Einstein’s equations from Einstein’s inertial motion and Newton’s law for relative acceleration Christoph Schmid∗ ETH Zurich, Institute for Theoretical Physics, 8093 Zurich, Switzerland (Dated: February 19, 2018) ˆˆ We show that Einstein’s equation for R00(P ) for nonrelativistic matter-sources in P and for arbitrarily strong gravitational fields, is identical with Newton’s equation for the relative radial ac- celeration of neighbouring freefalling test-particles, spherically averaged.— With Einstein’s concept of inertial motion (≡ freefalling-nonrotating), inertial worldlines (≡ geodesics) in Newtonian exper- ˆˆ iments can intersect repeatedly. This is evidence for the space-time curvature encoded in R00.— These two laws of Newton and Einstein are explicitely identical, if one uses (1) our adapted space- time slicing (generated by the radial 4-geodesics of the primary observer with worldline through P andu ¯obs(P ) =e ¯0ˆ(P ), (2) our adapted Local Ortho-Normal Bases, LONBs (radially parallel with the primary observer’s LONB), and (3) Riemann normal 3-coordinates (centered at the primary observer). Hats on indices denote LONB components.— Our result: Full general relativity follows from Newton’s law of relative acceleration by using Lorentz covariance and energy-momentum con- servation combined with the Bianchi identity.— The gravitational field equation of Newton-Gauss ˆˆ and Einstein’s field equation for R00(P ) are both linear in gravitational fields, if the primary observer (≡ worldline through P ) is inertial. Einstein’s principle of equivalence between fictitious forces and gravitational forces is formulated as a precise equivalence theorem with explicit equations of motion from general relativity. With this equivalence theorem, the gravitational field equation of 19th-century Newton-Gauss physics ˆˆ and Einstein’s field equation for R00(P ) are both bilinear in the gravitational forces for non-inertial primary observers. 0ˆ0ˆ ˆi0ˆ ˆi R = −divE~ g and R = −(curlB~ g/2) hold exactly in general relativity for inertial primary observers, if one uses our space-time slicing and our LONB’s. The gravitoelectric E~ g and the gravito- ~ magnetic Bg fields are defined and measured exactly with nonrelativistic test-particles via (d/dt)(pˆi) ~ ~ ~ ~ and (d/dt)(Sˆi) in direct correspondence with the electromagnetic (E, B) fields. The (Eg, Bg) fields are identical with the Ricci connection for any observer’s LONBs and for displacements along any observer’s worldline, (ωaˆˆb)0ˆ. In the explicit equations of particle-motion of general relativity (using our adapted space-time slicing and our adapted LONBs), there are precisely two gravitational forces equivalent to the two fictitious forces on the worldline of the observer: the force from E~ g, equivalent to the fictitious force measured by an accelerated observer, and the force from B~ g, equivalent to the ficitious Coriolis force for a rotating observer. ˆˆ The exact Ricci curvature component Ri0 can be measured with non-relativistic test particles. ˆi0ˆ 0ˆ0ˆ (R , R )P are linear in the gravitational fields (E~ g, B~ g) for inertial primary observers with world- ˆˆ ˆˆ lines through P . For non-inertial primary observers, (Ri0, R00) are bilinear in the gravitational ˆi0ˆ 0ˆ0ˆ fields (E~ g, B~ g), which are the only gravitational fields in (R , R ). ˆˆ Einstein’s R00 equation for non-relativistic matter and for inertial primary observers gives the 0ˆˆi Gauss law, divE~ g = −4πGNρmass, and Einstein’s R equation gives the gravito-magnetic Amp`ere arXiv:1607.08661v1 [gr-qc] 28 Jul 2016 law, curlB~ g = −16πGNJ~mass.— For relativistic matter and inertial primary observers, Einstein’s 0ˆ0ˆ R equation gives div E~ g = −4πGN(˜ρε + 3˜p), whereρ ˜ε is the energy density, and (3˜p) is the trace 0ˆˆi of the momentum-flow 3-tensor, both in the frame withu ¯obs =e ¯0ˆ. Einstein’s R equation for an inertial observer gives the relativistic gravito-magnetic Amp`ere law, curlB~ g = −16πGNJ~ε. ˆˆ The remaining six Ricci components, Rij , involve the curvature of space-space plaquettes, which are unmeasurable with non-relativistic particles in quasi-local experiments. With our primary-observer-adapted spacetime splitting and LONBs, the equations of motion of general relativity for particles without nongravitational forces and for a noninertial primary observer are (1) form-identical with the 19th-century equations of Newtonian mechanics for nonrelativistic particles, (2) form-identical with the equations of motion for special relativity with the obvious replacements (E,~ B~ ) ⇒ (E~ g, B~ g) and q ⇒ ε ≡ total energy of particle.— We formulate the precise theorem of equivalence of fictitious and gravitational forces in the equations of motion. PACS numbers: 04.20.-q, 04.25.-g, 04.20.Cv, 98.80.Jk ∗Electronic address: [email protected] 2 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.