The Gravito-Maxwell Equations of General Relativity in the local reference frame of a GR-noninertial observer

We show that the acceleration-difference of neighboring free-falling particles (= geodesic deviation) measured in the local reference frame of a GR-noninertial observer is not given by the Riemann tensor. With the gravito-electric field of GR defined as the acceleration of free-falling quasistatic particles relative to the observer, the divergence of the gravito-electric field measured in the reference frame of a GR-noninertial observer is different from the Ricci curvature $R^0_{\,\,0}$. We derive our exact, explicit, and simple gravito-Gauss law for the divergence of the gravito-electric field in our new reference frame of a GR-noninertial observer with his LONB (Local Ortho-Normal Basis) and his LONB-connections in his time and 3-directions: the sources of the divergence of the gravito-electric field are contributed by all fields including the GR-gravitational fields, gravito-electric and gravito-magnetic. In the reference frame of a GR-inertial observer our gravito-Gauss law coincides with Einstein's $R^0_{\,\,0}$ equation, which does not have gravitational fields as sources. We derive the gravito-Ampere law, the gravito-Faraday law and the law for the divergence of the gravito-magnetic field. The densities of energy, momentum, and momentum-flow of GR-gravitational fields are local observables, but depend on observer with his local reference frame: these quantities are zero if measured by a GR-inertial observer. For a GR-noninertial observer the sources of gravitational energy, momentum, and momentum-flow densities have the opposite sign from the electromagnetic and matter sources. In the gravito-Gauss law the sources contributed by gravitational energy and momentum-flow densities have a repulsive effect on the gravitational acceleration-difference of particles.

We show that the acceleration-difference of neighboring freefalling particles (= geodesic deviation) measured in the local reference frame of a GR-noninertial observer is not given by the Riemann tensor. With the gravito-electric field Eg of GR defined as the acceleration of freefalling quasistatic particles relative to the observer, div Eg measured in the reference frame of a GR-noninertial observer is different from the curvature R 0 0 . We derive our exact, explicit, and simple gravito-Gauss law for div Eg in our new reference frame of a GR-noninertial observer with his LONB (Local Ortho-Normal Basisēâ) and his LONB-connections (ωbâ)ĉ in his time-and 3-directions: the sources of div Eg are contributed by all fields including the GR-gravitational fields ( Eg, Bg). In the reference frame of a GR-inertial observer our gravito-Gauss law coincides with with Einstein's R 0 0 equation, which does not have gravitational fields as sources. We derive the gravito-Ampère law for curl Bg, the gravito-Faraday law for curl Eg, and the law for div Bg. The densities of energy, momentum, and momentum-flow of GR-gravitational fields ( Eg, Bg) are local observables, but they depend on the observer with his local reference frame: if measured by a GR-inertial observer on his worldline in his frame of LONB connections, these quantities are zero. For a GR-noninertial observer the sources of gravitational energy, momentum, and momentum-flow densities have the opposite sign from the electromagnetic and matter sources. The sources in the gravito-Gauss law contributed by gravitational energy and momentum-flow densities have a repulsive effect on the gravitational acceleration-difference of particles.

I. INTRODUCTION, METHOD, AND RESULTS
The gravito-Gauss law of GR gives div E g , where the gravito-electric field E g is identical with the gravitational acceleration of quasistatic particles: E g ≡ g (GR) quasistat . But acceleration cannot be measured without a reference frame: Newton's accelerations are defined and measured relative to a Newton-inertial frame. In classical mechanics accelerations are defined and measured also relative to reference frames which are accelerated and/or rotating relative to Newton-inertial.
In GR, gravitational acceleration (≡ acceleration of freefalling particles) is defined and measured relative to a chosen observer, g (GR) quasistatic ≡ E g . A GR-inertial observer is freefalling and nonrotating relative to spin-axes of gyroscopes comoving on his worldline, and he measures E g = 0. Einstein 1911: "gravitational acceleration is relative, it depends on the observer" [1][2][3][4]. -The difference between a GR-inertial versus noninertial observer's reference frame is crucial in the gravito-Maxwell equations but irrelevant in Einstein's equations.
The worldline of a free-falling test-particle defines a geodesic. The acceleration-difference of neighboring freefalling test-particles is equal to the geodesic deviation. The measured geodesic deviation depends on the local reference frame of the chosen observer which includes his * Electronic address: chschmid@itp.phys.ethz.ch neighboring auxiliary observers.
We now prove that the geodesic deviation measured in the local reference frame of a GR-noninertial observer is not given by the Riemann tensor. This is evident for the case of a vanishing Riemann tensor: the accelerationdifference of neighboring freefalling quasistatic particles measured in a rotating reference frame (relative to gyrospin axes) is non-zero because of centrifugal accelerations. Hence the geodesic deviation measured in a rotating reference frame is non-zero even if the Riemann tensor is zero. GR-texts [6][7][8][9] are wrong in concluding the opposite.
In our method the reference frame of the observer is crucial. Cartan's method works with a field of Local Ortho-Normal Bases, LONBsēâ(P ), which we identify with a field of observers. The LONB-components Eî g ≡ gî quasistat are measured by these observers on their worldlines. Hats on indices denote LONB-components. The gravito-magnetic field is defined and measured by the gravitational spin-precession of the observer's gyroscopes Bî g ≡ −Ωî gyros relative to the observer's LONB.
• Our new method uses our reference frame of LONBs for a noninertial observer (accelerated and spinning relative to GR-inertial) with his Ricci LONB-connections (ωbâ)ĉ in his time-and 3-spacedirections, Eqs. (21). The LONB-connections (ωbâ)î in his 3-space directions connect to his auxiliary observers, which are co-accelerated, cospinning, and co-orbiting.
No coordinate-frame exists adapted to a GR-rotating ob-server with his co-spinning and co-orbiting auxiliary observers as shown in Sect. V 2.
Measured in the reference frame of an observer with his LONB-connections (ωbâ)î to his auxiliary observers, Eqs. (21): the acceleration difference of neighboring quasistatic freefalling particles, the geodesic deviation, divided by separations and spherically averaged gives the 3-divergence of gravitational accelerations, div g (GR) quasistat ≡ div E g = ∂îEî g .
• Measured in the reference frame of LONBs for a GR-noninertial observer with his LONB connections (ωbâ)ĉ, the spherical average of geodesic deviations, div E g , is different from the Ricci curvature R00. Our new exact result, explicitely simple: This equation gives the correction for incorrect conclusions on geodesic deviations in GR-books [6][7][8][9]. On the right-hand side of Eq. (1) every term depends on whether the observer (with his frame of LONBs) is GR-inertial or not. In contrast, the Riemann tensor and its two parts, the Ricci and Weyl tensors, do not depend on whether the chosen observer is GR-inertial or noninertial.
The sources of div E g contributed by the gravitational (ρ ε + 3p) grav have the opposite sign from the matter and electromagnetic sources and have a repulsive effect on the gravitational accelerationdifference of test particles measured by div E g .
In the frame of a GR-noninertial observer our gravito-Gauss law, Eq. (2), differs from Einstein's R00 equation, In Einstein's equations the sources do not include the densities of gravitational energy, momentum, and momentum-flow: the symmetry of the three sources (matter, electromagnetic, gravitational) is missing.
In our gravito-Gauss law of GR, div E g and ( E 2 g + 2 B 2 g ) depend on the acceleration and rotation (relative to GRinertial) of the chosen local observer with his reference frame of LONBs, Eqs. (21). -In crucial contrast, all terms in Einstein's equations are independent of the acceleration and rotation of the observer with his reference frame. -In the frame and on the worldline of a GRinertial observer, R00 = div E g and E 2 g = 0, B 2 g = 0, and our gravito-Gauss law becomes identical with Einstein's R 0 0 equation from Eqs. (1)- (3). But local frames of GR-inertial observers (freefalling and nonrotating) are never used in the solar system nor for our inhomogeneous universe: • new result: in our strongly inhomogeneous universe the gravito-Gauss law in the local reference frame of LONBs of a non-inertial observer predicts repulsive gravity from (ρ ε + 3p) grav = −( E 2 g + 2 B 2 g )/(4πG). This contributes to the measured accelerated expansion of our universe today. It is important to determine the magnitude of this effect.
We derive our three remaining exact, explicit, and simple gravito-Maxwell equations of GR in the local reference frame of LONBs for a GR-noninertial observer: the gravito-Ampère and Faraday laws and the law for div B g , The gravito-Ampère law of GR has no term ∂t E g .
• The sources of div E g and curl B g include terms which must be identified with the densities of gravitational energy ρ (grav) ε , gravitational energy current J (grav) ε , and the trace of the gravitational momentum-flow 3p grav measured on the worldline and in the local frame of LONBs for the observer, • Our new result: our gravitational (ρ ε + 3p) grav and J (grav) ε are local observables, but they depend on the local observer with his reference frame of LONBs. These observables are zero if measured by a GR-inertial observer on his worldline and with his reference frame.
In dramatic contrast Landau and Lifshitz [10] gave the gravitational energy density in arbitrary coordinates: in a (3+1)-split it has thousands of terms.
Using Eq. (5) we can rewrite the gravito-Gauss law, In Sect. VIII F we prove from Einstein's concepts 1911 (without using Einstein's equations of 1915) that the source of the gravito-Gauss law contributed by the gravitational fields is ( E 2 g + 2 B 2 g ), hence repulsive as in Eq. (5). For Einstein's equations it is irrelevant, whether the observer with his frame is GR-inertial or non-inertial. But for the gravito-Maxwell equations of GR with their gravitational energy, momentum, and momentum-flow it is crucial, whether the observer with his frame is GRinertial or not.
The accelerated expansion of our inhomogeneous universe is quantified theoretically by the divergence of gravito-electric accelerations div E g given by the gravito-Gauss law and integrated over our Hubble volume. The sources of div E g include the repulsive gravitational E 2 g . Very many papers on gravito-electromagnetism have published equations for linear perturbations, which is no substitute for our exact equations.
For exact treatments, a very different approach from ours has been taken in most (maybe all) publications on gravito-electromagnetism: they work with coordinate bases, e.g. Ref. [11]. But (a) coordinate bases cannot be adapted to a rotating observer with his adapted auxiliary observers (co-orbiting and co-spinning) as shown in our Sect. V 2. This fact and (b) working with Riemann metric functions g µν (x λ ) makes the (fully explicit) gravito-Gauss equation highly nonlinear.
The simplicity of our exact and fully explicit gravito-Gauss law, Eq. (2), is due to our local reference frame of LONBs for a GR-noninertial observer with his Ricci LONB connections (ωbâ)ĉ in his time-direction and his 3-directions, Eqs. (21). This gives our exact and simple gravito-Gauss law: the derivative term is linear, and the source contributed by gravitational fields is ( E 2 g + 2 B 2 g ), which has a repulsive effect.
A. Concepts and methods for deriving our gravito-Maxwell equations of GR in observer's reference frame of LONBs Our paper is based on: (1) the crucial role of the observer, GR-inertial versus non-inertial, with his local reference frame of neighboring LONBs, (2) Cartan's method with his field of LONBs, which we identify with a field of observers. -Our concepts and methods are different from those presented in GR-texts and most research papers. Our entirely new methods are: (A) our Ricci LONB-connections (ωbâ)ĉ in the timedirection and the 3-directions of a non-inertial observer, Eqs. (21), which are directly given by ( E g , B g ), (B) our Golden Rule, Eq. (38), for the Riemann tensor in LONB-components in terms of the Ricci LONB connections (ωbâ)ĉ in the local frame of a non-inertial primary observer.
In Sect. II A we show how the gravito-electric field E (GR) g is operationally defined and measured by a chosen observer on his worldline: E (GR) g is identical with the gravitational acceleration g GR of quasistatic particles relative to the observer measured on his worldline, For a GR-inertial observer E (GR) g = 0 measured by him on his worldline. "One can no more speak of absolute acceleration" (Einstein).
In Sect. II B we give our new operational definition of the gravito-magnetic field B g measured by an observer on his worldline: the gravitational precession Ω of gyroscopes relative to the observer-LONB gives B A GR-inertial observer measures B (GR) g = 0 on his worldline. -GR-inertial motion of an observer or particle is freefalling and non-rotating relative to spin-axes of gyroscopes on its worldline.
Sect. III gives our new result that the Ricci LONBconnection (ωâb)0 of an observer's LONB along his worldline is given directly by the gravitational fields F (g) ab measured by the observer on his worldline, Inside the bracket of (ωâb)ĉ are Lorentz transformation indices, outside the bracket is the displacement index.
In Sect. IV we treat "Geometry without metric": affine geometry, affine connections, and parallel transport. The Ricci LONB-connection along the worldline of an inertial observer with his LONB carried along is, inertial observer: (ωbâ)0 = 0.
The LONB-connections in radial directions from an inertial primary observer to his adapted auxiliary observers at infinitesimal δr are given by auxiliary observers at rest and nonrotated relative to the primary observer, frame of inertial observer: (ωbâ) Our frame of Ricci LONB-connections of an inertial observer is our new tool, fundamentally different from the local inertial frame in GR-texts, which is independent of the choice of an observer. -Our new frame of LONBs is crucial for our remarkable result of Eq. (32).
Sect. V gives the construction of our most crucial new tool, our local reference frame of LONBs for a non-inertial observer with his neighboring LONBs. The LONB-connections for displacements from a non-inertial observer in his time-direction and radial directions are: These LONB-connections, Eqs. (21), are the key to our exact, explicit, and simple gravito-Maxwell equations. Sect. V A gives our GR-equations of motion for particle momenta and spins in our reference frame of LONBs for a non-inertial observer: exact, explicit, simple, Sect. VI gives a brief review of Cartan's method for computing Riemann curvature from the deficit LONB Lorentz transformation after a round trip along an infinitesimal coordinate plaquette [µ, ν]. This gives Cartan's curvature 2-form equation, Cartan's LONB-method is unavoidable for computing Riemann curvature from an observer's frame of LONBconnections.
In Sect. VII we consider the geodesic deviation, the freefall acceleration-difference, spherically averaged, in our frame of LONBs for an inertial observer, Eq. (18): Cartan's wedge terms vanish, (ωbĉ ∧ωĉâ) µν = 0, and the exact curvature is linear in (ωbâ)ĉ. In our frame of a GR-inertial observer, for nonrelativistic physics and neglecting electromagnetic fields, the partial differential equation for R00 is identical with (and follows from) the Gauss law for Newton-gravity, In Sect. VII C we prove that Einstein's equations follow from the Gauss-Newton law div g Newton = −4πGρ mass and the GR-concept of timelike geodesics (= freefall) plus Special Relativity and the contracted Bianchi identity. -The gravito-Gauss law of GR in the frame of a GR-inertial observer is, inertial obs: div E g = −4πG(ρ ε + 3p) matter+EM , where (3p) ≡ trace of 3-momentum-flow tensor.
In Sect. VIII A we derive our Golden Rule, Eq. (38), for the Riemann tensor in LONB-components in terms of the Ricci LONB-connections (ωâb)ĉ in the local frame of LONBs of a non-inertial primary observer.
In Sects. VIII B-VIII E we use this Golden Rule to derive our exact gravito-Maxwell equations in our local frame of LONB-connections (ωâb)ĉ for a non-inertial observer. -In our gravito-Gauss law the repulsive gravitational source (ρ ε + 3p) grav = −( E 2 g + 2 B 2 g )/(4πG) contributes to the accelerated expansion of our inhomogeneous universe.
Sect. VIII F is based on Einstein's concepts of 1911 without Einstein's equations of 1915: we give an elementary re-derivation of our exact and new repulsive source term ( E 2 g + 2 B 2 g ) in our gravito-Gauss law for div E g .

II. GRAVITO-ELECTRIC AND GRAVITO-MAGNETIC FIELDS OF GR
A. GR-gravito-electric field identical with gravitional acceleration of quasistatic particles relative to observer We show how in GR the gravito-electric field E g is operationally defined and measured: E g is identical to the gravitational acceleration g of quasistatic particles. The gravitational acceleration is measured relative to the chosen observer on his worldline: measured by a GR-inertial observer on his worldline E g = 0.
We start from the definition of the classical Maxwellelectric field E for negligible gravity: (1) via the electric force F el on a quasistatic testparticle with infinitesimal charge, lim q→0 ( F el /q) ≡ E, (2) with Newton's second law, F = m inertial a, and the acceleration a measured relative to an inertial frame of Newton resp. Special Relativity.
Hence the electric field E is defined and measured by: (rel.to inertial Newton resp.SR) particle released from rest,only ED . (7) The test-particle must be quasistatic, otherwise the acceleration by the magnetic Lorentz force would also contribute, a Lorentz = (q/m) [ v× B ]. -The electric field E, measured with quasistatic particles, is used in the equations of Special Relativity for relativistic particles.
Starting from the definition of the Maxwell-electric field in Eq. (7), three steps are needed for the definition of the gravito-electric field of GR: 1) Replace electric charge q in the electric force by gravitational mass m gravit in the gravitational force on quasistatic particles. Write m inertial in Newton's 2nd law.
2) Use that gravitational acceleration is independent of the substance, tested by Eöt-Wash to ±10 −13 [12]: the ratio of (m gravit /m inert ) is a universal constant, and units are chosen to make it equal to 1. -Going from electrodynamics to gravitodynamics of quasistatic particles, (q/m) ⇒ (m gravit /m inertial ) = 1.
3) Recognize that the definition of acceleration by a ≡ (d v/dt) alone is empty: accelerations must be measured relative to a reference frame.
Newton's gravitational accelerations are defined and measured in Newton-inertial frames, they are absolute. Einstein's gravitational accelerations are defined and measured relative to the frame of the chosen local observer.
These three steps give the general operational definition of the gravito-electric field in GR, quasistat ≡ a (rel.to local obs) freefall,quasistat .
A free-falling observer measures on his worldline: Einstein's gravitational acceleration g is relative, it depends on the chosen local observer's acceleration relative to freefall. Einstein's "happiest thought of my life" in 1907 [1][2][3][4]: "The gravitational field has only a relative existence: ... for an observer falling freely from the roof of a house, there exists, at least in his immediate surroundings, no gravitational field". GR-textbooks (and many papers) do not give a general definition of gravitational acceleration and gravitoelectromagnetic fields: their gravitational acceleration and gravito-electromagnetic fields are in the frame of observers at rest relative to the Newton-inertial frame for the solar system. The gravitational accelerations in GR-textbooks are absolute in contradiction to Einstein's "happiest thought of my life": for free-falling observers, who measure zero gravitational acceleration on their worldlines, GR-textbooks are wrong. -Inhomogeneous cosmology has no unique field of preferred local frames/observers, therefore gravitational accelerations and gravito-electromagnetic fields are necessarily relative, they depend on the chosen space-time slicing with observers at rest on that slicing.
The components of the gravito-electric field Eî g ≡ gî quasistat are defined and measured in the chosen observer's Local Ortho-Normal Basis, LONBēâ in the tangent spaces T M P for points P on his worldline. Hats on indices denote a LONB, Misner, Thorne, and Wheeler, MTW [5]. Our sign convention for the Minkowski metric is from MTW [5], ηâb = diag (−1, +1, +1, +1). -In aî = (dvî/dt ) the time-interval is measured on the observer's chronometer.
Einstein's equations alone do not predict gravitational accelerations and the gravito-electromagnetic fields: the choice of a field of observers (GR-inertial vs. non-inertial) with their LONBs is needed to predict gravitational accelerations and gravito-electromagnetic fields.
To first order in δt, a particle released by the observer into freefall will acquire a first-order velocity change δvî, but the particle will still be on the observer's worldline, since its displacement will be of second order in δt. -No role for Riemann metrics in tangent spaces with Minkowski metric on the worldline of the observer.
Test-particles must be quasistatic, otherwise the gravito-Lorentz acceleration g GR ∝ ( v × B g ) contributes.
In GR the gravitational acceleration gî quasistat ≡ Eî g cannot be a 3-vector, because it vanishes in the reference frame of a freefalling observer. Hence we use the mathfrak-notation gî and the cal-notation Eî g . We show in Sect. III C that gî quasistat ≡ Eî g = −(ωî0)0, the observer's Ricci LONB-connection for a displacement along his worldline.
The gravito-electric field E g measured by the acceleration of freefalling quasistatic test particles relative to the observer is valid in relativistic equations of motion and field equations in the local frame of this observer.

Einstein-equivalence of classical fictitious accelerations with GR-gravitational accelerations without matter sources
Measuring accelerations needs a reference frame. In contexts where Newton gravity and classical mechanics hold, if the acceleration of a freefalling particle is measured relative to a Newton-non-inertial frame, there are fictitious forces and fictitious accelerations, Einstein's gravitational acceleration g GR is the sum of g Newton , which is generated by mass sources, plus the classical fictious acceleration of the Newton-inertial frame relative to the chosen GR-observer, which is not generated by mass sources, Eq. (9).
Since GR has no Newton-inertial frames, one must use Einstein's equivalence between classical fictitious accelerations of freefalling particles relative to a Newton-noninertial frame with contributions to g GR without matter sources : "Two relatively accelerated systems K and K ′ have an equal title as systems of reference. We are able to produce a gravitational field merely by changing the system of reference". Einstein, 1916 [4].
In Einstein's definition of g GR , the observer is arbitrary. But natural fields of "Fiducial Observers" are chosen as discussed in the next subsection.

Field of Fiducial Observers: FIDOs
Cartan's method uses a field of Local Ortho-Normal Bases, LONBs, which we identify with a field of observers each with his LONB in the tangent spaces T M P at points on his world-line.
There is great freedom in choosing Cartan's LONBs. But one chooses a field of "Fiducial Observers", FIDOs, a concept introduced by Thorne et al [13]. Examples: (1) For the solar system with planets as test-particles, a field of adapted FIDOs is singled out: every FIDO remains at fixed measured distance from the Sun and at fixed (θ, φ) and with LONBs non-spinning relative to the perihelia of outer planets. These GR-Fiducial Observers for the solar system are Einstein-noninertial, they are identical with Newton-inertial observers of the solar system, and they are at rest and nonspinning in Schwarzschild coordinates.
(2) For our inhomogeneous universe a field of cosmic FIDOs is singled out: relative to the CMB-sky with the CMB-dipole removed, (a) our FIDOs remain fixed in angular position and orientation of their (ēθ,ēφ), (b) FIDOs have an isotropic expansion rate given by the angular average of the measured expansion rate of galaxies in a bin of luminosity distance. These cosmic FIDOs have an unperturbed Hubble expansion. -These cosmic FIDOs are needed to operationally define and measure the gravitoelectric field Eî g of GR in our inhomogeneous universe.
B. GR-gravito-magnetic field Bg measured by gravitational precession Ω (relat.obs) gyro The gravito-magnetic field B g of GR is operationally defined and measured by the gravitational precession of gyroscopes relative to the observer's LONB with gyros comoving on observer's worldline. If measured by a GRinertial observer on his worldline B g = 0. -Our operational definition of B g is new.
Measuring B g is analogous to measuring the Maxwellmagnetic field B by the precession of magnetic dipoles relative to inertial frames of Newton or Special Relativity.
The spin and the magnetic dipole moment µ mag point in different directions in general, e.g. for the Earth. For GR instructive is a classical magnetic dipole with identical constituent-distributions of charges and masses, and the angular velocity of spin-precession, The transition from the definition of B in classical magnetism to our operational definition of the gravitomagnetic field B g of General Relativity needs four steps: (1) We replace the electromagnetic coupling to the electric charge q by the gravitational coupling to the nonrelativistic gravitational mass m grav .
(2) We use the result that the ratio of gravitational mass to inertial mass is universal within experimental errors ±10 −13 , and units are chosen such that this ratio is set to 1. Hence, going from electrodynamics to gravitodynamics, (q/m) ⇒ (m grav /m inertial ) = 1, (3) The gyro-precession must be measured relative to a reference frame. The only reference frame available in GR is the reference frame of the chosen local observer with his LONB in the tangent spaces on his world line.
(4) Our normalization of B g is fixed by (E transforming under a change of the observer's acceleration and rotation as components of the gravitational field F (g) ab , Sect. III C. -Our normalization of B g differs from two unjustified normalizations in the literature: (a) requiring the GR-field equation for the gravito-magnetic field to have the same prefactor for J ε as in the classical Ampère law for J q , (b) requiring the equation of motion for particle-motion to have the same prefactor for B (GR) g as in the classical Lorentzacceleration a Lorentz = q v × B.
Our four steps give the operational definition of the gravito-magnetic field of GR by the gyro-precession an-gular velocity of two non-aligned gyros (comoving on observer's worldline) relative to the observer-LONB, precess.comoving gyros .
The Hodge-duals B (g) iĵ and Ωîĵ are given via the totally antisymmetric Levi-Civita 3-tensor εîĵk with ε123 ≡ +1, Gravito-magnetism is simpler than classical magnetism, where macroscopic bodies have non-identical constituent-motions of charges versus masses.
In gravito-magnetism the constituent-motions of m gravit and m inertial are identical.
A single observer cannot use the gravito-Lorentz acceleration a Lorentz ∝ v × B g to measure the gravito-magnetic field on his worldline. Measuring the gravito-Lorentz acceleration needs test-particles with finite velocity, and acceleration measurements need a second velocity measurement in a tangent space T M (P 2 ) away from the primary observer's worldline. This needs an appropriate choice of auxiliary observers with their LONBs, a task solved in our crucial Sect. V (where B g is needed as an input).
The gravitomagnetic B g of GR has been measured by Foucault in Paris in 1853 with gyroscopes precessing relative to his eî = (East, North, vertical). -The satelliteobservatory Gravity Probe B has measured B g by gyroscope precession relative to its local LONBē0 =ū GPB andēî determined by the lines of sight to two quasars.

C. Inertial motion in General Relativity
One cannot remove the pillar of Newton's inertial motion from a theory of gravity without putting in its place the new pillar of inertial motion in General Relativity. Einstein's revolutionary concept of inertial motion of point-particles is free-falling motion.
• Operational definition of GR-inertial motion: 1) free-falling motion, 2) non-rotating motion relative to spin-axes of two comoving gyroscopes with spin-axes not parallel.
Many GR-texts do not give the second part of the definition of GR-inertial motion: non-rotating.

III. RICCI LONB-CONNECTION (ωbâ)0 ALONG WORLDLINE OF OBSERVER
Cartan uses a field of Local Ortho-Normal Bases, LONBsēâ. We identify Cartan's LONB-field with a field of observers. -We derive our new result that ( E g , B g ) measured by an observer on his worldline equals his Ricci LONB-connection (ωbâ)0 along his worldline, Eq. (15).
Relative to GR-inertial motion, the derivative of the observer's LONB along his worldline is equal to the covariant derivative as discussed in Sect. IV A, Eq. (17), The covariant derivative (Dēâ/Dt) of the observer's LONB along his worldline is given by the Ricci LONBconnection of the observer's LONB along his worldline, Combining the last two equations to eliminate (Dēâ/Dt) obs.LONB gives, The observer's acceleration and rotation (Lorentz transformations) relative to GR-inertial motion are equal to the Ricci connection (ωbâ)0 of the observer's LONB along his worldline.
(Dū obs /Dt obs )î along the observer's worldline is equal to the acceleration of the observer relative to GR-inertial motion as discussed in Sect. IV A, ( duî obs dt obs ) (rel.to GR.inertial) = (ωî0)0.
In Sect. II A we have presented the opposite point of view: the acceleration of a freefalling particle relative to the observer with his LONB gives the operational definition of the gravito-electric field E g , Comparing the two points of view of the last two equations (acceleration of what? relative to what?) gives the relation between the gravito-electric field Eî g measured by the observer and the Ricci LONB-connection (ωî0)0 of the observer-LONB along his worldline, our new result, The LONB-connection (ωî0)0 = −(gî) (GR) quasistat = −Eî g is directly measurable and depends on the observer.
For gyroscopes carried by the chosen observer, the gyro-spin precession relative to the observer-LONB per measured observer-time is, The covariant derivative of a gyro-spin is zero, The last two equations give the Ricci connection, gyro precession . Our operational definition of the gravito-magnetic field of GR has been given in Eq. (11), precess.comov.gyros .
Comparing the last two equations we obtain the crucial relation between the gravito-magnetic field B (g) jî (measured by the observer on his worldline) and the Ricci LONB-connection (ωĵî)0 of the observer-LONB per dis-placementē0 =ū obs , our new result, (14) C. Gravitational fields F bâ , which is equal to minus the Ricci connection of the observer's LONB for displacements along his worldline, The electromagnetic fields are related to the field tensor Fbâ in an identical way, Ek = Fk0, Bkî = Fkî. -Eqs. (15) are new and fix our normalization of B g .
A single observer with his LONB in the Minkowski tangent spaces T M P for P on his worldline measures the acceleration of freefalling particles (released by him) and the precession of gyroscopes (comoving on his worldline): he measures his gravitational fields ( E g , B g ) on his worldline and his Ricci LONB connection (ωbâ)0 along his worldline. The gravitational fields ( E g , B g ) depend on the observer: they are zero when measured by a GR-inertial observer on his worldline.-In Minkowski tangent spaces on the worldline of an observer: no possible role for Riemann metrics for particular spacetime geometries.

IV. GEOMETRY WITHOUT RIEMANN METRICS
Affine connection with parallel transport in GR needs affine geometry but not Riemann metrics. -Different space-time geometries have different Riemann metrics.
A. Affine connection and parallel transport along worldlines of GR-inertial observers GR-inertial motion is identical with GR-parallel transport and GR-affine connection in a timelike direction.
The LONB-connection in a timelike direction is given by GR-inertial motion of an observer: 1) free-fall gives a GR-straight worldline (geodesic) and GR-parallel transport ofē0 ≡ū obs , 2) non-rotating, parallel transport ofē (obs) i , is given experimentally by spin-axes of two comoving gyros with spin-axes not parallel, on worldline of inertial observer: (ωbâ) The GR-covariant derivative in a timelike directionē0 is the derivative relative to the LONB of a GR-inertial observer withū obs =ē0, No Riemann metrics are needed for the GR-affine connection of vectors and tensors in LONB components.

B. Local reference frame of LONBs for GR-inertial observer
Our new tool is the local reference frame of LONBs for a primary inertial observer along his worldline, which includes a field of neighboring LONBs (of adapted auxiliary observers) at infinitesimal separations, Eq. (18) below. This new frame makes our exact and explicit equations remarkably simple. -Our new frame is entirely different from the "local inertial coordinate system" in the literature, which does not refer to a primary observer.

Affine connection and parallel transport for radial displacements from primary inertial observer
For an inertial primary observer withū on his worldline, we define his adapted auxiliary observers for 1st-order-infinitesimal radial displacements δr by two requirements: (1) adapted auxiliary observers are at relative rest with the primary inertial observer:ē (aux.obs) 0 andē (prim.obs) 0 are related by parallel transport under infinitesimal radial displacements. The first-order radial distance and 3-direction of an auxiliary observer relative toē (prim.obs) i are independent of time. Hence the affine connection for radial displacements from a primary inertial observer to his adapted auxiliary observers at relative rest is, (2) The 3-LONBsēî of auxiliary observers and of the primary observer must be parallel under infinitesimal radial displacements, i.e. relatively non-rotated, The last two conditions for radial displacements from the worldline of an inertial primary observer plus the condition for a displacement along his worldline gives, inertial primary observer: (ωbâ) The four covariant derivatives ∇â in the LONBdirectionsēâ of a primary GR-inertial observer are identical with the ordinary derivatives ∂â in the the local frame of LONBs of the GR-inertial observer, No role for a Riemann metric.
• Our spatially local frame along the worldline of an inertial observer with the Ricci LONB connections in his time and his spatial directions (ωbâ)ĉ of Eq. (18) is new and fundamentally different from the "local inertial frame" of GR-texts, which is independent of a primary observer.
We shall show that in our reference frame of LONBs for a GR-inertial observer, Eq. (18), for nonrelativistic physics and without electromagnetism, the partial differential equation for R00 is identical with the Gauss law for Newton-gravity, Sect. VII C, Note: along the worldline of a GR-inertial primary observer, his adapted auxiliary observers are not GRinertial because of tidal forces and relative torques.

V. FRAME OF NON-INERTIAL OBSERVER: LONBS OF NEIGHBORING OBSERVERS
The crucial concept and new tool for proving our exact, explicit, and simple gravito-Maxwell equations of GR in Sect. VIII is the local frame of LONBs for a noninertial primary observer with his neighboring LONBs, which can be interpreted as LONBs of his adapted auxiliary observers, Eqs. (21). This frame of LONBs needs: (A) the Ricci LONB-connection in the time-direction of the primary observer, (ωbâ)0, given in Eqs. (13,14), (B) the LONB-connections for radial displacements from the primary observer to his adapted auxiliary observers at infinitesimal δr, (ωbâ) (r=0) i , derived now.

Accelerated-nonspinning primary observer
Our auxiliary observers, at infinitesimal radial displacements from the primary observer and adapted to the accelerated-nonspinning primary observer satisfy two conditions: (a) Adapted auxiliary observers are at rest relative to the primary observer compared by parallel transport in an infinitesimal radial displacement, i.e. theirē0 are radially parallel, (b) Adapted auxiliary observer have spatial LONBs radially parallel to the primary observer'sē Hence the radial LONB-connections from the primary observer vanish, accelerated-nonspinning primary observer: (20)

Spinning-freefalling primary observer
For a spinning and freefalling primary observer, our adapted auxiliary observers at infinitesimal δr are in circular co-orbiting motion around and co-spinning with the primary spinning observer. The auxiliary worldlines are circular spirals winding around the primary worldline.
To first order in δr, the auxiliary observers have nonrelativistic velocities relative to the primary observer, Spatial LONB-vectorsēî of auxiliary observers in direction of motion have corresponding infinitesimal Lorentz-boosts in the time-direction. They generate spacelike spiral worldlines winding around the primary observer's worldline. Therefore: • no fully adapted coordinates exist for a rotating primary observer with his auxiliary observers (coorbiting and co-spinning).
There are no rotations of 3-LONBsē (aux.obs) i for our adapted auxiliary observers relative to the 3-LONBs of the primary observer, (ωîĵ)k = 0.
The results above, first for an accelerated-nonrotating primary observer, afterwards for a nonacceleratedrotating primary observer, are additive.
Our crucial result: in the spatially local reference frame of LONBs for a noninertial primary observer (with his adapted auxiliary observers), all 24 Ricci LONBconnections for displacements along his worldline and in his radial directions are either given by E g and B g or zero: • frame of LONBs for noninertial observer: displacement along worldline of primary observer, These LONB-connections (ωâb)ĉ in the frame of a noninertial primary observer are the key for proving our exact, explicit, and simple equations of motion, Eqs. (22), and our exact, explicit, and simple gravito-Maxwell equations, Sects. VIII B-VIII E.

A. Equations of motion in local frame of GR-noninertial primary observer
The equation of motion for the momentump of a testparticle in gravitational and electromagnetic fields is, In a general frame the equation of motion has 24 Ricci connection components (ωâb)ĉ resp. 40 Christoffelconnection components Γ α βγ . But in our frame of LONBs for a GR-noninertial primary observer and his auxiliary observers, the Ricci LONB-connections (ωbâ)ĉ in his time direction and his spatial directions are given by ( E g , B g ) in the very simple Eqs. (21). This gives the exact GR-equations of motion in our local reference frame of LONBs for a non-inertial primary observer and for particles starting on the primary worldline in arbitrarily strong gravitational and electromagnetic fields: exact, explicit, and strikingly simple, exact GR for relativistic test-particles where v ≡ (d x/dt ) is the 3-velocity measured by the primary observer in his Minkowski tangent space on his worldline. A particle's total relativistic energy measured by the observer is ε, the relativistic 3-momentum is p = ε v, for photons ε = ω, and (d/dt ) prim.obs refers to the time-difference measured by the primary observer on his chronometer. In contrast to Eq. (22), the explicit exact equations of motion in coordinate bases are complicated.
Eqs. (22) are also the equations of motion for Special Relativity in the local frame of a not-freefalling observer who is rotating relative spin-axes of gyros.
For our exact GR-equations of motion, an initial measurement of a particle momentump is made in the tangent space T M P with P on the worldline of the primary observer. A second measurement is made by a neighbouring observer (adapted to the primary observer) at an infinitesimal time δt later.
The gravitational fields ( E g , B g ) are not 3-vectors. The gravitational fields F For spins, the exact equation of motion for GR-inertial test-particles (i.e. influenced only by gravitational fields) in the local frame of a noninertial primary observer is, where we have used (S ·p) = 0, hence S0 = ( S · v).

Fictitious Coriolis acceleration GR-equivalent with contribution to gravito-magnetic acceleration
Since GR has no Newton-inertial frames, the classical fictitious Coriolis acceleration (in a frame rotating relative to Newton-inertial), : the gravito-magnetic field B g of GR is generated in part by energy currents and in part as a fictitious gravito-magnetic field.

B. Spatially Local Coordinates of GR-noninertial observer
For an inertial or noninertial primary observer with his LONBs in the tangent spaces on his worldline, we choose our slicing of spacetime by hypersurfaces Σ t for first-order radial separations from the primary observer: Σ t starts Minkowski-orthogonal to the primary worldline, and our Σ t is generated by radial 4-geodesics.
Σ t has as time-coordinate t the time measured on the wristwatch of the primary observer.
To first order in radial distance r from the primary worldline, the intrinsic geometry of Σ t is Euclidean, and we choose Cartesian coordinates x i oriented in the directions of the primary observer's LONBsēî.
The lines of fixed 3-coordinates x i are the worldlines of auxiliary observers adapted to the primary observer.
The lapse function α between slices Σ t is defined as the elapsed measured time δτ per increase in coordinatetime δt along the normaln on the hypersurface Σ t , The shift β is defined as shift of the time coordinate line (= worldline of auxiliary observer) from the normal on Σ t per coordinate time t, sign convention of MTW [5], )n (Σt) .

Local coordinates of GR-inertial observer
The spatially local coordinates for an inertial primary observer to first order in δr have, In the local frame of an inertial primary observer with his adapted auxiliary observers and with his adapted spatially local coordinates, the crucial result for computing space-time curvature is that the primary and auxiliary observers have, coordinate basesē a = LONBsēâ (24) On the entire worldline of a primary inertial observer, the metric is η µν , and the first derivatives of the metric in the observer's time-and 3-directions vanish, µν,λ = 0.
These coordinates are new and fundamentally different from the "local inertial coordinates" of GR-textbooks.

Local coordinates of GR-noninertial observer
We first consider a nonspinning primary observer accelerated in the positive x-direction, E x g = − (a x obs ) rel.to ff . In an infinitesimal time, the accelerated observer's time axis at (t 0 + δt) is tilted infinitesimally (relative to the observer's time axis at t 0 ) towards the direction of acceleration. The accelerated observer's x-axis, a line of constant time, is tilted towards the inertial positive t axis by the same positive amount (−E x g δt), For a freefalling primary observer, rotating or nonrotating, the lapse α is zero. Hence the last equation is valid for any primary observer. Now consider a spinning-freefalling primary observer. His adapted auxiliary observers are at time-independent separations, hence at fixed values of our r-coordinate. His auxiliary observers are co-orbiting with the 3-LONBs of the spinning primary observer, hence at fixed (θ, φ). In our coordinates, the time-coordinate lines coincide with the worldlines of our adapted auxiliary observers.
The shift vector β to first order in δ r P is equal to the nonrelativistic velocity of the auxiliary observer v aux.obs relative to the normals on Σ t and equal to his nonrelativistic orbital velocity relative to the primary observer, For our spatially local coordinates of a primary observer, the shift is zero for a nonrotating primary observer, freefalling or not freefalling. Hence the last equation is valid for any primary observer.
On the entire worldline of a chosen non-inertial observer (r = 0), the metric g (r=0) µν and its first derivatives in the observer's time direction are, To first order in δr, the intrinsic geometry of Σ t is Euclidean, and we have chosen Cartesian 3-coordinates, prim.obs.acceleration , an infinitesimal Lorentz boost of the primary observer relative to GRinertial tilts his local time-axis and spatial axes relative to GR-inertial. The resulting lapse and shift give,

VI. INTRINSIC CURVATURE WITH CARTAN'S LONB-METHOD
Cartan's LONB-method is seldom used in research papers and not taught in most graduate GR-courses, but we need it. -To introduce Cartan's LONB-method for computing curvature we start in a 2-space. A closed infinitesimal curve C, positively oriented, has an integral of LONB-rotation angles (dα/dx µ ) = ω µ (relative to parallel transport), where (δα) C is the deficit LONB-rotation angle. The Gauss curvature R Gauss is (δα) C divided by the measured area A C inside the boundary C for C → 0, A. Cartan's curvature equation from roundtrip by parallel transport In curved (3+1) space, the Riemann curvature tensor at P is defined by the infinitesimal deficit LONB Lorentz transformation (δ Lâb) C after a round trip by parallel transport along an infinitesimal closed curve C around P .
In parallel transport of a vector, its LONB-components transform, where (ωâb) ν is Cartan's LONB-connection with the displacement coordinate-component ν. We expand both factors around an initial point P 0 to first order in δx, In parallel transport, the expansion of the vector field components Vb gives, We insert the last two expansions into the first equation and integrate along the closed curve C.
The product of constant terms gives dx µ = 0. The products which are linear in (x − x 0 ) give the deficit Lorentz transformation δ C Lâb after an infinitesimal roundtrip by parallel transport around C, The Riemann curvature 2-form (Râb) µν is defined via the infinitesimal deficit Lorentz transformation δ C Lâb after the roundtrip around an infinitesimal C, The infinitesimal roundtrip integration in coordinate space [ µ, ν ] calls for the 2-form components (...) µν .
The antisymmetric derivative of a 1-formσ, called exterior derivative and denoted by dσ, produces a 2-form, The antisymmetric product of two 1-forms, called exterior product and wedge product, produces a 2-form, With the infinitesimal deficit Lorentz transformation after an infinitesimal roundtrip by parallel transport δ C Lâb and with the operational definition of the Riemann curvature 2-form (Râb) µν in Eq. (28) follows Cartan's Riemann curvature 2-form equation, Cartan's Riemann curvature 2-form (Râb) µν with LONBindices written in lower positions has: (1) the antisymmetric LONB-index pair [â,b ] for the infinitesimal deficit Lorentz transformation δ C Lâb after a roundtrip by parallel transport, (2) the antisymmetric coordinate-derivative pair [ µ, ν ] of the closed displacement plaquette for the round trip.
The Riemann tensor and its two parts, the Ricci and Weyl tensors, do not depend on whether the chosen observer is GR-inertial or non-inertial. But Cartan's LONB-connections (ωâb) µ depend on the observer's acceleration/rotation relative to GR-inertial.

VII. GRAVITO-GAUSS LAW OF GR IN FRAME OF INERTIAL OBSERVER
A. Freefall acceleration difference in local frame of inertial observer Our crucial new tool, the frame of an inertial primary observer, includes his adapted auxiliary observers with their LONBs, Eq. (18). We now show that in this frame (R01)01 = ∂1E We compute the geodesic deviation, the accelerationdifference of freefalling particles infinitesimally separated, initially at rest in the frame of the chosen inertial primary observer.
To compute the curvature component (R01) 01 , one needs the space-time coordinate-plaquette [ ∂ t , ∂ 1 ]. -In the local coordinates adapted to a GR-inertial primary observer, and going along the four sides of the space-time plaquette [ ∂ t , ∂ 1 ] starting on his worldline, the LONBsēâ and his coordinate basesē a ≡ ∂ a are equal, Eq. (24), in local frame of GR-inertial primary observer: eâ =ē a .
To compute curvature, one needs LONB-connections (ωbâ) c along the four sides of the coordinate plaquette. Along the worldline of the primary GR-inertial observer, the LONB-connection vanishes, (ωbâ) For the two spatial displacements starting on the primary worldline, the LONB-connections also vanish in the GRinertial frame, (ωbâ) (r=0) i = 0. Along the 4th side of the coordinate plaquette, along the worldline of the adapted auxiliary observer (who is at fixed radial distance from the primary observer and GR-noninertial because of tidal forces/torques), (ωbâ) (aux.obs) 0 = 0. Conclusion: • in Cartan's curvature equation on the worldline of a GR-inertial primary observer in his local frame with his adapted auxiliary observers and coordinates: the bilinear terms, wedge terms, vanish, This exact equation is linear in E g = g (GR) quasistat . In the frame of a GR-inertial observer, the freefall acceleration difference ∂1g (quasistat) 1 is given by the Riemann tensor.
• But in the reference frame of a GR-noninertial observer, the measured freefall acceleration-difference is not given by the Riemann tensor as demonstrated in Sects. VIII B and VIII F.
B. R00 = div Eg in frame of inertial observer In our local frame of LONBs for a GR-inertial observer the exact explicit differential expression for R 0 0 is extremely simple: R 0 0 is identical with ∂îEî = div E g . In the reference frame of a GR-inertial primary observer, the freefall acceleration-difference (= geodesic deviation) of a quasistatic particle and the primary inertial observer (divided by their infinitesimal separation) is given by Eq. (30). -The spherical average of the freefallacceleration difference gives ∂îgî quasistat = ∂îEî g = div E g .
In the local reference frame of a GR-inertial primary observer, div E g and R00 are given by the identical, exact, and linear differential expression in E g ≡ g (quasistat) GR , reference frame of GR-inertial primary observer: This simple exact equation depends on our new local reference frame of a GR-inertial observer, Eq. (18).
Later we shall prove the crucial fact: in the reference frame of a GR-non-inertial observer, R00 = div E (GR) g .

C. From Newton-Gauss law to Einstein equations
Einstein's R00 equation for nonrelativistic matter and neglecting electromagnetic fields is R00 = −4πG ρ mass . Our important result: • In our reference frame of LONBs for a GR-inertial observer, for nonrelativistic physics and neglecting electromagnetic fields, the partial differential equation for R00 is identical with the Gauss law for Newton-gravity, Nonrelativistically R00 = −4πGρ m holds in the frame of any observer, GR-inertial and noninertial.
• From the Gauss-Newton law and the concept of timelike GR-geodesics (freefalling) follows Einstein's R00 equation for nonrelativistic matter, Eq. (32). From this combined with Special Relativity and the contracted Bianchi identity follow (as explained in GR-texts) the full Einstein equations, where Tâb = energy-momentum tensor of matter and electromagnetic fields, and T is its trace.
Einstein's R00 equation for relativistic sources follows, where 3p ≡ trace of 3-momentum-flow tensorpîĵ. -The 3-momentum-flow tensorpîĵ must be distinguished from the pressure tensor pîĵ, which is measured in the instantaneous rest-frame of a fluid element. -Maxwell's stress tensor has the opposite sign from the 3-momentum-flow tensorpîĵ of electromagnetism. -In Einstein's equation for R00 the source from electromagnetic fields is, To obtain the Riemann tensor in LONB-components (Râb)ĉd in our frame of LONBs for a non-inertial primary observer, we start with Cartan's curvature equation, where the pair of displacement subscripts γδ denotes the antisymmetric pair (γδ − δγ) in the coordinate basis, But we need all four indices of the same type: (1) for contracting the Riemann tensor to obtain the Ricci tensor, (2) for the 1st Bianchi identity. -We want directly measurable components, hence LONB components.
We work in our frame of a non-inertial observer with the LONB-connections in his time-direction and his 3directions, Eqs. (21).
For the derivative ∂ γ acting on the first factor, the second factor is evaluated on the primary worldline, (eŝ) The derivative ∂ γ acting on the first factor at P on the primary worldline is equal to the derivative ∂ĉ, This gives the result for the LONB-derivative ∂ĉ acting on the Ricci LONB-connection (ωâb)d, For the derivative ∂ γ at P on the primary worldline acting on the second factor, ∂ (P ) γ (eŝ) δ , and after antisymmetrization [γ ⇔ δ], we use Cartan's implicit equation for the LONB-connectionωŝr, dẽŝ = −ωŝr ∧ẽr.
With displacement indices in the coordinate-basis, On the primary worldline, (er) δ = δrd and ∂ γ = ∂ĉ, This gives the result for the LONB-derivative ∂ĉ at P on the primary worldline acting on (eŝ) δ , The three terms for (Râb)ĉd from Eqs. (35-37) added give our Golden Rule for the Riemann tensor in LONBcomponents in our frame of a non-inertial primary observer with the Ricci LONB-connections (ωbâ)ĉ,
First consider the wedge term in Cartan's curvature equation which produces (R00) (ω∧ω) , Eq. (35), where the Ricci LONB-connections (ωâb)ĉ for our frame of a primary observer are from Eq. (21). Next consider the term in the Golden Rule where the LONB derivative ∂ĉ acts on the Ricci LONB connection, Eq. (36), and use the Ricci LONB connections (ωâb)d in the frame of the primary observer, Eq. (21), Finally consider the term in the Golden Rule where the LONB-derivative ∂ (P ) c acts on (ẽŝ) δ , Eq. (37), The sum of the three terms gives the exact explicit differential expression for R00 in our reference frame of any primary observer with his LONB-connections, where div E g = ∂îEî g in our frame of LONBs. The simplicity of Eq. (39) depends on our reference frame of an observer with his LONB connections of Eqs. (21). But in general coordinates, the explicit expression for R 0 0 is very complicated . -The structure of the right-hand side in Eq. (39) (without numerical prefactors and signs) follows from the structure of our Golden Rule and from J P : the derivative term must be linear in div E g , the bilinear terms must be proportional to E 2 g and to B 2 g . The acceleration-difference of neighbouring freefalling particles, spherically averaged, is given by div E g . Our exact Eq. (39) shows the crucial difference between R00 and the geodesic deviations given by div E g . The geodesic deviations depend on whether the observer (with his frame of LONBs) is accelerated and/or rotating. -Eq. (39) contradicts conclusions in GR-texts [6][7][8][9] that "geodesic deviations are given by the Riemann tensor".
We combine the last two equations to eliminate R00, gravito-Gauss law of General Relativity in LONB-frame of noninertial observer div E g = −4πG(ρ ε + 3p) matter div E g and ( E 2 g + 2 B 2 g ) depend on the observer (inertial versus non-inertial) and his frame of LONBs. -Our gravito-Gauss law of GR is exact, totally explicit, simple, and new. The simplicity of our exact gravito-Gauss law is due to our reference frame of LONBs for a GRnoninertial observer, Eq. (21), which is entirely new.
The sources of the GR-gravitoelectric field E g in the gravito-Gauss law are contributed by all sources, including the gravitational sources, In contrast, the sources in Einstein's R00 equation do not include the gravitational sources, The acceleration-difference of neighbouring freefalling particles, spherically averaged, is given by div g GR ≡ div E g and determined by the gravito-Gauss law of GR. -But R00 does not determine the acceleration-difference of freefalling particles in the frame of a noninertial observer.
• The sources of div E g include the gravitational bilinears ( E 2 g + 2 B 2 g ). This is a dark source for repulsive gravity in contrast to the gravitationally attractive matter and electromagnetic sources.
For the new terms E 2 g and 2 B 2 g we give independent and elementary derivations in Sect. VIII F, which are based only on Einstein's concepts of 1911, but not on Einstein's equations of 1915.
In the solar system, Fiducial Observers at fixed coordinates (r, θ, φ) measure E g = 0, hence they measure repulsive gravity in div E g in the halo of the Sun.
In our inhomogeneous-anisotropic universe, the Fiducial Observers adapted to any of the standard coordinates are not GR-inertial, and in our late universe the gravitomagnetic field is negligible in the gravito-Gauss equation: • repulsive gravity from (ρ ε + 3p) grav ≈ − E 2 g /(4πG) contributes to the observed accelerated expansion of our universe today. An important task is to determine the magnitude of this effect.
In the gravito-Maxwell laws of GR, the form of the bilinear expressions in ( E g , B g ) follows from covariance under J P , i.e. under rotations and space-reflections: the sources of div E g have J P = 0 + .
In the frame of a GR-inertial primary observer, R00 = div E g . But in the frame of a GR-noninertial primary observer, these two quantities are unequal: for R00 it is irrelevant, whether the primary observer is inertial or non-inertial. But for div E g , all depends on whether the primary observer is inertial versus non-inertial. 1. Gravitational (ρε + 3p) depends on observer's local frame Sources of div E g in the gravito-Gauss law are (ρ ε + 3p) of (1) matter, (2) electromagnetic fields, (ρ ε + 3p) EM = ( E 2 + B 2 )/(4π), plus (3) a term which must be identified with (ρ ε + 3p) of gravitational fields, The gravitational (ρ ε + 3p) P depends on the observer (noninertial vs inertial) with worldline through P and his frame with connections (ωbâ) (P ) c . In the frame of a inertial observer with worldline through P , (ρ ε + 3p) (P ) grav = 0. Landau and Lifshitz derived the density of gravitational energy, momentum, and momentum flow [10]. Explicit expressions are also given in MTW [5] and Weinberg [6]. These authors worked with general coordinates, not adapted to a chosen local observer with his local frame. For a (3+1)-split their results give thousands of terms instead of our simple results, Eqs. (42, 45).
2. Repulsive gravity: irrelevant in FLRW universe, but inescapable in strongly perturbed universe In a Friedmann-Lemaitre-Robertson-Walker universe natural observers are comoving and nonrotating relative to matter and electromagnetic energy-momentum: natural observers are GR-inertial, on their worldlines E = 0, B = 0, and there is no repulsive gravity.
But for the accelerated expansion of our strongly perturbed late universe measured on our past light-cone, the natural observers, FIDOs, are everywhere on the light rays between the supernova explosion at redshift z < 2 and us at z = 0 as defined in Sect. II A 2. Almost all of these FIDOs are GR-noninertial and measure contributions from repulsive gravity.

C. Gravito-Ampère law of General Relativity
For a primary observer, his local frame of LONBs gives his Ricci LONB-connections (ωâb)ĉ, Eqs. (21), in his time direction and his 3-directions. The Ricci LONBconnections determine Rî0, which is given by the three terms in the Golden Rule, Eqs. (35-37), The sum of these three terms give Rî0 expressed by the Ricci LONB-connections of the primary observer, The simplicity of Eq. (43) depends on our frame of a noninertial observer with its LONB connections, Eqs. (21). The Ricci Rî0 equation of General Relativity states, Eliminating Rî0 in the last two equations for Rî0 gives, gravito-Ampère law of General Relativity in local frame of GR-noninertial observer The gravito-Ampère law of GR has no term ∂t E g in contrast to ∂ t E in the Maxwell-Ampère law. If the gravito-Ampère law had a term ∂t E g , gravitational vector waves, which do not exist, would be predicted. The form of bilinear expressions in ( E g , B g ) follows from J P covariance: sources of curl B g have J P = 1 − .

Gravitational energy current density Jε
depends on observer's local frame The sources of curl B g in the gravito-Ampère law are the energy current density of (1) matter, (2) electromagnetic field (Poynting vector), plus (3) a term which must be identified with the energy current density of the gravitational field (gravito-Poynting vector), The gravito-Ampère law can be written, The gravitational energy current density ( J ε ) (P ) grav and (curl B g ) P depend on the observer with worldline through P withū If two of the three cyclically permuted indices (b,ĉ,d) are equal, the first Bianchi identity is empty.
The first Bianchi identity for two indices0 is, cyclic lower ind.
The derivative terms (∂ω), Eq. (36), give, The wedge term (ω ∧ω) in the Golden Rule, Eq. (35), with the Ricci LONB-connections of the primary observer, (ωâb) The form of bilinear expressions in ( E g , B g ) follows from covariance under J P : curl E g has J P = 1 + , therefore it cannot have a source bilinear in ( E g , B g ).
E. Divergence of gravito-magnetic field Bg in GR The first Bianchi identity for one index0 in the nonpermuted position, (R0î)ĵk, states that the sum of the cyclic permutations of (îĵk) gives zero.
The term (∂ẽŝ) in the Golden Rule, Eq. (37), in our local frame of the observer, Eq. (21), gives, The sum of the contributions gives, div B g law of General Relativity div B g + E g · B g = 0. (48) The form of bilinears in ( E g , B g ) follows from covariance under J P : the source of div B g has J P = 0 − . The Bianchi identity with one index0 in the cyclic permutation of the three lower indices is empty.
The gravito-Maxwell equations of GR for a chosen observer do not involve Einstein's Rîĵ equations for the intrinsic curvature of his 3-space. The acceleration of a freefalling particle is undefined unless one has a reference frame: Newton-inertial frames for Newton-gravity, non-inertial frames in classical mechanics, SR-inertial frames for Special Relativity, the local frame of LONBs of a chosen observer for GR.
In this Subsection we show directly our crucial result: the acceleration-differences of neighboring freefalling particles spherically averaged measured in the local frame of LONBs of a GR-noninertial observer is not given by the Ricci tensor, but this acceleration-difference involves ( E 2 g + 2 B 2 g ), which causes repulsive gravity. -Here we shall use only Einstein's concepts of 1911 without using Einstein's equations of 1915.

Fictitious centrifugal acceleration in rotating frame
gives repulsive Einstein-gravity from B 2 g We give an elementary argument which shows that the acceleration-difference of freefalling particles (measured in the frame of LONBs of a GR-rotating observer and spherically averaged) is not described by the Ricci R00.
If at a point P the sources of matter and electromagnetic fields are zero, (R00) (P ) = 0. -But in a rotating frame the centrifugal acceleration of classical mechanics gives relative accelerations of quasistatic freefalling particles which can be arbitrarily large, a centrifugal = Ω × ( r × Ω), div a centrifugal = 2 Ω 2 .
Ω is the angular velocity of the frame-rotation relative to Newton-inertial. -The classical Gauss law for accelerations of freefalling quasistatic particles in a rotating frame is, div a freefall = − 4πG ρ mass + 2 Ω 2 .
Without Newton-inertial frames, one cannot separate fictitious accelerations from classical gravitational accelerations. Einstein's gravitational acceleration g GR is the sum of g Newton , which is generated by mass sources, and the classical fictious acceleration of the Newton-inertial frame relative to the chosen GR-observer, which is not generated by mass sources, Eq. (9). -Einstein stated the equivalence of classical fictitious accelerations with additional gravitational accelerations in his approach.
In GR, the acceleration of quasistatic freefalling particles relative to the LONB of the chosen observer (FIDO) defines E (GR) g , Eq. (8). The precession angular velocity for spin-axes of comoving gyros relative to the LONB of the FIDO defines B (GR) g , Eq. (11). This completes the transcription of the Gauss law of Newton gravity to acceleration-differences of freefalling particles in the frame of a rotating-freefalling FIDO with Einstein's concepts of 1911, div E g = − 4πG ρ mass + 2 B 2 g .
Einstein: "We are able to produce a gravitational field merely by changing the system of reference". We have given an elementary re-derivation of repulsive gravity from the term 2 B 2 g in the gravito-Gauss law of GR for the frame of a freefalling-rotating observer. We have used Einstein's concepts of 1911 without using any concepts of curved space-time.
The acceleration-difference of freefalling particles depends on the observer (GR-inertial versus GRnoninertial) with his reference frame. -In contrast, the Riemann tensor does not depend on whether the observer is GR-inertial or not.
GR-texts, e.g. [6][7][8][9], imply: if the Riemann tensor at P is zero, the acceleration-difference of freefalling particles measured in the frame of a freefalling-rotating observer at P is also zero. This statement is wrong, as shown by the counter-example given here.

Fictitious repulsive tidal acceleration in accelerated frame of Special Relativity
Relative to a linearly accelerated observer (nonrotating) with his local reference frame of LONBs, we derive our new fictitious repulsive acceleration-difference of freefalling particles in Special Relativity. -In the gravito-Gauss law of General Relativity, this acceleration difference gives the repulsive E 2 g term, which contributes to the accelerated expansion of the universe.
In the local frame of an observer at x = 0, nonrotating but accelerated relative to inertial in the positive x-direction, two freefalling particles are initially at rest at x i A = 0 and x i B = (δx, 0, 0). For first-order δt, the freefalling particles have δs ∝ (δt) 2 = 0 in the observer's frame. Their velocities are first order infinitesimal, hence no length contraction and no time dilation. The velocity-difference of the particles remains zero in the SR-noninertial frame. -But we show that their acceleration-difference is nonzero in the reference frame of the accelerated observer.
Relative to the SR-inertial time-axis, the observer's time-axis gets tilted more and more (as time goes on) in the positive x-direction relative to the SR-inertial timeaxis. The observer's x-axis gets tilted relative to the SR-inertial x-axis in the positive time direction.
The time-lapse α in the local accelerated frame is the measured time elapsed between two equal-time slices Σ t of the primary accelerated observer separated by unit time on his wristwatch. -The lapse α at the position of particle A is α A = 1. The lapse at B is, α B = 1 − (ax A ) rel.to obs δx B .
In our example, the acceleration (ax A ) rel.to obs is negative, hence α B > 1. -The acceleration of freefall particle B relative to the observer is, The difference of accelerations of our neighbouring freefalling particles in the frame of the accelerated and nonrotating observer in Special Relativity is nonzero, This shows: the acceleration-difference of neighbouring free-falling particles, ∂xax, measured in the local frame of an accelerated-nonrotating observer is nonzero in Special Relativity, it is not given by the Riemann tensor: the contrary conclusions in [7,8] are wrong. For Special Relativity in an accelerated-nonrotating frame, the acceleration-difference of infinitesimally separated freefalling particles, ∂x ax ff , is positive, "repulsive".
The gravito-electric field E g measured in the local frame of an accelerated-nonrotating observer is defined equal to the acceleration a relative to the observer of freefalling particles (released by the observer). The last equation gives the acceleration-difference of neighboring freefalling particles spherically averaged. With c = 1, This is a law of Special Relativity: for accelerations of Newtonian order of magnitude, it reduces to div E g = 0.
Eq. (51) is our gravito-Gauss law of GR in the reference frame of an accelerated-nonrotating observer at a point free of matter and electromagnetic fields. We have rederived this law using Einstein's concepts of 1911, but without using Einstein's equations. 3. Derivation of gravito-Faraday law of GR from Euler's fictitious acceleration with Einstein's concepts of 1911 We have derived the gravito-Faraday law of GR using the tools of GR in Sect. VIII D. -We now re-derive this gravito-Faraday law with Einstein's concepts of 1911, but without using Einstein's equations of 1915.
In a frame with time-dependent angular velocity Ω relative to Newton-inertial, the fictitious Euler acceleration of freefalling particles is, Euler's frame is rotating relative to the Newtonnonrotating frame. But the Euler acceleration also holds for the frame rotating relative to local Einsteinnonrotating frames.
The operational definitions of ( E g , B g ) by the gravitational acceleration of freefalling particles and the gravitational precession of gyros relative to the observer are, and Ω (relat.to obs.LONB) gyro have opposite signs. Inserting the definitions of ( E g , B g ) into Euler's fictitious acceleration for the observer's frame gives, This is the re-derivation of the gravito-Faraday law of GR from Euler's fictitious acceleration using Einstein's concepts of 1911, but without using Einstein's equations.