**3. The space-time in the Solar System and near the Earth**

In this Section we will describe the conventions used to describe physics on the Earth's surface and space physics near the Earth and in the Solar System. Instead of starting from the Earth, where Newtonian gravity is dominating, we shall begin with the general relativistic description of the Solar System.

The IAU conventions (Soffel M.H. et al, 2003) for the Solar System identify a system of harmonic coordinates (a BCRF frame) centered on the solar system barycenter and a Post-Newtonian solution of Einstein's equations in a special harmonic gauge at the *O*(1/*c*3)

.

order, which can be interpreted as an asymptotically-Minkowskian Post-Newtonian Einstein space-time. With a suitable coordinate transformation this solution is transformed in a description of the same space-time with new harmonic coordinates centered on the center of the Earth (a GCRF frame). However the presentation of this Einstein's space-time is strongly special relativistic (just *O*(1/*c*2) for the NASA coordinates of spacecrafts (Moyer T.D., 2003)) and becomes Galilean when one makes the transition from the coordinates with origin in the center of the Earth to coordinates fixed on the crust of the Earth (a ITRF frame in the IERS conventions) (IERS, 2003).

#### **3.1 BCRS - Barycentric Celestial Reference System**

The resolution B1.3 of IAU2000 (Soffel M.H. et al, 2003) states that the Barycentric Celestial Reference System BCRS is a global reference system of barycentric space-time coordinates for the Solar System within the framework of general relativity. It is centered in the barycenter of the Solar System, which can be considered as a quasi-inertial Minkowski observer with a constant 4-velocity (the time axis of the barycentric time *tB* = *TCB*), because the effects of the Milky Way are negligible. Its spatial axes (in the instantaneous 3-spaces Σ*tB* with *tB* = *const*. with rectangular 3-coordinates *x<sup>i</sup> <sup>B</sup>*) are restricted to be *kinematically non-rotating*, namely they have no systematic rotation with respect to distant objects in the universe. For all practical applications the spatial axes are assumed to be oriented like the spatial axes of ICRS (see next Section). Therefore to each ICRF frame giving a materialization of ICRS is associated a BCRF frame.

The harmonic 4-coordinates and the retarded Post-Newtonian solution of Einstein's equations for the 4-metric *gBμν*(*xB*) given in the IAU2000 conventions are

$$\begin{aligned} \mathbf{x}\_{B}^{\mu} &= \left(\mathbf{x}\_{B}^{o} = \mathbf{c}\,t\_{B}; \mathbf{x}\_{B}^{i}\right)\_{\prime} \\\\ \mathbf{g}\_{Boo}(\mathbf{x}\_{B}) &= \epsilon \left[1 - \frac{2\,w\_{B}(\mathbf{x}\_{B})}{c^{2}} - \frac{2\,w\_{B}^{2}(\mathbf{x}\_{B})}{c^{4}} + O(c^{-5})\right], \\\ \mathbf{g}\_{Boi}(\mathbf{x}\_{B}) &= \epsilon \left[\frac{4\,w\_{Bi}(\mathbf{x}\_{B})}{c^{3}} + O(c^{-5})\right], \\\ \mathbf{g}\_{Bij}(\mathbf{x}\_{B}) &= -\epsilon^{3}\mathbf{g}\_{Bij}(\mathbf{x}\_{B}) = -\epsilon \left[\left(1 + \frac{2\,w\_{B}(\mathbf{x}\_{B})}{c^{2}}\right)\delta\_{ij} + O(c^{-4})\right]. \end{aligned}$$

The signature of the 4-metric is the same as for Minkowski metric *<sup>η</sup>μν* <sup>=</sup> *�* (+ − −−) ( *�* = + is the particle physics convention, *�* <sup>=</sup> <sup>−</sup> is the general relativity one). The 3-metric <sup>3</sup>*gBij*(*xB*) on the 3-spaces Σ*tB* is positive-definite.

See Appendix A of Ref. (Soffel M.H. et al, 2003) for the Post-Newtonian gravitational potentials *wB*(*xB*) and *w<sup>i</sup> <sup>B</sup>*(*xB*) generated by the Sun and the planets. These extended bodies are usually approximated with their center of mass (mass monopole) carrying, when needed like for Saturn, a spin dipole.

The barycenter of the Solar System has coordinates *x μ <sup>B</sup>*(*B*) = *xo B*; 0*<sup>i</sup>* and its world-line is a straight-line (the time axis) approximating a time-like geodesic of the 4-metric if we neglect galactic and extra-galactic influences. In each point of the barycentric world-line there is an *orthonormal tetrad* with the time-like 4-vector given by the barycenter 4-velocity *u<sup>μ</sup> <sup>B</sup>*(*B*) = 1;0  10 Will-be-set-by-IN-TECH

order, which can be interpreted as an asymptotically-Minkowskian Post-Newtonian Einstein space-time. With a suitable coordinate transformation this solution is transformed in a description of the same space-time with new harmonic coordinates centered on the center of the Earth (a GCRF frame). However the presentation of this Einstein's space-time is strongly special relativistic (just *O*(1/*c*2) for the NASA coordinates of spacecrafts (Moyer T.D., 2003)) and becomes Galilean when one makes the transition from the coordinates with origin in the center of the Earth to coordinates fixed on the crust of the Earth (a ITRF frame in the IERS

The resolution B1.3 of IAU2000 (Soffel M.H. et al, 2003) states that the Barycentric Celestial Reference System BCRS is a global reference system of barycentric space-time coordinates for the Solar System within the framework of general relativity. It is centered in the barycenter of the Solar System, which can be considered as a quasi-inertial Minkowski observer with a constant 4-velocity (the time axis of the barycentric time *tB* = *TCB*), because the effects of the Milky Way are negligible. Its spatial axes (in the instantaneous 3-spaces Σ*tB* with *tB* = *const*.

have no systematic rotation with respect to distant objects in the universe. For all practical applications the spatial axes are assumed to be oriented like the spatial axes of ICRS (see next Section). Therefore to each ICRF frame giving a materialization of ICRS is associated a BCRF

The harmonic 4-coordinates and the retarded Post-Newtonian solution of Einstein's equations

*<sup>B</sup>*) are restricted to be *kinematically non-rotating*, namely they

conventions) (IERS, 2003).

with rectangular 3-coordinates *x<sup>i</sup>*

frame.

**3.1 BCRS - Barycentric Celestial Reference System**

for the 4-metric *gBμν*(*xB*) given in the IAU2000 conventions are

*gBij*(*xB*) = <sup>−</sup>*�* <sup>3</sup>*gBij*(*xB*) = <sup>−</sup>*�*

The barycenter of the Solar System has coordinates *x*

*<sup>B</sup>* <sup>=</sup> *c tB*; *<sup>x</sup><sup>i</sup>*

<sup>1</sup> <sup>−</sup> <sup>2</sup> *wB*(*xB*)

4 *wBi*(*xB*)

*B* ,

*<sup>c</sup>*<sup>2</sup> <sup>−</sup> <sup>2</sup> *<sup>w</sup>*<sup>2</sup>

*<sup>c</sup>*<sup>3</sup> <sup>+</sup> *<sup>O</sup>*(*c*−5)

*<sup>B</sup>*(*xB*)

 ,

 (1 +

The signature of the 4-metric is the same as for Minkowski metric *<sup>η</sup>μν* <sup>=</sup> *�* (+ − −−) ( *�* = + is the particle physics convention, *�* <sup>=</sup> <sup>−</sup> is the general relativity one). The 3-metric <sup>3</sup>*gBij*(*xB*)

See Appendix A of Ref. (Soffel M.H. et al, 2003) for the Post-Newtonian gravitational

are usually approximated with their center of mass (mass monopole) carrying, when needed

straight-line (the time axis) approximating a time-like geodesic of the 4-metric if we neglect galactic and extra-galactic influences. In each point of the barycentric world-line there is an

*orthonormal tetrad* with the time-like 4-vector given by the barycenter 4-velocity *u<sup>μ</sup>*

*<sup>c</sup>*<sup>4</sup> <sup>+</sup> *<sup>O</sup>*(*c*−5)

2 *wB*(*xB*)

*<sup>B</sup>*(*xB*) generated by the Sun and the planets. These extended bodies

 *xo B*; 0*<sup>i</sup>* 

*μ <sup>B</sup>*(*B*) =  ,

*<sup>c</sup>*<sup>2</sup> ) *<sup>δ</sup>ij* <sup>+</sup> *<sup>O</sup>*(*c*−4)

 .

and its world-line is a

*<sup>B</sup>*(*B*) =

 1;0 

*x μ <sup>B</sup>* = *xo*

*gBoo*(*xB*) = *�*

*gBoi*(*xB*) = *�*

on the 3-spaces Σ*tB* is positive-definite.

potentials *wB*(*xB*) and *w<sup>i</sup>*

like for Saturn, a spin dipole.

and with the 3 mutually orthogonal spatial axes *� μ <sup>B</sup>*(*B*)*<sup>r</sup>* = 0;*�B*(*B*)*<sup>r</sup>* whose orientation is determined by ICRS. The instantaneous 3-spaces Σ*tB* are considered as nearly Euclidean inertial 3-spaces. However their extrinsic 3-curvature as 3-sub-manifolds of the space-time is not zero but of order *O*(*c*−2), so that strictly speaking they do not correspond to Einstein's clock synchronization convention.

The world-line *x μ B*(*G*) (*x<sup>o</sup> <sup>B</sup>*) = *xo B*;*xB*(*G*)(*x<sup>o</sup> B*) of the Earth's geo-center (a time-like geodesics of the Post-Newtonian 4-metric *gBμν*(*xB*)) is determined by the *JPL ephemerides as solution of the equations of motion of the solar system bodies*. The geo-center has the 4-velocity *u<sup>μ</sup> B*(*G*) (*x<sup>o</sup> <sup>B</sup>*) = *d x<sup>μ</sup> B*(*G*)(*x<sup>o</sup> B*) *dx<sup>o</sup> B* = 1;*vB*(*G*)(*x<sup>o</sup> B*) and carries spatial 3-axes *� μ B*(*G*)*r* (*x<sup>o</sup> <sup>B</sup>*) = 0;*�B*(*G*)*<sup>r</sup>* assumed parallel to the axes *� μ <sup>B</sup>*(*B*)*<sup>r</sup>* of the barycenter. For an arbitrary point in the solar system with coordinates *xo <sup>B</sup>*;*xB* we have *xB* = *xB*(*G*)(*x<sup>o</sup> B*) +*rB*(*G*)(*x<sup>o</sup> B*).

The global reference system BCRS is the reference system in which the positions and motions of bodies outside the immediate environment of the Earth have to be expressed. It is appropriate for the solution of the equations of motion of solar system bodies (the development of the solar system ephemerides). Within it the positions and motions of galactic and extra-galactic objects are most simply expressed. It is the system to be used for most positional-astronomy reference data, e.g. star catalogues.

#### **3.2 GCRS - Geocentric Celestial Reference System**

The resolution B1.3 of IAU2000 (Soffel M.H. et al, 2003) states that the Geocentric Celestial Reference System BCRS is a global reference system of space-time coordinates for Earth based measurements and the solution of the equations of motion of bodies in the near-Earth environment (artificial satellites) within the framework of general relativity. The GCRS is defined such that the transformation between BCRS and GCRS spatial coordinates *contains no rotation component*, so that GCRS is kinematically non-rotating with respect to BCRS. The equations of motion of an Earth satellite with respect to GCRS will contain relativistic Coriolis forces that come mainly from geodesic precession. The spatial orientation of the GCRS is derived from that of BCRS, that is by the orientation of the ICRS. Its origin is the world-line of the geo-center (fictitious observer at the center of the Earth): it is the time axis of the geocentric time *tG* = *TCG* and the instantaneous 3-spaces Σ*tG* with *tG* = *const*. are inertial hyper-planes (Einstein's convention for clock synchronization) only at the lowest order in 1/c.

GCRS has the following 4-coordinates and retarded Post-Newtonian solution of Einstein's equations for the 4-metric *gGμν*(*xG*)

$$\begin{split} \mathbf{x}\_{\mathrm{G}}^{\mu} &= \left( \mathbf{x}\_{\mathrm{G}}^{\rho} = \mathbf{c} \, t\_{\mathrm{G}}; \mathbf{x}\_{\mathrm{G}}^{a} \right)\_{\mathrm{M}} \\ \mathbf{g}\_{\mathrm{G}\mathrm{od}}(\mathbf{x}\_{\mathrm{G}}) &= \boldsymbol{\epsilon} \left[ 1 - \frac{2 \, w\_{\mathrm{G}}(\mathbf{x}\_{\mathrm{G}})}{c^{2}} - \frac{2 \, w\_{\mathrm{G}}^{2}(\mathbf{x}\_{\mathrm{G}})}{c^{4}} + O(c^{-5}) \right], \\ \mathbf{g}\_{\mathrm{G}\mathrm{od}}(\mathbf{x}\_{\mathrm{G}}) &= \boldsymbol{\epsilon} \left[ \frac{4 \, w\_{\mathrm{G}}^{a}(\mathbf{x}\_{\mathrm{G}})}{c^{3}} + O(c^{-5}) \right], \\ \mathbf{g}\_{\mathrm{G}\mathrm{d}}(\mathbf{x}\_{\mathrm{G}}) &= -\boldsymbol{\epsilon}^{3} g\_{\mathrm{G}\mathrm{d}}(\mathbf{x}\_{\mathrm{G}}) = -\boldsymbol{\epsilon} \left[ \left( 1 + \frac{2 \, w\_{\mathrm{G}}(\mathbf{x}\_{\mathrm{G}})}{c^{2}} \right) \delta\_{\mathrm{ab}} + O(c^{-4}) \right]. \end{split}$$

See Appendix A of Ref. (Soffel M.H. et al, 2003) for the GCRS Post-Newtonian gravitational potentials *wG*(*xG*) and *w<sup>a</sup> <sup>G</sup>*(*xG*). While *wG*(*xG*) generalizes the Newton potential, the components *gGoa*(*xG*) (i.e. *w<sup>a</sup> <sup>G</sup>*(*xG*)) are responsible for the *gravito-magnetic* effects near the Earth like the Lense-Thirring or frame-dragging effect (Ciufolini I. et al, 1995; Will C.M., 2011). The Post-Newtonian solution *gGμν*(*xG*) describes the *exterior* gravitational field outside the Earth surface, not inside.

The geo-center has coordinates *x μ <sup>G</sup>*(*G*) = *xo G*; 0*<sup>a</sup>* with tangent time-like vector (the unit 4-velocity) *u<sup>μ</sup> <sup>G</sup>*(*G*) = (1; 0*a*), while the spatial axes have the 3 orthogonal tangent space-like unit vectors *� μ G*(*G*)*r* . It is a time-like geodesic of the Post-Newtonian 4-metric *gGμν*(*xG*), if the Earth is approximated as a *mass monopole*. Otherwise the Earth mass and spin multipoles will create a deviation of the geo-center world-line from a time-like geodesic.

The tetrad carried by the geo-center is obtained from the BCRS tetrad with the tensorial transformation law of 4-vectors, i.e. with the matrix *<sup>∂</sup><sup>x</sup> μ G ∂x<sup>ν</sup> B* |*geocenter*. To evaluate it one needs the transformation between BCRS and GRCS coordinates. If the barycentric 3-velocity and 3-acceleration of the geo-center are *vB*(*G*) <sup>=</sup> *<sup>d</sup>xB*(*G*) *dtB* and *aB*(*G*) <sup>=</sup> *<sup>d</sup>vB*(*G*) *dtB* respectively and if we introduce the relative 3-vector *rB*(*G*)(*x<sup>o</sup> <sup>B</sup>*) = *xB* <sup>−</sup> *xB*(*G*)(*x<sup>o</sup> <sup>B</sup>*), the BCRS-GCRS coordinate transformation is (Soffel M.H. et al, 2003)

$$\begin{split} t\_G &= t\_B - \frac{1}{c^2} \left[ A(t\_B) + v^i\_{B(G)} r^i\_{B(G)} \right] + \\ &+ \frac{1}{c^4} \left[ B(t\_B) + B^i(t\_B) r^i\_{B(G)} + B^{ij}(t\_B) \, r^i\_{B(G)} \, r^j\_{B(G)} + \mathcal{C}(t\_b, \vec{x}\_B) \right] + O(c^{-5}) = \\ &= t\_B - \frac{1}{c^2} \left[ \int\_{t\_{Bo}}^{t\_B} dt \, \left( \frac{\vec{\sigma}^2\_{B(G)}}{2} + w\_{B\text{ext}}(\vec{x}\_{B(G)}) \right) + v^i\_{B(G)} r^i\_{B(G)} \right] + O(c^{-4}), \\ \mathbf{x}^a\_G &= \delta\_{ai} \left[ r^i\_{B(G)} + \frac{1}{c^2} \left( \frac{1}{2} \, \vec{v}^i\_{B(G)} \, v^j\_{B(G)} \, r^j\_{B(G)} + w\_{B\text{ext}}(\vec{x}\_{B(G)}) \, r^i\_{B(G)} \right) + \\ &+ r^i\_{B(G)} \, a^j\_{B(G)} \, r^j\_{B(G)} - \frac{1}{2} \, a^i\_{B(G)} \, r^2\_{B(G)} \right] + O(c^{-4}), \end{split}$$

The functions *A*(*tB*), *B*(*tB*), *B<sup>i</sup>* (*tB*), *<sup>B</sup>ij*(*tB*) and *<sup>C</sup>*(*tB*,*xB*(*G*)), depending on the BCRS gravitational potentials *wBext*(*xB*(*G*)) (the BCRS Newtonian potential of all solar system bodies apart from the Earth acting on the geo-center) and *w<sup>i</sup> Bext*(*xB*(*G*)) (the BCRS gravito-magnetic potential) are given in Ref. (Soffel M.H. et al, 2003).

As shown in Ref. (Soffel M.H. et al, 2003), this transformation reduces to a pure Lorentz boost without rotation modulo terms of order *O*(*c*−4) in the limit of no acceleration due to the gravitational field (i.e. with *xB*(*G*)(*tB*) = *vB*(*G*) *tB*, *vB*(*G*) = *const*., *vB*(*G*) = |*vB*(*G*)|, *βB*(*G*) = *vB*(*G*)/*C*, *<sup>γ</sup>B*(*G*) = (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> *B*(*G*) )−1/2)

$$\begin{aligned} t\_G &= \gamma\_{B(G)} \left( t\_B - \frac{\vec{v}\_{B(G)} \cdot \vec{x}\_B}{c^2} \right) + O(c^{-4}), \\ \vec{x}\_G &= \vec{x}\_B - \gamma\_{B(G)} \vec{v}\_{B(G)} \, t\_B + \frac{\gamma\_{B(G)} - 1}{v\_{B(G)}^2} \vec{v}\_{B(G)} \cdot \vec{x}\_B \, \vec{v}\_{B(G)} + O(c^{-4}). \end{aligned}$$

12 Will-be-set-by-IN-TECH

See Appendix A of Ref. (Soffel M.H. et al, 2003) for the GCRS Post-Newtonian gravitational

Earth like the Lense-Thirring or frame-dragging effect (Ciufolini I. et al, 1995; Will C.M., 2011). The Post-Newtonian solution *gGμν*(*xG*) describes the *exterior* gravitational field outside the

Earth is approximated as a *mass monopole*. Otherwise the Earth mass and spin multipoles will

The tetrad carried by the geo-center is obtained from the BCRS tetrad with the tensorial

the transformation between BCRS and GRCS coordinates. If the barycentric 3-velocity and

*<sup>G</sup>*(*G*) = (1; 0*a*), while the spatial axes have the 3 orthogonal tangent space-like

*<sup>B</sup>*) = *xB* <sup>−</sup> *xB*(*G*)(*x<sup>o</sup>*

*<sup>B</sup>*(*G*) *r j*

> + *v<sup>i</sup> B*(*G*)*r<sup>i</sup> B*(*G*)

*<sup>B</sup>*(*G*) <sup>+</sup> *wBext*(*xB*(*G*))*r<sup>i</sup>*

+ *O*(*c*−4),

. It is a time-like geodesic of the Post-Newtonian 4-metric *gGμν*(*xG*), if the

*μ G ∂x<sup>ν</sup> B*

*dtB* and *aB*(*G*) <sup>=</sup> *<sup>d</sup>vB*(*G*)

*<sup>B</sup>*(*G*) + *C*(*tb*,*xB*)

(*tB*), *<sup>B</sup>ij*(*tB*) and *<sup>C</sup>*(*tB*,*xB*(*G*)), depending on the BCRS

*vB*(*G*) · *xB vB*(*G*) <sup>+</sup> *<sup>O</sup>*(*c*−4).

*<sup>B</sup>*(*G*) +

+ *O*(*c*−4),

*Bext*(*xB*(*G*)) (the BCRS gravito-magnetic

 *xo G*; 0*<sup>a</sup>* 

*μ <sup>G</sup>*(*G*) =

create a deviation of the geo-center world-line from a time-like geodesic.

*<sup>B</sup>*(*G*) *<sup>r</sup><sup>i</sup> B*(*G*) +

2 *ai <sup>B</sup>*(*G*) *r* 2 *B*(*G*) 

*<sup>B</sup>*(*G*) <sup>+</sup> *<sup>B</sup>ij*(*tB*)*r<sup>i</sup>*

<sup>2</sup> <sup>+</sup> *wBext*(*xB*(*G*))

gravitational potentials *wBext*(*xB*(*G*)) (the BCRS Newtonian potential of all solar system bodies

As shown in Ref. (Soffel M.H. et al, 2003), this transformation reduces to a pure Lorentz boost without rotation modulo terms of order *O*(*c*−4) in the limit of no acceleration due to the gravitational field (i.e. with *xB*(*G*)(*tB*) = *vB*(*G*) *tB*, *vB*(*G*) = *const*., *vB*(*G*) = |*vB*(*G*)|, *βB*(*G*) =

+ *O*(*c*−4),

*γB*(*G*) − 1 *v*2 *B*(*G*)

transformation law of 4-vectors, i.e. with the matrix *<sup>∂</sup><sup>x</sup>*

*A*(*tB*) + *v<sup>i</sup>*

(*tB*)*r<sup>i</sup>*

3-acceleration of the geo-center are *vB*(*G*) <sup>=</sup> *<sup>d</sup>xB*(*G*)

we introduce the relative 3-vector *rB*(*G*)(*x<sup>o</sup>*

transformation is (Soffel M.H. et al, 2003)

*c*2 

*B*(*tB*) + *B<sup>i</sup>*

 *tB tBo dt <sup>v</sup>*<sup>2</sup> *B*(*G*)

> 1 *c*2 1 2 *vi <sup>B</sup>*(*G*) *v j <sup>B</sup>*(*G*) *r j*

apart from the Earth acting on the geo-center) and *w<sup>i</sup>*

potential) are given in Ref. (Soffel M.H. et al, 2003).

*B*(*G*)

*xG* = *xB* − *γB*(*G*) *vB*(*G*) *tB* +

*tG* = *γB*(*G*)

)−1/2)

*tB* <sup>−</sup> *vB*(*G*) · *xB c*2

*tG* <sup>=</sup> *tB* <sup>−</sup> <sup>1</sup>

<sup>=</sup> *tB* <sup>−</sup> <sup>1</sup> *c*2

> *ri <sup>B</sup>*(*G*) +

The functions *A*(*tB*), *B*(*tB*), *B<sup>i</sup>*

+ 1 *c*4 

+ *r<sup>i</sup> <sup>B</sup>*(*G*) *a j <sup>B</sup>*(*G*) *r j <sup>B</sup>*(*G*) <sup>−</sup> <sup>1</sup>

*vB*(*G*)/*C*, *<sup>γ</sup>B*(*G*) = (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup>

*xa <sup>G</sup>* = *δai* *<sup>G</sup>*(*xG*). While *wG*(*xG*) generalizes the Newton potential, the

*<sup>G</sup>*(*xG*)) are responsible for the *gravito-magnetic* effects near the

with tangent time-like vector (the unit


*<sup>B</sup>*), the BCRS-GCRS coordinate

+ *O*(*c*−5) =

*dtB* respectively and if

potentials *wG*(*xG*) and *w<sup>a</sup>*

Earth surface, not inside.

*μ G*(*G*)*r*

4-velocity) *u<sup>μ</sup>*

unit vectors *�*

components *gGoa*(*xG*) (i.e. *w<sup>a</sup>*

The geo-center has coordinates *x*

Without the kinematically non-rotating constraint, GCRS would have a *slow rotation* (≈ 1.9 arcsec/century) with respect to the BCRS, the largest component of which is the *geodetic (DeSitter-Fokker) precession*, i.e. it would be *dynamically non-rotating* (and Coriolis terms should be added to the equations of motion of bodies in GCRS). Instead in the *kinematically non-rotating* version the *motion of the celestial pole* is defined in GCRS and the *geodetic precession* appears in the *precession-nutation theory* rather than in the transformation between GCRS and BCRS.

In the GCRS of IERS2003 (IERS, 2003) there are small velocities allowing one to use Galilean calculations plus relativistic corrections. However the *ecliptic plane* was redefined only in the IAU2006 resolutions (IAU, 2006) and a nearly relativistic dynamical theory of Earth rotation appeared only in 2009 with IAU2006 (Coppola V., 2009).

Therefore there are still open problems in the relativistic formulation of angular variables (Kaplan G.H., 2005):

1) the algorithms for *space motion, parallax, light-time and gravitational deflection* (for the observer at the geo-center the gravity field of the Earth is neglected in evaluating the deflection(star catalogs and ephemerides use 3-vectors in BCRS) );

2) the series of rotations for *precession, nutation, Earth rotation and polar motion* (in this order) use 3-vectors in GCRS;

3) the *aberration* calculation connects the two systems because it contains the transformation between them: its input are two 3-vectors in BCRS and its output is a 3-vector in GCRS;

4) in the VLBI case *aberration* does not appear explicitly, but the conventional algorithm for the *delay* observable incorporates 3-vectors expressed in both systems.

## **3.3 ITRS - International Terrestrial Reference System**

The International Terrestrial Reference System ITRS is the Earth-fixed geodetic system which matches the *reference ellipsoid* WGS-84 (basis of the terrestrial coordinates latitude, longitude, height, obtainable from GPS; it has equatorial radius 6, 378.137 m and polar flattening 1/298.257223563) to several centimeters and is defined on the instantaneous 3-spaces Σ*tG* of constant geocentric time *tG* = *TCG* = *const*.. It uses geocentric rectangular 3-coordinates *xT* = *xITRS* on Σ*tG* connected to the geocentric ones *xG* by time-dependent rotations. It is centered on the geo-center like GCRS with the center of mass defined for the whole Earth including oceans and atmosphere. The coordinates of *ITRS* ≈ *WGS*84 ≈ *GPS* are *c tG*;*xITRS* . GCRS is obtained from ITRS with a series of *time-dependent rotations* fixed by the conventions in IERS2003 (IERS, 2003) for the precession-nutation theory of Earth rotation. In chapter 5 of Ref. (Kaplan G.H., 2005) there is the old precession-nutation theory, while in chapter 6 there is the new theory of Earth rotation (updated with IAU2006). In Ref. (Coppola V., 2009) there is a more dynamical version IAU2006. Therefore the quasi-inertial relativistic 3-spaces Σ*tG* of GCRS are replaced with quasi-Euclidean non-relativistic 3-spaces (still denoted Σ*tG* ) only by means of rotations.

The World Geodetic System WGS84 (WGS, 1984) is the latest revision (dated 1984 and revised in 2004) of a standard for use in cartography, geodesy and navigation. It comprises a standard coordinate frame for the Earth, a standard spheroidal reference surface (the reference ellipsoid) for raw altitude data and a gravitational equipotential surface (the *geoid*) that defines the minimal sea level. The measurement of the form and dimensions of the Earth, the location of objects on its surface and the Earth gravity field are done by means of artificial satellites like the GPS ones (Seeber G., 2003). Let us remark that the gravitational field inside the Earth is evaluated in geodesy with Newtonian gravity, while the external GCRS gravitational potential is evaluated with Post-Newtonian general relativity and the junction of the two approaches has still to be done.

In the description of Earth rotation *precession* and *nutation* are really two aspects of a single phenomenon, the overall response of the spinning oblate, elastic Earth to external gravitational torques from the Moon, Sun and planets. As a result of these torques, the orientation of the Earth's rotation axis is constantly changing with respect to a space-fixed (locally inertial) reference system. The motion of the celestial pole among the stars is conventionally described as consisting of a smooth *long term motion called precession* upon which is superimposed a series of *small periodic components called nutation*.

In the old theory precession and nutation are described by 3× 3 rotation matrices operating on column 3-vectors in a traditional equatorial celestial coordinate system.The 3-vectors have the

$$\begin{aligned} \text{form } \vec{x} = \begin{pmatrix} \mathbf{x}\_x \\ \mathbf{x}\_y \\ \mathbf{x}\_z \end{pmatrix} = \begin{pmatrix} d\cos\delta\cos a \\ d\cos\delta\sin a \\ d\sin\delta \end{pmatrix}, \text{ where } a = t\mathbf{g}^{-1} \frac{\mathbf{x}\_y}{\mathbf{x}\_x} \text{ is the right ascension, } \delta = t\mathbf{g}^{-1} \frac{\mathbf{x}\_z}{\sqrt{\mathbf{x}\_x^2 + \mathbf{x}\_y^2}} \text{ and } \mathbf{x}\_z = \begin{pmatrix} \mathbf{g}^{-1} & \mathbf{g}^{-1} & \mathbf{g}^{-1} \end{pmatrix} \end{aligned}$$

is the *declination* and *d* is the *distance*from the specific origin of the system. For stars and objects *at infinity* (beyond the solar system), *d* is often simply put to 1.

In these traditional systems the adjective *mean* is applied to quantities (pole, equator, equinox, coordinates) affected only by precession, while *true* describes quantities affected by both precession and nutation. Thus it is the *true* quantities that are directly relevant to observations; *mean* quantities now usually represent an intermediate step in the computation.

Let us now describe the rotations in the 3-spaces Σ*tG* connecting a GCRS 3-vector *xG* to a ITRS 3-vector *xT* according to the conventions of the new theory of Ref. (IERS, 2003). The new definitions were forced by the fact the the errors in the determination of the old quantities were too big.

A matrix **B**, called *frame bias matrix*, is required to convert ICRS data to the *dynamical mean equator and equinox J2000.0*: *xmean*(2000) = **B***xICRS*. The same matrix is used in geocentric transformations to adjust 3-vectors in the GCRS so that they can be operated on by the conventional precession and nutation matrices. The matrix **B** corresponds to a fixed set of very small rotations: **B** = *R*1(−*ηo*) *R*2(*ξo*) *R*3(*d αo*) with *dα<sup>o</sup>* = −14.6*mas*, *ξ<sup>o</sup>* = −16.6170*mas*, *η<sup>o</sup>* = −6.8192, all converted to radians (divide by 206264806.247).

If **B** is the frame bias matrix , *P*(*tG*) the GCRS matrix for precession and *N*(*tG*) the GCRS matrix for nutation, for a 3-vector *xG* in GCRS we have

$$\begin{array}{ll} \vec{x}\_{G} & \stackrel{\textbf{B}}{\rightarrow} \textsc{MEAN} \, E \, \textbf{Q} \, \textbf{I} \, \text{ATOR} \, \text{and} \, E \, \textbf{Q} \, \text{II} \, \text{NOX} \, \text{of} \, \text{J} \, \text{2000.} \, \text{M} \\\\ & \stackrel{P(t\_{G})}{\rightarrow} \, \textsc{MEAN} \, \text{E} \, \text{Q} \, \text{I} \, \text{ATOR} \, \text{and} \, \text{E} \, \text{Q} \, \text{II} \, \text{NOX} \, \text{at} \, t\_{G} \, \\\\ & \stackrel{\textbf{N}(t\_{G})}{\rightarrow} \, \textbf{E}\_{\mathcal{Y}} = \, \left( \text{TRUE} \, \text{E} \, \text{Q} \, \text{I} \, \text{ATOR} \, \text{and} \, \text{E} \, \text{Q} \, \text{II} \, \text{NOX} \, \text{at} \, t\_{G} \right) \end{array}$$

.

14 Will-be-set-by-IN-TECH

the minimal sea level. The measurement of the form and dimensions of the Earth, the location of objects on its surface and the Earth gravity field are done by means of artificial satellites like the GPS ones (Seeber G., 2003). Let us remark that the gravitational field inside the Earth is evaluated in geodesy with Newtonian gravity, while the external GCRS gravitational potential is evaluated with Post-Newtonian general relativity and the junction of the two approaches

In the description of Earth rotation *precession* and *nutation* are really two aspects of a single phenomenon, the overall response of the spinning oblate, elastic Earth to external gravitational torques from the Moon, Sun and planets. As a result of these torques, the orientation of the Earth's rotation axis is constantly changing with respect to a space-fixed (locally inertial) reference system. The motion of the celestial pole among the stars is conventionally described as consisting of a smooth *long term motion called precession* upon

In the old theory precession and nutation are described by 3× 3 rotation matrices operating on column 3-vectors in a traditional equatorial celestial coordinate system.The 3-vectors have the

is the *declination* and *d* is the *distance*from the specific origin of the system. For stars and objects

In these traditional systems the adjective *mean* is applied to quantities (pole, equator, equinox, coordinates) affected only by precession, while *true* describes quantities affected by both precession and nutation. Thus it is the *true* quantities that are directly relevant to observations;

Let us now describe the rotations in the 3-spaces Σ*tG* connecting a GCRS 3-vector *xG* to a ITRS 3-vector *xT* according to the conventions of the new theory of Ref. (IERS, 2003). The new definitions were forced by the fact the the errors in the determination of the old quantities

A matrix **B**, called *frame bias matrix*, is required to convert ICRS data to the *dynamical mean equator and equinox J2000.0*: *xmean*(2000) = **B***xICRS*. The same matrix is used in geocentric transformations to adjust 3-vectors in the GCRS so that they can be operated on by the conventional precession and nutation matrices. The matrix **B** corresponds to a fixed set of very small rotations: **B** = *R*1(−*ηo*) *R*2(*ξo*) *R*3(*d αo*) with *dα<sup>o</sup>* = −14.6*mas*, *ξ<sup>o</sup>* = −16.6170*mas*,

If **B** is the frame bias matrix , *P*(*tG*) the GCRS matrix for precession and *N*(*tG*) the GCRS

→ *MEAN EQUATOR and EQU INOX o f J*2000.0,

*TRUE EQUATOR and EQU INOX at tG*

� .

*<sup>P</sup>*(*tG*) <sup>→</sup> *MEAN EQUATOR and EQU INOX at tG*,

*mean* quantities now usually represent an intermediate step in the computation.

*xx* is the *right ascension*, *<sup>δ</sup>* <sup>=</sup> *tg*−<sup>1</sup>

<sup>√</sup> *xz x*2 *x*+*x*<sup>2</sup> *y*

<sup>⎠</sup>, where *<sup>α</sup>* <sup>=</sup> *tg*−<sup>1</sup> *xy*

which is superimposed a series of *small periodic components called nutation*.

⎞

has still to be done.

form*x* =

were too big.

⎛ ⎝ *xx xy xz* ⎞ ⎠ = ⎛ ⎝

*d cos δ cos α d cos δ sin α d sin δ*

*at infinity* (beyond the solar system), *d* is often simply put to 1.

*η<sup>o</sup>* = −6.8192, all converted to radians (divide by 206264806.247).

matrix for nutation, for a 3-vector *xG* in GCRS we have

*<sup>N</sup>*(*tG*) <sup>→</sup>

*<sup>E</sup>*<sup>Y</sup> = �

*xG* **B**

If the 3-vector *<sup>E</sup>*<sup>Y</sup> is decomposed on the basis ⎛ ⎝ 1 0 0 ⎞ ⎠, ⎛ ⎝ 0 1 0 ⎞ ⎠ and ⎛ ⎝ 0 0 1 ⎞ ⎠, then by definition the *true equinox at tG* in GCRS is the unit 3-vector

$$
\vec{\mathcal{Y}}\_G = \boldsymbol{\mathcal{B}}^T \boldsymbol{P}^T(t\_G) \boldsymbol{N}^T(t\_G) \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}.
$$

By definition the *true celestial pole at date tG* - the *Celestial Intermediate Pole CIP* - in GCRS is the unit 3-vector

$$\begin{aligned} \vec{n}\_G &= \boldsymbol{B}^T \boldsymbol{P}^T (t\_G) \, \boldsymbol{N}^T (t\_G) \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} X \\ Y \\ Z \end{pmatrix} = \begin{pmatrix} \sin d \cos E \\ \sin d \sin E \\ \cos d \end{pmatrix}, \\ \vec{n}\_G \cdot \vec{\mathcal{J}}\_G &= 0. \end{aligned}$$

By definition the *Earth's axis* is the line through the geo-center in direction of the CIP. The angle of rotation about this axis (*θ*, linear in UT1 and *independent from the precession-nutation model for the Earth*) must be measured with respect to some agreed-upon direction in space (CIO, see later on).

The reference point on the equator (origin of *θ*) must be defined in such a way that the rate of change of the Earth's rotation angle, measured with respect to this point, is the angular velocity of the Earth about the CIP. As the CIP moves, the point must move to remain in the equatorial plane (instantaneously orthogonal to the CIP axis); but the point motion must be such that the measured rotation angle is not contaminated by some component of the motion of the CIP itself. This leads to the concept of *Non-Rotating Origin* (NLO) on the equator: as the equator moves the point's instantaneous motion must always be orthogonal to the equator (whereas the equinox has a motion along the equator: the precession in right ascension). That is, the point motion at some time *tG* must be directly toward or away from the position of the pole of rotation at *tG*. The point is not unique.

The new conventions use the *Celestial Intermediate Reference System CIRS Eσ*, which has the NLO azimuthal origin at the *Celestial Intermediate Origin CIO or σ*, a well defined point on the equator of CIP with GCRS coordinates

$$\vec{\sigma}\_{\rm G} = \vec{\mathcal{Y}}\_{\rm G} \cos \mathcal{E}\_{o} - (\vec{n}\_{\rm G} \times \vec{\mathcal{Y}}\_{\rm G}) \sin \mathcal{E}\_{o} \qquad \vec{n}\_{\rm G} \cdot \vec{\sigma}\_{\rm G} = 0,$$

where E*<sup>o</sup>* is an angle, named *equation of the origins* (the arc on the instantaneous true equator of date *tG* from the CIO at equinox; it is the right ascension of the true equinox relative to the CIO; it is also the difference *θ* − *GAST*, where GAST is the angular equivalent of Greenwich apparent sidereal time), given at p.60 of Ref.(Kaplan G.H., 2005). Now we have an orthonormal triad: *nG*,*σ<sup>G</sup>* and *yG* = *nG* ×*σG*.

The coordinates in *E<sup>σ</sup>* are

$$\begin{aligned} \text{GCRS} & \stackrel{\text{C}}{\rightarrow} E\_{\sigma \prime} & \qquad \vec{\text{x}}\_{\sigma} = \text{C } \vec{\text{x}}\_{\text{G} \prime} \\ & \mathcal{C}^{T} = \begin{pmatrix} \vec{\sigma}\_{\text{G} \prime} \vec{y}\_{\text{G} \prime} \,\vec{n}\_{\text{G}} \end{pmatrix} = \begin{pmatrix} \sigma\_{1} \,\, y\_{1} \,\, \mathbf{x}\_{\text{CIP}} \\ \sigma\_{2} \,\, y\_{2} \,\, y\_{\text{CIP}} \\ \sigma\_{3} \,\, y\_{3} \,\, \mathbf{z}\_{\text{CIP}} \end{pmatrix} = \mathcal{R}\_{3}(-E) \,\, \mathcal{R}\_{2}(-d) \,\, \mathcal{R}\_{3}(E) \,\, \mathcal{R}\_{3}(s), \end{aligned}$$

with the angles *d* and *E* appearing in *nG*. The angle *s*, the CIO *locator*, given at p.62 of Ref. (Kaplan G.H., 2005), represents the difference between the length of the arc from the point N westward to the CIO (on the instantaneous equator) and the length of the arc from N westward to the GCRS origin of right ascension (on the GCRS equator).

On the celestial sphere, the Earth's instantaneous (moving) equator intersects the GCRS equator at two nodes. Let N be the ascending node of the instantaneous equator on the GCRS equator.

The matrix *C* is the CIO-based rotation taking into account nutation, precession and frame bias.

The Earth rotation angle *θ* with the origin CIO rotates the CIO equatorial axis to an instantaneous axis (the *Terrestrial Intermediate Origin TIO or ω*˜ ), which is a NLO azimuthal origin for the *Terrestrial Intermediate Reference System TIRS Eω*˜

$$E\_{\sigma} \stackrel{\mathcal{R}\_3(\theta)}{\xrightarrow{\mathcal{S}\_{\tilde{\mathcal{O}}\prime}}} E\_{\tilde{\mathcal{O}}\prime} \qquad \vec{\mathfrak{x}}\_{\tilde{\mathcal{O}}} = R\_3(\theta) \,\vec{\mathfrak{x}}\_{\sigma}.$$

Then to arrive to ITRS from *Eω*˜ we must take into account the *polar motion*

$$\begin{array}{ll} E\_{\vec{\omega}} & \stackrel{W^{\mathsf{T}}(t\_{\mathcal{G}})}{\rightarrow} \, \, ITRS\_{\prime} & \vec{\mathbf{x}}\_{T} = \mathcal{W}^{T}(t\_{G}) \, \vec{\mathbf{x}}\_{\mathcal{G}^{\prime}} \\\\ W(t\_{G}) & = R\_{\mathcal{3}}(-\operatorname{s}^{\prime}) \, R\_{\mathcal{2}}(\boldsymbol{\uppi}\_{p}) \, R\_{1}(\boldsymbol{y}\_{p})\_{\prime} \end{array}$$

where *WT*(*tG*) is the *polar motion* (wobble) matrix and *xp* and *yp* are the coordinates of CIP in ITRS. This rotation reorients the pole from the ITRS z-axis to the CIP and moves the origin of longitude very slightly from the ITRS x-axis to TIO (the angle *s* � ≈ −47*microarcsec*). See p.63 of Ref. (Kaplan G.H., 2005).

In conclusion we have the following sequence of rotations connecting GCRS to ITRS

$$\begin{array}{lcl} \text{GCRS} & \rightarrow & E\_{\sigma} \rightarrow & E\_{\tilde{\mathcal{O}}} \rightarrow & ITRS\_{\prime} \\\\ \vec{\mathfrak{x}}\_{G} & \rightarrow & \vec{\mathfrak{x}}\_{\sigma} = \mathsf{C} \ \vec{\mathfrak{x}}\_{G} \rightarrow & \vec{\mathfrak{x}}\_{\tilde{\mathcal{O}}} = R\_{3}^{T}(-\theta) \ \vec{\mathfrak{x}}\_{\sigma} \rightarrow & \vec{\mathfrak{x}}\_{T} = W^{T} \vec{\mathfrak{x}}\_{\tilde{\mathcal{O}}} \\\\ \vec{\mathfrak{x}}\_{T} = & W^{T} R\_{3}^{T}(-\theta) \ \mathsf{C} \ \vec{\mathfrak{x}}\_{G} .\end{array}$$

Therefore ITRS is defined by taking the instantaneous 3-spaces Σ*tG* of GCRS and by rotating the 3-coordinates in each 3-space to take into account the rotation of the Earth. However in this way all the clocks on the Earth surface have the same geocentric time *tG*. A more relativistic formulation should replace the final rotation matrix *R* with a Lorentz transformation Λ = *B R*, where the Lorentz boost *B* would imply the transformation of the global GCRS time *tG* into the different local coordinate times associated with the proper times (SI atomic seconds) of the atomic clocks in each point of the Earth surface.

#### **4. The space-time outside the Solar System**

Reference data for positional astronomy, such as the the data in astrometric star catalogs or barycentric planetary ephemerides, are specified in the International Celestial Reference 16 Will-be-set-by-IN-TECH

with the angles *d* and *E* appearing in *nG*. The angle *s*, the CIO *locator*, given at p.62 of Ref. (Kaplan G.H., 2005), represents the difference between the length of the arc from the point N westward to the CIO (on the instantaneous equator) and the length of the arc from N

On the celestial sphere, the Earth's instantaneous (moving) equator intersects the GCRS equator at two nodes. Let N be the ascending node of the instantaneous equator on the GCRS

The matrix *C* is the CIO-based rotation taking into account nutation, precession and frame

The Earth rotation angle *θ* with the origin CIO rotates the CIO equatorial axis to an instantaneous axis (the *Terrestrial Intermediate Origin TIO or ω*˜ ), which is a NLO azimuthal

*<sup>R</sup>*3(*θ*) <sup>→</sup> *<sup>E</sup>ω*˜ , *<sup>x</sup>ω*˜ <sup>=</sup> *<sup>R</sup>*3(*θ*)*<sup>x</sup>σ*.

*<sup>W</sup><sup>T</sup>* (*tG*) <sup>→</sup> *ITRS*, *xT* <sup>=</sup> *<sup>W</sup>T*(*tG*)*<sup>x</sup>ω*˜ ,

) *R*2(*xp*) *R*1(*yp*),

�

<sup>3</sup> (−*θ*)*<sup>x</sup><sup>σ</sup>* <sup>→</sup> *xT* <sup>=</sup> *<sup>W</sup><sup>T</sup> <sup>x</sup>ω*˜ ,

≈ −47*microarcsec*). See p.63

�

where *WT*(*tG*) is the *polar motion* (wobble) matrix and *xp* and *yp* are the coordinates of CIP in ITRS. This rotation reorients the pole from the ITRS z-axis to the CIP and moves the origin of

Therefore ITRS is defined by taking the instantaneous 3-spaces Σ*tG* of GCRS and by rotating the 3-coordinates in each 3-space to take into account the rotation of the Earth. However in this way all the clocks on the Earth surface have the same geocentric time *tG*. A more relativistic formulation should replace the final rotation matrix *R* with a Lorentz transformation Λ = *B R*, where the Lorentz boost *B* would imply the transformation of the global GCRS time *tG* into the different local coordinate times associated with the proper times (SI atomic seconds) of the

Reference data for positional astronomy, such as the the data in astrometric star catalogs or barycentric planetary ephemerides, are specified in the International Celestial Reference

westward to the GCRS origin of right ascension (on the GCRS equator).

origin for the *Terrestrial Intermediate Reference System TIRS Eω*˜

*Eσ*

longitude very slightly from the ITRS x-axis to TIO (the angle *s*

*GCRS* → *E<sup>σ</sup>* → *Eω*˜ → *ITRS*,

*xG* <sup>→</sup> *<sup>x</sup><sup>σ</sup>* <sup>=</sup> *<sup>C</sup>xG* <sup>→</sup> *<sup>x</sup>ω*˜ <sup>=</sup> *<sup>R</sup><sup>T</sup>*

<sup>3</sup> (−*θ*) *CxG*.

*xT* = *W<sup>T</sup> R<sup>T</sup>*

atomic clocks in each point of the Earth surface.

**4. The space-time outside the Solar System**

*Eω*˜

of Ref. (Kaplan G.H., 2005).

Then to arrive to ITRS from *Eω*˜ we must take into account the *polar motion*

*W*(*tG*) = *R*3(−*s*

In conclusion we have the following sequence of rotations connecting GCRS to ITRS

equator.

bias.

System ICRS (Kaplan G.H., 2005; Kovalevski J. et al, 1989; Sovers O.J. et al, 1998; Johnstone K.J. et al, 1999; Ma C. et al, 1998; Fey A. et al, 2009) with origin in the solar system barycenter and with spatial axes fixed with respect to space. A materialization as a ICRF is obtained by supposing that the origin is a quasi-inertial observer and that we have a quasi-inertial (essentially non-relativistic) reference frame with rectangular 3-coordinates (or equatorial geographical coordinates) in a nearly Galilei space-time whose 3-spaces are Euclidean. The directions of the spatial axes are effectively defined by the adopted coordinates (i.e. using the tabulated *right ascensions* and *declinations* and, in the case of a star catalogue, the *proper motions (ephemerides)*) of 212 extragalactic radio sources observed by VLBI . These radio sources (quasars and AGN, active galactic nuclei) are assumed to have no observable intrinsic angular momentum. At low accuracy one uses a star catalogue system such as the FK5 (Fey A. et al, 2009). At a more accurate level taking into account optical wavelengths, one has the Hipparcos Celestial Reference Frame HCRF, composed of the positions and proper motions of the astrometrically well-behaved stars in the Hipparcos catalog.

Thus, the ICRS is a *space-fixed* system, more precisely a *kinematically non-rotating* system, without an associated epoch. ICRS provides the orientation of BCRS and closely matches the conventional dynamical system defined by the Earth's mean equator and equinox of J2000.0: the alignment difference is at the 0.02 arcsecond level, negligible for many applications.

However if we take into account the description of the universe given by cosmology, the actual cosmological space-time cannot be a nearly Galilei space-time but it must be a cosmological space-time with some theoretical *cosmic time*. In the standard cosmological model (Bartelmann M., 2010; Bean R., 2009) it is a homogeneous and isotropic Friedmann-Robertson-Walker space-time whose instantaneous 3-spaces have nearly vanishing internal 3-curvature, so that may locally be replaced with Euclidean 3-spaces as it is done in galactic dynamics. However they have a time-dependent conformal factor (it is one in Galilei space-time) responsible for the Hubble constant regulating the expansion of the universe. Moreover the Hubble constant is also the negative of the trace of the external 3-curvature of the 3-space as 3-sub-manifold of the space-time. As a consequence the transition from the astronomical ICRS to an astrophysical description taking into account cosmology is far from being understood.
