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The result: the expression for Einstein’s R 0ˆ in erational definition of the ordinary electric field, where 0ˆ terms of quasi-static (or non-relativistic) test-particles, we replace the particle’s charge by its rest mass m, for gravitational fields of arbitray strength, and for ar- −1 d (g) bitrary (relativistic) source-matter, is exactly and ex- m pˆ ≡ E dt i ˆi plicitely identical with the Newtonian expression, and ⇔ ~a = E~ (g) = ~g, (4) this expression is exactly linear in the gravitational field, ff for freefalling, quasistatic test-particles. inertial primary observer, radially parallel LONBs: Local time-intervals dt are measured on the observer’s 0ˆ ~ ⇔ R 0ˆ = div Eg = div~g. (3) wristwatch. The measured 3-momentum is pˆi with re- spect to the observer’s LONB. For a freefalling test- In our exact operational definition in arbitrary (3+1)- particle, quasistatic relative to the observer, the mea- ~ spacetimes, the gravito-electric field Eg = ~g is the ac- sured gravitational acceleration relative to the observer celeration of quasi-static (or non-relativistic) freefalling (quasistatic) ~ is ~a ff = ~g = Eg, measured by Galilei. test-particles, measured by the chosen observer. But this The LONB-components paˆ are directly measurable. ~ Eg remains exactly valid for relativistic test-particles in This is in stark contrast to coordinate-basis components α the equations of motion and in curvature calculations. p , which are not measurable before one has obtained gαβ In a companion paper, we shall show in detail that by solving Einstein’s equations for the specific problem for non-inertial observers and for quasi-static (or non- at hand. relativistic) test-particles, (1) the exact explicit expres- LONBs off the observer’s worldline are not needed in sion for Einstein’s R 0ˆ and the 19th-century Newto- 0ˆ Eq. (4), because a particle released from rest (or qua- nian expression for relative acceleration of neighbour- sistatic state) will still be on the observer’s worldline af- ing freefalling particles, spherically averaged, are iden- ter an infinitesimal time δt, since δs ∝ (δt)2 ⇒ 0, while tical, if one uses Einstein’s equivalence of fictitious forces δv ∝ δt =6 0. ~ ~ and gravitational forces (Eg, Bg), which has been demon- For a freefalling observer, E~g = ~g is zero on his world- strated explicitely in [4], and (2) that the two identical line: Einstein’s “happiest thought of my life”. expressions are nonlinear in the gravitational fields. Gravito-electric fields E~g of arbitrary strength can be In the second (trivial) step for deriving Einstein’s R 0ˆ 0ˆ measured exactly with freefalling test-particles which are equation, we put non-relativistic source-matter on the quasistatic relative to the observer, Eq. (4). But this matter-side of Einstein’s equation: it follows that Ein- same measured E~g is exactly valid for relativistic test- stein’s R 0ˆ equation for nonrelativistic source-matter and 0ˆ particles in the equations of motion. for gravitational fields of arbitrary strength is exactly The gravito-magnetic field B~g has been postulated by identical with the Newtonian equation for the relative ra- Heaviside in 1893 [5]. Our exact operational definition of dial acceleration of neighbouring freefalling test-particles, B~ g is given in [4]. spherically averaged. The term “weak gravitational fields” for local discus- In a third, well-known step, given in textbooks, one de- sions is often used in textbooks. But “weak gravity” is rives the general Einstein equations from Einstein’s R 0ˆ 0ˆ meaningless locally, because the gravitational field ~g and equation for nonrelativistic source-matter by using local the gravitational tidal field R 0ˆ are not dimensionless. 0ˆ Lorentz covariance and energy-momentum conservation Cartan’s method with LONB-connection coefficients is combined with the Bianch identity. unavoidable for our computation of curvature from mea- These three steps complete our rigorous derivation of surements by non-inertial observers. But Cartan’s LONB Einstein’s field equations for general relativity. — Addi- method is not taught in almost all graduate programs in tional results in [4]. general relativity in the USA, and most researchers have The tools needed in this paper are: (1) our exact never used Cartan’s method to solve a problem. There- ~ operational definition of the gravito-electric field Eg, fore we introduce elements of Cartan’s method. (2) the Ricci connection coefficients for a Lorentz boost of Ricci’s LONB-connection coefficients are illustrated by LONBs under a displacement in time, (ωˆi ) , and (3) our 0ˆ 0ˆ an airplane on the shortest path (geodesic) from Zurich (g) to Chicago and the Local Ortho-Normal Bases (LONBs) identity E = −(ωˆiˆ)ˆ. ˆi 0 0 chosen to be in the directions “East” and “North”. These The gravito-electric field E~ measured by any local ob- g LONBs rotate relative to the geodesic (relative to parallel server (with his LONBs along his worldline) is given by transport) with a rotation angle δα per measured path our exact and general operational definition in arbitray lenght δs, i.e. with the rotation rate ω = (dα/ds). (3+1)-spacetimes, Eq. (4), which is probably new. — In For infinitesimal displacements δD~ in any direction, contrast to the literature, we use no perturbation the- the rotation angle δα of LONBs is given by a linear map ory on a background geometry, no weak gravitational encoded by the Ricci rotation coefficients ωcˆ, fields. — E~g is defined as the measured acceleration of quasistatic freefalling test-particles analogous to the op- δα = ωcˆ δDcˆ. 3
The Ricci rotation coefficients are also called connection The proof: from the point of view of the observer with coefficients, because they connect the LONBs at infinites- his LONBs along his worldline, the gravitational accel- imally neighboring points by a rotation relative to the eration g = a(ff particle) of freefalling quasistatic test- ˆi ˆi infinitesimal geodesic between these points. particles (starting on the observer’s worldline) is by def- Cartan’s LONB connection coefficients use displace- inition identical to the exact gravitoelectric field Eˆi of ments in the coordinates, general relativity, Eq. (4). — But from the point of view γ of freefalling test-particles, the acceleration of the qua- δα = ω δD . γ sistatic observer with his LONBs is by definition identical
In three spatial dimensions, the rotation of LONBs to the exact Ricci LONB-boost coefficients (ωˆi0ˆ)0ˆ, relative to the geodesic from P to Q must be given by a E(g) ≡ [(a ) (relat.to obs.)] = g rotation matrix. For a rotation in the (~exˆ, ~eyˆ)-plane, ˆi ˆi ff particle quasistatic ˆi (relat.toff) ~e cos α sin α ~e = − [(aˆ) ]quasistatic ≡ − (ωˆˆ)ˆ. xˆ = xˆ . i observer i0 0 ~eyˆ − sin α cos α ~eyˆ Q P Galilei measured exact Ricci connection coefficients of 2 For infinitesimal displacements, hence infinitesimal rota- general relativity: (ωˆi0ˆ)0ˆ = δˆizˆ (9.1m/s ) for LONBs in tions (first derivatives in α), the rotation matrix is, directions East, North, vertical. Our general, exact definition of the gravitomagnetic ~exˆ 0 1 ~exˆ ~ ~ (relat.to obs.) = 1 + α . field, Bg/2 ≡ Ω gyroscope , is discussed in [4]. The ~eyˆ −1 0 ~eyˆ Q P Ricci connection coefficients (ωˆiˆj )0ˆ equal minus the pre- cession rate of gyroscopes (comoving with the observer), The infinitesimal LONB-rotation matrix δRˆiˆj is given (ω ) = −Ω (gyro) ≡ −ε Ω (gyro). These exact Ricci by the linear map from the infinitesimal coordinate- ˆiˆj 0ˆ ˆiˆj ˆiˆjkˆ kˆ displacement vector Dcˆ, connection coefficients of general relativity were mea- sured by Foucault in 1853. cˆ δRˆiˆj = (ωˆiˆj )cˆ δD , In striking contrast, Christoffel connection coefficients α α ˆ ˆ (for coordinate bases), Γ ≡ (Γ )γ , have no direct ω1ˆ2ˆ = − ω2ˆ1ˆ = α1ˆ2ˆ = rotation angle in [ 1, 2 ] plane. βγ β physical-geometric meaning, and they cannot be known, The (ωˆiˆj )cˆ are the Ricci connection coefficients. until the metric fields gµν (x) have been obtained by solv- In (1+1)-spacetime, the Lorentz transformation of the ing Einstein’s equations for a given problem. chosen LONBs relative to a given displacement geodesic We write Christoffel connection coefficients with a is a Lorentz boost Laˆ , ˆb bracket: inside the bracket are the coordinate-basis transformation-indices (α, β), outside the bracket is the e¯ cosh χ sinh χ e¯ tˆ = tˆ , coordinate-displacement index γ. e¯ sinh χ cosh χ e¯ xˆ Q xˆ P For curvature computations there are two methods, (1) the standard method with coordinate bases and with tanh χ ≡ v/c, with χ called “rapidity”, and χ ad- Christoffel connections (Γα ) , (2) Cartan’s method ditive for successive Lorentz boosts in the same spatial β γ with Local Ortho-Normal Bases and LONB-connections direction. For infinitesimal displacements, the infinitesi- (ωaˆ ) . mal Lorentz boost Laˆ is, ˆb γ ˆb For a primary non-inertial observer, Cartan’s method is strongly preferred, because a radially parallel LONB- e¯tˆ 0 1 e¯tˆ = 1 + χ . vectore ¯ (P ) off the primary observer’s worldline, which e¯xˆ 1 0 e¯xˆ 0ˆ Q P is highly convenient for measuring relative radial acceler-
In (3+1)-spacetime, and with two lower indices, ωaˆˆb ation, does not point in the same direction as the natural is antisymmetric for Lorentz boosts (and for rotations), coordinate-basis vectore ¯0(P )= ∂t for a rotating or non- freefalling observer. cˆ δLaˆˆb = (ωaˆˆb)cˆ δD , (ωˆi0ˆ)cˆ = − (ω0ˆˆi)cˆ = (χˆi0ˆ)cˆ. Cartan’s curvature equation gives the Riemann curva- ture 2-form Raˆ with 2-form components (Raˆ ) [6, 7]. For a displacement in observer-time, the exact Ricci ˆb ˆb γδ 2-form components are antisymmetric covariant compo- connection coefficients (ωaˆˆb)0ˆ of general relativity can be measured in quasistatic experiments. But these Ricci nents in a coordinate basis, denoted by Greek letters. — connection coefficients predict the motion of relativistic For an inertial primary observer and with our LONBs radially parallel, all of Cartan’s LONB connection co- particles with the equations of motion. aˆ efficients (ω )γ vanish on the worldline of the primary Our gravito-electric field E~g is identical with minus ˆb the Ricci Lorentz-boost coefficients for a displacement in observer, Eqs. (1, 2). Therefore, in Cartan’s curvature time, equation, the term bilinear in the connection, the wedge product (antisymmetric in the suppressed coordinate- E(g) = − (ω ) . (5) basis displacement-indices) [ ωaˆ ∧ ωcˆ ] vanishes. Hence ˆi ˆi 0ˆ 0ˆ cˆ ˆb 4
Cartan’s curvature 2-form R aˆ is equal to the exterior But for non-inertial primary observers, Einstein’s R0ˆ ˆb 0ˆ derivative d of the LONB-connection 1-form ω aˆ in no- equation and the Newtonian relative acceleration equa- ˆb tation free of form-components, tion are both non-linear in the gravitational fields and identical, if one uses Einstein’s equivalence of gravita- R aˆ aˆ ˆb = dω ˆb, (6) tional forces and fictitious forces [4]. For a superficial reader, Gauss’s law in general relativ- where d denotes the antisymmetric ordinary partial ity, Eq. (9), is “nothing new”. However: (1) Our law of ˆ derivative, and (ˆa, b) are the Lorentz-transformation in- Eq. (9) is derived rigorously, and it is exactly linear for in- dices of the LONBs. ertial primary observers. We have not used the usual ap- Writing explicitely the antisymmetric 2-form- proximation of linearized gravity. The exact law is non- component indices [µ,ν] (plaquette indices) on the linear for non-inertial primary observers. (2) Our law of left-hand-side and the antisymmetric pair of derivative- Eq. (9) only holds for auxiliary observers with LONBs index and displacement-index on the right-hand side, parallel along radial geodesics to the LONBs of the pri- Eq. (6) reads, mary observer at a given time. (3) Our law of Eq. (9) aˆ aˆ aˆ does not hold for the Local Inertial Frame (LIF) and ( R ˆ )µν = ( dω ˆ )µν ≡ ∂µ (ω ˆ)ν − [ µ ⇔ ν ]. (7) b b b the Local Inertial Coordinate Systems (LICS) around P0 An instructive elementary derivation of the curvature (used in textbooks), where the basis vectors are parallel equations (6, 7) for 2-space is given in [4]. along geodesics radiating out from one point P0 in all Eqs. (6, 7) give our crucial curvature result for general spacetime directions. Our law of Eq. (9) cannot hold in relativity: a LIF, because (with curvature) LONBs cannot be par- allel on all three sides of the (geodesic) triangle: (i) from inertial primary observer, P0 along the worldline of the primary inertial observer, radially parallel LONBs: (ii) from P0 along the worldline of an inertial particle with nonzero velocity relative to the primary observer, 0ˆ R ˆ0 R 0ˆ = ( ˆi)0ˆˆi = (iii) from the primary to the auxiliary observer at a fixed ˆ 0 ~ ff time t = t0 + δt. (4) Our law of Eq. (9) only holds for = − ∂ˆi (ω ˆi)0ˆ = div Eg = ∂r