Are there (physically implementable)space-time coordinates that

-besides having the relativistic quality-are immediate ?

Implications for satellite constellations.

Albert TarantolaUniversité de Paris VI

Bartolomé CollObservatoire de Paris

Reference Systems and Positioning Systems

• When characterizing the situation of the points of a region with respect to one observer, we use a Reference System.

• When indicating to each point of a region its owncoordinates, we use a Positioning System.

Reference System Positioning System

observer

users

In Newtonian theory, where the speed of lightis infinite, the reference and positioning functionsare interchangeable.

In relativity, this is not true.

In Newtonian theory, where the speed of lightis infinite, the reference and positioning functionsare interchangeable.

In relativity, this is not true.

Reference systems are necessarily retarded.

Only the positioning systems are immediate.

When possible to build,positioning systems are preferable to reference systems.

Important epistemic result:

Relativistic positioning systems are scarce.

A typical positioning system is made offour sources emitting parameterized electromagnetic signals(e.g. clocks). The coordinate system thus realized is defined

by the coordinate (hyper) surfacesformed by the light cones of each signal.

THE Relativistic Positioning SystemFour clocks broadcast their proper time. Any observer in space-time receives, at any point along its space-time trajectory, fourtimes {τ1, τ2, τ3, τ4} , that are the space-time coordinates. Essen-tially, there is no other immediate coordinate system.

Any observer wandering in space-time with a receiver, can recordits sequence of positions

{τ10 , τ20 , τ30 , τ40}{τ11 , τ21 , τ31 , τ41}{τ12 , τ22 , τ32 , τ42}. . .

i.e., can know its trajectory in that coordinate system.

THE Relativistic Positioning SystemFour clocks broadcast their proper time. Any observer in space-time receives, at any point along its space-time trajectory, fourtimes {τ1, τ2, τ3, τ4} , that are the space-time coordinates. Essen-tially, there is no other immediate coordinate system.

Any observer wandering in space-time with a receiver, can recordits sequence of positions

{τ10 , τ20 , τ30 , τ40}{τ11 , τ21 , τ31 , τ41}{τ12 , τ22 , τ32 , τ42}. . .

i.e., can know its trajectory in that coordinate system.

If the four clocks (or “satellites”) have “cross-link” capabilities(i.e., if each one can receive the signals of the other three), theyknow their trajectory (in these coordinates).

So, instead of just regularly broadcasting their proper time, theycan regularly broadcast their position, i.e., in fact, they broadcasttheir trajectory. Any observer wandering in space-time then con-stantly receives 16 quantities (4 from each satellite).

Property: In special relativity (flat space-time, no gravitation),the observer can use these 16 quantities to compute the compo-nents of the (Minkowski) space-time metric in the coordinates{τ1, τ2, τ3, τ4} . So, not only we have a system of (immediate)coordinates, we know how they relate to the local unit of time(and of length).

THEOREM: Flat space-time (special relativity).

Assume that the 4 trajectories crossed at somepoint in space-time (there are weaker conditions).Then, if an observer has received the completesignals (trajectories) of the 4 satellites from thatmoment on, the observer not only has available a system of immediate space-time coordinates,she/he can express the components of the space-time metric (Minkowski) in these coordinates,so she/he does not need to bring a clock or a meter.

In general relativity this is not true.Redundant data from more than four “satellites”

is then to be used to model the space-time metric.In fact, such a system could be the ultimate

space “gravimeter”

Basic data are the arrival times at somesatellites of the signals emitted by othersatellites.

Basic data are the arrival times at somesatellites of the signals emitted by othersatellites. But satellites may have embarked accelerometers, gradiometers,gyroscopes, interferometers, etc.

Basic data are the arrival times at somesatellites of the signals emitted by othersatellites. But satellites may have embarked accelerometers, gradiometers,gyroscopes, interferometers, etc. Given the space-timemetric, and given the trajectories(in the emission coordinates), those observations can be predicted.

Basic data are the arrival times at somesatellites of the signals emitted by othersatellites. But satellites may have embarked accelerometers, gradiometers,gyroscopes, interferometers, etc. Given the space-timemetric, and given the trajectories(in the emission coordinates), those observations can be predicted.The inverse problem can be set as an optimization problem:

Basic data are the arrival times at somesatellites of the signals emitted by othersatellites. But satellites may have embarked accelerometers, gradiometers,gyroscopes, interferometers, etc. Given the space-timemetric, and given the trajectories(in the emission coordinates), those observations can be predicted.The inverse problem can be set as an optimization problem:Which is the space-time metric thatbest predicts the observations?

First Constraint on the MetricIn the ‘light-coordinates’ {τα} being used, the contravariant com-ponents of the metric must are have zeros on the diagonal,

{gαβ} =

0 g12 g13 g14

g12 0 g23 g24

g13 g23 0 g34

g14 g24 g34 0

,so the basic unknowns of the problem are the six quantities

{g12, g13, g14, g23, g24, g34} .

This constraint is imposed exactly, by just expressing all the rela-tions of the theory in terms of these six quantities.The covariant components of the covariant metric are not zero.

Second Constraint (Einstein Equation)The Einstein equation is

Tαβ =1χ

Eαβ ,

where Tαβ is the stress-energy tensor, and where Eαβ is the Ein-stein tensor,

Eαβ = Rαβ − 12 gαβ R ,

and where

Γαβγ = 12 gασ ( ∂β gγσ + ∂γ gβσ − ∂σ gβγ )

Rαβγδ = ∂γ Γαδβ − ∂δ Γαγβ + Γαµγ Γµδβ − Γαµδ ΓµγβRαβ = Rγαγβ .

And so on...

In fact, such a system would theultimate space gravimeter...

GOCE Gradiometer

GOCE (launched March 17, 2009) employs a three-axis electro-static gravity gradiometer that allows gravity gradients to be mea-sured in all spatial directions. The measured signal is the dif-ference in gravitational acceleration at the test-mass location in-side the spacecraft caused by gravity anomalies from attractingmasses of the Earth. Exploiting these differential measurementsfor all three spatial axes has the advantage that all disturbingforces acting uniformly on the spacecraft (i.e. drag, thrusters re-sulting in linear or angular accelerations) can be compensated for.The length of the baseline for an accelerometer pair is 50 cm.

A Galactic Positioning System

To define a Galactic Positioning System, we propose:

1. By convention, the four basic pulsars are:

0751+1807 (3.5 ms) , 2322+2057 (4.8 ms) , 0711-6830 (5.5 ms) , 1518+0205B (7.9 ms) .

2. The origin (t01,t02,t03,t04) = (0,0,0,0) of the space-time coordinates is the event 2001 January 1 0H 0’ 0” at the focal point of the Cambridge radiotelescope (the one used to discover the pulsars).

Puls

ar 1

Eart

h

Puls

ar 2

Plu

to

spac

ecra

ft

(0,0)

(3,5)

A Galactic Positioning System

The anomalies (in form and arrival time) allow the authentication of the origin by all users (no need to transport it).

Variations of the form of the pulses.

Space-time events located with an accuracy of the order of 4 ns,i.e., of the order of one meter

(not very accurate, but very stable and with a large domain of validity).

The 4 Causal Classes of Newtonian Frames

are respectively time-like , space-like coordinate

t , e

t , e

T , E

covectors

tangent vectors

planes

The 6 planes of every class X1X2X3X4X5X6 are generated restectivelyby the 4 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

eeeeTTTT

teee ttee ttte tttt

TTTTTE TTTTTT TTTTTT

TTTEEEteeeTTTE

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

covectors

tangent vectors

planes

The 6 planes of every class X1X2X3X4X5X6 are generated respectivelyby the 4 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

THEOREM (Coll et al.): 1+1 dimensions, flat space-time. If an observer has received the complete signals (trajectories) of the 2 satellites, and the acceleration of one of them is known during a causal interval, then, the observer not only has available a system of immediate space-time coordinates, she/he can compute the components of the space-time metric (Minkowski) in these emission coordinates, so she/he does not need to bring a clock or a meter.

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