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Einstein's $ R^{\Hat {0}\Hat {0}} $ Equation for Non-Relativistic Sources

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Einstein's $ R^{\Hat {0}\Hat {0}} $ Equation for Non-Relativistic Sources Einstein’s R0ˆ0ˆ equation for nonrelativistic sources derived from Einstein’s inertial motion and the Newtonian law for relative acceleration Christoph Schmid∗ ETH Zurich, Institute for Theoretical Physics, 8093 Zurich, Switzerland (Dated: March 22, 2018) With Einstein’s inertial motion (freefalling and nonrotating relative to gyroscopes), geodesics for nonrelativistic particles can intersect repeatedly, allowing one to compute the space-time curvature ˆˆ ˆˆ R00 exactly. Einstein’s R00 for strong gravitational fields and for relativistic source-matter is iden- tical with the Newtonian expression for the relative radial acceleration of neighbouring freefalling test-particles, spherically averaged. — Einstein’s field equations follow from Newtonian experiments, local Lorentz-covariance, and energy-momentum conservation combined with the Bianchi identity. PACS numbers: 04.20.-q, 04.20.Cv Up to now, a rigorous derivation of Einstein’s field structive. Newtonian relative acceleration in general La- equations for general relativity has been lacking: Wald [1] grangian 3-coordinates (e.g. comoving with the wind) writes “a clue is provided”, “the correspondence suggests has the same number of 106 uninstructive terms. the field equation”. Weinberg [2] takes the ”weak static The expressions for Einstein’s R 0ˆ (P ) and the New- 0ˆ limit”, makes a ”guess”, and argues with ”number of tonian relative acceleration are extremely simple and ex- derivatives”. Misner, Thorne, and Wheeler [3] give ”Six plicitely identical with the following choices: (1) We work Routes to Einstein’s field equations”, among which they with Local Ortho-Normal Bases (LONBs) in Cartan’s recommend (1) “model geometrodynamics after electro- method. (2) We use a primary observer (non-inertial or dynamics”, (2) ”take the variational principle with only inertial) with worldline through P , withu ¯obs =e ¯0ˆ, and a scalar linear in second derivatives of the metric and no with his spatial LONBse ¯ˆi along his worldline. (3) We higher derivatives”. use the primary observer’s spacetime slicing Σt by radial In contrast, we give a rigorous derivation of Einstein’s 4-geodesics starting Lorentz-orthogonal to his worldline, field equations for general relativity. and t ≡ time measured on his worldline. (4) Most crucial: for measuring relative accelerations of neighbouring test- The crucial input is Einstein’s concept of inertial mo- particles, we need auxiliary observers with LONBs radi- tion (freefalling and nonrotating relative to comoving ally parallel (at a given time) to the primary observer’s gyroscopes): the worldlines of freefalling nonrelativistic LONBs (to avoid unnecessary extra terms). Therefore test-particles, geodesics, can intersect repeatedly. Two the Ricci connection coefficients for radial displacements stones released one after the other from rest at the North from the primary observer’s worldline vanish, Pole freefalling into a vertical well through the center of the Earth to the South Pole and back through the center auxiliary observers of the Earth to the North Pole: Einstein’s geodesics cross with radially parallel LONBs at given time repeatedly, violating the spacetime analogue of Euclid’s ⇔ [(ωaˆ ) ] = 0, (1) axiom of parallels, evidence that space-time is curved. ˆb ˆi r=0 In our decisive first step, we prove that the exact for (a, b, ...) = spacetime indices, (i, j, ...) = 3-space in- arXiv:1609.02195v1 [gr-qc] 7 Sep 2016 ˆ0 space-time curvature encoded in R 0ˆ(P ) (curvature-side dices. Eq. (1) is a special case of Newtonian observers of Einstein’s equation) is fully determined by measuring at relative rest at a given time. (5) We use Riemann test-particles which are quasi-static (or non-relativistic) normal 3-coordinates centered on the primary observer relative to an observer with worldline through P and on the slices of fixed time: r-coordinate lines are radial u¯ (P )=e ¯ (P ). It is superfluous to re-measure R 0ˆ ≡ ≡ obs 0ˆ 0ˆ geodesics, r geodesic radial distance, (θ, φ)P start- with relativistic test-particles. — Further, we prove that ing angles of radial geodesics to P , and (x,y,z)P with R 0ˆ is identical with the Newtonian expression for the rel- 0ˆ the standard connection to spherical coordinates. ative radial acceleration of neighboring freefalling nonrel- In this first paper, we treat only inertial primary ob- ativistic test-particles, spherically averaged, for gravita- servers. Hence the Ricci connection coefficients for dis- tional fields of arbitrary strength and for arbitrary (rela- placements along the primary observer’s worldine vanish, tivistic) source-matter. — Hats over indices denote Local inertial primary observer ⇔ [(ωaˆ ) ] = 0. (2) Ortho-Normal Bases (LONB) following Misner, Thorne, ˆb 0ˆ r=0 and Wheeler, and we use their sign conventions [3]. With Eqs. (1, 2), all Ricci connection coefficients van- 0 The explicit expression for Einstein’s curvature R 0 ish on the worldline of the primary inertial observer. Our in general coordinates in terms of Christoffel connection auxiliary observers cannot be inertial, unless spacetime α α coefficients (Γ β)γ ≡ Γ βγ has 106 terms, utterly unin- is flat. 2 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.
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