PHYSICAL RELATIVISTIC FRAMES

Bartolomé Coll

Systèmes de référence relativistesDANOF - CNRS

Observatoire de Paris

bartolome.coll@obspm.frhttp://coll.cc

COLLABORATORS______

Some ideas and results on this subject come from a longcollaboration with

Lluís BEL

Joan Josep FERRANDO

Juan Antonio MORALES

Albert TARANTOLA

Coordinate systems are the basic pieces for the construction of

reference systems

and

reference frames

Particular studies of some of them exist, but their general studyhas never been considered

COORDINATES teee______

In classical space-time, one uses to take coordinate systems in whichone coordinate line is the time, and the other three, constants in time,may be more or less directly constructed by means of rods in theinstantaneous space:

In the space-time, the line determined by the clock is said time-like,and those determined by the rods are said space-like.

COORDINATES tttt______

In classical space-time, coordinate systems other than those of typeteee are also possible, for example those of type tttt constituted byfour megaphones shouting the time of a clock:

Coordinate systems of types teee and tttt are said to belong todifferent causal classes.

How many causal classes of coordinatesexist in Newtonian space-time?

The 12 Causal Classes of Newtonian Frames

are respectively time-like , space-like coordinate

t , e

t , e

T , E

hypersurfaces or covectors

lines or tangent vectors

surfaces or planes

The 6 surfaces or planes of every class X1X2X3X4X5X6 are generated restectivelyby the 4 hypersurfaces or covectors x1, x2, x3, x4 in the following order

X2 = x1 x3X4 = x2 x3X1 = x1 x2 X3 = x1 x4

X5 = x2 x4 X6 = x3 x4

teee ttee ttte tttt

TEEEEE TEEEEE TEEEEE TEEEEE EEEEEE EEEEEE EEEEEE EEEEEE

teee TTTEEE TTTEEE TTTEEE TTTEEE

eeee

The 199 Causal Classes of Space-time Frames

elee tlee ttee llle tlle ttle ttte llll tlll ttll tttl tttt

EEEEEE LEEEEE TEEEEE LLEEEE EEEEEE LEEEEE EEELEE TEEEEE EEEEEE LEEEEE EEEEEE LEEEEE EEEEEE EEEEEE EEEEEE EEEEEE EEEEEE EEEEEE EEEEEE EEEEEE EEEEEE EEEEEE EEEEEE TLEEEE TTEEEE LLLEEE TLLEEE LELEEE LEELEE EELLEE TLEEEE TEEEEE LLEEEE ELEEEE TEEEEE LEEEEE LEEEEE LEEEEE LEEEEE LEEEEE TTLEEE TTTEEE LLLLEE TLLLEE TEELEE EETLEE TTEEEE LLLEEE TLEEEE TTEEEE LLEEEE EELLEE ELEEEE TEEEEE LLEEEE LLEEEE eeee TTLLEE TTTLEE TTTTEE LLLLLE LLELEE LELLEE TLLEEE TLELEE LLLEEE TLLEEE ELELEE TLEEEE TEEEEE LLELEE TLLLLE TTLLLE TTTLLE TTTTLE TELLEE LETLEE TTLEEE TTELEE TTLEEE TTTEEE LLELEE LELLEE LLEEEE TTTTTE LLLLLL TLLLLL TTLLLL TETLEE TTTEEE LLLLEE TLLLEE TLELEE TELLEE TLEEEE TTTLLL TTTTLL TTTTTL TTTTTT LLTLEE TTLLEE TLTLEE TTTLEE

TTLEEE TTTEEE TTLLEE TTLELE TTLEEE TTTEEE TTLLEE TTLELE TLLEEE TTLEEE TTTEEE TLLEEE TLLLEE TLLEEE TTTLEE TTLTEE TTLETL TTTTEE TTTLEE TLLLEE TTLLLE TTLELL TTTLLE TTLTLE TLLTEE leee TTLLTE TTLETL TTTTLE TTLTTE TLLLLE TTLETT TTTTTE TTLLLL TTTLLL TLLTLE TTLTLL TTLLTL TTTTLL TTLTTL TLLTTE TTLLTT TTTTTL TTLTTT TTTTTT

TTTEEE TTTLEE TTTTEL TTTLLE TTTEEE TTTLEE TTTEEE teee TTTTLE TTTTTE TTTLLL TTTTLL TTTTTL TTTTTT

TTLTLE TTLLTE TTTTLE TTTTTE TTLLLE TTTLLE TLLLLE llee TTLTLL TTLLTL TTTTLL TTLLTT TTLTLT TTTTTL TTTTLT TTTTTT

TTTTLE TTTTTE TTTTLL TTTTTL TTTLLE TTTTLT TTTTTT

ttee TTTTTE TTTTTL TTTTTT

llle TTLTLL TTTTLL TTTTTL TTTTTT

tlle TTTTLL TTTTTL TTTTTT

ttle TTTTTL TTTTTT

ttte TTTTTT

llll TTTTTT

tlll TTTTTT

ttll TTTTTT

tttl TTTTTT

tttt TTTTTT

eeee leee teee llee

tlee

B. Coll and J.A. Morales, The 199 Causal Classes of Space-time Frames, Internat. J. Theo. Phys., 31, 6, p 1045-62 (1992)

are respectively time-like , light-like , space-like coordinate

t , l , e

t , l , e

T , L , E

hypersurfaces or covectors

lines or tangent vectors

surfaces or planes

The 6 surfaces or planes of every class X1X2X3X4X5X6 are generated respectivelyby the 4 hypersurfaces or covectors x1, x2, x3, x4 in the following order

X1 = x1 x2 X2 = x1 x3 X3 = x1 x4

X4 = x2 x3 X5 = x2 x4 X6 = x3 x4

Perhaps the origin of this sort of cultural alienation is in the fact thatour vision of the world is too polarized in the evolution point ofview of the space-time, a plausible one, of course, but a veryparticular one in the set of the 199 possible points of view.

____

The study of coordinate systems with potential interest inastronomy and physics, independently of their usual or unusualcharacter, seems largely incomplete.

Frequently, coordinate systems are used to situate points of a regionwith respect to one observer, giving rise to reference systems andreference frames.

But coordinate systems may also be used to indicate, to every point,its position.

In Newtonian theory, both functions may be directly assigned to asole coordinate system, but in relativity this is not possible. Thus,in relativity we are lead to consider separately both,

reference systems and reference frames

and

positioning systems and positioning frames

___________

We are here interested in positioning systems.

A relativistic positioning frame is intended to be

* generic* free* immediate

* generic (for a given class) means that the coordinate system mustexist in any space-time of this class.For example, harmonic frames are generic for the class ofall space-times.

* (gravity-)free means that its construction does not need theknowledge of the gravitational field.For example, harmonic frames are not free.

* immediate means that every point may know its coordinateswithout delay.For example, two-way signals from a focus do notgenerate an immediate system.

An important epistemic result is that the set of relativisticpositioning frames constitute a little class of frames, the simplestone of the class consisting of four satellites (point-like transmittersin free fall) broadcasting their proper times.

* In two dimensions (one time-like /one space-like):

THE PROJECT SYPOR(Système de positionnement relativiste)

______

Objective: to endow the Earth with a primary relativisticpositioning system

THE PROJECT SYPOR______

* The project SYPOR aims to use the constellation of satellites ofthe future GALILEO navigation system as an immediate, generic,free and primary relativistic positionning system for the Earth.

* Every four neighbouring satellites of the constellation constitute alocal chart, the constellation defining then the primary atlas of localcharts for the surrounding area of the Earth.

* Cartographic or geocentric coordinates appear as secondarycoordinate systems. The function of the control segment must beinversely read, as being to establish the transformation between theprimary and the secondary systems.

THE PROJECT SYPOR______

The primary character of the relativistic positioning system leadsto impose:

* a device, on four at least of the satellites, pointing to the ICRS(International Celestial Reference System) in order to define virtuallocal charts "at rest" with respect to the ICRS (external control ofthe system as a whole).

* a device, on every satellite, exchanging proper times, (internalcontrol of the parts of the system)

* a device, on every satellite, broadcasting over the Earth, aside itsproper time, those of their neighbours (control by the u s e r ssegment).