Coll Positioning systems: a two-dimensional approach

0.04 MB·English

by  Joan Josep Ferrando

Coll Positioning systems:
a two-dimensional approach
Joan Josep Ferrando
Departamentd’AstronomiaiAstrofísica,UniversitatdeValència,46100Burjassot,València,Spain
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0 Abstract. ThebasicelementsofCollpositioningsystems(nclocksbroadcastingelectromagnetic
0 signalsin a n-dimensionalspace-time)arepresentedin the two-dimensionalcase. This simplified
2 approachallows us to explain and to analyze the propertiesand interest of these relativistic posi-
tioningsystems.Thepositioningsystemdefinedinflatmetricbytwogeodesicclocksisanalyzed.
n
a TheinterestoftheCollsystemsingravimetryispointedout.
J
Keywords: relativisticpositioningsystems

7 PACS: 04.20.-q,95.10.Jk
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INTRODUCTION
v
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The relativistic positioning systems were introduced by B. Coll a few years ago at the
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Spanish Relativity Meeting celebrated in Valladolid [1]. Remember that these systems
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0 are defined by four clocks broadcasting their proper time. Here we will name them for
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short ’Coll systems’. In a ’long contribution’ to these proceedings B. Coll explains
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/ the interest, characteristics and good qualities of these relativistic positioning systems
c
in the generic four-dimensional case. In this short communication we present a two-
q
- dimensionalapproach totheCollsystems.
r
g Thetwo-dimensionalapproachshouldhelpusunderstandbetterhowtheserelativistic
:
v systemsworkandtherichnessoftheelementsoftheCollsystems.Indeed,thesimplicity
i ofthe2-dimensionalcaseallowsustousepreciseand explicitdiagramswhichimprove
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the qualitative comprehension of the positioning systems. Moreover, two-dimensional
r
a examplesadmitsimpleandexplicitanalyticresults.
Nevertheless, it is worth remarking that the two-dimensional case has particulari-
ties and results that cannot be generalized to the generic four-dimensional case. Con-
sequently, the two-dimensional approach is suitable for learning basic concepts about
positioningsystems,butit does not allowusto studysomespecificpositioningfeatures
thatnecessarilyneed a three-orafour-dimensionalapproach.
In the second section we introduce the basic elements of a Coll system: emission
coordinates and its other essential physical components. In the third section we explain
theanalyticmethodtoobtaintheemissioncoordinatesfromanarbitrarynullcoordinate
systemandweuseittodevelopthepositioningsystemdefinedinflatspace-timebytwo
geodesic clocks. We finish with some comments about other positioning subjects that
weareconsideringat present.
BASIC CONCEPTS AND ESSENTIAL PHYSICAL COMPONENTS
Inatwo-dimensionalspace-time,letg andg betheworldlinesoftwoclocksmeasuring
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their proper times t 1 and t 2 respectively. Suppose they broadcast them by means of
electromagnetic signals, and that these signals reach each other of the world lines. The
future light cones (here reduced to pairs of ’light’ lines) cut in the region between
both emitters and they are tangent outside. Thus, these proper times do not distinguish
differenteventson theemissionnullgeodesicsoftheexteriorregion.
Theinternalregion,boundedbytheemitterworldlines,definesacoordinatedomain,
the emission coordinate domain W . Indeed, every event on this domain can be distin-
guished by the times (t 1,t 2) received from the emitter clocks. In other words, the past
light cone of every event on the emission domain cuts the emitter world lines at g (t 1)
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and g (t 2) respectively: then {t 1,t 2} are the emission coordinates of this event. An
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importantproperty of theemissioncoordinates we havedefined is that they are nullco-
ordinates. The plane {t 1}×{t 2} in which the different data of the positioning system
can betranscribedis calledthegridofthepositioningsystem.
An observer g travelling throughout the emission coordinate domain and equipped
with a receiver which allows to read the proper times (t 1,t 2) at each point of his
trajectoryis auserofthispositioningsystem.
In defining the emission coordinates we have introduced the first essential physical
componentsofaColl system:
- Theprincipalemittersg , g , which broadcasttheirpropertimet 1, t 2.
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- Theusersg ,travellingintheemittercoordinatedomainW ,receivetheemittedtimes
{t 1,t 2} (theiremittercoordinates).
Theseelementsdefineageneric,freeandimmediatelocationsystem(itcanbedefined
inagenericspace-time;itcanbedefined withoutknowingthegravitationalfield; auser
knowshiscoordinateswithoutdelay).
Any user receiving continuously the user’s positioning data {t 1,t 2} may extract his
trajectory, t 2 = F(t 1), in the grid. Nevertheless, whatever the user be, these data are
insufficientto constructbothofthetwo emittertrajectories.
In order to give to any user the capability of knowing the emitter trajectories in the
grid, the positioning system must be endowed with a device allowing every emitter to
alsobroadcast thepropertimeit isreceivingfrom theotheremitter:
- The emitters g , g are also transmitters: they receive the signals (such as a user)
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and broadcastthem.
- Theusersg also receivethetransmittedtimes{t¯1,t¯2}.
Inotherwords,theclocksmustbeallowedtobroadcasttheiremissioncoordinatesand
then,anyuserreceivingcontinuouslytheemitter’spositioningdata{t 1,t 2;t¯1,t¯2}may
extractfromthemtheequationst¯2=j (t 1)andt¯1=j (t 2)oftheemittertrajectories.
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Apositioningsystemso endowedwillbecalled an auto-locatedpositioningsystem.
Eventually,thepositioningsystemcan beendowedwithcomplementarydevices.For
example,in obtainingthedynamicpropertiesofthesystem:
- Theemittersg , g can carry accelerometers and broadcast theiracceleration.
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- Theusersg can alsoreceivetheemitteracceleration data{a ,a }.
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In some cases, it can be useful that the users generate their own data: they can carry
a clock that measures their proper time t and an accelerometer that measures their
acceleration a .
Thus, a Coll positioning system can be performed in such a way that any user can
obtainasubsetoftheuserdata: {t 1,t 2;t¯1,t¯2;a ,a ;t ,a }.
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POSITIONING WITH GEODESIC EMITTERS IN FLAT METRIC
Let us assume the proper time history of two emitters to be known in a null coordinate
system{u,v}:
u=u (t 1) u=u (t 2)
g ≡ 1 g ≡ 2 (1)
1 (cid:26)v=v1(t 1) 2 (cid:26)v=v2(t 2)
Wecan introducethepropertimesascoordinates {t 1,t 2} as follows:
u=u (t 1), v=v (t 2) (2)
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This change defines emission null coordinates in the emission coordinate domain
W ≡ (u,v) / F−1(v) ≤ u, F (u) ≤ v . In the region outside W this change also
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determ(cid:8)ines null coordinates which are an ex(cid:9)tension of the emission coordinates. But in
this region the coordinates are not physical, i.e. are not the emitted proper times of the
principalemittersg , g .
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Now we use this procedure for the case of two geodesic emitters g , g in flat space-
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time. In inertial null coordinates {u,v} the proper time parametrization of the emitters
are:
u=l t 1 u=l t 2+u
g1 ≡(cid:26)v= l11t 1+v0 g2 ≡(cid:26)v= l12t 2 0 (3)
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Then,theemittercoordinates{t 1,t 2}are defined by thechange:
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u=u (t 1)=l t 1, v=v (t 2)= t 2 (4)
1 1 2 l
2
From here we can obtain the metric tensor in emitter coordinates {t 1,t 2} and we
obtain: ds2 = l dt 1dt 2,l ≡ l 1. On the other hand, in emission coordinates {t 1,t 2},
l
2
theequationsoftheemittertrajectoriesare:
g ≡ t 1 =t 1 g ≡ t 1 =j 2(t 2)≡ l1t 2+t 01 (5)
1 (cid:26)t 2 =j 1(t 1)≡ l1t 1+t 02 2 (cid:26)t 2 =t 2
Let g be a user of this positioning system. What information can this user obtain
from the public data? Evidently (t 1,t 2) place the user on the user grid, and (t¯1,t¯2),
t¯i=j (t j),placetheemittersontheusergrid.Ontheotherhand,themetriccomponent
j
could be obtainedfrom theemitter’s positioningdata {t 1,t 2;t¯1,t¯2} at two events. The
space-timeintervalis:
D t 1D t 2
ds2 = dt 1dt 2 (6)
rD t¯1D t¯2
DISCUSSION AND WORK IN PROGRESS
Wefinishthistalkwithsomecommentsaboutotherpositioningsubjectswearestudying
at present. Firstly, the interest of the Coll systems in gravimetry. If we suppose that the
userhas nopreviousinformationonthegravitationalfield, whatmetricinformationcan
a user obtain from the public and proper user data? Can a user do gravimetry by using
ourpositioningsystem?We haveshownthat[2]:
- The publicdata {t 1,t 2;t¯1,t¯2;a ,a } determinethe space-timemetricinterval and
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itsgradientalongtheemittertrajectories.
- The public-user data {t 1,t 2;t ,a } determine the space-time metric interval and its
gradientalongtheusertrajectory.
Thedevelopmentofageneralmethodthatofferagoodestimationofthegravitational
field from this information is still an open problem, but some preliminary results show
itsinterestindeterminingtheparameters in agiven(parameterized) model[3].
On the other hand, some circumstances can lead to take another point of view: the
userknowsthespace-timeinwhichheisimmersed(Minkowski,Schwarzschild,...)and
we want to study the information that the data received by the user offer. We have
undertaken this problem for the flat case an we have obtained interesting preliminary
results.In particular, wehaveshownthat[4]:
- If a user receives the emitter positioning data {t 1,t 2;t¯1,t¯2} along his trajectory
and the acceleration of one of the emitters during a sole echo interval (i.e., travel
time of a two-way signal from an emitter to the other), then this user knows: his local
unities of time and distance, the metric interval in emission coordinates everywhere,
hisownaccelerationandtheaccelerationoftheprincipalemitters,thechangebetween
emission and inertial coordinates, his trajectory and the emitter trajectories in inertial
coordinates.
ACKNOWLEDGMENTS
ThisworkhasbeensupportedbytheSpanishMinisteriodeEducaciónyCiencia,MEC-
FEDERproject AYA2003-08739-C02-02.
REFERENCES
1. B. Coll, "Elements for a theory of relativistic coordinate systems. Formal and physical aspects" in
ReferenceFramesandGravitomagnetism(WorldScientific,Singapore,2000),p.53.
2. B.CollB.,J.J.Ferrando,andJ.Morales,(submittedPhys.Rev.D).
3. B.CollB.,J.J.Ferrando,andJ.Morales,(tobesubmittedPhys.Rev.D).
4. J. J. Ferrando,andJ. Morales„"Two-dimensionalconstructions",in Relativistic Coordinates,Refer-
enceandPositioningSystems,NotesSchoolSalamanca2005.