker t e nsor of the

curve p lus a skew-sym metr ic t ensor or thog onal to the 4-veloc i ty , i. e. a 'gauge '

t e rm in the sense of Sect ion 2 , represent ing the ro ta t ion of the reference f rame

relat ive to an instantaneous comoving free-fal l ing frame. We also observe,

qui te genera l ly , an enlarged ro te for Lorentz t ransformat ions : Not only

iner ti a l t e t rads are re la ted b y mea ns of them, but acce lera ted te t rads as wel l,

provided these tet rads are r iding on a world f ine obeying one and the same law

of t ransport . I t i s clear that , in general , for tet rads r iding on a same curve, we

go f rom one 'v ierbe in ' to another by mean s of ( ins tantaneous) Lore ntz t rans-

format ions , but as proper t ime e lapses these t ransformat ions have to be

changed, unless the t e t rads are comoving. Hence we see tha t , f rom the point

of v iew of k inemat ics , i t is comot ion , and not iner t ia ( i . e. uni form rec t il inear

mot i on ) , t he fundam en t a l p rope r t y o f space - ti me r e f lec t ed in the Loren t z

group. In th i s man ner we conclude tha t the formal i sm of the or thon orm al

tetrads r iding on t ime-l ike curves provides al l the required tools to deal with

arbi t rary ro ta t ing loca l f rames of reference , wi th the i r or ig ins undergoing

genera l time- l ike m ot ion s in space- time.

A c k n o w l e d g e m e n t

The author wishes to than k Professor Rainer K. Sachs, and the D epartm ent o f

Mathematics, University of California at Berkeley, for their hosp itality during th e

period when this work was done.

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SOME REMARKS ON ORTHONORMAL TETRAD TRANSPORT 5

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