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I. GRAVITO-ELECTRIC AND field: B~ g is defined via the gravitational torque caus- ~ ~ GRAVITO-MAGNETIC FIELDS Eg, Bg ing the spin-precession of quasistatic freefalling test par- ticles with spin S~ and free of nongravitational torques Our exact and general operational definition in arbitray (or by quasistatic gyroscopes). In the gyro-magnetic ra- spacetimes of the gravitoelectric field E~g measured by tio q/(2m) of electromagnetism, the particle charge q any observer is probably new. must be replaced by the rest-mass m, therefore the gyro- In contrast to the literature, we use no perturbation gravitomagnetic ratio is (1/2), theory on a background geometry, no “weak gravitational d fields”, no “Newtonian limit”. ~ ~ Sˆi ≡ [ S × (Bg/2) ]ˆi (2) For an observer with his Local Ortho-Normal Bases dt (LONBs) on his worldline our exact operational defini- ⇔ Ω~ gyro = − (B~ g/2), quasistatic gyroscope, (3) tion of the gravitoelectric field E~g is by measuring the ac- 1/2 = gyro-gravitomagnetic ratio. celeration of quasistatic freefalling test-particles in anal- ogy to the operational definition of the ordinary elec- The gravitomagnetic field B~ g can also be measured by tric field. We must replace the particle’s charge q by its the deflection of freefalling test particles, nonrelativistic rest mass m. This gives the operational definition of the relative to the observer, gravito-electric field, d ~p = m [ E~g + (~v × B~g)]. (4) − d (g) dt m 1 p ≡ E dt ˆi ˆi ~ × ~ ⇔ ~a = E~ (g) = ~g, (1) The second term, Fg = m(~v Bg) is the Coriolis force ff of Newtonian physics, which arises in rotating reference for freefalling, quasistatic test particles. frames. It has the same form as the Lorentz force of electromagnetism with the charge q replaced by the rest The local time-interval dt is measured on the observer’s mass m for a nonrelativistic particle. — Our definition wristwatch. The measured 3-momentum is p , and the ˆi of B~ agrees with Thorne et al [2]. measured acceleration of the quasistatic particle relative g The gravitomagnetic field B~ g of general relativity has to the observer is ~aff = ~g ≡ gravitational acceleration. We use use the method of Elie´ Cartan, who gives vec- been measured by Foucault with gyroscopes precessing tors V¯ (and tensors) at any spacetime point P by their relative to his LONBs,e ¯0ˆ =u ¯obs ande ¯ˆi = (East, North, ~ components V aˆ in the chosen Local Ortho-Normal Ba- vertical). Recently, Bg has been measured on Gravity sis. — The LONB-components V aˆ are directly measur- Probe B by gyroscope precession relative to LONBs given able. This is in stark contrast to coordinate-basis com- by the line of sight to quasars resp. distant stars without ponents V α, which are not measurable before one has measurable proper motion. obtained gαβ by solving Einstein’s equations for the spe- In 1893, Oliver Heaviside, in his paper A Gravitational cific problem at hand. — Cartan’s method uses coordi- and Electromagnetic Analogy [3], gave the same opera- nates only in the mapping from an event P to the event- tional definition of the gravito-electric field E~g, and he µ coordinates, P ⇒ xP . postulated the gravito-magnetic field B~ g in analogy to It is important to distinguish LONB-components, de- Amp`ere-Maxwell. noted by hats, from coordinate-basis components, de- For an inertial observer (freefalling-nonrotating) with noted without hats in the notation of Misner, Thorne, worldline through P , (E~g, B~g)P are zero. and Wheeler [1]. We denote spacetime LONB-indices Arbitrarily strong gravito-magnetic field B~ of general ˆ ˆ ˆ g by (ˆa, b, ...), 3-space LONB-indices by (i, j, ...), and relativity can be measured exactly with the precession coordinate-indices by Greek letters. of quasistatic gyroscopes Eq. (2), or with the Coriolis- LONBs off the observer’s worldline are not needed in deflection of nonrelativistic test particles, Eq. (4). — But Eq. (1), because a particle released from rest will still be this same measured B~ is exactly valid for relativistic test- on the observer’s worldline after an infinitesimal time δt, g 2 particles in the equations of motion, Eqs. (25, 30), and since δs ∝ (δt) ⇒ 0, while δv ∝ δt =6 0. in the field equations. Arbitrarily strong gravito-electric fields E~g of general relativity can be measured exactly with freefalling test- particles which are quasistatic relative to the observer in Galilei-type experiments, Eq. (1). — But this same measured E~g is exactly valid for relativistic test-particles in the equations of motion of general relativity, Eqs. (25), and in the field equations. For the gravitomagnetic field B~g, we give the exact op- erational definition in arbitray spacetimes, which is anal- ogous to the modern definition of the ordinary magnetic 3

A. Ricci’s LONB-connection transport) is given by the rotation matrix,

~exˆ cos α sin α ~exˆ Relative to an airplane on the shortest path (geodesic) = .  ~eyˆ   − sin α cos α   ~eyˆ  from Zurich to Chicago, the Local Ortho-Normal Bases Q P (LONBs), chosen to be in the directions “East” and For infinitesimal displacements, the infinitesimal rotation “North”, otate relative to the geodesic (relative to par- matrix is, allel transport) with a rotation angle δα per path length ~e 0 1 ~e δs, i.e. with the rotation rate ω = (dα/ds). xˆ = 1 + α xˆ .  ~eyˆ    −1 0   ~eyˆ  For an infinitesimal displacement δD~ along a geodesic Q P in any direction, the infinitesimal rotation angle δα of The infinitesimal LONB-rotation matrix δRˆiˆj is given LONBs is given by a linear map encoded by the Ricci by the linear map from the infinitesimal coordinate- γ rotation coefficients ωcˆ displacement vector D , γ δRˆˆ = (ωˆˆ)γ δD , (6) δα = ωcˆ δDcˆ. ij ij ωˆˆ = − ωˆˆ = αˆˆ = rotation angle in [ 1ˆ, 2ˆ ] plane. The Ricci rotation coefficients are also called connection 12 21 12 coefficients, because they connect the LONBs at infinites- The coefficients (ωˆiˆj )γ are the connection 1-form com- imally neighboring points by a rotation relative to the ponents. infinitesimal geodesic between these points (i.e. relative The rotational change of a LONB vector relative to to parallel transport). parallel transport under a displacement in the coordinate γ For non-geodesic displacement curves, the tangents for x is called the covariant derivative of ~eˆi with respect to γ infinitesimal displacements are infinitesimal geodesics. the coordinate x and denoted with the symbol ∇γ , For the connection coefficients, it is irrelevant, whether γ (∂~eˆi/∂x )relat.to parall.trsp. ≡ ∇γ ~eˆi. (7) the displacement curve is geodesic or non-geodesic, only the tangent vectors matter, either the coordinate ba- Combining Eqs. (6) and (7) gives, sisvector, ∂ =¯e , or the LONB-vector, ∂ =e ¯ . γ γ cˆ cˆ ∇γ ~eˆj = ~eˆi (ωˆiˆj )γ . (8) For Ricci connections (LONB rotations), it is irrele- vant, whether we are in curved space or e.g. in the Eu- In 3-space, for an infinitesimal displacement from P clidean plane with polar coordinates (r, φ) and LONBs to Q, the LONBs ~eˆi rotate relative to parallel trans- port, which can be given by (1) the tangent vector (~erˆ, ~e ˆ), where for a displacement vector ~e ˆ the LONB- φ φ to the geodesic curve and (2) the spin-axes of two rotation angle is (ω ) =1/r. rˆφˆ φˆ transported gyroscopes. The infinitesimal rotations are The Ricci rotation coefficients ωcˆ are directly mea- given by the antisymmetric matrix with components surable because of their LONB-displacement-index. — (ω1ˆ2ˆ)γ , (ω2ˆ3ˆ)γ , (ω3ˆ1ˆ)γ. But for computations of curvature in Sect. IV B, a line- In curved (1+1)-spacetime, the Lorentz transformation C integral along a displacement curve calls for infinitesi- of the chosen LONBs relative to a given displacement mal displacement vectors with contravariant components geodesic is a Lorentz boost Laˆ , γ γ ˆb δxP P ′ , the infinitesimal difference of coordinates xP . This calls for a linear map from infinitesimal coordinate- e¯ˆ cosh χ sinh χ e¯ˆ γ t = t , displacement vectors δx ′ (input) to the corresponding e¯ sinh χ cosh χ e¯ P P  xˆ Q    xˆ P measured LONB-rotation angles δα (output),

γ rapidity ≡ χ = additive, tanh χ ≡ v/c. δα = ωγ δx . For infinitesimal displacements, the infinitesimal Lorentz This linear map is the LONB-connection 1 - formω, ˜ boost Laˆ , is, ˆb given explicitely by covariant components ωγ (≡ 1-form e¯ 0 1 e¯ components). In this paper, Greek indices always refer tˆ = 1 + χ tˆ . e¯ 1 0 e¯ to a coordinate basis. Using 1-form components makes  xˆ Q     xˆ P the line-integral free of metric factors g , µν In space-time, we denote vectors by a bar, V¯ , in 3-space, ~ γ we denote vectors by an arrow, V. α(C) = ωγ dx . (5) ZC In curved (3+1)-spacetime, and with two lower indices, ωaˆˆb is antisymmetric for Lorentz boosts and for rotations, γ A line-integral with dx cries out to have some 1-form γ δLaˆˆb = (ωaˆˆb)γ δD , ωˆi0ˆ = − ω0ˆˆi = χˆi0ˆ. σγ as an integrand. Vice versa, the connection 1-form ωγ cries out to be contracted with a contravariant dis- For a displacement in observer-time, the exact Ricci γ placement vector to give a rotation angle, δα = ωγ δx . connection coefficients (ωaˆˆb)0ˆ of general relativity can be This is index-matching. measured in quasistatic experiments. But these Ricci con- The rotation of the chosen LONBs (~exˆ, ~eyˆ) relative nection coefficients predict the motion of relativistic par- to the geodesic from P to Q (i.e. relative to parallel ticles with the equations of motion, Eqs. (25, 30). 4

B. (E~g, B~ g) identical with directly and exactly measured in arbitrarily strong grav- Ricci connection in time (ωaˆˆb)0ˆ itational fields of general relativity using freefalling test particles and gyroscopes which are quasistatic relative to the observer withu ¯ =e ¯ in Galilei-type and Foucault- Our gravitoelectric field E~g is identically equal to the obs 0ˆ negative of the Ricci Lorentz-boost coefficients for a dis- type experiments. Therefore, it is superfluous to use rel- placement in time, ativistic test-particles to measure the Ricci connection coefficients for a displacement in time. aˆ (g) Cartan’s LONB-connection 1-form (ω )0, with its dis- E = −(ωˆˆ)ˆ. (9) ˆb ˆi i 0 0 placement index in the coordinate basis, are measurable µ The proof is immediate, see the next equation: from the as soon as a coordinatization P ⇒ xP is chosen. No point of view of the observer with his LONBs along his metric coefficients gαβ are needed. worldline, the gravitational acceleration g = a(ff particle) In striking contrast, the connection coefficients for co- ˆi ˆi ordinate bases, the Christoffel symbols, (Γα ) ≡ Γα , of freefalling quasistatic test-particles (starting on the β γ βγ have no direct physical-geometric meaning, they are not observer’s worldline) is by definition identical to the measurable until the metric fields g (x) have been ob- exact gravitoelectric field E of general relativity from µν ˆi tained by either solving Einstein’s equations or going Eq. (1). — But from the point of view of freefalling test- out and measuring distances, time-intervals, angles, and particles, the acceleration of the quasistatic observer with Lorentz-boost-angles in a fine-mesh coordinate grid. — his LONBs is by definition identical to the exact Ricci We write Christoffel connection-1-form coefficients with LONB-Lorentz-boost coefficients (ωˆi0ˆ)0ˆ, α α a bracket: Γ βγ ≡ (Γ β)γ . Inside the bracket are the coordinate-basis transformation-indices (α, β), outside E(g) ≡ [(a ) (ff particle) ] ˆi ˆi relat.to obs. quasistatic the bracket is the coordinate-displacement index γ. (obs.) Conclusion: = − [(aˆi) relat.to ff ]quasistatic ≡ − (ωˆi0ˆ)0ˆ. (10) • Ricci connection coefficients (ωaˆ ) are directly ˆb dˆ The exact Ricci connection coefficients (ωˆi0ˆ)0ˆ of general relativity in arbitrarily strong gravitational fields can be measurable for given LONBs, directly measured in quasistatic experiments by the accel- • Cartan connection coefficients (ωaˆ ) are measur- ˆb µ eration of freefalling test particles relative to the LONBs able for given LONBs after a coordinatization- of the observer as Galilei did. mapping P ⇒ xµ is chosen, but no metric is For the gravitomagnetic 3-vector field B~ (g), we intro- (g) needed, duce its Hodge-dual antisymmetric 2-tensor Bˆˆ , jk α α • Christoffel connection coefficients (Γ β)δ ≡ Γ βν (g) (g) cannot be known until one has solved Einstein’s B ≡ ε B , (11) ˆiˆj ˆiˆjkˆ kˆ equations for the problem at hand in order to ob- tain the metric g for the chosen coordinate sys- where ε is the Levi-Civita tensor, totally antisymmet- αβ ˆiˆjkˆ tem. ric, whose primary definition is given in a LONB, The connection-coefficients for local ortho-normal ε1ˆ2ˆ3ˆ ≡ + 1 for LONB with positive orientation,(12) bases (Ricci and Cartan) are more efficient than Christof- fel’s connection-1-form coefficients: In two spatial di- B1ˆ2ˆ ≡ B3ˆ and cyclic permutations. mensions, Ricci and Cartan connections each need only The Levi-Civita tensor in a coordinate-basis in 3 dimen- one rotation angle for a given displacement. In contrast, α sions, εαβγ, cannot be known before Einstein’s equations the Christoffel-connection 1-form (Γ β)γ for a given dis- have been solved for the specific problem at hand. placement needs four numbers for the coordinate-basis Recall from Eq. (3) that the gyro-precession relative transformation, two for stretching/compressing basis vec- to the observer equals (−B~ g/2), hence tors, one for skewing them, and one for rotating them.

(− B(g)/2) ≡ [(Ω )(gyro) ] ˆi ˆj ˆiˆj relat.to obs. quasistatic (obs) = − [(Ωˆiˆj )relat.to gyro]quasistatic ≡ − (ωˆi ˆj )0ˆ. (13)

The exact Ricci connection coefficients (ωˆiˆj )0ˆ of general relativity can be directly measured in quasistatic exper- iments by the precession of gyroscopes relative to the LONBs of the observer as Foucault did. All Ricci connection coefficients for a displacement in time, (ωaˆˆb)0ˆ, which have all indices in LONBs, can be 5

C. Equations of motion with II. NEWTON-INERTIAL MOTION VERSUS general LONBs and coordinates GENERAL-RELATIVITY-INERTIAL

We obtain the equations of motion in terms of Ricci This paper is based on two pillars: coefficients from (1) the equations of motion in terms of (E~g, B~ g) for quasistatic freefalling-nonrotating test- 1. Einstein’s revolutionary concept of inertial motion (g) (g) as freefalling-nonrotating, which replaces the first particles, Sect. I, (2) the identity of (E ,B ) with ˆi ˆiˆj law of Newton, his law of inertia, the Ricci connection for a displacement in time, (ωaˆˆb)0ˆ, Sect. I B, 2. The Newtonian law on relative acceleration of neighbouring freefalling nonrelativistic test parti- −1 d m pˆ + (ωˆˆ)ˆ = 0, i =1, 2, 3, cles, spherically averaged. dtˆ i i0 0 d We never assume a “Newtonian limit of general relativ- Sˆ + (ωˆˆ)ˆ Sˆ = 0. (14) dtˆ i ij 0 j ity”, and we never assume “weak gravity”. From the above two pillars, it immediately follows At the level of first derivatives of vectors, there can- that space-time is curved, i.e. the curvature of [t, xi]- not be curvature effects, we are at the level of spe- plaquettes does not vanish, a straighforward, but revolu- cial relativity, and the Lorentz-covariant extension of tionary insight demonstrated in Sec. II C. Eqs. (14) is trivial: we must include spatial displace- Incorporating the local Minkowski metric of special rel- ments in (dxµ/dt) and spatial components of the vector ativity leads directly to Riemannian geometry. ˆb p . This Lorentz-covariant extension of Eq. (14) gives From these inputs alone, the relativistic geodesic equation and Fermi spin trans- port equation, We derive the exact curvature component R0ˆ0ˆ (P ) µ of general relativity d aˆ aˆ ˆb dx p + (ω )µ p = 0, from Newtonian relative acceleration experiments dt ˆb dt µ of freefalling test-particles, d ˆ dx Saˆ + (ωaˆ ) Sb = 0. (15) dt ˆb µ dt 0ˆ0ˆ ∂ relative quasistatic [R ]GR exact = − [ spherical av.]r=0. The covariant derivative ∇µ of vector- and tensor-fields ∂r can be defined by where one takes the spherical average of the relative ra-

(∇µV¯ )aˆ ≡ ∂µ(V¯aˆ) dial acceleration of freefalling quasistatic test-particles starting at radial distance r. for LONBs parallel in direction ∂ . (16) µ Einstein’s Ricci-0-0 curvature of general relativity is The covariant derivative along the worldline of any in- exactly determined by an experiment with quasistatic dividual particle is denoted by (Dp/Dt¯ ) resp. (DS/Dt¯ ). test-particles. Hence,

freefalling-nonrotating particles: A. Newton’s method to find inertial motion Dp¯ DS¯ = 0, = 0. (17) Dt Dt Newton’s first law, the law of inertia, states that force- free particles move in straight lines and at constant veloc- For arbitrary LONBse ¯ , the covariant derivative of aˆ ities. The same law of inertia holds in special relativity. vector fields follows directly from the definition of the Newton’s law of inertia is valid only relative to in- Ricci connection coefficients in Sect. I A, ertial reference frames. Inertial frames are defined as ˆ those frames in which the law of inertia is valid. Iner- (∇ V¯ )aˆ = ∂ V aˆ + (ωaˆ ) V b, γ γ ˆb γ tial frames can be constructed operationally either using ∇ aˆ γ e¯ˆb =e ¯aˆ (ω ˆb)γ. (18) three force-free particles moving in non-coplanar direc- tions, or using one force-free particle plus two gyroscopes The fundamental observational rocks for general rela- with non-aligned spins axes. tivity are (E~g, B~ g), measured by the acceleration resp. precession of freefalling-nonrotating test-particles resp. • There is a grave problem with this standard formu- gyroscopes which are quasistatic relative to the observer, lation of Newton’s first law: force-free particles do as in the experiments of Galilei resp. Foucault. Apart not exist e.g. for solar-system dynamics, gravita- from a minus sign and a factor 2, these measured fields tional forces are always present, therefore the op- are identical with Ricci’s LONB-connection for a dis- erational construction of an inertial frame is a very placement along the observer’s worldline. difficult task. 6

In a few cases, gravity is irrelevant: (1) for particle purely geodesic means), or by repeating the experiment physics, gravity is negligible, since the relative magni- of Foucault’s pendulum. The absolute rotation of this tude of the gravitostatic compared to electrostatic force planet might be clearly shown in this way.” between two protons is extremely small, 10−36, (2) for On such a planet with permanent dense cloud cover, it the motion of polished balls on a polished horizontal ta- is impossible to use Newton’s method “to draw evidence ble, gravity cannot act, it is orthogonal to the table. ... from the forces that are the causes ... of the true Newton’s solution for this problem: He wrote in the motions,” since the Sun is invisible. On such a planet, it Principia [4] that for finding “true motion” equivalent is impossible to decide, whether the planet is on an orbit to “absolute motion” (i.e. motion relative to inertial around the Sun, i.e. in non-inertial motion in the sense frames), distinguished from “relative motion”, and in of Newton, or moving in a straight line with constant particular true (absolute) acceleration (i.e. acceleration velocity, i.e. in inertial motion in the sense of Newton. relative to inertial frames), distinguished from relative If the planet was sufficiently small, tidal effects from the acceleration, one must first subtract the gravitational ef- Sun would be undetectable. It follows that fects of all celestial bodies. In the Scholium on space • and time, at the end of the initial “Definitions”, Newton Newton’s first law is operationally empty for a per- writes on p. 412 of [4]: “True motion is neither generated son on a very small planet with permanent cloud nor changed except by forces impressed upon the moving cover: One pillar of Newtonian physics is destroyed. body.” On p. 414: • “It is certainly very difficult to find out the true C. Einstein’s revolutionary concept of motions of individual bodies. ... Nevertheless, the inertial motion case is not hopeless. For it is possible to draw ev- idence ... from the forces that are the causes ... of But one cannot remove one pillar from a theory with- the true motions.” out putting in its place a new pillar: In order to find inertial frames, Newton invokes the forces • The new pillar is Einstein’s revolutionary re- from his second law. Newton implies that without his sec- definition of inertial motion: Inertial motion and ond law, his first law would be “hopeless”. — On p. 415, the local inertial frame in general relativity are op- at the very end of this Scholium, Newton concludes: “In erationally defined by a freefalling particle together what follows, a fuller explanation will be given of how to with the spin axes (not parallel) of two comoving determine true motion from their causes ... For this was gyroscopes, i.e. freefalling-nonrotating motion, the purpose for which I composed the following treatise.” Essentially by this method, the local inertial center-of- Freefalling particles determine and define the straight- mass frame for the solar system is determined today, the est timelike worldlines, geodesics. They have maximal “International Celestial Reference Frame” of 1998. proper time between any two nearby points on the world- Note the dramatic difference between the two aspects line, dτ = maximum. of inertial frames: (1) It is easy to establish the non- SpacetimeR curvature, i.e. curvature of space-time pla- rotating frame by using two gyroscopes (Foucault) or, quettes [t, x] etc. in distinction to space-space plaque- much less precisely, with the rotating-bucket experiment ttes [x, y] etc., follows directly from Newtonian experi- of Newton. (2) It is very work-intensive to establish a ments on relative acceleration of neighbouring freefalling frame without linear acceleration relative to an inertial test particles combined with Einstein’s concept of iner- frame in the mechanics of the solar system. tial motion and geodesics: In a Gedanken-experiment, It is remarkable that most textbooks on classical me- consider a vertical well drilled down through an ideal chanics pretend that Newtonian inertial frames can be ellipsoidal Earth from the North Pole through the cen- constructed operationally using force-free particles, al- ter of the Earth to the South Pole. Drop a pebble and though force-free particles do not exist for the mechanics afterwards another pebble. The pebbles will fall freely of the solar system. (geodesic worldlines) from the North Pole to the center of the Earth, rise to the surface of the Earth at the South Pole and fall back again. The two geodesics will cross B. Poincar´e’s man under permanent cloud cover again and again, direct evidence of space-time curvature from Newtonian experiments. In Sect. IV, An entirely different point was discussed by Henri • we compute the space-time curvature component Poincar´ein “Science and Hypothesis” in 1904 [5]: “Sup- R0ˆ = (R0ˆ ) of general relativity for [0i] space- 0ˆ ˆi 0ˆˆi pose a man were transported to a planet, the sky of which time plaquettes exactly from Newtonian exper- was constantly covered with a thick curtain of clouds, so iments on relative acceleration of neighbouring that he could never see the other stars. On that planet freefalling test particles, quasistatic relative toe ¯0ˆ, he would live as if it were isolated in space. But he spherically averaged, using the revolutionary con- would notice that it rotates, either by measuring the cept of inertial motion as freefalling-nonrotating. planet’s ellipsoidal shape (... which could be done by 7

III. EQUIVALENCE OF magnetic field in electromagnetism, and for general rela- FICTITIOUS AND GRAVITATIONAL FORCES tivity, the gravito-magnetic field is defined in Eqs. (2 - 4). IN EQUATIONS OF MOTION Both forces are written here for nonrelativistic particles. For an observer in a windowless spaceship, a fictitious We do not trace the history of the equivalence principle Coriolis force and a gravito-magnetic Lorentz-force are in Einstein’s writings beginning in 1907. fundamentally indistinguishable, hence equivalent. Starting with Einstein’s theory of relativity of 1915, For an observer in a windowless spaceship, the mass- we prove the two equivalence theorems with their exact sources of gravitational fields (Sun, etc) are unknown. explicit equations of motion, Therefore, for such an observer the gravitational field equations are useless, in 19th-century physics Gauss’s law 1. fora non-inertial observer the equivalence and exact for gravity, div E~g = −4πGN ρmass. equality of gravitational forces and fictitious forces Newton has emphasized that two reference systems ro- in the equations of motion, Sect. IIID, tating relative to each other are equivalent, unless one considers the forces, Sec. II A. Einstein’s principle of 2. for an inertial observer the exact explicit vanishing equivalence is no longer valid, as soon as (1) the source- of (d/dt)(p ) and (d/dt)(S ) for inertial particles in ˆi ˆi masses of gravity are seen (observer not “under a perma- Sect. III C. nent cloud cover”) and (2) the gravitational forces are ex- Our explicit equalities presented hold, if and only if plicitely known by solving the gravitational field equation one uses our adapted spacetime slicing and our adapted of Gauss-Newton or Einstein. Conclusion: the Principle LONBs presented in Sect. IIIB. of Equivalence is not valid at the level of the gravitational The reader might want to first read our results field equations. in Sects. IIID and IIIC, afterwards our method in • The equivalence principle states that fictitious Sect. III B, and finally our discussion of fundamentals forces are equivalent to gravitational forces in the in Sect. III A. equations of motion. The equivalence principle holds for first time-derivatives of particle momenta and spins on the worldline of the primary observer. A. Fundamentals • But the equivalence cannot hold for relative acceler- Fictitious forces, e.g. centrifugal forces seen by a ro- ation (tidal acceleration) and for relative precession tating observer in his rotating reference frame, have been of two particles at different positions, i.e. the equiv- important since Huygens, Newton, Leibniz, and Hooke. alence cannot hold for second derivatives of particle Fictitious forces are also called inertial forces. — Because momenta and spins, one derivative in time and one the centrifugal force vanishes on the worldline of a ro- derivative in space (to compare particles at differ- tating observer, this force will not play any role for the ent locations): The equivalence cannot hold for the equivalence between fictitious and gravitational forces in curvature tensors. — For an accelerated observer in the equations of motion. Minkowski spacetime, there are no tidal forces, no Equivalence principle for E~g: For an observer acceler- relative acceleration of freefalling particles, and no ated with ~aobs relative to freefall, who is inside a win- spacetime-curvature. dowless elevator or windowless space-ship or in fog, and (relat.to obs) • The equivalence cannot hold, if in the equations measures the acceleration ~aff particle of freefalling test- of motion at P , the particle is initially, at tP , off particles at his position, it is impossible to distinguish the primary observer’s world-line, because the tidal between (1) particle-acceleration caused by the fictitious forces would be relevant. force due to the observer’s own acceleration relative to freefall, and (2) particle-acceleration caused by gravi- • The equivalence cannot hold in the gravitational tational forces due to the attraction by matter-sources field equations of Einstein or Gauss, divE~g = (Earth, Sun, Moon, etc). Therefore, this fictitious force −4π GN ρmass, with their matter-sources. — For an and this gravitational force are equivalent in the equations accelerated observer in Minkowski space, there are of motion. no matter sources. Equivalence principle for B~ : For an observer rotating g We focus on one primary observer. In the equations of with angular velocity Ω~ obs relative to comoving gyro-spin motion, e.g. in [(d/dt) (pˆ)](t1) = limδt→0[pˆ(t1 + δt) − axes at his position, who is in a windowless elevator or i i pˆi(t1)]/δt, spacetime curvature is absent, if and only if spaceship or in fog, it is impossible to distinguish (1) a particles start at t on the worldline of the primary ob- ~ ~ 1 fictitious Coriolis force F = 2 m [~v × Ωobs] due to his server. own rotation, from (2) a gravito-magnetic Lorentz-force Many auxiliary local observers are needed to measure ~ ~ ~ F = m[~v × Bg] from the gravitational field Bg gener- the momenta ~p and spins S~ of test-particles at t = t1 +δt, ated by mass-currents. The gravitomagnetic field B~ g has when they are no longer on the worldline of the pri- been postulated by Heaviside in 1893 [3] in analogy to the mary observer. We shall see that our auxiliary observers 8 cannot be inertial, even if the primary observer is iner- point on the observer’s worldline. — The equiva- tial (unless spacetime curvature vanishes). Every aux- lence principle holds for the equations of motion, iliary local observer, with his worldline through Q and where curvature is invisible. The equivalence prin- u¯obs(Q)=¯e0ˆ(Q) and with his local spatial axese ¯ˆi, is in ciple makes no statements about relative accelera- a one-to-one relationship with the Ricci connection coef- tion and curvature. ficients (ωaˆ ) at Q. ˆb 0ˆ 2. Equations of relative acceleration of neighbouring An observer can be non-inertial in exactly two funda- freefalling test-particles starting at two different po- mental ways: sitions with second derivatives of momenta around 1. observer’s acceleration relative to freefall, a plaquette with one time-derivative and one spa- tial derivative to obtain relative acceleration, hence 2. observer’s rotation relative to spin axes of comoving space-time curvature. gyroscopes.

Correspondingly, there are exactly two fictitious forces in B. The primary-observer-adapted the equations of motion on the worldline of a non-inertial space-time slicing and LONB-field observer, equivalent to two gravitational forces: 1. the fictitious force measured by an observer acceler- For the theorem of equivalence between fictitious ated relative to freefall, which is equivalent to New- forces and gravitational forces, the equations of motion ∇ p¯ = 0 and ∇ S¯ = 0 for inertial particles are ton’s force in a gravito-electric field, ~g ≡ E~ , p¯ p¯ g not instructive by themselves, because they do not make 2. the fictitious Coriolis force measured by an observer explicit the fundamental distinction between inertial and rotating relative to gyro axes, which is equivalent to non-inertial observers. The more explicit equations of motion for in- the gravito-magnetic Lorentz force due to B~ g pos- tulated by Heaviside in 1893. ertial particles, the geodesic equation, (d/dt) pˆi + aˆ µ (ωˆiaˆ)µ p (dx /dt) = 0, and the Fermi spin-transport aˆ µ The remaining two of the four fictitious forces [6] can- equation, (d/dt) Sˆi + (ωˆiaˆ)µ S (dx /dt)=0, are not not contribute in the equations of motion on the world- instructive by themselves, because they contain, depend- line of a primary observer, because these forces vanish on ing on the formalism used, 6 × 4 = 24 Ricci LONB- his worldline. Therefore these two fictitious forces cannot connection coefficients (ωaˆˆb)cˆ, resp. Cartan LONB- occur in the equivalence theorem, connection coefficients (ωaˆˆb)γ , since they are antisym- metric in the lower-index pair [ˆaˆb], hence 6 pairs, and 1. the fictitious centrifugal force, they have 4 displacement indices. — There are 4×10= 40 α Christoffel connection coefficients (Γ )γ , since they are F/m~ = [ Ω~ obs × [ ~r × Ω~ obs ]], β symmetric in the last index-pair (βγ), hence 10 pairs, and there are 4 values for α. — These 40 resp. 24 connection 2. the fictitious force due to an observer’s non- coefficients are highly uninstructive. uniform rotation, For the exact equations of motion demonstrating ex- plicitely the equivalence theorems for both non-inertial ~ × ~ F/m = [ ~r ( dΩobs/dt )], for inertial observers, we need a primary-obserer-adapted spacetime splitting and LONB-field: both vanish on the worldline of the primary rotating observer, r =0. • our primary-observer-adapted space-time splitting uses fixed-time slices which are generated by ra- To put this subsection “Fundamentals” in perspective, dial 4-geodesics starting Lorentz-orthogonal to the we print the criticism of the equivalence principle by worldline of the primary observer, J.L. Synge [7]: “Does the principle of equivalence mean space-time slicing by radial 4-geodesics that the effects of the gravitational field are indistinguish- from primary observer, able from the effects of the observer’s acceleration? If so, it is false. ... Either there is a gravitational field or there • the time-coordinate t all over the fixed-time slice is none, according to the Riemann tensor. ... This has Σt is defined as the time measured on the wrist- nothing to do with the observer’s worldline. ... The Prin- watch of the primary observer, ciple of equivalence ... should now be buried.” Synge mixes up two entirely different levels, first time- • our Local Ortho-Normal Bases, LONBs, are radi- derivatives of momenta versus second derivatives of mo- ally parallel, i.e. parallel along radial 4-geodesics menta around a plaquette: from the primary observer’s LONB, 1. Equations of motion, acceleration with first time- LONBs radially parallel derivatives of momenta for particles starting at one to primary observer’s LONB, 9

• 3-coordinates on the slices Σt are not needed in the Our extended fields of LONBs define an extended ref- equations of motion. erence frame for an inertial or for a non-inertial primary observer along their entire worldlines. — Spatially, this In the equations of motion, for (d/dt) (pˆi), and for the holds outwards until radial 4-geodesics cross, e.g. from equivalence theorems, we work with LONBs exclusively, multiply-imaged quasars. hence with Ricci connection coefficients (ωaˆ ) . We do ˆb dˆ so, because the LONB-components pˆi can be measured directly, while the coordinate-basis components pi cannot C. Equivalence theorem for inertial observers be measured without having first solved Einstein’s equa- tions to obtain the metric in the coordinate basis, g . µν The equivalence theorem for inertial observers states: Our slicing and LONB-field are needed for the pri- Along the entire worldline of a primary inertial observer mary observer’s direct determination of momentum com- (freefalling-nonrotating), there exists an adapted slicing ponents p (t + δt) and spin components S (t + δt) off ˆi 1 ˆi 1 and adapted LONBs, which is our slicing and our LONBs his worldline: The primary observer measuring ~p (t) ofa given in Sect. III B, such that the equations of motion for particle off his worldline, needs a parallel transport of the relativistic inertial test-particles, the geodesic equation particle’sp ¯(t) on the radial geodesic on Σ to his own t and Fermi transport equation, Eqs. (15), give zero on LONB at the time t. In this parallel transport, the com- the right-hand-side, they have the same form as for free ponents p stay unchanged, because we have chosen the ˆi particles in special relativity with Minkowski coordinates, LONBs radially parallel. This makes the explicit equa- tions simple. d d Our crucial result: with our LONBs radially parallel, inertial particles: (pˆ) = 0, (Sˆ)=0, (23) dt i dt i • the connection coefficients for spatial displacements where t is the time measured on the worldline of the vanish on the entire worldline of the primary ob- primary observer. server, For an inertial observer and with our adapted slic- ing and adapted LONBs, all Ricci connection coefficients [ (ω ˆ)ˆ ] (r→0, t) = 0, (19) aˆ b i (ω ) , vanish. For an inertial observer with a general ˆ aˆˆb cˆ ˆi = (1, 2, 3), a,ˆ b = (0, 1, 2, 3). (20) non-adapted slicing and non-adapted LONBs, there are 6 × 4 = 24 non-vanishing Ricci connection coefficients • For a non-inertial observer, the connection coeffi- (ωaˆˆb)cˆ, which would be needed in the geodesic equation cients for displacements in time are non-zero, and the Fermi spin-transport equation with non-adapted LONBs and non-adapted spacetime-slicing. (ω ) = E(g), ˆi 0ˆ 0ˆ ˆi For an inertial observer with our adapted slicing and (g) LONBs, the equivalence theorem of general relativity (ω ) = B . (21) ˆi ˆj 0ˆ ˆiˆj states that all gravitational fields vanish which is equiv- alent with the vanishing of all 24 Ricci connection coef- • For an inertial primary observer, all connection co- ficients (ωaˆˆb)cˆ. efficients vanish on the primary observer’s entire worldline, ˆ ˆ D. Equivalence theorem for noninertial observers [(ωaˆˆb)dˆ](r→0, t) = 0, a,ˆ b, d = (0, 1, 2, 3). (22) Our three fundamental results for the equivalence the- A non-inertial and an inertial observer have necessar- orem in the equations of motion of general relativity for ily different worldlines (displacement curves), but only non-inertial observers are given first, the derivations fol- the tangent vector in P matters in the connection-1-form low afterwards: coefficients (ωaˆˆb)cˆ. The tangent-vector in P is the same, if the two observers are instantaneously comoving in P . • Our results are explicitely valid, if and only Our radially parallel spatial LONB-fielde ¯ˆi in curved if one uses our primary-observer-adapted slic- spacetime is consistent with the 3-globally parallel ba- ing of spacetime, generated by radial 4-geodesics sis vector-field ∂i in Newton’s Euclidean 3-space with starting Lorentz-orthogonal to the primary ob- Cartesian coordinates. server’s worldline, and our primary-observer- In the equations of motion, evaluated on the world- adapted LONBs, which are radially parallel to the line of the observer, it is irrelevant that our LONB’s LONB of the primary observer as described in (¯e0ˆ, e¯xˆ, e¯yˆ, e¯zˆ) are not parallel in the (θ, φ) directions. Sec. III B. Our radially parallel e¯0ˆ field in curved spacetime is unique in being consistent with Newtonian observers at • First result: For any test-particle and an observer a given time being at relative rest, hence basis vectors who is non-relativistic relative to this particle, our ∂t spatially parallel in the language of affine geometry of exact equations of motion in arbitrarily strong grav- Felix Klein of 1872. itational fields of general relativity are identical with 10

the 19th-century Newton-Heaviside-Maxwell equa- It is remarkable that these simple exact equations of tions of motion, motion for general relativity with arbitrarily strong gav- d itational fields are missing in textbooks. (pˆ) = (24) Eqs. (24, 25) are exact in curved spacetime. But equa- dt i tions of motion have only first derivatives of particle ~ ~ m [Eg + ~v × Bg]ˆi gravity in general relativity, momenta, therefore they are independent of curvature, ~ ~ which needs second derivatives of vectors arond a space- + q [E + ~v × B]ˆi, el.mag. in general relativity. time plaquette. Every term in this exact equation of general rela- tivity is identical to the terms known in the 19th • Third result: Theorem of equivalence between fic- century: The conceptual framework is different in titious forces and gravitational forces in the exact general relativity, but the equations are the same. equations of motion of general relativity for rela- (1) Newton’s gravitational force, the gravito-electric tivistic particles in arbitrarily strong gravitational fields: force due to E~g = ~g in Heaviside’s notation of 1893, defined for general relativity in Eq. (1), (1) the equivalence between the fictitious force mea- ~ sured by an accelerated observer and a gravito- (2) Heaviside’s gravito-magnetic force due to Bg, ~ postulated 1893 [3], defined for general relativity in electric field Eg generated by sources of matter, Eq. (2), and form-identical with the Lorentz mag- (2) the equivalence between the fictitious Cori- netic force due to B~ of Amp`ere and Maxwell, olis force measured by a rotating observer and a gravito-magnetic Lorentz force from a gravito- (3) Maxwell’s electromagnetic forces due to (E,~ B~ ). magnetic field B~g generated by matter-currents. Our exact equation of motion of general relativity in our primary-observer-adapted spacetime slicing and Our fundamental definitions of E~g and B~ g in Eqs. (1) LONBs, Eqs. (24), has only two gravitational fields, and (3), inserted in the exact equation of motion of E~g and B~ g, versus 40 Christoffel symbols, utterly non- general relativity, Eq. (25), for particles free of non- instructive, for general coordinates (both in general rela- gravitational forces give, tivity and in Newtonian physics with its universal time). d (p ) = ε [ −~a + 2 ~v × Ω~ ] • Second result: For relativistic test-particles, the ex- dt ˆi obs obs ˆi act equations of motion of general relativity in ar- = ε [ E~g + ~v × B~ g ]ˆ. (26) bitrarily strong gravitational fields and electromag- i netic fields are, These equations hold for relativistic particles in general d relativity. — The first of these two equations agrees with (pˆ) = ε [E~g + ~v × B~ g]ˆ gravity in GR (25) Landau and Lifshitz, Classical Mechanics [6], Eq. (39.7) dt i i for the nonrelativistic case. ~ ~ + q [E + ~v × B]ˆi, el.mag. in general relativity The exact equivalence between gravitational forces and d fictitious forces in the equation of motion for general rel- ε = ε ~v · E~ gravity in general relativity dt g ativity and for relativistic particles is evident by compar- + q ~v · E,~ el.mag. in general relativity. ing the two right-hand-sides in Eq. (26), The energy and 3-momentum of the test-particle, equivalence of ε = total particle-energy = γm, γ = (1 − v2/c2)−1/2, fictitious and gravitational forces: ~ energy of photon = εγ = hνγ, ~a obs = − Eg, 1 ~p = 3-momentum = γ m ~v, for photon ~p = hνγ ~evˆ. Ω~ = B~ . (27) obs 2 g With our primary-observer-adapted space-time slicing ~ and LONBs, It is a crucial fact that the exact Eg = −~aobs, and B~ /2 = Ω~ , can be measured with freefalling test- 1. the terms due to electromagnetic fields in general g obs particles and gyroscopes which are quasistatic relative to relativity are explicitely identical with the equa- the observer, Eqs. (1, 3), but they predict the motion of tions in special relativity, relativistic particles in Eq. (26). 2. the terms due to gravitational fields in general rela- Arbitrarily strong gravito-electric and gravito- tivity are form-identical with the terms for electro- magnetic fields can be measured by quasi-static methods magnetic fields in Minkowski spacetime of special in the instataneous comoving inertial frame, since only relativity, except for the replacements, infinitesimally small velocities and gyro-precession angles arise after an infinitesimally short time. ~ ~ ⇒ ~ ~ (E, B) (Eg, Bg) Our results in this subsection look very familiar. But q ⇒ ε. our results are entirely new : 11

(1) Our exact results for general relativity with arbi- d aˆ pˆ = ± (ωˆ )ˆ p , (29) trarily strong gravitational fields are new. No “Newto- dt i iaˆ 0 nian approximation”, no weak field limit, no perturba- (g) (ωˆi0ˆ)0ˆ = − Eˆ , tion theory. i 1 (g) (2) The exact definition of E~ in arbitrarily strong (ωˆˆ)ˆ = − B , g ij 0 2 ˆiˆj gravitational fields is new, Sec. I. ~ ~ 0ˆ ˆi (3) The identity of (Eg, Bg) with the Ricci connection Using the 4-momentum components p = ε, and p = i for a displacement in time, (ωaˆˆb)0ˆ, is new, Sec. I B. εv , we obtain Eqs. (25). (4) Our exact equations of motion in arbitrarily strong To derive the equations of motion for a non-inertial gravitational fields, Eqs. (24, 25), are new. observer, Eqs. (25), the Local Inertial Coordinate Sys- (5) Our equations Eqs. (26, 27) for the equivalence tem and the associated Local Inertial Frame used in all of fictitious and gravitational forces in the equations of textbooks are useless. motion are new. The time-evolution equation for spin, the Fermi spin It is remarkable that most textbooks on general rela- transport equation, Eqs. (15) simplify in our primary- tivity do not discuss the equivalence of fictitious forces observer-adapted LONBs and slicing. Since the spin 4- and gravitational forces in the equations of motion. The vector in the rest frame of a particle has by definition no aˆ few textbooks, which do dicuss this equivalence, are not time-component, we have p Saˆ = 0, and we can elimi- specific about which fictitious force is equivalent to which nate S0ˆ gravitational force. But most important: these textbooks aˆ 0ˆ do not give equations. Words without equations is un- p Saˆ = 0 ⇒ S = ~v · S,~ ususal for an important topic in theoretical physics. d (S ) = (S~ × B~ /2) + (~v · S~) E(g), (30) It is also remarkable that in textbooks there is no dis- dt ˆi g ˆi ˆi cussion of measurements by rotating observers, no equa- tion demonstrating the equivalence of the fictitious Cori- where ~v ≡ (d~x/dt) is the 3-velocity of the particle. For olis force with the gravitomagnetic Lorentz force, no dis- a comoving gyroscope, the second term in Eq. (30) van- cussion of the fundamental impossibility to distinguish ishes, and this equation for relativistic gyroscopes be- these two forces in the equations of motion. comes identical to the equation for nonrelativistic gyro- The old paradigm for fundamentals in general relativity scopes, Eq. (2). For a non-comoving gyroscope, the sec- has been that one should work with general coordinates. ond term is the gravitational analogue of the Thomas Only for applications to special situations, Schwarzschild, precession (1927) in classical electrodynamics [8]. Kerr, Friedmann-Robertson-Walker, should adapted co- ordinates be used. E. Connection-1-forms versus curvature tensor The new paradigm for fundamentals in general rela- tivity is that one must choose adapted spacetime slicing and adapted LONBs to exhibit (1) the identity of 19th- Textbooks give an asymmetric status to connections, e.g. Ricci’s (ω aˆ ) , versus curvature tensors, e.g. Rie- century Newtonian equations of motion and the equa- ˆb cˆ mann’s (R ˆa ) . tions of motion of general relativity for nonrelativistic ˆb cˆdˆ test-particles, (2) the identity of the equations of motion For dynamics, the equations of motion and the gravita- of special relativity in an electromagnetic field and the tional field equations are two equally important legs. In equations of motion of general relativity with the obvi- Wheeler’s words: “matter tells spacetime how to curve, ous replacements (E,~ B~ ) ⇒ (E~g, B~ g) and q ⇒ ε. curved spacetime tells matter how to move.” The derivation of our equations of motion in gravita- In the equations of motion, the measured forces are encoded in the Ricci connection coefficients (ω aˆ ) with tional plus electromagnetic fields, Eqs. (25), follows from ˆb 0ˆ (1) the geodesic equation plus the electromagnetic term our adapted slicing and LONBs. These observables are and (2) our primary-observer-adapted space-time slicing E~g and B~ g, directly measured, the observational rock, on and LONBs, which the equations of motion are built. — The equations of motion do not involve the Riemann tensor. d Observers are in a one-to-one relation with Ricci’s con- p = − (ω ) paˆ (dxµ/dt) + q F uaˆ ˆi ˆiaˆ µ ˆiaˆ nection for a time-like displacement, (ω aˆ ) . Hence, ob- dt ˆb 0ˆ aˆ aˆ servers and their worldlines are as fundamental as Ricci’s = − (ωˆiaˆ)0ˆ p + q Fˆiaˆ u . (28) connection for a time-like displacement. With our primary-observer-adapted slicing and LONBs, radially parallel LONBs, of Sec. IIIB:

connection coefficients nonzero only for displacement in time along worldline of primary observer, 12

IV. SPACETIME CURVATURE FROM The intrinsic Gauss curvature RG at any point P of NEWTONIAN EXPERIMENTS any 2-dimensional space can be operationally defined by the deficit angle δ divided by the measured area A inside In Sect. IIC we have considered the Newtonian exper- the curve C for the length of the curve around P going iment of dropping pebbles down a well drilled all the way to zero, through the Earth. We have explained, how this New- −1 RG = lim ( A δC ). (32) tonian experiment together with Einstein’s revolutionary C→0 concept of inertial motion and geodesics gives direct ev- idence of space-time curvature. For our chosen geodesic triangle on the surface of a We now show by explicit computation, how the space- spherical Earth, the area of A can be measured entirely time curvature-coefficient R0ˆ0ˆ of general relativity is com- on the surface of the Earth, an intrinsic measurement on 2 pletely determined by Newtonian experiments alone. the surface. The result is A = rEarth (π/2). The deficit In general relativity, (3+1)-spacetime is not embedded angle is δ = π/2, hence the Gauss intrinsic curvature is −2 in some higher dimensional spacetime, and there is no ob- RG = rEarth. servational evidence for such an embedding. Therefore, The primary definition of intrinsic curvature in elemen- curvature of (3+1)-spacetime always means intrinsic cur- tary geometry is given by Eqs. (31) and (32) in terms of vature. Generally, intrinsic curvature is determined by the deficit rotation angle for geodesic triangles and poly- measurements intrinsic to the space considered. The in- gons. trinsic curvature of a surface of an apple at a point P can be determined by measuring angles and lengths on the surface of the apple near P . B. Curvature from LONB-rotation angle around closed curves

A. Curvature from deficit tangent-rotation angle Cartan used a different method to compute the same around closed curves deficit angle δC around a curve C.

0 Cartan’s method for computing curvature is missing in The sum of inner angles in a triangle is always 180 in many textbooks on general relativity, and it is not taught the Euclidean plane. in most graduate programs in gravitation and cosmol- For N-polygons as boundaries of simply connected ar- ogy. Therefore, we give a short introduction to Cartan’s eas in the Euclidean plane, the sum of inner angles is not method. useful, it is (N −2)π and goes to infinity for N → ∞. It is Instead of using as a path an arbitrary closed curve much more useful to consider the sum of tangent-rotation C = ∂A, and instead of considering the rotation of the angles αi. For boundart-curves C of simply connected ar- tangent vectors along this curve C, Cartan used (1) the eas A, the integral of tangent-rotation angles is always a rotation of the chosen LONB-field, already crucial for full rotation, 2π, in the Euclidean plane, Secs. I and III, (2) the closed path along a coordinate plaquette [¯eα, e¯β ] ≡ [ ∂α, ∂β ] in the chosen coordinate (tangents) dα (tangents) αi ⇒ ( ) ds = 2π. system. Therefore, Cartan’s method needs the LONB- IC ds Xi rotation for a displacement in a coordinate, (ωˆiˆj )µ. — Cartan needs a coordinatization, i.e. a mapping P ⇒ In a non-Euclidean 2-space, e.g. on the surface of an xµ, but the metric g is not needed to compute the apple, N-polygons and curves C which are the boundary µν Riemann tensor with Cartan’s method. The metric gµν of a simply connected area A, C = ∂A, have a sum, resp. cannot be known until Einstein’s equations are solved for integral, of tangent-rotation angles different from a full the specific problem. rotation of 2π, In curved 2-space, LONB-rotations are given by an angle, no need for a rotation matrix, (ω ˆ)µ ⇒ ωµ. (tangents) dα (tangents) aˆb αi ⇒ ( ) ds ≡ 2π − δC, In the Euclidean plane, the total LONB-rotation angle IC ds Xi along any closed curve C = ∂A is always zero, δC ≡ deficit rotation angle. (31) dα (LONBs) ≡ µ ( µ ) ωµ : ωµ dx = 0. As an example, we compute the deficit angle δ for a dx IC geodesic triangle on the surface of a spherical Earth: (1) start at the North-pole and go South along the For a Cartesian LONB-field in Euclidean 2-space, the zero-degree meridian to the equator, LONB-rotation angle is identically zero. For the LONB- (2) go East along the equator to the 900 meridian, field aligned with polar coordinates in the Euclidean 2- (3) follow the 900 meridian back to the North-pole. space, (1) the LONB rotation angle is nonzero for a dis- This is a geodesic triangle with three tangent-rotation placement in the direction ∂φ, (2) the LONB-field is sin- angles of π/2, the total tangent-rotation angle is 3π/2, gular at the origin, hence it is not admitted, if the origin hence the deficit angle is δ = π/2. is inside A. 13

In curved 2-space, the total rotation angle of LONBs We now need Cartan’s tools: connection 1-form, exte- relative to infinitesimal geodesic pieces along the posi- rior derivative, and Cartan’s curvature 2-form. tively oriented boundary C = ∂A gives the deficit angle, In curvilinear coordinates (in curved space or flat space), the antisymmetric covariant derivative of a vector µ ¯ ωµ dx ≡ − δC ≡ − deficit angle. (33) field V in covariant components is equal to the ordinary IC antisymmetric derivative, The line-integral calls for the 1-form ω˜ with covariant [ ∇µVν − ∇ν Vµ ] = [ ∂µVν − ∂ν Vµ ]. components ωµ. Using 1-form components ωµ causes ?? the absence of metric factors gµν in the line-integral ( ) This holds, because the Christoffel connection coefficients over curved space. α (Γ β)δ are symmetric in the last two indices. For two areas touching along a piece of common bound- We shall see that covariant derivatives play no role in ary, the total deficit angle is additive, because the LONB- Cartan’s method of computing curvature. rotation angles cancel along the common boundary due In the calculus of forms (here for Riemannian geom- to the opposite direction of the boundary curves. The etry, which has a metric), a vector, if and only if it is areas are also additive. Therefore, the total deficit an- given by its covariant components is called a 1-form and gle divided by the total area, (δC/A) is the same for any denoted by a tilde (instead of an arrow or a bar),σ ˜ with (simply connected) area on a perfect sphere. — For the 1-form components σµ. The index structure of differen- surface of an apple, an infinitesimal curve C around any tial forms is trivial, therefore one often drops indices, point P gives the local measure of intrinsic curvature, the Gauss curvature at P of Eq. (32), 1-formσ ˜ ⇔ 1-form components σµ, −1 RG = lim ( A δC ). The antisymmetric derivative of a 1-formσ ˜ is called C→0 exterior derivative, denoted by the symbol d in d σ˜, As an example of Cartan’s method of LONB-rotaton angles around a closed curve, we compute the intrinsic exterior derivative: curvature of the surface of a spherical Earth We choose d σ˜ ⇔ (d σ˜)µν ≡ ∂µ σν − ∂ν σµ. the LONB field (~eEast, ~eNorth). For the closed boundary- path ∂A, we must avoid the singular points of this LONB The exterior derivative of a 1-formσ ˜ produces d σ˜, a co- field, North pole and South pole: variant antisymmetric 2-tensor, which by definition is a (1) go South along 00 meridian, start an infinitesimal 2-form. — General 2-forms are defined as antisymmetric δθ away from North pole, end at equator: LONB rotation covariant 2-tensors. angle = 0, In 3-space, the Hodge-dual star-operation of d σ˜ is (2) go East along equator to 900 meridian: LONB ro- formed with the Levi-Civita tensor, which is defined by tation angle = 0, (1) totally antisymmetric tensor, (2) ε1ˆ2ˆ3ˆ ≡ +1 in a (3) go North along 900 meridian to δθ before North LONB with positive orientation, as discussed in Eq. (12). pole: LONB rotation angle = 0, Applying the Hodge-dual star-operation to the exterior (4) go West along δθ = infinitesimal constant, from derivative of a 1-form in 3-space, ∗ d σ,˜ gives the curl of 900 meridian to 00 meridian: the vector field ~σ, The total LONB-rotation angle is negative, α = − π/2, the deficit angle is positive, δ = π/2, measured (∗ d σ˜)λ = (curl ~σ)λ. area A = (π/2) r2 , intrinsic curvature R = r−2 . Earth G Earth Hence, Cartan’s curl equation for curvature in two spa- tial dimensions, Eq. (35), can be re-written in the calcu- C. Cartan’s curvature equation in two dimensions lus of forms, Cartan’s curvature scalar in 2 dim.: We convert the line-integral for the deficit angle δ of Eq. (33) using Stoke’s theorem, 1 R = ∗ d ω˜ = εµν [ ∂ ω − ∂ ω ]. (36) 2 µ ν ν ν µ δC = − ωµ dx = − (curl ~ω) · dA.~ (34) IC=∂A ZA Cartan’s equation for the curvature 2-form components in 2 dimensions, In two dimensions, curl ~ω and the Gauss curvature RG are both scalars under rotation and space reflections Rµν = ∂µ ων − ∂ν ωµ. (37) (even under parity). The rotation deficit angle δ and the oriented area A are both pseudoscalars. The geometric meaning of the two antisymmetric co- variant indices in (dω)µν : (1) find the infinitesimal Lorentz transformation of Cartan curvature eq. in 2 dimensions LONBs under a first displacement along an observer’s RG = − curl ~ω, (35) worldline,e ¯ν ≡ ∂ν , to obtain ων , 14

ˆˆ (2) compare this quantity on a neighbouring worldline D. Exact Ricci curvature R00 from separated by a second displacement ∂µ to obtain ∂µ ων , relative acceleration of quasistatic particles (3) take the difference with the same in opposite order in (1+1)-spacetime µ ⇔ ν to obtain (∂µ ων − ∂ν ωµ), which means going around the closed displacement curve C = ∂A, the coor- We prove the crucial new theorem stated in the title dinate plaquette [∂µ, ∂ν ]. of this section, valid in arbitrarily strong gravitational The displacement indices must necessarily be an anti- fields. As a preparation, we start with (1+1)-dimensional symmetric covariant index-pair. Only in this case is the spacetime in this subsection . displacement plaquette closed. Example on Earth: if you We compute the curvature in (1+1)-spacetime with our o o o go East by 10 , go North by 10 , go West by 10 , go spacetime slicing Σt and our radially parallel LONBs. We o South by 10 , you have a closed plaquette. If instead, also need a spatial coordinate P ⇒ xP . We choose the generated by ∂aˆ, you go East 100 km, North 100 km, Riemann normal 1-coordinate x with gxx ≡ 1, i.e. the go West 100 km, go South 100 km, you have no closed coordinate xP is equal to the measured distance from path. A coordinate-displacement plaquette, formed by the observer to P on Σt. [∂µ, ∂ν ], is closed, as it must be for Eq. (33). But a • plaquette formed by [∂µˆ, ∂νˆ] is not closed. This is the For curvature computations in this first paper, we reason, why the displacement-plaquette indices must be consider an inertial primary observer. This is suf- ˆˆ coordinate indices. ficient to derive Einstein’s R00 equation from New- The curvature 2-form Rµν in 2 dimensions, Eq. (37), tonian experiments. written suppressing the displacement index-pair, i.e. without the plaquette index-pair [µν], is denoted by In a second paper, spacetime curvature will be computed script-R, for a non-inertial observer. We choose the spacetime coordinate-plaquette [∂t, ∂x] curvature from deficit rotation angle in 2-dim, formed on one side by the worldline of the primary ob- server at x =0. without LONB-rotation indices: (1) For a displacement along a worldline of an inertial R = d ω,˜ (38) primary observer at r = 0, which is Cartan’s curvature equation in 2 dimensions. along inertial worldline : [(ωrˆtˆ)t]x=0 = 0. Up to now in Sect. IV, an LONB-rotation in 2 dimen- sions has been given by one number, the rotation an- (2) With LONBs spatially parallel, the LONB Lorentz- gle. We now start using again the matrix notation for in- boost angles vanish in the x direction, t = fixed, aˆ finitesimal LONB-rotations and Lorentz-boosts ω ˆ for b along spatial geodesics : (ωxˆtˆ)x = 0. a given displacement in the coordinate basis ∂ν ≡ e¯ν , which gives Cartan’s connection 1-form (ωaˆ ) . Taking ˆb ν (3) For an auxiliary observer at x = fixed and infinites- the exterior derivative with ∂µ as in Eq. (37), but now imal, the LONB-Lorentz-boost angle for a displacement in the notation of an infinitesimal LONB-transformation- along his worldline, i.e. the relativistic connection coef- matrix, gives, ficients (ωaˆˆb)0, are determined exactly by experiments with quasistatic inertial test particles initially at rest rel- Raˆ Cartan’s curvature 2-form = ( ˆb)µν . ative to the auxiliary observer, as in Eqs. (1, 9), ˆ LONB transformation indices = [a, ˆ b ], along worldline δx constant: plaquette for displacements in coord. = [ µ, ν ].(39) (g) (ωxˆtˆ)t = − Ex dv This completes our derivation of Cartan’s curva- quasistatic = − ( )inertial particle = − ax. ture equation in (1+1)-spacetime and in 2-space, which dt started with the total rotation angle of LONBs around a With our choice of the spatial coordinate, the coordinate closed curve, Sec. IV B, distance x is the measured geodesic distance. Around the infinitesimal [ t, x ]-plaquette of the observer, the co- Raˆ aˆ ( ˆb)µν = ( dω ˆb)µν , ordinate time t is the time measured by the observer, to aˆ aˆ first order in x. The measured velocity of quasistatic R ˆ = dω ˆ. (40) b b particles is v = dx/dt. The relative (tidal) acceleration of two neighbouring inertial quasistatic test particles, one constantly at rest relative to the primary inertial observer, the other one starting at rest relative to the primary and to the auxil- iary observers, dv ∂ [(ω ) ] = − div E~ = − ∂ [( )quasistatic], (41) x xˆtˆ t g x dt freefalling 15

(g) where ∂xEx = div E~g, because the LONBs are radially [ µ, ν ], divided by the plaquette area, is given by, parallel at all times. Raˆ The space-time curvature is obtained by considering ( ˆb)µν : (46) the deficit Lorentz-boost angle δ of LONBs around our a,ˆ ˆb = LONB indices of deficit Lorentz-transf., infinitesimal [ t, x ]-plaquette, µ, ν = coord. plaquette displacement-indices.

µ δ = − (ωxˆtˆ)µ dx , (42) Dropping the displacement indices around the plaquette I[x,t] gives the curvature-2-form Raˆ . ˆb In more than two dimensions, Cartan’s curvature for- where the displacements in x do not give a contribution, mula has an extra term, which vanishes for our primary- since the LONBs are radially parallel, and the displace- inertial-observer adapted LONBs. — We report Cartan’s ment along the primary observer’s worldline at x = 0 curvature equation from [1], does not give a contribution, since the primary observer is inertial, hence the LONBs along his worldline are self- aˆ aˆ aˆ sˆ R ˆ = dω ˆ + ω sˆ ∧ ω ˆ. (47) parallel. b b b The deficit Lorentz-boost angle δ per measured plaque- The second term vanishes on the worldline of a inertial tte area is per definition Cartan’s (1+1) Riemann curva- primary observer with his radially parallel LONBs, be- tˆ cause all connection coefficients vanish on his worldline. ture (R xˆ)tx. With coordinate indices equal to LONB indices at P0,e ¯a =e ¯aˆ, the contraction of the two spatial tˆ indices gives the Ricci component R ˆ ˆˆ t F. Exact Ricci curvature R00 from inertial primary observer, radially parallel LONBs relative acceleration of quasistatic particles in (3+1) space-time tˆ tˆ ~ quasistatic R tˆ = (R xˆ)tˆxˆ = −div Eg = −∂x(ax)freefalling , (43) Along the worldline of a primary observer, his LONBs The relationship between curvature and measurements and our coordinate bases are identical,e ¯ =e ¯ . by non-inertial observers will be given in a second paper. aˆ a Away from the worldline of a primary inertial observer, e¯aˆ =¯ea continues to hold to first order in r for our choice • One can measure the curvature R0ˆ0ˆ exactly in arbi- of slicing Σt, LONBs, and 3-coordinates. trarily strong gravitational fields with test-particles In curved (3+1) spacetime, a primary inertial observer which are quasi-static relative to the inertial ob- measures the spherical average < ... >spherical of the ra- server withu ¯ =e ¯ as shown in Eq. (43). obs 0ˆ dial accelerations aradial = (dvr/dt) of freefalling test- particles initially at measured radial distance r. The test-particles at r can be chosen quasi-static relative to E. Cartan’s curvature equation the primary observer at r = 0, and they will remain qua- in more than two dimensions sistatic within an infinitesimal time-interval. The spher- ical average is equal to the average over the three prin- For Cartan’s curvature equation in three and more di- cipal directions (~exˆ, ~eyˆ, ~ezˆ). Hence, the generalization of mensions, we need the antisymmetric product of two 1- Eq. (43) to (3+1) spacetime is, formsσ ˜ andρ ˜, called the exterior product and denoted by a wedge,σ ˜ ∧ ρ˜, for inertial primary observer: ˆ00ˆ relative freefalling (R )exact = (−∂r spherical)quasistatic exterior product: [˜σ ∧ ρ˜]µν ≡ σµ ρν − σν ρµ. (44) = (div E~g)inertial primary observer. (48) The Hodge-star-dual of the exterior product of two 1- Both equations are wrong for: forms in three spatial dimensions, [ ⋆ (˜σ ∧ ρ˜)]i, is equal to the vector product [ ~σ × ~ρ ]i , • a non-inertial primary observer, • [ ⋆ (˜σ ∧ ρ˜)]ˆi = εˆiˆjkˆ σˆj ρkˆ = [ ~σ × ~ρ ]i. (45) secondary observers (to obtain the relative acceler- ation) whose LONBs are not radially parallel with We omit all equations which are not needed for Cartan’s the LONBs of the primary observer. curvature equation. Conclusion: In dimensions n higher than 2, a connection Lorentz transformation of LONBs is given by a (n x n)-matrix. • For a freefalling-nonrotating primary observer with For a coordinate-displacement µ, it is given by Cartan’s u¯obs =e ¯0ˆ, the exact Ricci component R0ˆ0ˆ of gen- connection-1-form (ωaˆ ) . ˆb µ eral relativity for arbitrarily strong gravity fields is The curvature, the deficit Lorentz transformation un- der a displacement around the coordinate plaquette unambigously determined 16

by measuring the relative acceleration of neigh- For R0ˆ0ˆ, a tensor component, it is irrelevant, whether bouring freefalling test-particles which are qua- the primary observer is inertial or non-inertial, and it sistatic relative to the observer. is irrelevant, whether we choose LONBs away from the primary worldline radially parallel or not (which dictates R0ˆ0ˆ is linear in the gravitational fields. the choice for auxiliary observers). — But div E~g in- Observers which are not freefalling-nonrotating will be volves Ricci connections, all depends on inertial versus treated in a separate paper. — For a uniformly rotating- non-inertial primary observer (field equation linear in E~g freefalling observer, there is only one fictitious force, the versus bilinear) and it depends on LONBs radially par- centrifugal force, causing the radial acceleration of quasi- allel or not (two fields, E~g and B~ g, versus more than a static test-particles, hundred connection coefficients). ~ ~ It has often been emphasized that a fundamental dif- ~acentrifugal = [ Ωobs × [ ~r × Ωobs ]] ference between general relativity and Newtonian physics 1 is the non-linearity of Einstein’s equations versus the = [ B~ g × [ ~r × B~g ]]. (49) 4 linearity of the Newton-Gauss field equation div E~g = − The radial derivative of the centrifugal acceleration at 4πGN ρmass. — Nothing could be farther from the truth: 0ˆ0ˆ r = 0 gives an additional term for R , ˆˆ • Einstein’s R00(P ) equation with nonrelativistic source-matter and the gravitational field equation R0ˆ0ˆ = (−∂ extra )freefalling extra r radial spherical quasistatic of 19th-century Newton-Gauss physics are ~ 2 = − const. Bg . (50) explicitely identical.

0ˆ0ˆ G. Einstein’s R equation Both field equations are: for nonrelativistic matter-sources identical with Newton’s equation of relative acceleration 1. linear for a freefalling-nonrotating observer with worldline through P , withu ¯obse =e ¯0ˆ, and with our radially parallel LONBs, The exact equality between the space-time curvature R 0ˆ0ˆ(P ) and the relative acceleration of freefalling par- 2. nonlinear for non-freefalling and/or rotating ticles, spherically averaged, for a freefalling-nonrotating observers with worldlines through P , because fictitious forces are equivalent to gravitational observer with worldline through P andu ¯obs(P )=¯e0ˆ(P ), Eq. (48), gives the proof that, forces, Sec. III, and e.g. the centrifugal term is quadratic in B~ g, Eq. (50). • In arbitrarily strong gravitational fields, Newton’s law for acceleration of freefalling particles at δr, measured relative to a freefalling-nonrotating ob- H. From nonrelativistic source-matter to the full server and spherically averaged, Einstein equations angular av. −2 = − GN Mδr (δr) , (51) inertial obs. Using, is explicitely identical 1. Lorentz covariance,

0ˆ0ˆ with Einstein’s R equation for source-matter 2. the contracted second Bianchi identity to sat- non-relativistic relative to this inertial observer, isfy the covariant conservation of the energy-

0ˆ0ˆ momentum-stress tensor, R = 4πGN ρmass. (52) one directly obtains Einstein’s equations, as explained in Only after choosing an inertial primary observer and all textbooks, LONBs radially parallel, is div E~g uniquely defined for a given physical situation. With these two choices and aˆˆb aˆˆb only with these choices, we obtain, G = 8 π GN T , (54)

aˆˆb aˆˆb 1 aˆˆb 0ˆ0ˆ ~ LONBs rad.parallel G ≡ R − η R. (55) R = − ( div Eg )inertial primary obs. = 4 π GN ρmass (53) 2 The explicit field equation for non-inertial observers is This completes our derivation of Einstein’s equations of bilinear in the gravitational fields (E~g, B~ g). These ex- general relativity from Newtonian experiments on rela- plicit field equations will be given in a second paper. tive acceleration. 17

I. Comparison with “heuristic derivations” this Einstein equation is identical to the 19th-century of Einstein’s equations Newtonian field equation: both equations are identical and linear for an inertial observer with worldline through There are two important differences between our paper P , both equations are identical and non-linear for non- and the literature: inertial observers. The first crucial difference is that our paper gives a rig- orous derivation of Einstein’s R0ˆ0ˆ equation from nothing more than Newtonian phsics and Einstein’s concept of inertial motion, while the literature only gives “heuristic J. Outlook derivations” of Einstein’s equations. Straumann [9], for his Sec. 3.2.1 gives the title: Heuris- In this paper, the field equation for R 0ˆ0ˆ (P ) has been tic “Derivation” of the Field Equations (“ ... “ marks by treated only for an inertial primary observer with world- Straumann). line through P and withu ¯obs (P )=¯e0ˆ (P ). This field Wald [10], in contrast to a rigorous derivation, writes equation of Einstein is linear in the gravitational field in Section 4.3, “a clue is provided”, “the correspondence E~g, and it is explicitely identical with the Newton-Gauss suggests the field equation”, but Wald gives no equation field equation, div E~g = −4πGN ρmass. for the correspondence. A first extension: The field equation for non-inertial Weinberg [11] in Sec. 7.1 takes the ”weak static limit”, observers will be given in a companion paper. Again, makes a ”guess”, and argues with ”number of deriva- 0ˆ0ˆ tives”. the 19th-century Newtonian equation and Einstein’s R Misner, Thorne, and Wheeler [12] give ”Six Routes to equation for non-relativistic source-matter are exactly Einstein’s field equations” in Box 17.2: (1) to model ge- and explicitely identical for arbitrarily strong gravita- ometrodynamics after electrodynamics, (2) to take the tional fields. But for non-inertial observers, both the variational principle with only a scalar linear in sec- field equation of 19th-century Newtonian physics and 0ˆ0ˆ ond derivatives of the metric and no higher derivatives, Einstein’s R equation are bilinear in the gravitational (3) “again electromagnetism as a model”, (4) superspace, fields. (5) field equation for spin 2, (6) Sakharov’s view of grav- A second extension: One asks the question, can this itation as an elasticity of space. linearity for inertial observers be extended to an equation In contrast, this paper gives a rigorous derivation of and its solution all over the universe? — In the paper [13], Einstein’s equations from Newtonian physics, not heuris- we have derived and solved exactly the linear field equa- tic arguments. tion for Einstein’s angular-momentum constraint at any The second crucial difference between the literature point P0 from the angular-momentum input of an arbi- 0ˆ0ˆ and our paper: For the R field equation at a given trary distribution of matter on the past light-cone of P0. point P , one can distinguish the matter-source “non- The output is B~g at P0, given as a simple explicit lin- relativistic versus relativistic”. But the term “Newtonian ear retarded integral over the matter-angular-momentum 0ˆ0ˆ limit” is meaningless for the R field equation, because input on the past light-cone of P0 back to the big bang.

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