Coordinates and frames from the causal point of view

0.05 MB·English

by  Juan Antonio Morales Lladosa

Coordinates and frames from the causal
point of view
Juan Antonio Morales Lladosa
Departamentd’AstronomiaiAstrofísica,UniversitatdeValència,46100Burjassot,València,Spain
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0 Abstract. Lorentzianframesmaybelongtooneofthe199causalclasses.Ofthesenumerouscausal
0 classes, people are essentially aware only of two of them. Nevertheless, other causal classes are
2 present in some well-knownsolutions, or presenta strong interest in the physicalconstructionof
coordinatesystems.Hereweshowtheunusualcausalclassestowhichbelongsofamiliarcoordinate
n
a systems as those of Lemaître, those of Eddington-Finkelstein,or those of Bondi-Sachs. Also the
J causalclassesassociatedtotheColllightcoordinates(fourcongruencesofrealgeodeticnulllines)
1 andtotheCollpositioningsystems(lightsignalsbroadcastedbyfourclocks)areanalyzed.Therole
3 thattheseresultsplayin thecomprehensionandclassificationofrelativisticcoordinatesystemsis
emphasized.

1
Keywords: space-timeframes,causalstructure
v
PACS: 04.20.-q
8
3
1
The main purpose of this short communication is to gain stimulus in the investiga-
1
0 tion of relativistic coordinate systems from the causal point of view. This issue might
6
be relevant in several situations. For example, to investigate the coordinates which are
0
/ appropriatedtodealwithevolutionproblemsthroughouthorizons.Current3+1numeri-
c
calcodesinrelativistichydrodynamicsarebeingimplementedusingsuchacoordinates.
q
- Also, to consider admissiblecuts of the space-time others than the very usual “space ⊕
r
g time” decomposition. The causal classification of frames may help us to better under-
:
v stand other aspects of the space-time, for instance the "light ⊕ light ⊕ light ⊕ light"
i decomposition.
X
Firstly,werememberthecausalclassificationofLorentzianframes(alsopresentedat
r
a theERE-88celebrated inSalamanca). Then,wegiveseveralexamplesofcausal classes
ofrelativisticcoordinates:thoseassociatedwiththecoordinatesintroducedbyLemaître,
byBondi andSachs, andby Coll.
In dimension n =4, the causal class of a frame {v ,v ,v ,v } is defined by a set of
1 2 3 4
14characters:
{c c c c ,C C C C C C ,c c c c }
1 2 3 4 12 13 14 23 24 34 1 2 3 4
c being the causal character of the vector v, C (i6=j) being the causal character of
i i ij
the adjoint 2-plane {v v }, and c being the causal character of the covectors of the
i j i
dual coframe {q i}, q i(v ) = d i. The covector q i is time-like (resp. space-like) iff the
j j
3-plane generated by {v} is space-like (resp. space-like). This applies for both the
j j6=i
Newtonian and the Lorentzian causal structures. In addition, for the later, the covector
q i is light-like iff the 3-plane generated by {v } is light-like. Elsewhere (see [1] and
j j6=i
[2])wehavepresented thefollowingresult:
Theorem: i) In the 4-dimensional Newtonian space-time there exist 4, and only 4,
causal classes of frames, and ii) In the 4-dimensional relativisticspace-time there exist
199,andonly199,causalclassesofLorentzianframes.
This result provides the causal classification of coordinate systems in Newtonian
and in Relativistic physics. A coordinate system {x1,x2,x3,x4} belongs to the causal
class {c c c c ,C C C C C C ,c c c c } if the cobasis {dx1,dx2,dx3,dx4} has
1 2 3 4 12 13 14 23 24 34 1 2 3 4
causal type (c c c c ) and has associated four families of coordinate 3-surfaces whose
1 2 3 4
mutual intersections give six families of coordinate 2-surfaces of causal characters
(C C C C C C ) and four congruences of coordinate lines of causal characters
12 13 14 23 24 34
(c c c c ).
1 2 3 4
Asamatterofnotation,romanicletters(e,t,l)representthecausalcharacterofvectors
(space-like, time-like, light-like , respectively), and capital (E,T,L) and italic letters
(e,t,l) denotethe causal character of 2-planes and covectors, respectively.Accordingly,
thefourNewtoniancausal classesare denotedas:
{teee,TTTEEE,teee},{ttee,TTTTTE,eeee},{ttte,TTTTTT,eeee},{tttt,TTTTTT,eeee}.
Among the 199 Lorentzian classes, four of them have the same set of causal characters
astheNewtonianones[1]. Next,weconsideranotherexamplesofrelativisticclasses.
1.- The three causal classes of Lemaître coordinates. The familiarmetric form of the
Schwarzschildsolutionin coordinates{t,r,q ,f }iswrittenas
2m 1
ds2 =− 1− dt2+ dr2+r2dW 2
(cid:18) r (cid:19) 2m
1−
r
wherer >2m and dW 2 =dq 2+sin2q df 2. Thecoordinatebasis{¶ t,¶ r,¶ q ,¶ f } belong
to the causal class {teee,TTTEEE,teee}. Lemaître [3] extended the Schwarzschild so-
lutionat theregion0<r≤2m,obtaininga metricform
2m 2m
ds2 =− 1− dT2+2e dTdr+dr2+r2dW 2 (e =±1)
(cid:18) r (cid:19) r r
that is regular at r = 2m. The causal character of the coordinate lines are given by the
sign of the four diagonal elements gaa of the metric gab in this basis. The causal
character of the coordinate 2-surfaces is given by the sign of the principal second or-
der minors gaa gbb −(gab )2. And the causal character of the coordinate 3-surfaces
aa
is related to the sign of the diagonal elements g of the contravariant metric ex-
pression gab . Consequently, the Lemaître coordinate basis {¶ T,¶ r,¶ q ,¶ f } belong to
the causal class {teee,TTTEEE,teee} if r > 2m, {leee,TLLEEE,tlee} if r = 2m, or
{eeee,TEEEEE,ttee}ifr<2m.
2.- The thirteen causal classes of Bondi-Sachs coordinates. Any space-time metric
maybeexpressedin theform [4]:
g g g g
00 01 02 03
¶ ¶
g 0 0 0
gab ≡g(cid:18)¶ xa ,¶ xb (cid:19)= g0012 0 g22 g23 
 g 0 g g 
 03 23 33 
Fortheassociatedcontravariantmetriconehasg00=g02=g03=0.Therefore,{x0=l }
(withl arealparameter)arenullhypersurfaces.Wheng g −g2 >0,thecoordinates
22 33 23
are called (generalized) Bondi-Sachs coordinates [4] [5], and are usually denoted by
x0 ≡ u, x1 ≡ r, x2 ≡ q , x3 ≡ f . We have that the Bondi-Sachs coordinate systems are
classifiedin 13causalclasses:
{tlee,TTTLLE,leee}
{llee,TTTLLE,leee} {elee,TTTLLE,leee}
{llee,TTLLLE,leee} {elee,TTLLLE,leee}
{llee,TLLLLE,llee} {elee,TLLLLE,leee}
{leee,TLLLEE,llee} {elee,TTELLE,leee}
{leee,TLLEEE,llee} {elee,TLELLE,leee}
{leee,TLLEEE,tlee} {elee,TEELLE,leee}
For example, a coordinate system of the causal class {elee,TEELLE,leee} has asso-
ciated a family of null coordinate 3-surfaces and three families of time-like 3-surfaces.
Their mutual cuts give one family of time-like surfaces, two families of null 2-surfaces
and three families of space-like 2-surfaces. The intersections of these surfaces give a
congruenceofnulllines andthreecongruences ofspace-likelines.
NotethattheBondi-SachscoordinatesareageneralizationofthefamiliarEddington-
FinkelsteincoordinatesusedintheSchwarzschildspace-time.Thesecoordinatesbelong
to the class {tlee,TTTLLE,leee} outside the horizon, to the class {llee,TLLLLE,llee}
atthehorizonr=2m,andtotheclass{elee,TEELLE,ltee}insidethehorizon.Thelater
is the class {leee,TLLEEE,tlee} (as it results when the first and the second vectors are
changed).
3.-Thetwo causalclassesof Collcoordinates.Let usconsidertheclasses
{eeee,EEEEEE,llll} and {llll,TTTTTT,eeee}
A coordinate systems of the first class has associated four families of null 3-surfaces
whose mutual cuts give six families of space-like 2-surfaces and four congruences of
space-like lines. This class includes the emission coordinates of the Coll positioning
systems [6] [7]. A coordinate system of the second class has associated four families
of time-like 3-surfaces whose mutual cuts give six families of time-like 2-surfaces and
four congruences of null lines. This class includes the Coll light coordinates built from
theintersectionoffourbeamsoflight[8]. Wecall themtheColl causalclasses.
The 199 causal classes of relativisticcoordinates may be wholly visualizedin a table
thatwecallthe199-Table(fordetails,see[1]and[2]).Here,wereproducethe199-Table
showingthelocationoftheCollcausal classes.
Each causal class has univocally associated its dual causal class [2]. For example
the Coll causal classes are dual each other. The same occurs for the causal classes of
the Eddington-Finkelstein coordinates at r > 2m and r < 2m. When a causal class and
its dual are equal, the class is said self-dual (as the class of the Eddington-Finkelstein
F
I e e e e l e e e elee t e e e l l e e tlee ttee llle tlle ttle ttte llll tlll ttll tttl tttt
G
U EEEEEE LEEEEE TEEEEE LLEEEE TTLEEE TTTEEE TTLLEE TTLELE TTTEEE TTTLEE TTLTLE TTLLTE TTTTLE TTTTTE TTLTLL TTTTLL TTTTTL TTTTTT TTTTTT TTTTTT TTTTTT TTTTTT TTTTTT
R TLEEEE TTEEEE LLLEEE TLLEEE TTTLEE TTLTEE TTLETL TTTTEE TTTTEL TTTLLE TTTTLE TTTTTE TTTTTE TTTTTL TTTTLL TTTTTL TTTTTT
E TTLEEE TTTEEE LLLLEE TLLLEE TTLLLE TTLELL TTTLLE TTLTLE TTTTLE TTTTTE TTLTLL TTLLTL TTTTLL TTTTTT TTTTTL TTTTTT
eeee TTLLEE TTTLEE TTTTEE LLLLLE TTLLTE TTLETL TTTTLE TTLTTE TTTLLL TTTTLL TTTTLL TTLLTT TTTTTL TTTTTT
1
. TLLLLE TTLLLE TTTLLE TTTTLE TTLETT TTTTTE TTLLLL TTTLLL TTTTTL TTTTTT TTLTLT TTTTTL TTTTLT
TTTTTE LLLLLL TLLLLL TTLLLL TTLTLL TTLLTL TTTTLL TTLTTL TTTTLT TTTTTT TTTTTT
T TTTLLL TTTTLL TTTTTL TTTTTT TTLLTT TTTTTL TTLTTT TTTTTT
h
e EEEEEE LEEEEE EEELEE TEEEEE TTLEEE TTTEEE TTLLEE TTLELE TEELLE TTTEEE TTTLEE TTLLLE TTTLLE TTTLLE
C LELEEE LEELEE EELLEE TLEEEE TTTLEE TLELLE
oll l e e e TLLEEELLEEEE ELEETLLLEEEE TTTLLEEEEEEE LTLLLEELEEEE TTTLELLLLLEE
c
a TELLEE LETLEE TTLEEE TTELEE TTLLLE
u TETLEE TTTEEE LLLLEE TLLLEE TTTLLE
s
a LLTLEE TTLLEE TLTLEE TTTLEE
l
c
la EEEEEE LEEEEE TEEEEE LLEEEE TTLEEE TTTEEE TTTEEE
s teee TLEEEE TTEEEE LLLEEE TLLEEE
se TTLEEE TTTEEE
s
{ EEEEEE LEEEEE ELEEEE TEEEEE TLLEEE TLLLEE TLLLLE
llll, l l e e LLLLEEELEEEE ELEELLLLEEEE ETLLEELLEEEE TTLEELELEEEE
T
T EEEEEE LEEEEE ELEEEE TEEEEE TLLEEE
T tlee
LLEEEE TLEEEE
T
T
T ttee EEEEEE LEEEEE TEEEEE
e
e llle EEEEEE LEEEEE LLEEEE LLELEE
e
e
} tlle EEEEEE LEEEEE LLEEEE
a
n
d ttle EEEEEE LEEEEE
{e ttte EEEEEE C o l l c a u s a l c l a s s e s
e
e
e llll EEEEEE
,
E
E tlll EEEEEE
E
E ttll EEEEEE
E
E tttl EEEEEE
,
}llll tttt EEEEEE
.
coordinatesatr=2m).Orthonormalframesandrealnullframesbelong,respectively,to
theself-dualclasses{teee,TTTEEE,eeee}and{llee,TLLLLE,llee}.Thereexisteleven
self-dual causal classes which are easily located looking at the diagonal region of the
199-Table.Theyare:
{eeee,TTTEEE,eeee} {leee,TTTEEE,leee}
{eeee,TTLLEE,eeee} {leee,TTLLEE,leee}
{eeee,TLLLLE,eeee} {leee,TTLELE,leee}
{eeee,LLLLLL,eeee}
{elee,TTELLE,leee} {teee,TTTEEE,teee}
{elee,TLLLLE,leee} {llee, TLLLLE,llee}
To gain more comprehension of the role that coordinates play in Relativity, the 199
causalclassificationwouldbeinvestigatedthroughandthrough.Thecausalclassesmay
be associated to admissible cuts of the space-time others than the well-known three-
space ⊕ one-time usual in the at present evolution conception of physics. Other cuts
(among the other 198 possible ones) may help us to better understand other aspects of
the space-time, and even to wake up our interest for other variations of physical fields
thanthetime-likeonesassociatedtotheevolutionformalism.Alotofworkstillremains
tobedonein thisdirection.
ACKNOWLEDGMENTS
ThisworkhasbeensupportedbytheSpanishMinisteriodeEducaciónyCiencia,MEC-
FEDERproject AYA2003-08739-C02-02.
REFERENCES
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causales de referencialesde espacio-tiempo" in Actas de los E. R. E-88, edited by J. Martín and E.
Ruíz(UniversidaddeSalamanca,1989),171–180.
3. G.Lemaître,Gen.Rel.andGrav.29,641–680(1997).
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7. J. J. Ferrando,"Coll Positioning Systems: a two-dimensionalapproach"(in these proceedings).See
alsogr-qc/0601117.
8. B.Coll,"CoordenadasLuzenRelatividad"inTrobadesCientífiquesdelaMediterrània:ActesE.R.E-
85,editedbyA.Molina(ServeidePublicacionsdel’ETSEIB,Barcelona,1985),29–39.AnEnglish
translationofthisarticleentitledLightcoordinatesinrelativityisavailableinhttp//www.coll.cc/.