Measuring gravito-magnetic effects by multi ring-laser gyroscope

F. Bosi,a G. Cella,b and A. Di Virgilioc INFN Sez. di Pisa, Pisa, Italy

A.Ortoland Laboratori Nazionali di Legnaro, INFN Legnaro (Padova), Italy

A. Porzioe and S. Solimenof University of Naples and CNR-SPIN, Naples, Italy

M. Cerdoniog and J. P. Zendrih INFN Sez. di Padova, Padova, Italy

M. Allegrini,i J. Belfi,j N. Beverini,k B. Bouhadef,l G. Carelli,m I. Ferrante,n E. Maccioni,o R. Passaquieti,p and F. Stefaniq University of Pisa and CNISM, Pisa, Italy

M. L. Ruggieror and A. Tartaglias Polit. of Torino and INFN, Torino, Italy

K. U. Schreibert and A. Gebaueru Technische Universitaet Muenchen, Forschungseinrichtung Satellitengeodaesie Fundamentalstation Wettzell, 93444 Bad K¨otzting,Germany

J-P. R. Wellsv Department of Physics and Astronomy, University of Canterbury, Christchurch 8020, New Zealand

Draft Draft

We propose an under-ground experiment to detect the general relativistic effects due to the curvature of space-time around the Earth (de Sitter effect) and to the rotation of the planet (dragging of the inertial frames or Lense-Thirring effect). It is based on the comparison between the IERS value of the Earth rotation vector and corresponding measurements obtained by a tri-axial laser detector of rotation. The proposed detector consists of six large ring-lasers arranged along three orthogonal axes. In about two years of data taking, the 1% sensitivity required for the measurement of the Lense-Thirring drag can√ be reached with square rings of 6 m side, assuming a shot noise limited sensitivity (20prad/s/ Hz). The multi-gyros system, composed of rings whose planes are perpendicular to one or the other of three orthogonal axes, can be built in several ways. Here, we consider cubic and octahedral structures. It is shown that the symmetries of the proposed configurations provide mathematical relations that can be used to ensure the long term stability of the apparatus.

PACS numbers: 42.15.Dp, 42.30.Sy, 42.55.Lt, 91.10.Nj

a Electronic address: [email protected] m b Electronic address: [email protected] Electronic address: [email protected] n c Electronic address: [email protected] Electronic address: [email protected] oElectronic address: [email protected] dElectronic address: [email protected] pElectronic address: [email protected] eElectronic address: [email protected] qElectronic address: [email protected] fElectronic address: [email protected] rElectronic address: [email protected] gElectronic address: [email protected] sElectronic address: [email protected] hElectronic address: [email protected] tElectronic address: [email protected] iElectronic address: [email protected] uElectronic address: [email protected] v jElectronic address: belfi@df.unipi.it Electronic address: [email protected] kElectronic address: [email protected] lElectronic address: [email protected] 2

I. INTRODUCTION between in-space and on-ground measurements could be very valuable. The general theory of relativity is the most satisfactory description of gravitational phenomena. The theoretical breakthrough came with Einstein’s geometrical represen- II. DETECTION OF GRAVITO-MAGNETIC EFFECTS tation of gravity: as different test masses fall in the same way in a gravitational field, gravity must be a property of space and time rather than of the masses themselves. Gravito-magnetism (GM) is a general relativistic phe- Until now, almost all successful tests of general rel- nomenon related to the presence of mass currents in the ativity (Shapiro time delay [1], light deflection by the reference frame of a given observer. In the case of celestial sun [2], perihelion shift of the orbit of Mercury [3]) have bodies, including the Earth, and excluding translational been probing the gravitational field of the Sun, without motion with respect to the center of the body, gravito- considering its proper rotation. However, general relativ- magnetic effects are due to the absolute rotation of the ity predicts that the stationary field of a rotating body massive source with respect to distant stars. When the is different from the static field produced by the same Einstein equations in vacuum are applied to this kind of non-rotating mass. The difference is known as gravito- symmetry and are linearised (weak field approximation) magnetism and consists of a drag of space-time due to GM is accounted for by the analogue of a magnetic field the mass currents. The rotational frame-dragging effect of a rotating spherical charge. In practice at the lowest is also known as the Lense-Thirring (LT) [4] effect. approximation level, a dipolar GM field is obtained, with the dimensions of an angular velocity. Its explicit form A direct experimental evidence of the existence of the in a non-rotating reference frame centred on the source GM field has been obtained so far by Ciufolini [5] and (in our case the Earth center), is (see e.g. [17]) by Francis Everitt and the GP-B group [6]. The Lense- Thirring effect, averaged over several orbits, has been recently verified by analysing the node orbital motion of 2G B = [J − 3(J · u )u ] (1) two laser ranged freely falling satellites (LAGEOS-1 and c2R3 ⊕ ⊕ r r LAGEOS-2) which orbit the Earth. In the measurement presented in Ref. [5] the two LAGEOS satellites were where R ≡ Rur is the position of the laboratory with used to confirm the LT effect with an accuracy of the respect to the center of the Earth and J⊕ is the angu- order of 10%. However, the launch of a third properly lar momentum of the Earth, whose modulus is of course designed satellite LARES will give the opportunity to given by the product of the moment of inertia of the measure the LT effect with an accuracy of the order of planet multiplied by its angular velocity. 1% [7]. The effect produced by a field like (1) on a massive test body moving with velocity v looks like the one produced The possibility to detect Lense-Thirring with ring by a magnetic field on a moving charge: in fact, the lasers has been discussed in the past [8, 9]. Recently geodesic equation in weak field approximation reads it has been already pointed out that a multi-gyros sys- tem is able to test locally the Lense-Thirring effect [10]: dv = G + v ∧ B (2) an array of six, 6 m side, square ring-lasers have enough dt sensitivity for this purpose. The rings must have different 2 orientation in space. In the present paper we concentrate where G = −GM/R ur is the Newtonian gravitational the attention on the symmetries of the rings arranged field, so that the effect can be described in terms of a on the faces of a cube or along the edges of an octa- gravito-electromagnetic Lorentz force, where the New- hedron, extracting the relevant relations important for tonian gravitational field plays the role of the gravito- the diagnostics of the system. At the end we summarize electric field (GE). and sketch the proposed experiment. For completeness Furthermore, the rotation of the source of the grav- we must mention that an experiment of the type we are itational field affects a gyroscope orbiting around it, in planning and preparing could also be made in principle such a way that it undergoes the so-called Lense-Thirring using matter waves instead of light. This possibility has precession, or dragging of the inertial frames of which been proved experimentally for various types of parti- the gyroscope defines an axis[17, 18]. This phenomenon cles such as electrons [11], neutrons [12], Cooper pairs shows up also when one considers a freely falling body [13], Calcium atoms [14], superfluid He3 [15] and super- with local zero angular momentum (ZAMO: Zero Angu- fluid He4 [16] . Cold atoms interferometry, in particular, lar Momentum Observer): it will be seen as rotating by yields very high sensitivity and it is suitable for space a distant observer at rest with the fixed stars [19]. experiments because of the apparatus small size. How- ever, atoms interferometry experiments in space do not provide an independent measurement of the Earth an- A. Mechanical gyroscopes gular velocity, are affected by the mass distribution of the Earth, and test the average of the relativistic effect Gravito-magnetic effects can in principle be measured rather than the local one. Eventually, the comparison applying different methodologies. The one that has most 3 often been considered is focused on the behaviour of a space. What is closed from the view point of the labo- gyroscope, that can be either in free fall (on board an ratory is not so for a fixed-stars-bound observer, but the orbiting satellite) or attached to the rotating Earth. The essential is that the two directions are not equivalent and axis of the gyroscope is affected in various ways by the that the two times required for light to come back to the presence of a gravitational field. As for GM, a little me- active region are (slightly) different. As it happened al- chanical gyroscope is the analogous of a small dipolar ready in the case of the mechanical gyroscopes, here too magnet (a current loop), so that it behaves as magnetic the difference in the two times of flight is made up of var- dipoles do when immersed in an external magnetic field. ious contributions depending on the rotation of the axes When studying the motion around the Earth of a gy- of the local reference frame with respect to distant stars, roscope whose spin vector is S , one is led to the formula on the fact that the local gravitational (Newtonian) po- [3, 17]: tential is not null, and of course on the GM drag (which is our main interest). What matters, however, is that the dS = Ω0 ∧ S (3) final proper time difference (a scalar quantity) is invari- dt ant: it does not depend on the choice of the reference frame or of the coordinates. In Appendix A we work out the explicit expression of Ω0 Performing the calculation in linear approximation for in general relativity and, more in general, in metric the- an instrument with its normal contained in the local ories of gravity, using the Parametrized Post-Newtonian meridian plane (see the Appendix A details) we find (PPN) formalism[20]: we show that it is related to the gravito-magnetic components g0i of the metric tensor · and its expression is given by (see Eqs. (A6)-(A10)) 4A GM 0 cδτ = Ω⊕ cos (θ + α) − 2 sin θ sin α Ω = ΩG + ΩB + ΩW + ΩT , so that we can distin- c c2R ¸ guish four contributions, namely the geodetic term ΩG, GI⊕ the Lense-Thirring term ΩB, the preferred frame term + 2 3 (2 cos θ cos α + sin θ sin α) (4) 0 c R ΩW , the Thomas term ΩT . All terms in Ω are called relativistic precessions, but properly speaking only the where A is the area encircled by the light beams, α is the second is due to the intrinsic gravito-magnetic field of angle between the local radial direction and the normal 1 the Earth, namely it is ΩB = − 2 B, and manifests the to the plane of the instrument, measured in the meridian Lense-Thirring drag. plane, and θ is the colatitude of the laboratory; Ω⊕ is Ciufolini [21] deduced the relativistic precession of the rotation rate of the Earth as measured in the local the whole orbital momentum of two LAGEOS satellites reference frame (which includes the local gravitational whose plane of the orbit is dragged along by the rotating time delay). Earth. Again on Eq. (3) was based the GP-B exper- Eq. (4) can also be written in terms of the flux of an iment, whose core were four freely falling spherical gy- effective angular velocity Ω through the cross section of roscopes carried by a satellite in polar orbit around the the apparatus: Earth [6]. While time goes on and the available data grow 4 it is expected that the Lense-Thirring drag will emerge δτ = A · Ω, (5) from the behaviour of the unique (so far) double pulsar c2 system [22]. where A = Aun is the area enclosed by the beams and oriented according to its normal vector un. In particular, 0 it is Ω = Ω⊕ + Ω , and the term proportional to Ω⊕ is B. Using light as a probe the purely kinematic Sagnac term, due to the rotation of 0 the Earth, while Ω = ΩG + ΩB + ΩW + ΩT encodes the A different experimental approach consists in using relativistic effects (see Appendix A) light as a probe. In this case the main remark is that For a ring laser in an Earth-bound laboratory, the the propagation of light in the gravitational field of a geodetic and Lense-Thirring terms are both of order ∼ rotating body is not symmetric. The coordinated time 10−9 with respect to the Sagnac term, while the Thomas duration for a given space trajectory in the same sense as term is 3 orders of magnitude smaller. As for the pre- the rotation of the central source is different from the one ferred frame term, the best estimates [25, 26] show that obtained when moving in the opposite direction. This this effect is about 2 orders of magnitude smaller than asymmetry would for instance be visible in the Shapiro the geodetic and Lense-Thirring terms. Consequently, to time delay of electromagnetic signals passing by the Sun leading order, the relativistic contribution to the rotation 0 (or Jupiter) on opposite sides of the rotation axis of the measured by the ring laser turns out to be Ω ' ΩG+ΩB, star (or the planet) [23][24]. which we aimed at measuring in our experiment. In other This property of the propagation of light is the one words, the goal of our experiment will be the estimate of which we wish to exploit in our Earth-bound experiment Ω0 (see Fig. 1) which embodies the gravito-magnetic ef- using a set of ring lasers. In a terrestrial laboratory, light fects in a terrestrial laboratory. circulating inside a laser cavity in opposite directions is In particular, the proposed experiment can also pro- forced, using mirrors, to move along a closed path in vide high precision tests of metric theories of gravity 4

length of the loop P : cτ+ = P = Nλ+. The same hap-

0,10 pens with the left handed beam, but being the total time different, also the wavelength of the corresponding stand- 0,08 ing wave will be different: cτ− = Nλ−. The two modes 0,06 of the ring can have different N, a situation usually called 0,04 ’split mode’, but the higher accuracy of the measurement 0,02 has been obtained so far with the two modes with equal

0,00 N. Considering the time of flight difference in terms of -0,02 the wavelengths of the two standing waves we see that:

-0,04

-0,06

Rotation Rate [prad/sec] f− − f+ δf

-0,08 cδτ = N (λ − λ ) = Nc = P λ (8) + − f 2 c

-0,10 0 45 90 135 180 The ring laser equation [27] relates the frequency split- [deg] ting δf of the two optical beams inside the ring interfer- ometer with the experienced rotation rate of its mirrors FIG. 1: The amplitude of the relativistic effects on the sur- face of the Earth, according to the theory of general relativity, 4A in units of prad/s, as a function of the colatitude θ. The con- δf = un · Ω, (9) 0 tinuous, dashed and dotted lines correspond to Ω = ΩG +ΩB λP 0 projected along the directions: i) parallel to Ω⊕ (i.e. Ωk); ii) where P is the perimeter and λ is the laser wavelength. ur (local radial or zenithal direction); and iii) uθ (local North- The response R of a ring laser to the rotation rate Ω, in South direction), respectively. To evaluate the contribution units of rad/sec, is simply a rescaling of the frequency to Ω0 from Ω , we have projected Ω along Ω (continuous G G ⊕ splitting by the scale factor S ≡ 4A , i.e. line plus triangles) and uθ (dotted lines plus squares). We λP note that the gravito-electric term has only the uθ compo- R ≡ δf/S = u · Ω. (10) nent and therefore along the radial direction we have a pure n gravito-magnetic term. The scale factor S plays a crucial role in the accuracy of the measurement of Ω and to estimate the relativistic 4A −10 effects the ratio λP must be known and kept at 10 which are described in the framework of (PPN) formal- accuracy level for months. The requirements to keep the ism. In fact, from Eqs. (A22-A23), we see that, on set- apparatus in the optimal working conditions will be dis- ting for the rotating Earth J = I⊕Ω⊕, we obtain cussed in section IV. Since the effective angular velocity as well as the GM gravito-magnetic one is of the order of 10−9Ω , angles Ω = −(1 + γ) sin ϑΩ u , (6) ⊕ G c2R ⊕ ϑ between vectors must be measured at the corresponding 1 + γ + α1 GI accuracy level. Unfortunately, the absolute measurement Ω = − 4 ⊕ [Ω − 3 (Ω · u ) u ] (7) B 2 c2R3 ⊕ ⊕ r r of un in the fixed stars reference system with the accu- racy of nano-radians can hardly be achieved. However, where α1 and γ are PPN parameters (e.g. α1 = 0 we can relax this requirement by using M ≥ 3 ring lasers and γ = 1 in general relativity) which account for the ef- oriented along directions uα (α = 1 ...M), where not fect of preferred reference frame and the amount of space all uα lie in the same plane. In fact, Ω can be com- curvature produced by a unit rest mass, respectively. pletely measured by means of its projections on at least As shown in Sect. VII B, from a high precision mea- 3 independent directions (e.g.defining a tri-dimensional surement of the vector Ω0 in the meridian plane, we Cartesian system) and the redundancies of the measure- should be able to place new constraints on the PPN pa- ment can be used as a monitor and control of the stability α rameters α1 and γ. of the directions u . We further assume that ring lasers have identical sensitivity and noise parameters. From an experimental point of view this can be easily satisfied III. THEORY OF THE MEASUREMENT: by building the devices with scale factors that differ less COMBINING TOGETHER THE RESPONSE OF than %. SEVERAL RINGS In order to simplify the sensitivity calculations of the system one can consider multi-axial configurations en- A. The response of a ring laser dowed with symmetries. As all the ring laser normals uα are equivalent in space, symmetric configurations should A ring laser converts time differences into frequency be more efficient in the rejection of spurious effects and differences. In fact, since the emission is continuous, the in the control and monitoring of the relative orientation right handed beam adjusts itself to give a standing wave of the ring lasers. The natural choice is to take advan- whose wavelength is an integer sub-multiple of the space tage of space symmetries of regular polyhedra, setting 5 one ring for each plane parallel to their faces. If we do follows the crucial assumption is that systematic errors not consider the degeneration between opposite faces, we (scale factors and relative alignment of uα) are negligi- have M=3 in the case of the cube, 4 for tetrahedron and ble with respect to statistical errors, i.e. |Ω · δuα| < σΩ octahedron, 6 for dodecahedron, and 10 for icosahedron. or equivalently |δuα| < σΩ/Ω, while the dihedral angles There is a peculiar geometry with M = 3, obtained by ar- arccos(uα ·uβ) can nearly approximate a regular polyhe- ranging the rings along the edges of an octahedron, where dron configuration. the different rings can be nested together, sharing 2 by Redundancy of responses, if M > 3 rings are involved, 2 the same mirrors. We will refer to it in the following can be used to control systematic errors projected along by speaking of “octahedral configuration”. The M = 3 the direction of Ω. In fact, the rigidity of the configu- is the minimum number of rings necessary to reconstruct ration imposes some linear kinematic constraints among the rotational vector, but a redundancy is very appropri- different estimates of the laboratory rotation. In general, ate to enhance statistics and to have control tests on the any linear combination of 3 responses Rα gives an esti- geometric accuracy. mate of the local rotation Ω and we can test the consis- In general, by simple arguments, one can demonstrate tency among different estimates by means of the ordinary that, for regular polyhedra configuration least square fit. A very simple linear constraint can be found for regular polyhedral configurations XM M uα = 0 (M > 3) (11) X α=1 L = Rα (14) α=1 and that and we will illustrate its statistical property as an exam- ple of the power of the method. XM M (Ω · u )2 = |Ω|2 (M ≥ 3). (12) From the definition of L immediately follows that it is α 3 α=1 Gaussian distributed√ with zero mean and standard devi- ation σL = MσΩ. In addition, possible misalignments As a consequence, one can study linear and quadratic δϑα ∧uα or scale factor fluctuations δSαuα are amplified combinations of ring lasers responses Rα which are invari- by a factor of Ω in the mean value of L Pant under permutationsP of the ring laser labels α, i.e. L = 2 XM α Rα and Q = α Rα. For non-symmetricP configura- < L > = Ω · δuα (15) tions we can generalize their definition as L = α LαRα P α=1 and Q = α QαβRαRβ , where Lα and Qαβ are suitable XM constants which depend on uα. The interest in such lin- k ear or quadratic forms relies on their behaviours in the = Ω (δSαuα + δϑα ∧ uα) , (16) presence of noise fluctuations or variations of the geom- α=1 etry of the configuration. 2 without affecting the corresponding variance σL. Thus They also allow us to carry out analytical estimates of < L > can be used as a “null constraint” which is mini- the overall sensitivity of a tri-axial system of ring laser mum when the configuration geometry is a regular poly- to relativistic effective rotation rates. hedron, and so the√ overall mean error parallel to Ω can −10 be monitored at ∼ MσΩ/Ω ' 10 accuracy level.

B. Requirements for the geometry of the configuration C. Estimate of the parallel component of the relativistic effective rotation vector The response of each ring laser can be conveniently written as An estimate of Ω2 for symmetric configurations readily follows from Eq. (12) Rα = Ω · (uα + δuα) + εα , (13) 3 XM Q = R2 (17) where δuα ≡ δSα uα + δϑα ∧ uα account for system- M α atic errors in the scale factors and orientations in space α=1 M M and εα represents the additive noise that affects the ro- 6 X 3 X = Ω2 + ε Ω · uα + ε2 . (18) tation measurement Rα, that we assume averaged on the M α M α observation time T ' 1 day. We assume as well that α=1 α=1 εα are Gaussian distributed random variables with zero Its mean value and standard deviation read (see App. B) 2 mean and variance σ . Modulus |δuα| ' δSα and di- Ω 1 rection δϑα ∧ uα represent the deviations from regular 2 2 < Q > = Ω + σΩ (19) polygon geometry in the plane and from polyhedra ge- r 3 ometry in the space, due to scale factor fluctuations δS α 18 4 12 2 2 σQ = σ + Ω σ . (20) and infinitesimal rotations δϑα, respectively. In what M Ω M Ω 6

In addition, one can demonstrate that Q is non-central χ2 The parallelism of v ∧ w with the normal to the distributed with M degrees of freedom and non-centrality meridian plane can be tested under the hypothesis that 2 parameter Ω . In order to estimate the relativistic effec- the rotation signal is fully located in the PM subspace tive rotation, we must subtract Ω⊕ from the rotation rate while the QM subspace contains only noise. The test estimated in the laboratory. To this end we calculate the can be easily performed over the norms of the two 2 2 difference ∆ ≡ (Q − Ω⊕) that, in the limit of high SNR projections EP (v, w) ≡ ||Pv,wR|| and EQ(v, w) ≡ (|Ω|/σ >> 1), tends to be Gaussian distributed with ||Q R||2, where we have introduced the symbol ||R|| = Ω Pv,w mean M 2 1/2 ( α=1 Rα) to indicate the L-2 norm in the M- 0 dimensional Euclidean response space. The M × M pro- < ∆ >' 2 Ω⊕ Ω (21) k jection matrices Pv,w and Qv,w can be written explicitly as functions of the unit vectors v and w and standard deviation √ √ T Pv,w = NV (NV ) (23) σ∆ ' (2 3/ M)Ω⊕ σΩ , (22) T Qv,w = I − NV (NV ) , (24) where we have neglected terms of the order of σΩ/Ω. The SNR =< ∆ > /σ∆ of the parallel component of where V is a 3 × 2 matrix with columns v and w, and relativisticp effective rotation is increased by a factor of I is the M × M identity matrix. As shown in appendix 2 M/3 with respect to the sensitivity of each ring laser. B , the probability distribution of EP is non-central χ The advantage of this approach is that we compare with 2 degrees of freedom and non-centrality parameter 2 2 scalar quantities (moduli of rotation vectors) measured Ω , while EQ should be χ distributed with M-2 degrees with respect to the local and distant stars reference sys- of freedom. tems. Its drawback is the very poor sensitivity to the The best estimate of v and w is then obtained by perpendicular component Ω⊥ of the relativistic effective 2 2 0 0 2 2 rotation. In fact, Ω − Ω = 2Ω · Ω⊕ +|Ω | , and the (ˆv, wˆ) = arg max ||Pv,wR|| , (25) ⊕ v,w ratio between the second term (which is associated to the perpendicular component as |Ω0|2 = Ω02 + Ω02) and the k ⊥ where the maxu,w is taken over the unit sphere. The first term is ∼ GM/c2R ' 10−10. direction of the Earth rotation axis can be estimated as It is worth noticing that statistical fluctuations of L a particular case of Eq. (10). In fact, the projectors Pw (control of geometry by redundancy) and Q (measure of and Qw can be obtained by substituting the matrix V relativistic effects) are uncorrelated, and that they tend for the 3 × 1 matrix W with columns w. The difference to be independent in the limit of high SNR. lies in the dimension of the corresponding subspaces, i.e. PW and QW have dimension 1 and M-1 respectively. It is worth noticing that the maximum of Eq. (25) can D. Estimate of the components of the relativistic be computed by an analytical formula both for the loca- effective rotation vector tion of the meridian plane and the direction of the Earth rotation axis. In fact, if we introduce in the local ref- By arranging the response of ring lasers Rα as erence frame the (local) spherical coordinate (R, Θ and M-tuples in a M-dimension vector space R = Φ) (we use capital letters to avoid confusion with Sect. (R1,R2,R3,...,RM ), we can easily define projection op- II) and parametrize the unit vectors v and w with these erators that allows the estimate of local meridian plane angles, for instance w = (cos Φ sin Θ, sin Φ sin Θ, cos Θ) M and also the direction w of Ω in the physical space. and v = (cos Φ cos Θ, sin Φ cos Θ, − sin Θ) , we have that 2 2 Moreover, the norm of projected random vectors are de- the maximum of ||Pv,wR|| and ||PwR|| is achieved for scribed by remarkably simple statistics. According to the µ ¶1/2 definition of the matrix product we have R = NΩ + ε, RT FR tan Θb = where N is a M × 3 matrix whose elements are Nαi = RT HR (u ) and ε = (ε , ε , ε , . . . , ε ). Thus, the random vec- α i 1 2 3 M µ ¶1/2 tors R can be projected on the linear subspaces P and RT KR M tan Φb = (26) QM of dimensions 2 and M − 2, which represent respec- RT JR tively a plane in the physical space and its complemen- tary space. The physical symmetry of the rotating Earth where F , H, K, J are M ×M symmetric matrices which imposes that the relativistic effective rotation vectors and are functions of the uα alone. Ω⊕ lie in the same plane, i.e. the meridian plane, and In general, there are no analytical calculations for therefore the knowledge of the orientation of this plane is mean and variance of vˆ, wˆ and one must run Monte Carlo crucial if we want to measure not only the modulus but simulations to get their estimates. However, in the limit the whole vector. We recall that a plane is defined as the of high SNR EP and EQ tend to be Gaussian distributed, set of the points sv + tw, where s and t range over all as well as fluctuations of vˆ and wˆ around their mean val- real numbers, v and w are given orthogonal unit vectors ues. The same reasoning holds true also for the estima- in the plane. tion of Θb and Φ.b 7

The validity of the proposed experimental configura- respectively. Thus a ∼ 10% accuracy can be achieved tion has been checked by a numerical simulation over a in 3 months by simply comparing the square modulus period of 1 year of the six responses of the octahedral of rotation vectors. In order to give a full estimate of configuration oriented as in Fig. 23 of subsection VC. the vector Ω0, we have also explicitly calculated day by In order to simplify the calculations we assume that the day the angles Θb and Φb for describing the orientation of laboratory colatitude is θ = π/4 and that the normal the meridian plane and the direction of the Earth rota- to the plane of a ring forms a π/4 angle with respect tion vector. The results are summarized in Fig. 3 and 4, to the Earth axis, and another normal is orthogonal to where we report the time evolution of these angles and the former and forms again a π/4 angle with the west- in Fig. 5 where we show the corresponding annual polar east direction. This configuration is close to a possible motion. experimental arrangement at the Gran Sasso National Laboratories (LNGS) within few degrees. The directions 2.5 ´ 10-6 of the unit vector uα in the local reference frame are  ³ ´ 1 1 1 -  u = u = , √ , D ´ 6  1 4 ³ 2 2 2 ´ 2. 10 rad u = u = − 1 , √1 , − 1 (27) @  2 5 ³ 2 2 2´  Q -  u = u = − √1 , 0, √1 1.5 ´ 10 6 3 6 2 2 and the rotation√ signal for the 6 rings are equal within ´ -6 a factor 2. We assume one mean sidereal day TS = 1. 10 0 50 100 150 200 250 300 350 70 Sidereal days

60 FIG. 3: Change of the angle Θ due to polar motion as mea- 50 sured by the ring laser responses in one year.

40 By synchronizing the polar motion measured in the lo- cal reference system with the polar motion measured by

Counts 30 IERS in the fixed star reference system, the two reference 20 frames will coincide within the accuracy of the measure- 10 ment of Ω and Ω⊕, say 1 part of 10 . As a final remark, 10

0 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1

rotation rate difference @pradsD 0,8

2 0,6 FIG. 2: Histograms of the difference ∆ between Q and Ω⊕, normalized with the mean sidereal day, collected for 3 months

0,4 (dark histogram) and one year (light histogram). [rad]

0,2 86164.0989 s of integration and a noise standard devi-

−2 0,0 ation σΩ = 7 × 10 prad/s. The variance σΩ is ex- trapolated from present ”G” sensitivity at 104 s and scaling by a factor 5, due by the increase of the ring -0,2 size and the power of the laser of a factor 1.5 and 0 50 100 150 200 250 300 350

10, respectively. The relativistic rotation contributions Sidereal days 0 −2 0 −2 Ωr = −2.8 × 10 prad/s and Ωθ = −5.6 × 10 prad/s have been added to the Earth rotation vector Ω⊕, as esti- FIG. 4: Change of the angle Φ due to polar motion as mea- mated by IERS [28]. The component√ of relativistic effects sured by the ring laser responses in one year. Note the large 0 0 0 −2 parallel to Ω⊕ is Ωk = (Ωθ +Ωr)/ 2 = 5.9×10 prad/s. variation of Φ which correspond to a nearly complete preces- Using Eq. (10) we calculated the responses of the 6 rings sion cycle of the Earth axis in one year. and then we injected the Gaussian noise. In Fig. 2 we 0 show the histograms of TS∆/(2π) accumulated for 90 we point out that the full measurement of the vector Ω 0 −2 and 366 sidereal days. The corresponding mean val- allow us for the estimate of Ω⊥ ' 2×10 prad/s with a ues of the parallel component of relativistic effects are standard deviation of the same order of magnitude of the −2 −2 0 −6.0 × 10 prad/s and −6.2 × 10 prad/s with stan- estimate of Ωk. This represent an increase of the relativis- dard deviations 4.7×10−3 prad/s and 2.6×10−3 prad/s, tic rotation signal of ∼ 30 %. However, the estimate of 8

Ω0 is crucial to separate the geodetic from Lense-Thirring contributions and/or to measure the PPN parameters α1 0.002 and γ.

0.5

0.001 0.4

0.3 (sec) LoD

0.000 rad Μ

0.2

-0.001

0.1 53500 54000 54500 55000

Time (days) -0.1 0.0 0.1 0.2 0.3 Μ rad FIG. 6: The change of the Length of the Day (LoD) over the last 6 years from the IERS 05C04 time series. Notice that FIG. 5: The estimated polar motion from the 6 ring laser estimated errors of LoD decreased in the last years to a level −14 responses. which correspond to 10 rad/sec, i.e. 0.1 ppb Ω⊕.

0.6

E. The Earth motion and feasibility of the experiment 0.5

Since our goal is the estimate of the Lense-Thirring ef- 0.4 fect at few % accuracy, the independent measurement of

Ω + Ω , which represents the rotation of the labora- 0.3 ⊕ REL rad tory with respect to distant stars, must be determined to −10 10 Ω⊕. Due to tidal forces and to the exchange of an- gular momentum between the solid Earth and geophysi- 0.2 cal fluids, the angular velocity of the Earth varies in time,

both in direction and modulus. Changes in modulus cor- 0.1 respond to a variation of the Length of the Day (LoD)

of few milliseconds with respect to atomic clocks. The -0.2 -0.1 0.0 0.1 0.2 0.3

direction of the rotation axis of the Earth varies with re- rad spect to both the fixed stars and the Earth-fixed reference frames. Nowadays, the best Earth rotation monitoring is FIG. 7: The change of the direction of the Earth rotation provided by the IERS 05C04 time series [28] which are axis (i.e. pole position) over the last 6 years from the IERS routinely obtained using the geodetic space techniques 05C04 time series. Estimated errors are also plotted. VLBI (Very Long Baseline Interferometry), SLR (Satel- lite Laser Ranging), GPS ( Global Positioning System) and DORIS (Doppler Orbitography and Radioposition- it is expected to be sufficiently small to contribute to Ω⊕ ing Integrated by Satellite). only trough Ω⊕k. However, ΩREL is still largely unknown In Figs. 6 and 7 we report the Length of the Day due to possible micro-rotations of the crust of the Earth. (LoD) and the pole position with the corresponding er- This is one of the causes limiting the performances of G in rors of the last six years. It is worth to noticing that the Wettzell: the Earth crust motion caused by atmospheric achieved precision is 0.001 ms in the LoD and 0.1 µarcsec changes. It is assumed that an underground facility is in the pole position. less sensitive to this kind of noise sources. It is as well Further improvements are expected in the next few important to keep the experiment close to VLBI stations. years and the overall errors in LoD and pole position The underground Gran Sasso Laboratories is placed half should decrease of a factor 10 that is crucial for a 1% way between two relatively close VLBI stations, Medic- measurement of the relativistic rotation terms. However, ina [29] and Matera [30] which can provide estimates of the IERS 05C04 time series is already sufficient to get the crustal motion of the Adriatic plate [31]. A signifi-

|Ω⊕| with 3% accuracy. cant contribution to ΩREL comes from the ”diurnal polar For what concerns the differential rotation of the lab- motion” (periodic motion of the Earth crust due to tides) −7 oratory with respect to the rotation estimated by IERS, and consists in periodic changes of amplitude ∼ 10 Ω⊕. 9

This effect has been already measured by large ring laser gyroscopes [32], and can be accurately modeled and then subtracted from ring lasers responses. We conclude that by means of available geodesics and geophysics techniques, provided that the experiment is located in an area with very low relative angular motion

(ΩREL ), a suitable tri-axial detector of rotation can in principle detect Ω0 with % precision.

IV. THE ’REAL APPARATUS’, THE PRESENT SENSITIVITY OF G IN WETTZELL

Sensor properties FIG. 8: Approximately eight days of raw G data taken with A closer look at equation 9 reveals that there are three 30 minutes of integration time. One can clearly see the con- basic effects one has to carefully account for. These are: tributions from diurnal polar motion, solid Earth tides and local tilt. • scale factor stability (4A/λP )

• orientation of the gyroscope with respect to the in- data point was taken by integrating over 30 minutes of stantaneous axis of rotation of the Earth measurement data. There are several distinct signal con- tributions in the data, which come from known geophys- • instantaneous rate of rotation of the Earth – Length ical effects. The most prominent signal is caused by di- of Day (LoD) urnal polar motion [33]. The polar motion data is super- The scale factor for all practical purposes has to be held imposed by a tilt signal caused by the semi-diurnal and constant to much better than 1 part in 1010. Otherwise diurnal tides of the solid Earth, distorting the otherwise the frame-dragging parameter cannot be determined un- sinusoidal diurnal frequencies slightly. At the Geodetic ambiguously. For G, the base of the gyroscope has been Observatory in Wettzell the tilt effects of the solid Earth manufactured from Zerodur, a glass ceramic with a ther- tides can be as large as 40 nrad in amplitude. In Fig. 8 mal expansion coefficient of α < 5 × 10−9/oC. Further- the diurnal signal is dominated by the polar motion [34]. more the instrument is located in a thermally insulated Less evident in Fig. 8 are the effects from local tilt, which and sealed environment with typical temperature varia- contains periodic signals of tidal origin as well as non- tions of less than 5 mK per day. However, because the periodic signals. The latter are non-periodic and usually underground laboratory is only at a depth of 5 m, there change slowly over the run of several days. High res- is still a peak to peak temperature variation of about 1 olution tiltmeters inside the pressure stabilizing vessel degree per year, accounting for the change of seasons. of the G ring laser keep track of these local effects and Changes in the atmospheric pressure also affect the di- the data is corrected for gravitational attraction (atmo- mensions of the ring laser structure by changing the com- sphere, sun and moon)l [33]. Large non-periodic local pression of the Zerodur block and cannot be neglected. tilts occur most prominently after abundant rainfall, in- Hence G is kept in a pressure stabilized enclosure. A feed- dicating hydrological interactions with the rock and soil back system based on the determination of the current beneath the ring laser monument. Fig. 9 shows the east value of the optical frequency of the lasing mode of one component of three tiltmeters installed i) on a gravime- sense of propagation allows for active control of the pres- ter pillar at the surface, ii) in 6 m depth, and iii) in 30 sure inside the steel vessel such that an overall geometric m depth. scale factor stability of better than 10−10 is routinely ob- While the tilmeter in 30m depths clearly shows the tained. At the same time the design of the instrument is periodic signal of the solid earth tides, the tilt record made as symmetric as possible. So changes in area and of the instruments near to the surface is dominated perimeter are compensated with a corresponding change by large non-periodic signals hydrological, thermoelastic in wavelength as long as no shear forces are present and and barometric origin. Several investigations have shown the longitudinal mode index stays the same. that the site and the installation depths of tiltmeters has A typical eight day long measurement sequence of rota- a major impact on environmental noise mainly coming tion rate data from the G ring laser is shown in Fig. 8. In from hydrology [35], [36], [37], [38] has shown that even order to demonstrate the obtained sensor sensitivity we in 100 m depths effects caused by hydrological changes have subtracted the mean Earth rotation rate from the are detectable, but strongly reduced in comparison to a gyroscope data. The y-axis gives the measured variation 50 m deep installation. First investigations related to to- of the rate of rotation, while the x-axis shows the time pographic and temperature induced effects were carried expressed in the form of the modified Julian date. Each out by [39] and [40]. Detailed investigations using the 10

FIG. 9: Measurement of local tilts as a function of depth in the Earth.

finite-element method have shown that these effects can FIG. 11: Resolution and stability of G, compared with Earth amount to more than 10 nrad ([41], [42]), while the dis- signals tance between the source and the location of observation can be several hundred meters. Additionally, recent work using the G ring laser data reveals that effects caused by wind friction at the Earth surface yields to high frequency rotations of large amplitudes. The large seasonal temperature effect on the G ring laser as well as the substantial local tilt signals and the rather high ambient noise level of our near soil surface structures give reasonable hope of much better perfor- mances of a ring laser installation in a deep underground laboratory such as the Gran Sasso Laboratories. For the detection of fundamental physics signals one has to remove all known perturbation signals of the Earth FIG. 12: Six rings arranged on the faces of a Cube, using the from the ring laser time-series. Furthermore we have ap- GEOSENSOR design, which has been successfully used so far plied 2 hours of averaging of the data in order to reduce for middle size rings, as our prototype G-Pisa the effect from short period perturbations. Fig. 10 shows an example. In Fig. 11 we show the current sensitivity ex- resolution tiltmeters have been removed, averaging as in- dicated above was applied to a series of 30 days of data collection, including the period shown in Fig. 8. It can be expected that a similar data set from the Gran Sasso laboratory would become substantially smoother, since most of the perturbations, caused by ambient atmosphere - topsoil interaction still contained in the data of Fig. 10 would no longer exist in the deep underground facil- ity. Changing hydrologic conditions presumably causing small local rotation and temperature variations, atmo- spheric pressure and wind loading are among the sources for the systematic signatures in the residual data.

V. CONFIGURATION OF A TRI-AXIAL FIG. 10: The rotation rate of the Earth measured with the DETECTOR G ring laser as a function of time. Averaging over 2 hours was applied to a corrected dataset, where all known geophysical From now on, we will restrict our analysis to 24 m signals have been removed. perimeter rings, arranged in two configurations that are of some experimental interest, i.e. 6 ring lasers rigidly pressed in term of Allan deviation of the G, the expected mounted on the faces of a cube, as shown in Fig. 12, sensitivity of each ring laser at Gran Sasso Laboratories and 3 ring lasers oriented along the edges of an octahe- and the relevant geophysical signal. dron, see Fig. 13. The cubic configuration requires 24 In order to reduce the local orientation uncertainties, mirrors forming 6 independent rings and the extension which remain after local tilts measured with the high of the GEOSENSOR design is straightforward (see sub- 11

A. Ring-laser sensitivity

2 The rotation sensitivity σΩ, for noise fluctuations which are dominated by laser shot noise over an inte- gration time T, reads r cP hf σ2 = , (28) Ω 4AQ WT where Q is the quality factor of the optical cavity, f = c/λ is the laser frequency, h is the Plank constant and W is the power of the laser [43]. The limiting sensitivity can be conveniently calculated scaling the parameters of the Wettzell ”G” ring laser µ ¶ µ ¶ µ ¶ P 16 m2 3 × 1012 σ = 2.910−13 × Ω 16 m A Q FIG. 13: Three rings are formed using 6 mirrors located on Ãr !Ãr ! the vertices of an octahedron 20 nW 105 s rad/s (29) W T section VC ); while the octahedral configuration require In order to obtain in few weeks a 10% accuracy level 6 mirrors only to form 3 orthogonal rings. By itself the in the measurement of the relativistic effective rotation configuration which uses a cube is redundant, each ring rates, we must achieve the sensitivity goal of σΩ = has a parallel companion, which can be used for the study 7 × 10−14 rad/s (or equivalently a rotation noise level of systematics. For the octahedron configuration the im- 20 prad/sec/Hz1/2 at a frequency of 1 day−1). From Eq. plementation of the GEOSENSOR design needs further (29) we have that a system of 6 rings with P = 24 m, development. Redundancy can be easily obtained con- Q = 3 × 1012 and W = 200 nW can fulfill this require- structing a second octahedron with planes parallel to the ment. other one. The two structures should be built very close to each other, in order to keep as much as possible the whole apparatus compact; in this way 6 rings are avail- B. Expected performances of not optimally able, analogously to the cube configuration, see Fig. 14. oriented rings This configuration has the advantage that there are con- We assume that the ring lasers are identical in the sense described in Sect. IIIE and that the dihedral an- gles arccos(uα · uβ) are measured better than one part in 1010 in order to estimate Ω independently from the reference frame. Note that only the stability of dihedral angles can be monitored by means of the Earth signal it- self only for short times (few days), while their measure- ments and controls must be performed independently in the laboratory. For instance, assuming that the scale fac- tors are controlled to the 10−10 accuracy, the responses of two parallel rings is statistically different from noise when their parallelism is modified. From an experimental point of view, to arrange in the Cartesian planes several rings and keep the configuration stable over the integration time T ' 1 day is a demanding FIG. 14: Six rings, two by two parallel, with mirrors on the task. However, we can relax such a demanding require- vertices of two octahedron, constructed very close one to the ment by means of data analysis procedures that account other in order to reduce the dimension of the apparatus. for slightly non-orthogonal dihedral angles. For instance, we can use the measured dihedral angles straints in the relative angle between rings, since each to estimate directly Ω. In fact, we can substitute the mirror is in common between two rings, and three linear quadratic combination of ring laser responses in Eq. (17) Fabry-Perot cavities are available using the three diago- with the equivalent bilinear combination nals of the rings. Those linear cavities have the capability XM of monitoring the relative angles between different rings, Q = QαβRαRβ (30) and as well the length of each diagonal. α=1,β=1 12 where Qαβ are the elements of the M × M matrix Q = N(NN T )−2N T . The statistics of Q is no longer non- central χ2; however, we can easily compute (see App. B for details) its mean

2 2 < Q >= |Ω| + MσΩ (31) and variance X X σ2 = 2σ4 Q2 + 4σ2 Ω2 Q2 uk uk (32) Q Ω αβ Ω αβ α β αβ αβ

In the limit of high SNR, fluctuations of Q tend to be Gaussian distributed, and so we recover the results in Eq. (22) for the overall sensitivity of the system. If we start with dihedral angle close to π/2 (say 1 part in 105), then FIG. 15: Drawing of G-Pisa, based on the GEOSENSOR design sensitivity loss is very small since it is of the same order.

C. Guidelines of the Experimental Apparatus seems rather difficult. Later on, a more flexible and less expensive design has been realized, called GEOSENSOR, The best performing ring, so far, is G which is a four which so far has been employed especially for smaller size mirrors ring. This is one of the reasons why the present rings. This design allows a very good relative alignment scheme uses a square ring geometry. In principle a trian- of the mirrors, it is relatively easy to change mirrors and gular ring, with 3 mirrors could be preferable since the tools to move each mirrors along different degrees of free- three mirrors are always inside a plane, and the losses will dom have been implemented. So far this kind of instru- be minimized as well, reducing the number of mirrors. ments have been done in steel. Fig. 15 shows a drawing It could be advantageous in principle, but a triangular of G-Pisa, our prototype. The optical cavity vacuum ring is less sensitive. For instance, let us compare the chamber has a stainless steel modular structure: 4 tow- performance of two rings inscribed in a circle of radius ers, located at the corners of the square and containing r; for a regular polygon with different number of sides the mirrors holders inside, are connected by pipes, in or- r2 2π N, the area is A = N 2 sin( N ) and the perimeter is der to form a ring vacuum chamber with a total volume π −3 3 P = 2N r sin( N ); it is straight forward to demonstrate of about 5 · 10 m . The mirrors are rigidly fixed to the that the triangular ring has 0.7 times the signal than the tower. The cavity alignment can be adjusted by moving square one, which is equivalent to say that the triangular the towers with respect to the slab through a lever system ring needs 2 times more time to reach the same level of that allows 2 degrees of freedom of movements. No win- accuracy as the square one. dow delimits the active region and the vacuum chamber The ring-laser response is proportional to the Scale Fac- is entirely filled with a mixture of He and a 50% isotopic tor ”S”. For a perfect square ring this proportionality mixture of 20Ne and 22Ne. The total pressure of the gas factor is equivalent to N the number of wavelength inside mixture is set to 560 Pa with a partial pressure of Neon of the ring: when the length of the ring changes, because of 20 P a. The active region is a plasma produced in a cap- a change in the temperature, the laser changes its wave- illary pyrex tube inserted at the middle of one of the ring length in order to keep N constant. This is true as long sides by a radio frequency capacitively coupled discharge. as the perimeter change is below a wavelength, 632 nm In a non monolithic device, temperature changes could in our case, and in this conditions the gain factor of the interrupt the continuous operation, and the perimeter is instrument guarantees a very high accuracy of the mea- actively controlled by acting on the mirrors and using as surement. For example: if the laboratory has δT = 1o reference a stabilized laser; very highly stabilised lasers degree temperature excursion, the ring perimeter is 36 are commercially available, for instance wavelength sta- m, in order to guarantee the operation of the ring-laser bilization at the level of 2.5 × 1011 using iodine line can with a fixed number of wavelengths N, it is necessary be obtained. G-Pisa is kept in continuous laser opera- to realize the whole apparatus using materials with tem- tion trough a perimeter stabilization servo system which perature expansion coefficient of the order of 10−8 K−1. acts along the diagonal direction, for two opposite placed This is the concept used for G in Wettzell: a structure mirrors, through piezoelectric actuators [44]. realised with material as Zerodur, with a design which The GEOSENSOR design has other advantages as well: can be defined monolithic, i.e. relative motions of the the mirrors are under vacuum and are not affected by the mirrors are not allowed. G has a very high stability, but outside pressure changes, they can be very easily aligned is rather expensive, and not very flexible with regards to and the cost is pretty much reduced compared with the changing the mirrors and align the laser cavity. More- monolithic design. The experience of G-Pisa has shown over the extension of this design to a large array of rings so far that it can work with different orientations. In fact 13

granite and 1 degree themperature change granite and 1 degree themperature change GainFactor relative change GainFactor relative change

4. ´ 10-7

3. ´ 10-7 2. ´ 10-10

2. ´ 10-7 1.5 ´ 10-10 1. ´ 10-7 ∆ x m -10 -0.0001 -0.00005 0.00005 0.0001 1. ´ 10 -1. ´ 10-7

- ´ -11 -2. ´ 10 7 5. 10

-3. ´ 10-7 ∆x m -0.0001 -0.00005 0.00005 0.0001

FIG. 16: Macc, for a rectangular ring, with sides 6 × 6.6 m, in function of a misalignment of one of the four mirrors with re- FIG. 17: Macc in function of a misalignment of one of the spect to the ideal position, the perimeter control acts on four four mirrors with respect to the ideal position, the perimeter mirrors, maximum thermal excursion of 1 degree and the sup- control acts on four (thick line) or two (dashed line) mirrors, port of the GEOSENSOR has thermal expansion coefficient maximum thermal excursion of 1 degree and the material has −6 of 7 × 10−6 K−1 (granite) 7 × 10 m/mK (granite)

Thermal expansion 10 times less than granite G-Pisa has worked both horizontally and vertically ori- 1 degree themperature change ented. It is in steel, inside the thermally stabilized room GainFactor relative change in the central area of Virgo, in order to improve thermal 8. ´ 10-11 stability, it has been mounted on top of a granite table (thermal expansion coefficient about 5 × 10−6 m/mK). To find the guidelines of the mechanical project, we have 6. ´ 10-11 used a simple program which consists in considering the ring as four points (the light spot on the mirrors) which 4. ´ 10-11 can be moved from the ideal position, both inside the plane or outside the plane. The model takes into account thermal expansion and the perimeter is kept constant by 2. ´ 10-11 acting diagonally on pairs of mirrors; the use of 2 mirrors or 4 mirrors for the feedback correction have been inves- o ∆x m tigated; the thermal excursion is considered of 1 degree. -0.0001 -0.00005 0.00005 0.0001 The scale factor S in presence of misalignments is com- pared with S0 ( scale factor at the optimal configuration); FIG. 18: Same as Fig. 17, but with expansion coefficient 10 S0−S this comparison is expressed as Macc = , which gives times lower S0 the accuracy limit induced by misalignments. The re- 10 −10 quired level of accuracy of 1 part of 10 is Macc = 10 . Fig. 16 shows Macc for a rectangular ring, with sides 6 each coordinate. In summary, if the thermal excursion is m and 6.6 m, in function of a misalignment of one of the 1 K, the position of the mirrors is an ideal square within four mirrors. ±50µm, the support has a thermal expansion coefficient −7 −1 Fig. 16 clearly shows that the Gain Factor changes a lot below 7×10 K , Macc remains in the range necessary with small change of mirrors positions. The situation for the needed accuracy using four or two mirrors active strongly improves by considering a perfect square ring. control of the perimeter. In fact, for a closed figure with a fixed number of sides, Misalignments which bring the light spots outside the the area over perimeter ratio has a maximum when the plane of the ring do not have appreciable effect on the polygon is regular one, as for example a ’perfect’ square gain factor, but they change the orientation of area vec- ring. Fig. 17 and 18 show Macc with 100 µm construction tor uα; in this case the effect for Macc depends on the precision and two possible choices of the thermal expan- relative angle between the ring and the Earth rotational sion coefficient. For instance, let us assume that each axis. Figs. 20 and 21 shows how the accuracy changes mirror position is in the ideal position within a quan- for two different ring orientations: parallel to the axis of tity δ which depends on the precision of the construc- the Earth and at 45o degrees respectively. The first is tion. Fig. 19 shows Macc when 3 out of the 4 mirrors are almost insensitive, while the other is sensitive to nano- positioned with an error, 10000 points have been evalu- metric misalignments. ated pseudo-randomly distributed between ±50µm along Fig. 22 shows Macc for a nm vertical misalignment of 14

50 micrometers error in the position of each mirror thermal expansion coefficient 7 10−6 /K Area vector 45 degrees with respect to the Earth axis 800 four mirrors perimeter granite thermal expansion 600 control Relative Accuracy Error 400

200

0 - −1 −0.5 0 0.5 1 1.5 2 5. ´ 10 10 −10 Macc x 10 800

600 two mirrors perimeter control ∆y HmL - ´ -8 - ´ -9 ´ -9 ´ -8 400 1. 10 5. 10 5. 10 1. 10

200

-10 0 -5. ´ 10 −1 −0.5 0 0.5 1 1.5 2 2.5 −10 Macc x 10

FIG. 19: Histogram of Macc when the position of 3 mirrors are within ±50 µm close to the ideal position, 10000 points FIG. 21: Accuracy change in percentage for vertical misalign- have been evaluated by randomly extracting the position error ments and area vector close to 45 degrees with respect to the (±50µm). The thermal expansion coefficient is 7 × 10−7K−1, Earth rotational axis. The ring geometry is not perfect in the thermal excursion 1 K, top histogram shows the case with 4 plane, there is a misalignment of 100µm, a maximum temper- ature change of 1 K, and the thermal expansion coefficient is mirrors control of the perimeter, bottom curve with 2 mirrors −6 control. 7 × 10 m/m/K

Relative accuracy of the Earth angular rotation Area vector parallel to the Earth axis in function of the inclination angle granite thermal expansion Relative Accuracy Error Relative Accuracy Error

1.88 ´ 10-12 4. ´ 10-10 1.86 ´ 10-12 2. ´ 10-10

1.84 ´ 10-12 Θ HradL -1.5 -1.0 -0.5 0.5 1.0 1.5 1.82 ´ 10-12 -2. ´ 10-10

1.8 ´ 10-12 -4. ´ 10-10

∆y HmL -0.0010 -0.0005 0.0005 0.0010 FIG. 22: Relative limit of the accuracy in the measurement FIG. 20: Accuracy change in percentage for vertical misalign- of the Earth angular rotation induced by a nm change in ments and area vector close to the parallel alignment to the the position of one of the mirrors with respect to its original Earth rotational axis. The ring geometry is not perfect in the position, in function of angle with the Earth rotation axis. plane, there is a misalignment of 100µm, a maximum temper- The area vector of the ring lays in the meridian plane. The ature change of 1 K, and the thermal expansion coefficient is −6 accuracy loss is zero when the Earth axis and the area vector 7 × 10 m/m/K are parallel, and is very high in the orthogonal alignment one of the rings in function of the angle with respect to to constantly monitor the relative angle between rings the Earth rotational axis. with nrad precision (only the relative alignment mat- In summary: The Gain Factor of each ring can be kept ters). This can be accomplished looking at the modal constant at the level of 1 part in 1010, if the positions of structure of the FP cavities formed along the diagonals. the mirrors are constructed and kept within +/ − 50 µm Moreover the orientation of each ring with respect the error close to the ideal square ring; the relative position Earth rotation axis should be such to avoid alignment between mirrors can be rigidly constrained with gran- too sensitive to the relative angle (relative angle with the ite, super-invar or similar low thermal expansion coeffi- Earth rotation axis below 60o). Using the Earth angular cient spacers, it is preferable to use all the four mirrors velocity rotation, which is perfectly stable for few days, for the perimeter active stabilization, but two mirrors the whole apparatus can be calibrated at the beginning; control could be acceptable as well if the structure has the relative angle, or the area of each ring could be not thermal coefficient better than granite. It is necessary perfectly planar or exactly 900, but it is important to 15 monitor the geometry of the structure during the whole measurement time (years). The mirror holders plays an important role, it can be advantageous to build them in Zerodur or similar material, in order to avoid displace- ments out of the plane. The mirrors holder should be designed in order to provide the tools to align the cav- ities; in principle each mirrors should have 5 degrees of freedom: three translations and two tilts, the rotation around the axis orthogonal to the mirror itself does not play a role; but since the mirrors are spherical only three motions are fundamental: we may have one translation along the diagonal and two mirrors tilts or three transla- tions. Let us consider now an octahedral geometry, containing the three rings. Fig. 22 shows that the relative angle between the differ- ent rings must be monitored at the level of nrad, and that it should be avoided to put one of the rings with an angle larger than 600 with respect the Earth rotation axis. We have done the exercise to fit the octahedron, with rings of 24 m perimeter, inside the node B of LNGS, considering that this node is 8 m tall, and imposing the constrains discussed in Fig. 22. The exercise is done with the oc- tahedron since it needs more space. Considering that the latitude of LNGS is 42o 2700N two configurations are given: to have the octahedron straight up (8.4 m tall) or laying on one side, one ring is respectively horizontally or vertically oriented and the other two symmetrically positioned with respect to the meridian plane. Let us consider the maximum size 9 m: 8.48 m, the diagonal of the octahedron, plus 0.6 m necessary to hold mirrors FIG. 23: Plan of LNGS laboratory close to node B and optics in general necessary for the read out. This octahedron can be contained inside each of the big halls of LNGS, in both orientation, but inside node B, which is the most isolated room of LNGS, see Figs. 23 and 25, the only possibility is the shorter configuration, with the longer side parallel to the floor. So, the octahedron, with rings of 24 m perimeter, can be contained inside node B, where the ceiling is 8 m tall, while for node C the structure should be scaled, probably no more than 20 m perimeter can be contained inside node C, since the ceiling there is 6 m tall. Fig. 24 shows the octahedron inside node B.

VI. DIAGNOSTICS OF DIHEDRAL ANGLES AND SCALE FACTORS FIG. 24: the octahedron inside node B To reduce the influence of systematics in long–term measurements, the control of the geometrical stability of ring laser system is of paramount importance. In partic- of three optical resonators: the ring itself and two linear ular, it is crucial to monitor the deviations from planarity ones oriented along the diagonals. These latter can be of each ring laser and their mutual orientations. used to monitor the geometrical stability of the whole A square ring consists of four spherical mirrors with the ring system. Deviations from a square geometry result same curvature radius R, placed at the corners. Square in tilting and/or displacements of the diagonal vectors, geometry guarantees that opposite mirrors are parallel so which in turn change the cavity eigenmodes. that they form two extra linear Fabry–Per`otcavities (see A linear symmetric FP cavity with spherical mirrors in z = ± 1 d = ± √1 L (L being the square ring arm) and Fig. 26). As a consequence, each square ring is made M 2 2 16

2 2 |C`m| / |C00| ` = 0 ` = 1 ` = 2 7 2 15 4 m = 0 1 5.16 × 10 Θx 1.33 × 10 Θx 7 2 15 2 2 m = 1 5.16 × 10 Θy 2.66 × 10 ΘxΘy 15 4 m = 2 1.33 × 10 Θy

TABLE I: Power coupled to the first cavity higher modes (` + m = 0, 1, 2) as a fraction of the external laser power for qc = qx = qy and X = Y = 0. The value are obtained for a ratio between the cavity length and the mirror radius of curvature of 1.5

FIG. 25: The ring laser system inside node B of LNGS, side view showing that passage between the two entrances transmitted though the output mirror M2. A modal de- composition of such a pattern gives a suitable set of coef- 2 ficients |C`m| which can be used for estimating the posi- centers on the z-axis, supports the Gaussian modes tion and angular misalignment of the cavity with respect to the reference beam E (x, y). à √ ! à √ ! in 1 2x 2y Supposing that at the begining (t = 0) the cavity ex- E`,m (x, y, z) = H` Hm ternal laser is perfectly aligned to a symmetric cavity wc (z) wc (z) wc (z) · µ ¶¸ (if the two mirror show equal transmittivity then the x2 + y2 2z cavity transmission is 1) so that all the incoming power ×exp −ik − ikz + i (` + m + 1) arctan , 2qc(z) b Pin is coupled to the TEM00 mode. The measurement (33) procedure we have devised is a tunable laser, showing a linewidth narrower than the cavity linewidth, tuned over ³ ´−1 1 λ a cavity FSR in a time interval ∆t so that each mode is where qc(z) = z − ib = − i 2 and b = ∆t Rc(z) πw (z) spanned in a time τ = , where F is the cavity finesse. p c F d (2R − d) are the complex curvature of the Gaussian The number of photons in the `m mode are given by (we beam and the confocal parameter, respectively; here the are now assuming a rectangular line shape instead of a curvature radius Rc (z) and the spot-size wc (z) read Lorentzian profile)

d2 − 2dR − 4z2 Pin 2 R (z) = n`m = k τ |C`m| . c 4z hv 2 2 2 λ 4z + 2dR − d This number of photons must be higher than the noise wc (z) = p . π 2 d (2R − d) equivalent number of photons hitting the detector in the√ same time interval. The noise equivalent power in The eigenmodes E`,m (x, y, z) form a complete set which W/ Hz, is connected to the equivalent number of pho- can be used for representing a generic field confined be- tons by tween the two generally misaligned mirrors of the cavity √ X NEP B n = τ , E (x, y, z) = C`,mE`,m (x, y, z) , NEP hν η `,m where η and B are the quantum efficiency and the detec- where tion bandwidth, respectively. Z Z To overcome the photon noise, we have to satisfy the C`m = dx dyEin (x, y) E`,m (x, y, zM ) inequality and E (x, y) is the beam illuminating the input mirror n n |C |2 in NEP = NEP 00 < 1 M1. If we suppose the mirror tilted by Θx and Θy and 2 n`m n00 |C`m| displaced by X and Y with respect to cavity axisz ˆ, we have In particular, looking at C01 coefficient we have · ¸ (x−X)2 (y−Y )2 √ −ik +Θxx+ +Θy y 2qc(z ) 2qc(z ) 1 NEP B Ein (x, y) ∝ e M M . n > τ ; 00 5.16 × 107Θ2 hν η 2 2 As an example, the relative intensities |C`m| / |C00| Pin for the first modes ` + m = 0, 1, 2 and X = Y = 0 are further, by assuming n00 ' hv τ we obtain reported in Tab. I. √ It is clear that the cavity axes misalignment can be NEP B P > . detected by looking at the intensity pattern of the beam in 5.16 × 107Θ2η 17 √ For typical silicon detectors NEP ∼ 10−14W/ Hz, B ∼ 10−14 rad/sec; here we have used GM = 3.986004418 × 6 14 3 2 6 10 Hz, η ∼ 0.9, so that 10 m /s , R = 6.378137 × 10 m and TS = 86164.0989 s. Assuming one year of data taking with −19 −2 Pin > 2.2 × 10 Θ . the same ring laser parameters used for the simulations in Sect. IIID we have that the standard deviation of Ωc0 For a tilt sensitivity of Θ ∼ 10−9 the power required at θ and Ωc0 is σb ' 0.03, and therefore upper limits of the input is r Ωr,θ some interest can be put on α1 and γ at the Gran Sasso colatitude θ ' π/4. Pin > 220 mW .

B. Interdisciplinary: Geodesy and Geophysics

Earth rotation rate and the orientation of the rota- tional axis of the Earth in space are the linking quantities between the terrestrial (ITRF) and the celestial (ICRF) reference frames. Currently a set of quasars, forming an external set of markers, provide the only way of deter- mining the rotational velocity and the variations of the orientation of the rotational axis of the Earth with suf- ficient accuracy. As already mentioned, 10 µs for the measurement of length of day (LOD) and 0.1 mas for the pole position are routinely achieved by a network of VLBI radio telescopes as one of the services (IERS) of FIG. 26: In a square ring configuration passive Fabry–Per`ot the International Association of Geodesy (IAG). The op- cavities are formed along the square diagonals (dashed line in eration of such a network requires expensive equipment the sketch). In the case of an octahedron each of these three and a lot of maintenance effort. Huge amounts of data passive cavities is shared by two rings. are recorded in each measurement session, which require physical transport over large distances for the correlation in the analysis centers. Data latency and the fact that there is no continuous measurement coverage are suggest- VII. MORE ABOUT THE IMPORTANCE OF ing the investigation of alternative methods for the pre- THIS MEASUREMENT cise estimation of Earth rotation. Furthermore it is desir- able to develop an independent measurement technique, A. Post Newtonian Parameters in order to identify intra-technique biases if they exist. Ring lasers are possible candidates for such an alterna- The proposed experimental apparatus is well suited for tive measurement technique. They measure the earth performing optical test of metric theory of gravitation. rotation locally and within much shorter time intervals. We start from the statement that the vector Ω0 should be Such gyros are widely used in aircraft navigation and can entirely contained in the meridian plane if the preferred measure rotations absolute, i.e. independent of an exter- frames effect, determined by W (see Eq. (A9)), can be nal reference frame. Therefore also local contributions to neglected. Indeed the currently available best estimates earth rotation are contained in the measurements. The [20] suggest that this effect is about 2 orders of magnitude effects of earth tides, strain, crust deformation, seismic smaller than the geodetic and Lense-Thirring contribu- events, polar motion are contained in the ring laser mea- tions. As a consequence, we expect that the measured surements due to their contribution to earth rotation or components of Ω0 outside the meridian plane should be due to variations in the orientation of the respective ring compatible with noise. In this case, our results could laser. However, the demands on such instruments are ex- be used to obtain new constraints, independent from the tremely high and cannot be met by existing commercial available ones, on the preferred frames parameters. In devices. They can be summarized as: addition, we can write the PPN parameters α1 and γ as 0 a function of the ur and uθ components of Ω • sensitivity to rotation 0.01 prad/s at about 1 hour ³ i of integration c0 c0 α1 = −4Ωθ csc θ − 8Ωr sec θ 10 ³ ´ • sensor stability of 1 part in 10 over several month c0 c0 to years (Chandler Wobble) γ − 1 = Ωθ csc θ − Ωr sec θ/2 − 2 , (34) • resolution in sensor orientation ≈ 1 nrad. This d0 0 where Ωr,θ ≡ Ωr,θ/w and w is the very precisely mea- corresponds to polar motion of around 1cm at the 2 sured constant w ≡ 2πGM/(c RTS) ' 5.0747798 × pole. 18

i This means that a reasonable improvement in sensor sen- tionary metric in the form[51] gµν = gµν (x ) an observer i i sitivity and stability is still required in order to make ring at rest at x = x0 measures the proper-time difference lasers viable tools to be applied to space geodesy. The δτ = τ+ − τ− between the right handed beam propaga- design of the G ring laser is one way of approaching these tion time (τ+) and the left handed one (τ−): demands and it not too far away from reaching this goal [45, 46]. Operating several such ring laser gyroscopes in q I q I i g0i i i geophysical independent regions simultaneously offers a δτ = −2 g00(x0) ds = −2 g00(x0) H · ds, S g00 S unique possibility to distinguish global from local (mon- (A1) umentation related) signal contributions through their where S is the spatial trajectory of the beams, whose independent data streams. g0i tangent vector is ds, and we set Hi = . g00 In order to evaluate the proper-time difference (A1), Discussions and Conclusions we need to know the space-time metric in our laboratory, that is to say the gravitational field nearby the world- line of the observer which performs measurements with The feasibility of the experiment for the measurement the ring laser. To this end, we consider an observer in of relativistic effective rotation rates appears to rest only arbitrary motion in a given background space-time, and on a tri-axial dynamical sensor of local rotation of enough write the corresponding local metric in a neighborhood sensitivity. Despite the fact that large ring lasers as G are of its world-line (see e.g. [3]) very stable platforms and with the provision of tight feed- back systems to stabilize the scale factor (cold cavity, as 2 well as the active cavity), currently ring laser gyroscopes g(0)(0) = 1 + 2A · x + O(x ), (A2) (k) 2 are not able to determine the DC part of the Earth rota- g(0)(i) = Ω(i)(k)x + O(x ), (A3) tion rate with a sensitivity compatible with the require- g = η + O(x2). (A4) ments for detection of the Lense-Thirring effect. While (i)(j) (i)(j) the contribution of the varying Earth rotation itself pre- sumably can be removed with sufficient accuracy from It is worth pointing out that the Eqs. (A2)-(A4) hold the C04 series of VLBI measurements, there remains the only near the world-line of the observer, where quadratic problem of determining the actual null-shift offsets from displacements terms are negligible. Here we suppose that the laser functions in the ring laser gyroscope. Since the observer carries an orthonormal tetrad (parentheses the gravito-magnetic effect is small and constant, a good refer to tetrad indices) e(α), whose four-vector e(0) co- discrimination against laser biases, such as for example incides with his four-velocity U, while the four-vectors ‘Fresnel drag’ inside the laser cavity must be achieved. e(i) define the basis of the spatial vectors in the tan- Therefore it will be advantageous to locally add one or gent space along its world-line. By construction we have several ring laser cavities in addition to the described e(α)e(β) = η(α)(β), where η(α)(β) is the Minkowski ten- structures for sufficient redundancy. sor. The metric components (A2)-(A4) are expressed Moreover, even if not strictly necessary for getting rid in coordinates that are associated to the given tetrad, (i) of all the systematics, it would be helpful to compare namely the space coordinates x and the observer’s (0) data taken at distant stations for having a more precise proper time x . In the above equations, A is the spa- discrimination of local effects from regional and global tial projection of the observer’s four-acceleration, while changes. In particular, we wish to operate the G ring the tensor Ω(i)(k) is related to the parallel transport of laser structure in parallel to the here proposed one. Pos- the basis four-vectors along the observer’s world-line: (β) sibly a second large ring laser located at the Cashmere ∇U e(α) = −e(β)Ω (α). In particular, if Ω(i)(j) were zero, facility in Christchurch, New Zealand, will be useful on the tetrad would be Fermi-Walker transported. Let us this respect provided that it can be run with sufficient remark that the metric (A2)-(A4) is Minkowskian along resolution and stability. the observer’s world-line (x(i) = 0); it is everywhere flat iff A = 0, i.e. the observer is in geodesic motion and the tetrad is non rotating (i.e. it does not rotate with respect APPENDIX A: RING LASER MEASUREMENTS to an inertial-guidance gyroscope). In the latter case, the IN THE LABORATORY FRAME first corrections to the flat space-time metric are O(x2) [3]. In this Appendix we evaluate the response to the grav- In order to explicitly write the local metric, which itational field of a ring laser in an Earth bound labora- through its gravito-magnetic (g0i) and gravito-electric tory and, to know the space-time metric in the laboratory (g00) components enables us to evaluate the proper-time frame we shall use the construction of the “proper refer- difference (A1), we must choose a suitable tetrad by tak- ence frame” as described in Ref. [3, 17]. ing into account the motion of the Earth-bound labora- As we discussed in Section III, a ring laser converts a tory in the background space-time metric. To this end, time difference into a frequency difference (see e.g. Eq. we consider the following PPN background metric which (8)). It is possible to show that (see e.g. [47]) in a sta- describes the gravitational field of the rotating Earth (see 19 e.g. [20]): Lense-Thirring precession ΩB is due to the angular mo- mentum of the Earth; iii) ΩW is due to the preferred ds2 = (1 − 2U(R))dT 2 − (1 + 2γU(R)) δ dXidXj + · ij¸ frames effect; and iv) the Thomas precession ΩT is re- (1 + γ + α /4) lated to the angular defect due to the Lorentz boost. 2 1 (J ∧ R) − α U(R)W dXidT, R3 ⊕ i 1 i It is worth noticing that for a laboratory bounded to the Earth (A5) dV where −U(R) is the Newtonian potential, J is the an- A ' − ∇U(R), (A12) ⊕ dT gular momentum of the Earth, Wi is the velocity of the reference frame in which the Earth is at rest with re- and the acceleration A can not be eliminated. Taking spect to mean rest-frame of the Universe; γ and α1 are into account Eq. (A12) and substituting in Eqs. (A7) post-Newtonian parameters that measure, respectively, and (A10) it is possible to write the two precessions in the effect of spatial curvature and the effect of preferred the form frames. The background metric (A5) is referred to an µ ¶ 1 Earth Fixed Inertial (ECI) frame, where Cartesian geo- Ω = − + γ ∇U(R) ∧ V , (A13) G 2 centric coordinatesp areP used, such√ that R is the position . 2 2 2 2 vector and R = i Xi = X + Y + Z . Then, we choose a laboratory tetrad which is related to the and background coordinate basis of (A5) by a pure Lorentz 1 Ω = A ∧ V . (A14) boost, together with a re-normalization of the basis vec- T 2 tors: in other words the local laboratory axes have the same orientations as those in the background ECI frame, In particular, for a geodetic motion (e.g. a free fall satel- and they could be physically realized by three orthonor- lite) A ≡ 0 and Eq. (A13) gives the geodetic preces- mal telescopes, always pointing toward the same distant sion for a gyroscope in free fall, while Thomas precession stars. (A14) is zero: strictly speaking, it is just in this case that In this case, one can show that the gravito-magnetic ΩG describes a geodetic effect, however the term can be contribution in the local metric reads [3, 17, 48, 49] also referred to the precession due to the Newtonian field (k) 0 Ω(i)(k)x = − (Ω ∧ x)(i), where the total relativistic of the source. contribution Ω0 is the sum of four terms, with the di- All terms in (A7)-(A10) must be evaluated along the mensions of angular rotation rates laboratory world-line (hence, they are constant in the lo- cal frame), whose position and velocity in the background 0 Ω = ΩG + ΩB + ΩW + ΩT (A6) frame are R and V , respectively. However, if we consider an actual laboratory fixed on the Earth surface, the spa- defined by tial axes of the corresponding tetrad rotate with respect to the coordinate basis of the metric (A5), and we must Ω = − (1 + γ) ∇U(R) ∧ V , (A7) G µ ¶ take into account in the gravito-magnetic term (A3) the 1 + γ + α /4 J 3J · R 1 ⊕ ⊕ contribution of the additional rotation vector Ω⊕, which ΩB = − − R , (A8) 2 R3 R5 corresponds to the Earth rotation rate, as measured in 1 the local frame[52]. Ω = α ∇U(R) ∧ W , (A9) W 1 4 As a consequence, it is possible to show that, up to lin- 1 dV ear displacements from the world-line, the relevant local Ω = − V ∧ . (A10) T 2 dT gravito-magnetic potential turns out to be 0 The vector Ω represents the precession rate that an g(0)(i) = (Ω ∧ x)(i) , (A15) inertial-guidance gyroscope, co-moving with the labora- 0 tory, would have with respect to the ideal laboratory spa- where Ω = −Ω⊕ − Ω , while the gravito-electric g(0)(0) tial axes (see e.g. [3, 17]) which are always oriented as one remains the same. those of the ECI frame; if the spin vector of the gyroscope Now, we are able to evaluate the proper-time difference is S, its precession is hence defined by q I i dS δτ = −2 g00(x0) H · ds. (A16) = Ω0 ∧ S (A11) S dt Without loss of generality, we suppose that the observer Differently speaking, we may say that the local spatial is at rest in the origin of the coordinates, so that, ac- basis vectors are not Fermi-Walker transported along the i cording to (A2), g00(x ) = 1. As a consequence, we have laboratory world-line. In particular the total precession 0 rate is made of four contributions: i) the geodetic or de I Sitter precession Ω is due to the motion of the labora- (Ω ∧ x) G δτ = −2 · ds. (A17) tory in the curved space-time around the Earth; ii) the S (1 + 2A · x) 20

Now, on taking into account the expression of acceler- where 4Ω⊕ · A is the purely kinematic Sagnac term, due ation of the laboratory frame (A12) and evaluating the to the rotation of the Earth, while 4Ω0 · A is the gravi- magnitude of the various terms, the leading contribution tational correction due to the contributions (A7)-(A10). to (A17) con be written, applying Stokes theorem According to Section IIIA, from Eq. (A19), it is then Z possible to write the ring laser equation in the form δτ = −2 [∇ ∧ (Ω ∧ x)] · dA, (A18) A where A = Aun is the area enclosed by the beams and 4A δf = un · Ω. (A21) oriented according to its normal vector un. On evaluat- λP ing the curl, taking into account that Ω is constant, we eventually obtain Z To further clarify Eqs. (A7)-(A10) it is useful to use an δτ = −4 Ω · dA = −4Ω · A. (A19) orthonormal spherical basis ur, uϑ, uϕ in the ECI frame, A such that the ϑ = π/2 plane coincides with the equa- 0 On substituting Ω = −Ω⊕ − Ω in (A19), we see that torial plane. As a consequence, the position vector of the proper-time delay can be written in the form the laboratory with respect to the center of the Earth is R = Rur and the kinematic constraint V = Ω⊕ ∧ R 0 δτ = 4Ω⊕ · A + 4Ω · A, (A20) holds, i.e. V = Ω⊕R sin θuϕ.

Thus, the components of Ω0 in physical units read GM Ω = − (1 + γ) sin ϑΩ u , (A22) G c2R ⊕ ϑ 1 + γ + α /4 G Ω = − 1 [J − 3 (J · u ) u ] , (A23) B 2 c2R3 ⊕ ⊕ r r α GM Ω = − 1 u ∧ W , (A24) W 4 c2R2 r 1 Ω = − Ω2 R2 sin2 ϑΩ , (A25) T 2c2 ⊕ ⊕

Moreover, we assume the general relativistic values of the and, to leading order, the total rotation rate which PPN parameters, γ = 1, α1 = 0, and use for the Newto- enters the Eq. (A20) is nian potential of the Earth its monopole approximation, i.e. U(R) = GM/R. Thus, the components (A22)-(A23) read GM GM G Ω = −2 sin ϑΩ u , (A26) Ω = −Ω + 2 sin ϑΩ u + [J − 3 (J · u ) u ] G c2R ⊕ ϑ ⊕ c2R ⊕ θ c2R3 ⊕ ⊕ r r G (A30) Ω = − [J − 3 (J · u ) u ] , (A27) B c2R3 ⊕ ⊕ r r ΩW = 0, (A28) 1 Ω = − Ω2 R2 sin2 ϑΩ , (A29) T 2c2 ⊕ ⊕

If we denote by α the angle between the radial direction ur and the normal vector un, on setting un = cos αur + sin αuθ in (A20), and using (A30), we may express the proper-time delay in the form · ¸ 4A GM GI δτ = Ω cos (θ + α) − 2 Ω sin θ sin α + ⊕ Ω (2 cos θ cos α + sin θ sin α) (A31) c2 ⊕ c2R ⊕ c2R3 ⊕ where we have written J⊕ = I⊕Ω⊕, in term of the I⊕, the moment of inertia of the Earth.

APPENDIX B: PROBABILITY DISTRIBUTION if x is a multivariate Gaussian random vector with mean OF QUADRATIC FORMS

The statistics of quadratic forms of Gaussian random vectors x are well known in the literature. In particular, 21 s and covariance matrix Σ, the mean and the variance where P (x) = exp[(x − s)T (x − s)/(2σ2)]/(2πσ2)M/2 of a quadratic form Q = xT Qx are given by is the Gaussian probability density of one sample of the random vector x. We can use the integral representation < Q > ≡ < xT Qx >= T r(QΣ) + sT Qs of the Dirac’s δ-function 2 T 2 2 σQ ≡ < (x Qx) > − < Q > = 2T r(QΣQΣ) + 4sT QΣQs (B1) Z +∞ T iω(Q−xT x) T δ(Q − x x) = e dω (B5) where Q is a square symmetric matrix, and T r are the −∞ transpose and trace operators, respectively. The statis- tics of Q in general is not known, unless QΣ is an idem- potent matrix [50]. In the case were x represents the response of ring lasers in a regular polyhedral configura- and write tion Q = I, with no common noise source and the same sensitivity Σ = σ2I, where I is the identity matrix, the Z above formulas greatly simplifies +∞ 1 P (Q) = dωeiωQ Q (2πσ2)M/2 < Q > = Mσ2 + E (B2) Z −∞· ¸ 1 σ2 = 2Mσ4 + 4Eσ2 , (B3) exp iω xT x − (x − s)T (x − s) dx Q 2σ2 where E = sts = ||s||2 is the signal energy. In this case also the statistics of Q readily follows. In fact, starting from its definition we have By re-arranging the exponent, the last integral can be re- Z cast as a M-dimensional Gaussian integral and calculated T PQ(Q) ≡ P (x)δ(Q − x x) dx (B4) explicitly

Z ·µ ¶ ¸ 1 1 1 E exp[iωσ2E/(1 − 2iωσ2)] exp iω − xT x + sT x − dx = (B6) (2πσ2)M/2 2σ2 σ2 σ2 (1 − 2iωσ2)M/2

where in the last expression one recognizes the moment have that the joint probability density P (EP ,EQ) reads generating functions of a non-central χ2 distributions with M degrees of freedom and non-centrality param- eter E. The probability density function of Q can be Z T T found using the tables of Fourier Transform pairs P (EP ,EQ) = P (x)δ(EP − x P x)δ(EQ − x Q x) dx (C1) Z ½ ¾ +∞ exp[iωσ2E/(1 − 2iωσ2)] where P (x) is the probability density of one sample of P (Q) = dω eiωQ Q (1 − 2iωσ2)M/2 the random vector x. The two Dirac δ-functions can be −∞ written using their Fourier transforms, µ ¶ M−2 4 p 1 2 Q 2 = exp[−(Q + E)/(2σ )] IM/2−1( QE/σ )) 2 E Z T [uEP +vEQ−x (uP +vQ)x] P (EP ,EQ) = P (x)e du dv dx where I (x) are the modified Bessel functions of order k. k (C2) where the integrals in du and dv are performed along the imaginary axis (i.e. u = iω1 and v = iω2 are purely imaginary complex numbers). Now assume the noise is APPENDIX C: PROBABILITY DISTRIBUTION Gaussian distributed, uncorrelated between different de- OF PROJECTORS tectors and with identical variance σ2 in every detector, namely The norm of complementary projection operators P and Q acting on Gaussian random vectors x are de- µ ¶ scribed by remarkably simple statistics. In fact, starting 1 1 T 2 2 P (x) = M exp − 2 (x − s) (x − s) (C3) from the definition of EP = ||P x|| and EQ = ||Qx|| we (2πσ2) 2 2σ 22

2 where s ≡ (Ω · u1,... Ω · uM ) is the rotation signal in where α = 1/2σΩ. Writing x as s + ε and switching the vectorial form. Then, integration variable to ε yields

³ ´ M Z α 2 £ ¤ P (E ,E ) = exp −α(x − s)T (x − s) × P Q π £ ¤ × exp −xT (uP + vQ)x dx · × euEP evEQ du dv, (C4)

³ ´ M α 2 P (EP ,EQ) = × (C5) Z π £ ¤ £ ¤ × exp −nT (αI + uP + vQ)n − 2nT (uP + vQ)s dε exp −sT (uP + vQ)s euEP evEQ du dv.

The integration in dn can be done by noting that it is a with standard M-dimensional Gaussian integral with the lin- Z µ ¶ 1 −sp u ear term, and in general, for any M × M symmetric ma- uEP P (EP ) = 2 exp 2 e du trix A and M-vector b, 1 + 2σ u 1 + 2σ u Z µ ¶ M µ ¶ 2 −1 Z µ ¶ 1 −sq v ¡ ¢ πM/2 bT A−1 b P (E ) = exp evEQ dv exp −nT A n + bT n dn = p exp . Q 1 + 2σ2v 1 + 2σ2v det(A) 4 (C6) (C13) In our case, T T and sp = s P s, sq = s Qs. The transformed functions is the moment generating functions of two non-central χ2 A = αI + uP + vQ distributions, with 2 and M−2 degrees of freedom respec- b = 2(uP + vQ)s. (C7) tively, and whose non-centrality parameters are sT P s T Now we exploit the properties of P and Q. Using their and s Q s respectively. Thus, Ã ! complementarity, we can write µ ¶ p 1 EP + sp EP sp P (EP ) = exp − I0 A = (α + u)P + (α + v)Q (C8) 2 2σ2 σ2 µ ¶ µ ¶ M Ãp ! 4 −1 and from the fact that they are orthogonal and idempo- 1 EQ + sq EQ EQsq P (E ) = exp − I M tent we also have Q 2 −2 2 2 2σ sq 2 σ A−1 = (α + u)−1P + (α + v)−1Q, (C9) (C14) hence where In(x) is the modified Bessel function of the first µ 2 2 ¶ kind. Some interesting conclusions can be drawn about T −1 u T v T b A b = 4 s P s + s Q s . (C10) the virtual channels EP and EQ, which make them in- α + u α + v teresting for the identification of meridian plane and the Furthermore, as P and Q are projection matrices, their estimate of Ω. eigenvalues are {0, 1} with multiplicities respectively 2 1. EP is distributed as a non-central χ with 2 degrees {M − 2, 2} for P and {2,M − 2} for Q. Then, writing of freedom and non-centrality parameter equal to A in diagonal form is trivial and leads to sT P s, i.e. the magnitude of the signal projection in the P subspace. det(A) = (α + u)2(α + v)M−2, (C11) 2. E is distributed as a non-central χ2 with M − 2 determinants being independent from the basis. By us- Q degrees of freedom and non-centrality parameter ing C10 and C11 in C6 one can see that the Gaussian equal to sT Q s, i.e. the magnitude of the signal integral splits into the product of factors involving either projection in the Q subspace. u or v. By further substituting in C6, the remaining in- tegrals separate and the probability density remarkably 3. EP and EQ are statistically independent processes. factorizes as 4. In the limit of high SNR, EP and EQ are Gaus- T P (EP ,EQ) = P (EP ) P (EQ) (C12) sian distributed with means < EP >= s P s, 23

< E >= sT Q s and variances σ2 = 4σ2 sT P s, Q EP Ω σ2 = (M − 2)σ2 sT Q s, respectively. EQ Ω

[1] I.I. Shapiro et al., Phys.Rev. Lett., 26, (1971) 1132. [25] J.F. Bell, F. Camilo, T. Damour, A Tighter Test of Local [2] N. Straumann, General Relativity and Relativistic Astro- Lorentz Invariance using PSR J2317+1439, Astrophys. physics, Springer-Verlag, Berlin (1991). J., 464, 857 (1996). [3] C. W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, [26] T. Damour, D. Vokrouhlick´y,Testing for gravitationally Freeman, S. Francisco (1973). preferred directions using the lunar orbit, Phys. Rev. D, [4] H. Thirring, Phys. Z., 19, 204 (1918). 53, 6740 (1996). [5] I. Ciufolini, E. Pavlis, Nature 431, 958 (2004); I. Ciufolini [27] G.E. Stedman, Ring-laser tests of fundamental physics et al., Space Sci. Rev., 148, 71-104 (2009). and geophysics, Rep. Prog. Phys., 60, 615–688 (1997). [6] C.W.F. Everitt, et al.. Gravity Probe B: Final Results [28] Time series of the daily estimate of Earth of a Space Experiment to Test General Relativity, Phys. rotation vector can be downloaded from Rev. Lett., 106, 221101 (2011). http://data.iers.org/products/176/11165/orig/ [7] L. Iorio, D.M. Lucchesi, I. Ciufolini, Class. Quant. Grav., eopc04.62-now 19, 4311-4326 (2002). [29] See e.g. http://www.med.ira.inaf.it/index_EN.htm [8] G. E. Stedman, K. U. Schreiber and H. R. Bilger, On [30] See e.g. http://www.asi.it/en/flash_en/observing/ the detectability of the Lense-Thirring field from rotating space_geodesy_center laboratory masses using ring laser gyroscope interferom- [31] E. Mantovani, D. Albarello, C. Tarnburelli, M. Viti, An- eters, Class. Quantum Grav. 20 (2003). nals of Geophysics 38, 67 (1995); R. Haas, E. Gueguen, [9] M.O. Scully, M.S. Zubairy, M.P. Haugan, Proposed opti- H. Scherneck, A. Nothnagel, and J. Campbell, Earth cal test of metric gravitation theories, Phys. Rev. A, 24, Planets Space, 52, 759-764, (2000). 2009 (1981). [32] K.U. Schreiber, et al., J. Geophys. Res., 109, B06405 [10] A. Di Virgilio, et al., A laser gyroscope system to detect (2004). the Gravito-Magnetic effect on Earth, Int. J. Mod. Phys. [33] K. U. Schreiber, A. Velikoseltsev,M. Rothacher, T. D, 19, 2331-2343 (2010). Kl¨ugel,G. E. Stedman, D. L. Wiltshire; Direct measure- [11] F. Hasselbach, and M. Nicklaus, Sagnac experiment with ment of diurnal polar motion by ring laser gyroscopes, J. electrons: Observation of the rotational phase shift of Geophys. Res., 109, B06405, (2004). electron waves in vacuum, Phys. Rev. A, 48, 143 (1993). [34] A. Brzezinski, Contribution to the theory of polar motion [12] S.A. Werner, J.L. Staudenmann and R. Colella, Effect of for an elastic Earth with liquid core, Manuscr. Geod., 11, Earth’s Rotation on the Quantum Mechanical Phase of 226-241 (1986). the Neutron, Phys. Rev. Lett., 42, 1103 (1979). [35] G. Jentzsch, M. Liebing, A. Weise, Deep Boreholes for [13] J.E. Zimmermann and J.E. Mercereau, Compton Wave- High Resolution Tilt Recording. Bulletin Inf. Marees Ter- length of Superconducting Electrons, Phys. Rev. Lett., restres, 115, 8498-8506 (1993). 14, 887 (1965). [36] H.-J. Kumpel, Verformung in der Umgebung von Brun- [14] F. Riehle, Th. Kirsters, A. Witte, J. Helmcke, and Ch.J. nen. Habilitationsschrift, Univ. Kiel, 198p (1989). Bord´e,Optical Ramsey spectroscopy in a rotating frame: [37] A. Weise, G. Jentzsch, A. Kiviniemi, J. Kaariainen, Com- Sagnac effect in a matter-wave interferometer, Phys. Rev. parison of long period tilt measurements: results from Lett., 67, 177 (1991). two clinometric stations Metsahovi and Lohja. Finland [15] R.W. Simmonds et al., Nature, 412, 55 (2001). J. Geodyn. 27, 237-257 (1999). [16] E. Hoskinsons et al., Phys. Rev. B74, 100509(R) (2006). [38] T. Jahr, G. Jentzsch, A. Weise, Natural and man-made [17] I. Ciufolini, J.A. Wheeler, Gravitation and Inertia, induced hydrological signals, detected by high resolu- Princeton University Press, Princeton (1995). tion tilt observations at the Geodynamic Observatory [18] L.I. Schiff, Motion of a gyroscope according to Einstein’s Moxa/Germany, Geodynamics, 48, 126-131, (2009). theory of gravitation, Proc. Nat. Acad. Sci., 46, 871 [39] J.C. Harrison, Cavity and topographic effects in tilt and (1960). strain measurements, JGR, 81/2, 319-328 (1976). [19] D. Raine, and E. Thomas, Black Holes - An Introduction, [40] J.C. Harrison, K. Herbst, Thermoelastic strains and tilts World Scientific, Singapore (2009). revisited. Geophysical Research Letters, 4/11 (1976). [20] C.M. Will, Living Rev. Relativity 9, 3 (2006), [41] A. Gebauer, C. Kroner, T. Jahr, The influence of to- http://www.livingreviews.org/lrr-2006-3. pographic and lithologic features on horizontal deforma- [21] I. Ciufolini, The 1995-99 measurements of the Lense- tions, Geophys. J. Int., 177, 586-602 (2009). Thirring effect using laser-ranged satellites, Class. Quan- [42] A. Gebauer, H. Steffen, C. Kroner, T. Jahr, Finite ele- tum Grav. 17, 2369 (2000). ment modelling of atmosphere loading effects on strain, [22] M. Bourgay, et al., An increased estimate of the merger tilt and displacement at multi-sensor stations, Geophys. rate of double neutron stars from observations of a highly J. Int., 181 (3), (2010). relativistic system, Nature, 426, 531-533 (2003). [43] K.U. Schreiber, J-P. R. Wells and G.E. Stedman; Noise [23] A. Tartaglia, Class. Quant. Grav., 17, (2000). 2381-2384. processes in large ring lasers; General Relativity and [24] A. Tartaglia, A. Nagar, M.L. Ruggiero, Phys. Rev. D, Gravitation, 40, No. 5, 935-943, (2008). 71, (2005). [44] J. Belfi, N. Beverini, F. Bosi, G. Carelli, A. Di Vir- 24

gilio, E. Maccioni, A. Ortolan and Fabio Stefani, Perime- the dragging of inertial frame in the HYPER project, ter actively stabilized ring laser gyroscope as nano- Class. Quantum Grav., 36, 411 (2004). rotational motion sensor, Applied Physics B (2011). doi: [49] N. Ashby, B. Shahid-Saless, Geodetic precession or drag- 10.1007/s00340-011-4721-y ging of inertial frames?, Phys. Rev. D, 42, 1118 (1990). [45] K. U. Schreiber, T. Klugel, A. Velikoseltsev, W. Schlater, [50] A.M. Mathai, S.B. Provost, Quadratic Forms in Random G.E. Stedman, and J-P. Wells, The Large Ring Laser Variables: Theory and Applications, Marcel Dekker, New G for Continuous Earth Rotation Monitoring. Pure and York (1992). Applied Geophysics, 166, 301 498 (2009). [51] Greek and Latin indices denote space-time and spatial [46] K. U. Schreiber, T. Klgel, J.-P. R. Wells, R. B. Hurst, and components, respectively; letters in boldface indicate A. Gebauer How to Detect the Chandler and the Annual spatial vectors, while letters in italic indicate four-vectors Wobble of the Earth with a Large Ring Laser Gyroscope and four-tensors; summation and differentiation conven- Phys. Rev. Lett., 107, 173904 (2011) tions are assumed. In this Appendix, if not otherwise Highlighted paper: http://physics.aps.org/synopsis- stated, we use units such that G = c = 1. for/10.1103/PhysRevLett.107.173904. [52] For an Earth-bound laboratory, it is Ω⊕ ' £ 1 2 2 ¤ [47] E. Kajari, M. Buser, C. Feiler, W.P. Schleich, Rotation 1 + U(R) + 2 Ω0R sin ϑ Ω0, where R is the ter- in relativity and the propagation of light, Riv. Nuovo restrial radius, ϑ is the colatitude angle of the laboratory Cimento, 32, 339 (2009). and Ω0 is the terrestrial rotation rate, as measured in [48] M.C. Angonin-Willaime, X. Ovido, Ph. Tourrenc, Gravi- an asymptotically flat inertial frame. tational perturbations on local experiments in a satellite: