Relativistic versus Newtonian frames: emission coordinates

Abstract

Only a causal class among the 199 Lorentzian ones, which do not exists in the Newtonian spacetime, is privileged to construct a generic, gravity free and immediate (non retarded) relativistic positioning system. This is the causal class of the null emission coordinates. Emission coordinates are defined and generated by four emitters broadcasting their proper times. The emission coordinates are covariant (frame independent) and hence valid for any user. Any observer can obtain the values of his(her) null emission coordinates from the emitters which provide him his(her) trajectory. Comment: 6 pages, 3 figures, added references, minor changes, Talk presented by one of us (JFPS) at the 2nd Colloquium "Scientific and Fundamental Aspects of the Galileo Programme", Padua, Italy, 14-16 October 2009, submitted to the Proceedings

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RELATIVISTIC VS NEWTONIAN FRAMES: EMISSION COORDINATES
J.-F. Pascual-S´ anchez, A. San Miguel, and F. Vicente
Dept. Matem´ atica Aplicada, Facultad de Ciencias, Universidad de Valladolid, Valladolid, 47005, Spain, E.U.
ABSTRACT
The number of possible 4D coordinate systems is larger
in a relativistic Lorentzian spacetime compared with a
Newtonian one. In the 4D Newtonian spacetime, in
which the speed of light is infinite, there exist 4 and only
4, causal classes of reference frames. Only one of them is
usedintheGPSandGLONASSsystems. However, inthe
relativistic 4D Lorentzian spacetime, due to the freedom
introduced by the finite propagation of light, there exist
199, and only 199, causal classes of reference frames and
the Newtonian ones are a small subclass.
A causal class is defined by a tangent frame, the dual
coframe and the 2D surfaces generated by the vectors of
the frame. Only a causal class among the 199 Lorentzian
ones, which do not exists in the Newtonian spacetime, is
privileged to construct a generic, gravity free (the previ-
ous knowledge of the gravitational field is not necessary)
and immediate (non retarded) positioning system. This
is the causal class of the emission coordinate system. In
this causal class, the coframe is non-orthogonal, real and
null and the four vectors of the frame are spacelike.
Emission coordinates are defined and generated by four
emitters broadcasting their proper times. The emission
coordinates are covariant (frame independent) and hence
valid for any user. Any observer can obtain the values
of the emission coordinates of the emitters which provide
him his trajectory.
If the emitters also broadcast to the users the proper times
they are receiving by cross-link autonavigation from the
other emitters, the system is called autonomous. In this
case, the trajectories of the emitters can be also known
by the users. This exact relativistic positioning system is
the alternative to considering post-Newtonian relativistic
corrections in a classical Newtonian framework, which is
the customary approach yet now used in the GPS and the
GLONASS.
Key words:
causal class, positioning system, gravimetry.
Emission coordinates, location systems,
1.INTRODUCTION
Globally, the current situation in the Global Navigation
Satellite Systems (GNSS) is almost analogous to the fo-
llowing one: imagine that a century after Kepler, the as-
tronomers were still using Kepler’s laws as algorithms
to correct the Ptolemaic epicycles by means of “Keple-
rian effects”. Similarly, a century after Einstein, one still
uses the Newtonian theory and corrects it by “relativistic
or Einsteinian effects” instead of starting with Einstein’s
gravitational theory from the beginning.
To show this, we will focus on the essential differences
between an old newtonian plus “relativistic corrections”
coming from the post-Newtonian framework, as in the
current operating systems which use only the usual class
ofNewtonianframes, andthenewfullyrelativisticframe-
work which use a new class of relativistic frame: emis-
sion coordinates. Note that there are not “relativistic co-
rrections” in relativity, as they are not “Newtonian cor-
rections” in Newtonian theory.
At present, the GNSS functioning as global positioning
systems, are the GPS and the GLONASS. In general, the
satellites (SVs) of the GNSS are affected by Relativity
in three different ways: in the equations of motion, in
the signal propagation and in the beat rate of the satellite
clocks. We will only briefly comment on the clock effects
because they are only the measurable ones in the present
GNSS and in the future Galileo.
Among the relativistic effects on the rate of the satellite
clocks with a time accuracy of nanoseconds and 10−12
of frequency accuracy, the most important ones (to first
post-newtonian order 1/c2) are: the Einstein effect or
gravitational frequency blue shift of the atomic clocks
of the satellites (Equivalence principle of General Rel-
ativity) with respect to Earth bound clocks due to their
position in the Earth gravitational field, time dilation or
Doppler shift of second order due to the speed of the
satellites (Special Relativity) and the kinematical Sagnac
effect due to the rotation of the Earth (Special Relativ-
ity), see Refs. [1] and [2] for reviews. If they were not
corrected by imposing an offset, the GNSS would not be
operative after few minutes.
However, with the present and future more accurate
clocks (pico and even femtosecond), it would be neces-

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sary in the Newtonian framework to consider other “rela-
tivistic corrections” at post-post-Newtonian order as well
as metric spatial curvature effects, tidal effects or delay
effects of gravity in the light propagation as the Shapiro
time delay.
In this situation, it can be wondered if it would not be
more convenient to change the present Newtonian frame-
work to an exact formulation in full General Relativi-
ty. This would imply to abandon the classical post-
newtonian framework for the description of GNSS. The
root of this radical change is the consideration of a new
4D proper relativistic frame (emission coordinates) ins-
tead of the usual Newtonian frame, which uses 3D spatial
reference systems, such as the ECI (Earth Centered In-
ertial system) or the ECEF (Earth Centered Earth Fixed
system), and a time reference (GPS time), separately.
Emission coordinates were firstly introduced by B. Coll
in a pioneering proposal presented at the ERE2000 Spa-
nish Relativity Meeting and published in [3]. To discuss
and understand the meaning of the emission coordinate
system is necessary to introduce previously some new
definitions, as such location systems or causal classifi-
cation of frames, and mathematical physics tools, mainly
geometrical. These new definitions and tools provide a
clear way to understand the differences among the spe-
cial subclass of Newtonian frames and the general class
of relativistic frames.
2.LOCATION SYSTEMS
Location systems are physical realizations of 4D coordi-
nate systems. Hence there is a differentiation of a coordi-
nate system as a mathematical object from its realization
through physical objects and protocols. A location sys-
tem is thus a precise protocol on a set of physical fields
allowing to materialize a coordinate system. However,
different physical protocols, involving different physical
fields, maybegivenforauniquemathematicalcoordinate
system.
A location system must include the protocols for the
physical construction of the coordinate lines, coordinate
surfaces or coordinate hypersurfaces of specific causal
orientations of the coordinate system that it realizes.
Thus, for instance, these coordinate elements may be
realized by means of clocks for timelike lines, laser
pulses for null lines, synchronized inextensible threads
for spacelike lines, laser beams or inextensible threads
for time like surfaces and light-front surfaces for null hy-
persurfaces. The different protocols involved in the con-
struction of location systems give rise to coordinate e-
lements (lines, surfaces and hypersurfaces) of different
causal orientations, i.e., they realize coordinate systems
of different causal nature.
2.1. Reference systems
Location systems are of two different types: reference
systems and positioning systems. The first ones are 4D
reference systems which allow one observer, considered
at the origin, to assign four coordinates to the events of
its neighborhood by means of electromagnetic signals. In
relativity due to the finite speed of light, this assignment
is retarded with a time delay.
A paradigmatic reference system in relativity is the radar
system which is based in the Poincar´ e protocol of syn-
chronization which uses two-way light signals from the
observer to the events to be coordinated. Unfortunately,
the radar system suffers from the bad property of be-
ing constructed from a retarded protocol due to the finite
speed of light.
2.2.Positioning systems
The second kind of location systems are 4D positioning
systems, which allow to every event of a given domain
to know its proper coordinates in an immediate or instan-
taneous way without delay. In addition to be immediate,
the positioning systems must verify other two conditions,
they must be generic and free of gravity. A positioning
system is generic, if it can be constructed in any space-
time and, it is free of gravity, if the knowledge of the
gravitational field is not necessary to construct it. Re-
ference systems privilege one specific observer among
all others, whereas in positioning systems no observers
are necessary at all and hence there is no necessity of any
synchronization procedure between different observers.
In relativity, a (retarded) reference system can be con-
structed starting from an (immediate) positioning system,
it is sufficient that each event sends its coordinates to the
observer at the origin of the reference system, but not
the other way around. In contrast, in Newtonian theory,
3Dreferenceandpositioningsystemsareinterchangeable
and as the velocity of radio signals, i.e. the speed of light,
is infinite, the Newtonian reference systems are not re-
tarded but immediate. The reference and positioning sys-
tems defined here are 4-dimensional objects, including
time location.
3.CAUSAL CLASSIFICATION OF FRAMES
In the Lorentzian spacetime of general relativity, direc-
tions and planes or hyperplanes of directions at any event
are said to be spacelike, lightlike (or null or isotropic) or
timelike oriented if they are respectively exterior, tangent
or secant to the light-cone of this event. These causal
orientations can be extended in a natural way to vectors,
covectors and volume forms on these sets of directions.
Thus, every one of the vectors eAof a frame of the tan-
gent space {eA} (A = 1,...,4) has a particular causal
orientation cA.

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However, the causal orientations CAB (A < B) of the
six different associated or adjoint planes Π(eA,eB) of
the frame {eA} are not determined by the specific causal
orientations cAof the vectors of the frame. For instance,
the plane associated to two spacelike vectors may have
any causal orientation. So, in general, the causal charac-
ters cAand CABare independent. Moreover, in order to
give a complete description of the causal properties of the
frames, one needs also to specify the causal orientations
cAof the four covectors or 1-forms θAgiving the dual
coframe {θA}, i.e. θA(eB) = δA
best way to visualize and characterize a spacetime coor-
dinate system is to start from four families of coordinate
3-surfaces, then, their mutual intersections give six fami-
lies of coordinate 2-surfaces and four congruences of co-
ordinate lines.
B. Following [4], the
Alternatively, one can use the related covectors or 1-
forms θA, instead of the 3-surfaces, and the vectors of a
coordinatetangentframe{eA}, insteadoffourcongruen-
ces of coordinate lines which are their integral curves.
The covector θAis timelike (resp.
3-plane Π(eB,eC,eD) generated by the three vectors
{eB}B?=Ais spacelike (resp. timelike). This applies for
both Newtonian and Lorentzian spacetimes. In addition,
forthelatter, thecovectorθAislightlike(ornull)iffthe3-
plane generated by {eB}B?=Ais lightlike (or null). Thus,
to specify the causal orientations of hyperplanes is not
necessary because is redundant with the causal orienta-
tion of the covectors.
spacelike) iff the
In this way, for a specific domain of a Lorentzian or New-
tonian spacetime, each frame {eA} is fully characterized
by its causal class. The causal class of a frame is the set
of all the frames that have same causal signature, which
is defined by a set of 14 causal characters:
{c1c2c3c4, C12C13C14C23C24C34, c1c2c3c4},
As notation for the causal characters, we will use lower
case roman types (s,t,l) to represent the causal charac-
ter of vectors (resp. spacelike, timelike, lightlike), and
capital types (S,T,L) and lower case italic types (s,t,l)
to denote the causal character of 2-planes and covectors,
respectively.
(1)
3.1.Relativistic frames
This new degree of freedom (lightlike) in the causal char-
acter, which is proper of Lorentzian relativistic space-
times but which does not exist in Newtonian spacetimes,
allows to obtain (see [4]), as it has been commented in
the abstract, the following theorem: In a relativistic 4-
dimensional Lorentzian spacetime, there exists 199, and
only 199, causal classes of frames. These 199 causal
classes have been completely classified.
We shall see that among the 199 Lorentzian causal
classes, only one is privileged to construct a generic gra-
vity free and immediate positioning system.
The notion of causal class extends naturally to the set of
coordinate lines of the coordinate system and so, to the
coordinate system itself. By definition, the causal class
of a coordinate system {xα}4
class {cα, Cαβ, cα} of its associated natural frame at the
events of the domain. In relativity, a specific causal class,
among the 199 ones, can be assigned to any of the dif-
ferent coordinate systems used in all the solutions of the
Einstein equations. However, for the same coordinate
system and the same solution, the causal class can change
depending on the region of the spacetime considered and
the coordinate system in this case is said to be inhomoge-
neous.
α=1in a domain is the causal
In fact, in any spacetime, every coordinate xαplays two
extreme roles: that of a hypersurface for every constant
value xα= const, of gradient dxα, and that of a coor-
dinate line of tangent vector ∂α, when the other coordi-
nates remain constant. This simple fact shows that, in
spite of the historical custom of associating to a coordi-
nate a causal orientation, saying that it is timelike, light-
like or spacelike, this appellation is not generically coher-
ent. Causal orientations are generically associated with
directions of geometric objects, but not with spacetime
coordinates associated to them. In the case of a coordi-
nate xα, this generic incoherence appears because its two
natural variations in the coordinate system, dxαand ∂α,
have generically different causal orientations. Only when
both causal orientations coincide, it is possible to extend
to the coordinate xαitself the character of the common
causal orientation of its two mentioned variations.
3.2.Newtonian frames
The
Lorentzian and Newtonian frames come from the
causal structure induced by the different metric descrip-
tions of Lorentzian and Newtonian spacetimes.
main difference comes essentially from the absence of
the lightlike character in the Newtonian case, because
the infinite speed of light has flattened the light cone
at any event. In relativity, the spacetime metric is
non-degenerate and defines a one-to-one correspon-
dence between vectors and covectors at the tangent and
cotangent space of every event.
differencesin thegeometric descriptionof
The
In contrast, in a Newtonian space-time no non-degenerate
metric structure exists and one have two different me-
trics, see [6]. This degenerate metric structure is given
by a rank one covariant time metric T = dt2and an or-
thogonal rank three contravariant space metric γ. In the
time metric appears t which is a absolute time scale and
the hypersurfaces t = const constitute the instantaneous
or simultaneity spaces. A vector e is spacelike if it is in-
stantaneous, i.e. if dt(e) = 0. Otherwise, it is is timelike.
So, itisclearthataframecanhaveatmostthreespacelike
vectors so there only exist four causal types of Newtonian
frame bases, namely: {tsss}, {ttss}, {ttts}, {tttt}.
Correspondingly, a covector θ ?= 0 is timelike if it has no
instantaneous part with respect to the contravariant space

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metric γ, i.e. if γ(θ) = 0 and it is necessarily of the
form θ = adt with a ?= 0, being future (resp. past)
oriented if a > 0 (resp. a < 0). Otherwise, the covector
θ is spacelike. Thus, attending to the causal orientation
of their covectors, there only exist two causal types of
Newtonian coframes bases: {tsss},{ssss}.
In summary, it can be shown that one has the fol-
lowing implications valid only for Newtonian frames:
{cA} ⇒ {CAB,cA},
{cA},
The simplicity of the Newtonian causal structure with
respect to the Lorentzian one lies in the fact that the
causal type of a Newtonian frame determines completely
its causal class. This is related to the fact that, in
Newtonian space-time, any set of spacelike vectors al-
ways generates a spacelike subspace. As a consequence,
the number of causally different Newtonian classes of
frames is equal to the dimension of the space. Hence,
in the 4-dimensional Newtonian space-time there exist
four, and only four, causal classes of frames.
are:
{tsss, TTTSSS, tsss}, {ttss, TTTTTS, ssss},
{ttts, TTTTTT, ssss} and {tttt, TTTTTT, ssss}.
For instance, the standard spatial coordinates ECI and
ECEF used in the GPS more the GPS time, i.e. those
that are locally realized with three rods and one clock,
belong to the same causal class {tsss, TTTSSS, tsss},
the first one above. The history of the clock is a timelike
coordinate line. The other coordinate lines are spacelike
straight lines tangent to the rods at every time. Also the
reference systems adopted by the I.A.U. for the Earth and
the barycenter of the Solar system as, respectively, the 3-
dimensional International Terrestrial Reference System
(ITRF), which is also an ECEF, plus the International
Atomic Time (TAI) and the 3-dimensional International
Celestial Reference Frame (ICRF) plus the TCB time be-
long to this usual causal Newtonian class.
{CAB} ⇒ {cA}, but {CAB} ?
{cA} ? {cA,CAB}
.
They
4. RELATIVISTIC POSITIONING
4.1. Coll positioning system
As it has been commented above, among the 199
Lorentzian causal classes, in which the four Newtonian
ones are included, only one is privileged to construct a
generic (valid for a wide class of spacetimes), gravity free
(the previous knowledge of the gravitational field is not
necessary) and immediate relativistic positioning system.
This is the causal class {ssss,SSSSSS,llll} of the
Coll homogeneous coordinate system [3, 2, 4]. In this
causal class the emission coordinates of the Coll posi-
tioning system are included. These emission coordinates
have been also studied in [7, 8, 9] in the special case of a
flat Minkowski spacetime without gravity.
The coordinate system of this causal class is always ho-
mogeneous and it has associated four families of null
3-surfaces or equivalently a real non-orthogonal null
Figure 1. Relativistic emission coordinates: intersection of the
four future light cones of the SVs with the past light cone of a
receiver. In the Figure only 3 light cones of the SVs are drawn
in a Lorentzian spacetime of 3 dimensions.
coframe, whose mutual intersections give six families of
spacelike 2-surfaces and four congruences of spacelike
lines. Such a coordinate system does not exist in a New-
tonian space-time where the light travels at infinite speed.
One satellite clock broadcasting its proper time is des-
cribed in the spacetime by a timelike line γA(τA) in
which each event of proper time τAis the vertex of a
future light cone. The set of these light cones of a emit-
ter constitutes a one-parameter (proper time) family of
null hypersurfaces. So, four satellite clocks broadcasting
their proper times determine four one-parameter families
of lightlike 3-surfaces (future light cones), see Figure1.
Thus, the Coll positioning system makes use of the ma-
thematical fact that four future light cones generically in-
tersect in an unique event, which is just the spacetime
position of the receiver or user.
In this relativistic positioning system, any receiver or user
at any event in a given spacetime region can know its
proper coordinates. The four proper times of four sate-
llites ({τA}; A = 1,2,3,4) read at an event by a receiver
or user constitute the null (or light) proper emission coor-
dinates or user positioning data of this event, with respect
to four SVs, see Figure2. These four numbers or pa-
rameters can be understood as the “distances” between
the reception event and the four satellites.
In a certain domain Ω ⊂ R4of the grid of parameters
{τA}, any user receiving continuously his emission co-
ordinates from four satellites may know his trajectory in
the grid of parameters. If the observer has his own clock,
with proper time denoted by σ, then he can know his tra-
jectory with proper time parametrization, τA= τA(σ),
and his four-velocity, uA(σ) = dτA/dσ.
For positioning out a GNSS constellation, i.e. for inter-
planetary missions in the Solar system, a “pulsar” Coll
relativistic positioning system can be conceived, based
on the X-ray signals of four properly selected stable mi-
llisecond pulsars and a conventional origin of the emi-
ssion coordinates. On the other hand, a navigation project
calledXNAV(basedinpulsars)isbeingdevelopedduring
the last years by DARPA and NASA but unfortunately,
see[11], isbasedinthesameNewtonianconceptsthatthe

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Figure 2. Past light cone of an event in 3 dimensions, the
proper time parameterized paths of 3 SVs (in violet) and the
lightlike geodesics (in green) followed by the signal from each
sate-llite to a event of the trajectory of a receiver.
GPS or Galileo. However, in this case, it is more compli-
cated because post-post-Newtonian corrections must be
implemented.
4.2. Contravariant metric in emission coordinates
As the emission coordinates belong to the causal class
{ssss,SSSSSS,llll} There is not space-time asym-
metry like in the standard Newtonian coframe (t s s s)
(one timelike “t” and three spacelike “s”). In emission
coordinates obtained from a general real null coframe
(l l l l) = {dτ1,dτ2,dτ3,dτ4}, which is neither or-
thogonal nor normalized, the contravariant spacetime
metric is symmetric with null diagonal elements and it
has the general expression [14, 2]:


where gAB> 0 for A ?= B. Four null covectors can
be linearly dependent although none of them is propor-
tional to another. To ensure that the four null covectors
are linearly independent and span a 4-dim spacetime, it
is sufficient that det(gAB) ?= 0. Finally, this metric has a
Lorentzian signature (+,−,−,−) iff det(gAB) < 0.
The expression (2) of the metric is observer independent
and has six degrees of freedom. In the terminology of
[7], the proper times τAare partial observables, while
the components of the metric gABare complete obser-
vables, i.e, gauge independent or invariant quantities un-
der diffeomorphisms in the Lorentzian spacetime.
gAB= dτA· dτB=

0
g12
0
g23
g24
g13
g23
0
g34
g14
g24
g34
0
g12
g13
g14


, (2)
A splitting of this metric can be considered, see [14],
changing from the six independent components (ten com-
ponents minus four gauge degrees of freedom of coordi-
nate transformations) of gABto a more convenient set,
which neatly separates two shape parameters depending
only on the direction of the covectors dτAor equiva-
lently depending exclusively on the trajectories of the
emitters, from other four scaling parameters depending
on the length of the covectors or depending on the proper
time of each satellite.
4.3.SYPOR project: autolocated positioning system
SYPOR project is the anagram (in French) of Relativis-
tic Positioning System project. The basic idea of this
project as it is exposed in [5, 2] is the following one:
A satellite constellation provided with clocks that inter-
change its proper time among them (interlinks) and with
Earth receivers, is a fully relativistic autonomous or au-
tolocated positioning system. Note that, nowadays, this
procedure of proper time auto navigation can be techni-
cally fulfilled.
In the SYPOR, the segment of Control is in the constella-
tion of satellites, see Figure3. The function of this new
Control segment is not to determine the ephemerides of
the satellites with respect to geocentric coordinates as in
the newtonian GNSS, but to determine the emission coor-
dinates of the receivers with respect to the constellation
of SVs. Therefore, the procedure used until now in the
newtonian GNSS is inverted.
Let us define properly what means autonomous or autolo-
cated. Four satellites emitting, without the necessity of a
synchronization convention, not only its proper times τA,
but also the proper times τABof the three close satellites
received by the satellite A in τA(in total sixteen emitter
positioning data {τA,τAB}; A ?= B; A,B = 1,2,3,4),
constitute an autolocated positioning system.
In an autolocated positioning system, the receivers can
know not only its spacetime path but also the trajectories
of the four satellites in the grid R4of emission coordi-
nates.
5.GRAVIMETRY AND POSITIONING
In General Relativity, the gravitational field is described
by the spacetime metric. If this metric is exactly known
a priori, the system just described will constitute an ideal
positioning system. In practice, the actual spacetime me-
tric (i.e., the gravitational field) is not exactly known (in
theGPSitissupposedtobeessentiallytheSchwarzschild
one) and the satellite system itself has to be used to infer
it. This problem arises when a satellite system is used for
both positioning and gravimetry.
To solve this joint problem, the considered satellites
should have more than one clock: they may carry an ac-
celerometer providing information on the spacetime con-
nection. Of course, in first approximation the satellites
are in free-fall and consequently have zero acceleration.
However, we are considering here the realistic case where