**Relativistic Metrology: From Earth to Astrophysics**

Luca Lusanna *Sezione INFN di Firenze, Polo Scientifico, Sesto Fiorentino (FI) Italy* 

### **1. Introduction**

394 Modern Metrology Concerns

USSR Astronomy Yearbook (AE SSSR), 1970 pp. 672, Leningrad: Publishing House

"Science"

Till twenty years ago the basic concepts of special and general relativity were introduced by speaking of *clocks* and *rods* with an unspecified structure. Then the advances in atomic physics and in space navigation led to a revolution in metrology with the elimination of these old idealized notions and their replacements with realistic standards. For instance one can compare the 1965 point of view of Ref. (Basri S.A., 1965) with the 1997 one of Ref. (Guinot B., 1997).

However also today many scientists still think in terms of the *absolute notions of time and space* present in the Galilei space-time used in Newtonian physics, due to the fact that on Earth non-relativistic quantum mechanics is able to treat consistently problems ranging from molecular physics till quantum information without taking into account gravity (when needed Newtonian gravity is used). Only the description of light in atomic physics requires relativity (the trajectories of photons do not exist in Galilei space-time). Therefore most of the problems are formulated in inertial frames centered on inertial observers (having a constant velocity) in Galilei space-time (they are connected by the group of Galilei transformations containing space and time translations, spatial rotations and boosts) and, if needed, extended to non-relativistic accelerated frames taking into account the associated inertial apparent forces. The rotation of the Earth and its motion around the Sun are negligible effects for this type of physics.

Instead particle physics must face high speed objects and needs the Minkowski space-time of special relativity. Now the notions of space and time are no longer absolute: only the *global space-time is an absolute notion*. The Large Hadron Collider LHC particle accelerator at CERN is described with the coordinate time and the coordinate position of an inertial frame of Minkowski space-time centered on an inertial observer. To get the description with respect to another inertial observer one needs the group of Poincaré transformations (space and time translations plus Lorentz transformations Λ, i.e. spatial rotations and boosts; *x* � *<sup>μ</sup>* = *a<sup>μ</sup>* + Λ*μν xν*). In the new inertial frame the new coordinate time (and also the coordinate position) depends on both the old coordinate time and positions. This has generated an endless (and still going on) philosophical discussion on the meaning of time. Since the Lorentz-scalar line element joining two nearby points in an inertial frame of Minkowski space-time is *ds*<sup>2</sup> = (*dxo*)<sup>2</sup> <sup>−</sup> <sup>∑</sup>*<sup>r</sup>* (*dxr*)<sup>2</sup> (with the particle physics conventions; *ds*<sup>2</sup> <sup>=</sup> <sup>−</sup>(*dxo*)<sup>2</sup> <sup>+</sup> <sup>∑</sup>*<sup>r</sup>* (*dxr*)<sup>2</sup> with the general relativity ones; *<sup>x</sup><sup>μ</sup>* are inertial time, *<sup>x</sup><sup>o</sup>* <sup>=</sup> *ct*, and space, *xr*, coordinates), space and time increments have different sign (Lorentz signature). The only intrinsic structure of Minkowski space-time is that in each point A there is a light-cone (or null cone) defined by *ds*<sup>2</sup> = 0, which is the locus of the ray of light (traveling with the velocity of light) arriving in that point from the past or emanating from that point towards the future. The points inside the light-cone in A have a time-like distance from A and can be reached (if in the future) by traveling with a velocity less than the velocity of light. The points outside the light-cone of A have space-like distance from A and could be reached only with super-luminal velocity. If there is an atomic clock in A moving along a time-like curve towards the future its Lorentz-scalar *proper time* is defined as *<sup>d</sup>τ*<sup>2</sup> <sup>=</sup> *ds*<sup>2</sup> (*dτ*<sup>2</sup> <sup>=</sup> <sup>−</sup>*ds*<sup>2</sup> with the other convention) and coincides with the coordinate time *x<sup>o</sup>* only in the inertial rest frame of the clock. However, since time is not absolute, there is no intrinsic notion of 3-space and of synchronization of clocks: both of them have to be defined with some convention. As a consequence the *1-way velocity of light* from one observer A to an observer B has a meaning only after a choice of a convention for synchronizing the clock in A with the one in B. Therefore the crucial quantity in special relativity is the *2-way (or round trip) velocity of light c* involving only one clock: the observer A emits a ray of light which is reflected somewhere and then reabsorbed by A so that only the clock of A is implied in measuring the time of flight. It is this velocity which is isotropic and constant in special relativity.

In Minkowski space-time the Euclidean 3-spaces of the inertial frames centered on an inertial observer A are identified by means of Einstein convention for the synchronization of clocks: the inertial observer A sends a ray of light at *x<sup>o</sup> <sup>i</sup>* towards the (in general accelerated) observer B; the ray is reflected towards A at a point P of B world-line and then reabsorbed by A at *xo <sup>f</sup>* ; by convention P is synchronous with the mid-point between emission and absorption on A's world-line, i.e. *x<sup>o</sup> <sup>P</sup>* <sup>=</sup> *<sup>x</sup><sup>o</sup> <sup>i</sup>* <sup>+</sup> <sup>1</sup> <sup>2</sup> (*x<sup>o</sup> <sup>f</sup>* <sup>−</sup> *<sup>x</sup><sup>o</sup> <sup>i</sup>* ) = <sup>1</sup> <sup>2</sup> (*x<sup>o</sup> <sup>i</sup>* <sup>+</sup> *<sup>x</sup><sup>o</sup> <sup>f</sup>*). This convention selects the Euclidean instantaneous 3-spaces *x<sup>o</sup>* = *ct* = *const*. of the inertial frames centered on A. Only in this case the one-way velocity of light between A and B coincides with the two-way one, *c*. However, if the observer A is accelerated, the convention breaks down and we need a theory of non-inertial frames in Minkowski space-time as the one developed in Ref.(Alba D. et al, 2010, 2007). In this theory the transition from an inertial to a non-inertial frame (with its relativistic inertial forces and its non-Euclidean 3-spaces) can be described as a gauge transformation connecting two different generalized conventions for clock synchronization: therefore physics does not change, only the appearances of phenomena change.

However the International Space Station ISS near the Earth and space navigation in the Solar System require general relativity (at least its Post-Newtonian approximation) to take into account the effects of the gravitational field which is missing in special relativity. Now also space-time is no longer an absolute notion but is dynamically determined by Einstein's equations. Einstein's space-times have Lorentz signature but the structure of the light-cones changes from a point to another one. However rays of light, moving along null geodesics, are assumed to have the same 2-way velocity of light *c* as in special relativity, being an eikonal approximation to Maxwell equations. The equivalence principle implies that global inertial frames cannot exist: only locally near a particle in free fall we can have a local inertial frame and a local special relativistic approximation. Again there is the problem of clock synchronization for the definition of the non-Euclidean 3-spaces: even if the space-time is dynamically determined by Einstein's equations, each solution can be presented in arbitrary systems of 4-coordinates, since this is the gauge freedom of general relativity (form invariance of Einstein's equations under general coordinate transformations).

2 Will-be-set-by-IN-TECH

only intrinsic structure of Minkowski space-time is that in each point A there is a light-cone (or null cone) defined by *ds*<sup>2</sup> = 0, which is the locus of the ray of light (traveling with the velocity of light) arriving in that point from the past or emanating from that point towards the future. The points inside the light-cone in A have a time-like distance from A and can be reached (if in the future) by traveling with a velocity less than the velocity of light. The points outside the light-cone of A have space-like distance from A and could be reached only with super-luminal velocity. If there is an atomic clock in A moving along a time-like curve towards the future its Lorentz-scalar *proper time* is defined as *<sup>d</sup>τ*<sup>2</sup> <sup>=</sup> *ds*<sup>2</sup> (*dτ*<sup>2</sup> <sup>=</sup> <sup>−</sup>*ds*<sup>2</sup> with the other convention) and coincides with the coordinate time *x<sup>o</sup>* only in the inertial rest frame of the clock. However, since time is not absolute, there is no intrinsic notion of 3-space and of synchronization of clocks: both of them have to be defined with some convention. As a consequence the *1-way velocity of light* from one observer A to an observer B has a meaning only after a choice of a convention for synchronizing the clock in A with the one in B. Therefore the crucial quantity in special relativity is the *2-way (or round trip) velocity of light c* involving only one clock: the observer A emits a ray of light which is reflected somewhere and then reabsorbed by A so that only the clock of A is implied in measuring the time of flight. It is this velocity which is

In Minkowski space-time the Euclidean 3-spaces of the inertial frames centered on an inertial observer A are identified by means of Einstein convention for the synchronization of clocks:

B; the ray is reflected towards A at a point P of B world-line and then reabsorbed by A at

Euclidean instantaneous 3-spaces *x<sup>o</sup>* = *ct* = *const*. of the inertial frames centered on A. Only in this case the one-way velocity of light between A and B coincides with the two-way one, *c*. However, if the observer A is accelerated, the convention breaks down and we need a theory of non-inertial frames in Minkowski space-time as the one developed in Ref.(Alba D. et al, 2010, 2007). In this theory the transition from an inertial to a non-inertial frame (with its relativistic inertial forces and its non-Euclidean 3-spaces) can be described as a gauge transformation connecting two different generalized conventions for clock synchronization:

However the International Space Station ISS near the Earth and space navigation in the Solar System require general relativity (at least its Post-Newtonian approximation) to take into account the effects of the gravitational field which is missing in special relativity. Now also space-time is no longer an absolute notion but is dynamically determined by Einstein's equations. Einstein's space-times have Lorentz signature but the structure of the light-cones changes from a point to another one. However rays of light, moving along null geodesics, are assumed to have the same 2-way velocity of light *c* as in special relativity, being an eikonal approximation to Maxwell equations. The equivalence principle implies that global inertial frames cannot exist: only locally near a particle in free fall we can have a local inertial frame and a local special relativistic approximation. Again there is the problem of clock synchronization for the definition of the non-Euclidean 3-spaces: even if the space-time is dynamically determined by Einstein's equations, each solution can be presented in arbitrary systems of 4-coordinates, since this is the gauge freedom of general relativity (form invariance

*<sup>f</sup>* ; by convention P is synchronous with the mid-point between emission and absorption

*<sup>i</sup>* ) = <sup>1</sup>

<sup>2</sup> (*x<sup>o</sup> <sup>i</sup>* <sup>+</sup> *<sup>x</sup><sup>o</sup>*

*<sup>i</sup>* towards the (in general accelerated) observer

*<sup>f</sup>*). This convention selects the

isotropic and constant in special relativity.

on A's world-line, i.e. *x<sup>o</sup>*

*xo*

the inertial observer A sends a ray of light at *x<sup>o</sup>*

*<sup>P</sup>* <sup>=</sup> *<sup>x</sup><sup>o</sup>*

*<sup>i</sup>* <sup>+</sup> <sup>1</sup> <sup>2</sup> (*x<sup>o</sup> <sup>f</sup>* <sup>−</sup> *<sup>x</sup><sup>o</sup>*

of Einstein's equations under general coordinate transformations).

therefore physics does not change, only the appearances of phenomena change.

Therefore the presentation (gauge choice) of a solution of Einstein's equations is nothing else that a *metrology choice of a standard of space-time*, i.e. a choice of the time and space 4-coordinates.

Moreover this choice is fundamental for the description of macroscopic matter (its energy-momentum tensor is the source term in Einstein's equations) at the experimental level: physicists, space engineers and astronomers use an *intrinsically coordinate-dependent* (i.e. dependent on the chosen conventions) description of the trajectory of every macroscopic body (from spacecrafts to satellites, planets, stars,....).

Around the Earth *GPS (Global Positioning System) is a space-time standard* (Ashby N., 2003), relying on the time and length standards on the Earth surface. There is an array of 24 satellites around the Earth each one with an atomic clock with an accuracy which for modern commercial devices is today less than 30 nanoseconds. The satellites are at an altitude of 20,000 Km and have a mean velocity of 14,000 km/hr. Special relativity implies that these clocks tick more slowly (about 7 microseconds per day) than clocks on Earth. But general relativity implies that they tick faster (about 48 microseconds per day), so that a satellite clock advances faster than a clock on ground by about 38 microseconds per day. If we forget general relativity the precision of GPS localization (less that 15 meters) is lost within two minutes.

While in non-inertial frames in Galilei and Minkowski space-times there is a good understanding of the apparent inertial forces, in general relativity the gravitational field is described by the 4-metric tensor <sup>4</sup>*gμν*(*x*) in an arbitrary 4-coordinate system centered on an arbitrary observer (the line element is now *ds*<sup>2</sup> = <sup>4</sup>*gμν*(*x*) *dx<sup>μ</sup> dxν*) and it is not clear how to introduce a distinction between gravitational and inertial effects. However this is possible at the Hamiltonian level for the globally hyperbolic, asymptotically Minkowskian space-times, where it is possible to define global 3+1 splittings of the space-time, namely a foliation with 3-spaces evolving in time. The study of the 4-metric tensor in this framework allows one to disentangle the two physical degrees of freedom (or *tidal* variables) of the gravitational field (the two polarizations of gravitational waves in the linearized theory) and the gauge (or *inertial* variables) degrees of freedom describing the arbitrariness in the choice of the 4-coordinates. As shown in Ref. (Lusanna L., 2011) among the inertial variables there is the so-called York time (the trace of the extrinsic curvature of the 3-space as a 3-sub-manifold of space-time): it describes the remnant of the special relativistic gauge freedom in clock synchronization in this class of general relativistic space-times.

For the physics in the Solar System one assumes that the relevant Einstein space-times are globally hyperbolic (namely admitting a global definition of time) and asymptotically flat (namely tending to Minkowsky space-time at spatial infinity) space-times containing *N* bodies (the Sun and the planets) treated as point-like objects carrying multipoles (spin, moment of inertia,... of the extended body). A Post-Newtonian approximation is used in solving Einstein's equations in harmonic gauges and the gravitational waves inside the Solar System are shown to be negligible.

We have spoken only about Einstein general relativity. See Ref. (Will C.M., 2006, 1993) for the status of alternative theories of gravity inside the Solar System and Refs. (Soffel M.H., 1989; Brumberg V.A., 1991; Damour T. et al, 1991, 1992, 1993, 1994) for a treatment going from geodesy to celestial mechanics.

In what follows there will be a sketch, with update bibliography, of relativistic metrology inside the Solar System. It includes

A) Space and time standards.

B) The conventions needed for the description of satellites around the Earth: it is done by means of NASA (USA National Aeronautics and Space Administration) coordinates (Moyer T.D., 2003) firstly in the International Terrestrial Reference System ITRS (with an associated frame ITRF fixed on the Earth surface; see Ref. (IERS, 2004) for the IERS2003 conventions of the International Earth Rotation and Reference System Service IERS) and then in the Geocentric Celestial Reference System GCRS (with an associated non-rotating frame GCRF centered on the Earth center; see Ref. (Soffel M.H. et al, 2003) for the International Astronomic Units IAU2000).

C) The conventions needed for the description of planets and other objects in the Solar System uses the Barycentric Celestial Reference System BCRS (with an associated quasi-inertial Minkowski frame BCRF, if perturbations from the Milky Way are ignored, centered in the barycenter of the Solar System and with the axes specified by means of fixed stars (quasars) in the Hypparcos catalog (Kovalesky J. et al, 1989; Sovers O.J. et al, 1998; Ma C. et al, 1998; Johnstone K.J. et al, 1999; Fey A. et al, 2009)) and ephemerides (see IAU2000 in Ref. (Soffel M.H. et al, 2003)).

Ref. (Kaplan G.H., 2005) contains all the relevant aspects of these conventions.

While ITRF is essentially realized as a non-relativistic non-inertial frame in Galilei space-time, BCRF is defined as a *quasi-inertial frame*, *non-rotating* with respect to some selected fixed stars, in Minkowski space-time with nearly-Euclidean Newton 3-spaces. The qualification *quasi-inertial* is introduced to take into account general relativity, where inertial frames exist only locally. It can also be considered as a Post-Minkowskian space-time with 3-spaces having a very small extrinsic curvature. GCRF is obtained from BCRF by taking into account Earth's rotation around the Sun with a suitable Lorentz boost with corrections from Post-Newtonian gravity. By taking into account the extension of the geoid and Earth revolution around its axis one goes from the quasi-Minkowskian GCRF to the quasi-Galilean ITRF.

New problems emerge by going outside the Solar System. In astronomy the positions of stars and galaxies are determined from the data (luminosity, light spectrum, angles) on the sky, i.e. on a 2-dimensional spherical surface around the Earth with the relations between it and the observatory on the Earth done with GPS.

Then one needs a description of stars and galaxies as living in a 4-dimensional nearly-Galilei space-time with the International Celestial Reference System ICRS (see Refs. (Kovalesky J. et al, 1989; Sovers O.J. et al, 1998; Ma C. et al, 1998; Johnstone K.J. et al, 1999; Fey A. et al, 2009)), whose materialization ICRF is considered as a "quasi-inertial frame" in a "quasi-Galilei space-time", in accord with the assumed validity of the cosmological and Copernican principles. Namely one assumes a homogeneous and isotropic cosmological Friedmann-Robertson - Walker solution of Einstein equations (the standard ΛCDM cosmological model). In it the constant intrinsic 3-curvature of instantaneous 3-spaces is nearly zero as implied by the CMB data (Bartelmann M., 2010; Bean R., 2009), so that Euclidean 3-spaces (and Newtonian gravity) can be used (all galactic dynamics is Newtonian gravity). See Ref. (Lindegren L. et al, 2003) for the IAU conventions for defining the astrometric radial velocity of stars taking into account astrometric positions, spectroscopy of star light and light propagation in gravitational fields.

However, to reconcile all the data with this 4-dimensional reconstruction one must postulate the existence of dark matter and dark energy as the dominant components of the classical universe (Durrer R., 2011; Bonvin C. et al, 2011; Garret K. et al, 2011; Ross M., 2010) after the recombination 3-surface (a kind of Heisenberg cut between quantum cosmology and classical astrophysics)!
