arXiv:0801.2907v2 [astro-ph] 3 Sep 2009 ∗ pnae of axes spin frame non-rotating hshsbe rtdmntae yLo ocutin Foucault Leon by demonstrated first 1852. been has This enmaue.()Teeteeysalframe-dragging has small which extremely The and (2) metric, mo- Schwarzschild measured. gyroscope’s curved been the the to due in is tion The which (1) precession, Sitter Relativity: de General by to predicted relative corrections two named precess been not has do zero” fact observational Earth This from quasars. away far scopes lcrncades [email protected] address: Electronic h ieeouinof time-evolution The ti an is It yBniadSme 1.—Na at hr are there Earth Near — [1]. Samuel and Bondi by .SMAYADCONCLUSIOS AND SUMMARY I. bevtoa fact observational gyroscopes, oa nrilae rltv ogvnlclfiuilae)i axes) fiducial local FR given of velocity to perturbations angular (relative linear axes for inertial Relativity local General in hold to rvtmgei field: gravitomagnetic vrg ftecsooia nrycret via currents energy cosmological the of average the hseuto saaoost m`r’ a,bti od for holds it but Amp`ere’s law, to analogous is equation This pc.—I h ouinfra pnuies h 1 the universe open an for solution the In — space. nvre rirrl ls oteMleUies n h de the and Universe Mach Milne Ernst the by cos to formulated a close of arbitrarily presence the universes in and Einste tensors of energy-momentum-stress and universes (FRW) Friedmann-Robertson-Walker force sthe is on.Tesaeo h xoeta uo sthe gyr is cutoff the exponential at the centered of scale sphere The the point. of circumference measured the omltdb ah l fte r xlctyflle you by with fulfilled sector explicitly vorticity are toroidal them the of in currents all energy Mach: by formulated nvre n nEnti’ lsdsai nvre elis We — universe. static closed Einstein’s in and universes Here eiaiewt epc ocsi ie and time, cosmic to respect with derivative eso httehroi eopsto ftria vortici toroidal of decomposition harmonic harmonics the that show We n h ruh Laplacian “rough” the differenc and the ∆ Weitzenb¨ock for the formula differential of of derivation calculus the the sphe Laplacian connecting Rham-Hodge the equations de by with the spaces gives form-identical Appendix are The — and functions. 3-space, Euclidean and ASnmes 42.q 42.g 98.80.Jk 04.25.-g, 04.20.-q, numbers: PACS etre remn-oeto-akruiesswith universes Friedmann-Robertson-Walker perturbed ahspicpe xc rm-rgigvagravitomagnet via frame-dragging Exact principle: Mach’s eso htteeis there that show We oetmconstraint, momentum r Y seprmnal eemndb the by determined experimentally is , eRa-og Laplacian, Rham-Hodge de µ stemaue edscdsac rmtegrsoet h c the to gyroscope the from distance geodesic measured the is .Conclusions A. ( r ( = ) X ~ ..b nrilgiac systems. guidance inertial by e.g. T uih nttt o hoeia hsc,89 uih S Zurich, 8093 Physics, Theoretical for Institute Zurich, ETH ℓm − oa nrilaxes, inertial local a ailfntoswihare which functions radial has − Ω ~ d/dr httesi xso gyro- of axes spin the that gyro )[(1 B ~ flclgrsoe,wihi ungvsteoeainldefi operational the gives turn in which gyroscopes, local of g xc dragging exact /R − ≡ bu the about Einstein’s exp( ) 2 ∇ Ω ~ 2 gyro ..telocal the i.e. nvco fields. vector on − n = ∆ and µr Dtd coe 0 2018) 30, October (Dated: . hsclcause physical G h rvtmgei edi asdb nrycurrents energy by caused is field gravitomagnetic The )] hitp Schmid Christoph 0 ˆ fteai ietoso local of directions axis the of ˆ “Mach , i qain ( equation, form-identical µ 2 form-identical − = H ulcr o h otct etri imnin3- Riemannian in sector vorticity the for curl curl gravitomagnetism − dtrdu,where radius, -dot 4( yterttn at adas h eieinsitof shift perihelion the also (and effect is Earth dragging Mercury) The rotating — the [3]. in by LAGEOS reported the been by measurement Grav- has whose by satellites and taken which [2], data year, B the Probe per from ity milli-arc-sec extracted effect, be 43 Lense-Thirring will be hopefully the to predicted Earth, is rotating which the by effect eto h bevtoa at“ahzero”. “Mach hand fact other observational the the on of Relativity, test General a Einstein’s of test et f“ahzero”? observational theory “Mach precise Can of extremely tests axes? the inertial explain the to resp. to i.e. predict orders gyroscopes, marching of J.A. the axes In gives spin zero”? Who “Mach words: fact Wheeler’s observational the axes, scope cause cal ae yEntMach Ernst by lated ESstlie adamaueeto h perihelion LA- the the of and measurement a B is a shift) Probe (and Gravity satellites by GEOS measurement a fore o h ieeouino nrilae sshown is axes inertial of time-evolution the for − dH/dt /r ℓ u h udmna usinremains: question fundamental the But nase oti udmna usinwas question fundamental this to answer An + ∆ a ffc h rcsino gyroscopes. of precession the affect can 1 = 2 freo m`r srpae ya by replaced Amp`ere is of -force o R akrud with backgrounds FRW for om ihtecr oain eas give also We notation. curl the with forms ie xeietlyb h precession the by experimentally given s ewe h eRa-og Laplacian Rham-Hodge de the between e ∗ ) i udmna et o h principle the for tests fundamental six t µ o h -pee h yeblc3-space, hyperbolic the 3-sphere, the for . nvre.—Tetm-vlto of time-evolution The — universes. W 2 ogclcntn.Ti nldsFRW includes This constant. molgical nlgu eut odi lsdFRW closed in hold results Analogous nssai lsduies,adfor and universe, closed static in’s ) ia esl,Nuan,adHankel and Neumann-, Bessel-, rical soeadgigtruhtesource the through going and oscope yfilsi em fvco spherical vector of terms in fields ty itruies.Hnethe Hence universe. Sitter A xlisteosre ieeouino gyro- of time-evolution observed the explains eto w hnscombined, things two of test ~ ouin.—W hwta only that show We — solutions. r g nvriiyfilsi imnin3- Riemannian in fields vorticity on all esrdrltv odsatquasars. distant to relative measured = for H sooia ore n 2 and source, osmological − time-dependent nrilframes inertial all 16 steHbl ae o sthe is dot rate, Hubble the is πG ierprubtosof perturbations linear 4 ]i his in 5] [4, N witzerland J ~ ε with K iutos ∆ situations. yaweighted a by B K ~ iino the of nition ( = g s in ism ( = postulate curl = postulate ± Yukawa − 1 J nteoehn a hand one the on ~ πR 1 ε , , A 0) via ~ 0) all all g is . . htphysi- What htinertial that formu- There- 2 axes exactly follow an average of the motion of masses that all FRW models with arbitrary linear perturbations in the Universe. Mach’s postulate demands exact frame also satisfy Mach’s postulates. This means that Cosmo- dragging, not merely a little bit of frame dragging as in logical General Relativity (at the level of linear perturba- the Lense-Thirring effect. — Mach did not have a mech- tions of FRW universes) is a true ”Theory of Relativity”. anism for dragging. The new force came with General We have shown in [6, 7] that the axis directions of Relativity: Gravitomagnetism, exemplified in the Lense- local inertial frames everywhere in the universe exactly Thirring effect. — Mach also did not know, what average follow a weighted average of the energy currents in the of the cosmological mass currents should be taken. universe, i.e. there is exact frame-dragging, for linear Mach’s starting postulate was: No absolute space, in perturbations of a Friedmann-Robertson-Walker (FRW) modern parlance no absolute element in the input. Only background with K = 0 (i.e. spatially flat). The weight relative motion of physical objects is significant. — From function has an exponential cutoff at the H-dot radius, this requirement follows Mach’s postulate of frame in- where H is the Hubble rate, and dot is the derivative variance of observable predictions under time-dependent with respect to measured cosmic time. Hence we have rotations of the reference frame. In Mach’s words [5]: demonstrated the validity of the hypothesis formulated by Mach about the physical cause for the time-evolution “Obviously it does not matter, whether we of inertial-frame axes for linear perturbations of a FRW think of the Earth rotating around its axis, or background with K = 0. The proof follows from Ein- if we imagine a static Earth with the celestial stein’s G0ˆˆi equation, the momentum constraint. The dy- bodies rotating around it.” namical mechanism is cosmological gravitomagnetism. In the present paper we show that results analogous From these requirements follows Mach’s postulate [4]: to those derived for K = 0 in [7] are valid for linear Centrifugal forces and the rotation of the plane of Fou- perturbations of FRW backgrounds with K = 1, i.e. cault’s pendulum are totally determined by the relative backgrounds which are spatially spherical (S3) resp.± hy- motion of masses in the universe with respect to Earth- perbolic (H3). fixed axes. In modern parlance: The time-evolution of Many alternative and inequivalent postulates have local inertial axes is totally determined by the energy cur- been proposed under the heading “Mach’s principle” by rents in the universe including the effect of gravitational many authors after E. Mach. An incomplete list of 10 in- waves. — The demands “no absolute element,” “frame equivalent versions of Mach’s principle has been given by invariance under rotational motion,” and “totally deter- Bondi and Samuel [1] (based in part on [8]). While the mined” necessitate the postulate of exact dragging of in- versions “Mach 9” and “Mach 10” listed by [1] are closely ertial axes by energy currents in the universe. With only related to our “Fundamental Tests 3 and 4” in Sec. V H, partial dragging it would be impossible to fulfill any of their versions “Mach 1-8” definitely do not correspond the three postulates ”no absolute element in the input”, to anything in the writings of Mach. Furthermore each ”frame invariance of the solutions”, and ”time evolution one of their versions “Mach 1-8” is either not true in cos- of inertial axes totally determined by cosmological energy mological General Relativity, as stated already in [1], or currents”. irrelevant, as we explain in Sec. I J. For a purely local analysis, if one has only laboratory A more detailed introduction to the conceptual issues experiments available (like Newton’s bucket experiment), of the postulate formulated by Mach and discussions of the local nonrotating frame is an absolute element. — publications [9]-[16] (mostly cosmological as postulated But, in contrast, for a global analysis, if one considers by Mach) are given in Secs. I and II of the companion observations of the entire universe, the local nonrotating paper [7]. Additional references are given in [8, 12, 13, frame is not an absolute element in the input, because 17, 18]. Comparisons of our methods and results with (according to Mach) the nonrotating frame is totally de- the recent papers by Biˇc´ak, Lynden-Bell, and Katz [16, termined by mass motions (energy currents) in the uni- 19] are at the end of Sects. IF and VH. — We use the verse, it is nonrotating relative to an average of the energy conventions of Misner, Thorne, and Wheeler [20]. currents in the universe. The “relativists” (Huyghens, Berkeley, Leibniz, Mach, and others) wanted a theory without absolute space B. Vorticity perturbations (without an absolute element in the input), a theory, where only relative motion is significant. After Ein- We now summarize the main results of the present pa- stein’s theory of General Relativity it was soon recog- per for the case when only vorticity perturbations (= nized that this theory, when applied to the solar system, 3-vector perturbations, by definition divergenceless) are is not a “Theory of Relativity” in the sense that the solu- present. tions violate the postulates “no absolute element”, “solu- Two important results from cosmological perturbation tions frame invariant”, “totally determined”, and “exact theory [21] for K = (0, 1) are needed to understand the dragging”, see Sec. I H. This is why Einstein and oth- following summary. In± the vorticity sector: ers started to focus on cosmological models. The un- perturbed Friedmann-Robertson-Walker models satisfy 1. The slicing of space-time in slices Σt of fixed time, Mach’s postulates. In this paper we derive the new result i.e. 3-spaces, is unique. The lapse function (elapsed 3

measured time between slices) and g00 are unper- world line. In the spirit of Mach one could fix the axis di- turbed. rections either along the world-line of the Earth to Earth- fixed axes, or (in an open, asymptotically unperturbed 2. The intrinsic geometry of each slice Σt, i.e. of 3- universe) to asymptotic quasars. space, remains unperturbed. Details and proofs are given in Sec. III of [7]. Note that Secs. III - V of [7] are written for FRW backgrounds with C. Gravitomagnetic and gravitoelectric fields K = (0, 1). For the± 3-geometry, which remains unperturbed (E3, The general operational definitions (valid beyond per- resp. S3, resp. H3), and for our fixed-time problem (the ~ momentum constraint), the uniquely appropriate coor- turbation theory) of the gravitomagnetic field Bg and ~ dinate choice (gauge choice) is comoving spherical coor- the gravitoelectric field Eg are given via measurements dinates, xi = (χ,θ,φ), where (aχ) is the measured ra- by fiducial observers (FIDOs) with nonintersecting world dial distance with a(t) = scale factor. With this gauge lines and with their local ortho-normal bases, LONBs (3) choice the coefficients of the 3-metric gij are also un- [22]. These FIDOs measure the first time-derivatives of perturbed. The spatial coordinate system is geometrically the LONB components of the 3-momentum, pˆi, for free- rigid apart from the uniform Hubble expansion. Also g00 falling quasistatic test particles, resp. of the spin-vector, is unperturbed, and we choose the time-coordinate t to be Sˆi, for comoving gyroscopes, the measured cosmic time. Hence, with our gauge choice d the only quantity referring to vorticity perturbations is free-falling quasistatic test particle: (p ) mEg, (2) i ˆi ˆi the shift 3-vector β (resp βi = g0i), dt ≡ d 1 2 2 2 2 2 2 2 2 ~ ~ ds = dt + a [ dχ + Rcom(dθ + sin θ dφ )] gyro comoving with FIDO: (Sˆi) (Bg S)ˆi, (3) − i dt ≡−2 × +2βi dx dt, (1) 1 Ωgyro = Bg, (4) ˆi 2 ˆi where Rcom stands for Rcomoving, and Rcom(χ) = − (χ, sin χ, sinh χ) for K = (0, +1, 1). The index of the where Ω are the LONB components of the precession shift 3-vector is lowered and raised− by the 3-metric, e.g. ˆi rate of the spin axes of a gyroscope. Hats over indices de- β = g βj . i ij note components in a local ortho-normal basis (LONB), Our choice of unperturbed comoving spherical coordi- and t is the local time measured by the FIDO. Note that nates for (E3,S3,H3) is singled out by the fact that our LONB components, e.g. p , are directly measurable (for radial coordinate lines (θ and φ fixed) are geodesics in ˆi a given field of LONBs), while coordinate-basis compo- 3-space, an important property in discussions of Mach’s nents, e.g. p , are not measurable without prior knowl- principle. In all other gauges, where perturbed 3-metrics µ edge of the metric g . g are used, the radial coordinate lines wind up more µν µν ~ ~ and more into “spirals” (as time goes on) relative to The operational definitions of Bg and Eg are identical with the operational definitions of the ordinary electric geodesics on Σt. Hence all other gauges (particularly time-orthogonal) give a very awkward way to coordina- and magnetic fields via measurements on charged point tize unperturbed (E3,S3,H3) spaces. test-particles and on charged spinning test particles, ex- Up to now the coordinatization (gauge) is not yet com- cept that the charge q is replaced by the mass m for pletely fixed, there still exists the freedom of residual quasistatic test particles. ~ ~ gauge transformations by time-dependent, spatially rigid The (Bg, Eg)-fields depend on the choice of FIDOs. rotations of the coordinates for 3-space. Rigid rotations For free-falling FIDOs E~ 0, and for nonrotating FI- g ≡ are crucial in Mach’s statement on frame-invariance [5]: DOs (relative to axes of local gyroscopes) B~ g 0, which See the quote in Sec. IA, ”Obviously it does not matter are the twin conditions in the equivalence principle.≡ — ...”. The (E~g, B~ g) fields are directly related to connection We shall show in Sec. IH that our solution of the equa- g 1 g coefficients: (ωˆˆ)ˆ E , (ωˆˆ)ˆ B , where tions of cosmological gravitomagnetism is explicitly form- i0 0 ≡ − ˆi ij 0 ≡ − 2 ˆiˆj B ε B , Ω ε Ω , and Ω = (ω ) . More de- invariant under time-dependent, spatially rigid rotations ˆiˆj ≡ ˆiˆjkˆ kˆ ˆiˆj ≡ ˆiˆjkˆ kˆ ˆiˆj ˆiˆj 0ˆ of the reference frame, i.e. frame-independent (valid in tails about the general operational definitions of (E~g, B~ g) all reference frames connected by time-dependent rigid are given in Sec. IV of [7]. rotations). In contrast, the solutions of the equations Our specific choice for a field of FIDOs: Our FIDOs of General Relativity for the solar system are not form- are at fixed values of our coordinates xi, and we fix the invariant under spatially rigid, time-dependent rotations. spatial axis directions of our FIDOs in the directions of Measuring the angular velocity of a given celestial body our coordinate basis vectors (∂/∂χ, ∂/∂θ, ∂/∂φ). — We requires that one first fixes the residual gauge freedom, recall from Sec. IB that our choice of unperturbed co- i.e. one must first state, relative to what one wants to moving spherical coordinates for (E3,S3,H3) is singled measure the angular velocity. This can be done by mak- out by the fact that the radial coordinate lines (θ and φ ing a definite choice of spatial axis directions along one fixed) are geodesics in 3-space. Additional details about 4 our specific choice of FIDOs are given in [7] at the end in the given background, H = a−1da/dt, H˙ = dH/dt. — of Sec. IV. Eq. (5) has the same form as Amp`ere’s law for stationary ~ 2 The gravitomagnetic vector potential Ag in the vortic- magnetism except for the µ A~g term. — The analogue of ~ ~ ity sector is uniquely determined by Bg curl Ag, be- the Maxwell term (∂ E~ ) in the Amp`ere-Maxwell equation ≡ t cause div A~ 0. — With β g and Ag defined is absent for the time-dependent context of gravitomag- g ≡ i ≡ 0i i independently above, it follows that the shift vector β~ is netism. No inconsistency arises when taking the diver- ~ gence of Eq. (5), since div A~ 0 and div J~ 0 in the equal to the gravitomagnetic vector potential Ag, see also g ≡ ε ≡ Sec. V of [7]. — Our gravitomagnetic vector potential vorticity sector. The analogue of a Maxwell term can- not be present in gravitomagnetism, this would produce A~g is directly proportional to Bardeen’s gauge-invariant amplitude Ψ as shown in Eq. (13). gravitational vector waves, which is impossible. Addi- In Sec. III we shall consider gravitomagnetism on a tional explanations about the momentum constraint are Minkowski background and show that our FIDOs (and given after Eq. (36) in Sec. VI of [7]. Other perturbed Einstein equations and the perturbed hence our E~ and B~ fields) are singled out by the prop- g g evolution equations for matter will not be needed in the erty that all 13 nonzero components of the Riemann ten- context of this paper. The full set of field equations for sor are uniquely determined by spatial and time deriva- ~ ~ linear cosmological gravitomagnetism is given in Secs. V tives of our Bg field alone. Our Bg field gives an efficient and VI of [7]. and physically transparent way to represent all informa- tion in the Riemann tensor. — In the opposite direction: ~ We shall show that our Bg field is determined by the Ricci E. The de Rham-Hodge Laplacian on vector fields ~ components Rkˆ0ˆ up to a homogeneous Bg field, which is in Riemannian 3-spaces equivalent to a time-dependent rigid rotation of the co- ordinates and FIDO axes according to Eq. (4). The absence of a 3-curvature term in Eq. (5) is con- nected to our use of the de Rham-Hodge Laplace oper- ator ∆, which is called “the” Laplace operator in the D. The momentum constraint literature on differential forms in Riemannian geometry [23]-[26]. Except when acting on scalar fields, the de The crucial equation for Mach’s principle is the mo- Rham-Hodge Laplace operator ∆ must be distinguished mentum constraint, Einstein’s G0ˆ equation, for cosmo- 2 µν ˆi from = g µ ν , named the “rough Laplacian” in logical vorticity perturbations. The momentum con- [24, 25].∇ — Unfortunately∇ ∇ 2 has been used in all publi- straint for perturbations of all three FRW background cations on cosmological vector∇ perturbations (as far as we geometries, K = ( 1, 0), is given by one form-identical know), e.g. in [21, 27], and moreover 2 has been called ± equation without curvature terms: Laplacian and/or denoted by ∆ in many∇ publications on vector perturbations, e.g. in [28]-[31]. Using 2 has the ( ∆+ µ2) A~ = 16πG J~ , (5) ∇ − g − N ε unfortunate consequence that curvature terms appear, where (µ/2)2 (dH/dt) (Hubble-dot radius)−2. where none are present when the de Rham-Hodge Lapla- Derivation of Eq.≡ − (5) in Sec.≡ II. Our momentum con- cian ∆ is used. The most important example is electro- straint has the form of the inhomogeneous Helmholtz magnetism in curved space-time, where the equivalence equation with the sign of the µ2-term opposite to the principle forbids curvature terms, if the de Rham-Hodge usual sign. The sign in Eq. (5) causes an exponential Laplacian resp d’Alembertian is used [32]. cutoff, which becomes important beyound the Hubble- In Riemannian 3-space, the de Rham-Hodge Laplacian dot radius. — The source J~ with J ε T 0ˆ for all types of acting on vorticity fields, i.e. fields with div ~a = ⋆d⋆ a˜ = ε ˆi ˆi matter is the directly measurable energy-current≡ density, δa˜ =0, is given by which is equal to the measurable momentum density (for (∆~a) = (curl curl~a) = (⋆d⋆d a˜) , (6) c = 1), hence the name “momentum constraint”. “Di- µ − µ − µ rectly measurable” means that J ε is measurable without ˆi where we have given both the notation of elementary prior knowledge of g0i, which is the output of solving ~ vector calculus and the notation of differential forms with the momentum constraint. — The energy current Jε is d exterior derivative, ( δ) d∗ adjoint operator to ~ analogous to the charge current Jq in Amp`ere’s law. For the≡ exterior derivative, ⋆− Hodge≡ ≡ dual, tilde˜ denoting ~ ≡ perfect fluids and linear perturbations Jε = (ρ + p)~v. — p-forms, and (~a)µ (˜a)µ. Note that (curl~a)µ = (⋆ d a˜)µ. All symbols in Eq. (5) refer to the physical scale, they Details are given in≡ the Appendix . are not rescaled to a comoving length scale. Therefore The de Rham-Hodge Laplacian is singled out among the scale factor a(t) does not appear in Eq. (5). the elliptic differential operators by the following prop- It is remarkable that Eq. (5) holds for time-dependent erties: (1) If in some region all types of sources are zero, gravitomagnetodynamics. All the same, it is an elliptic div ~v = 0 and curl ~v = 0, then also ∆ ~v = 0. (2) partial differential equation: No partial time-derivatives ∆ commutes with d, δ and ⋆. Therefore applying curl, of the perturbation field A~g. Time derivatives only appear div, and grad to an eigenfield of the Laplacian produces 5 again an eigenfield of the Laplacian with the same eigen- other gauge-invariant velocity amplitude vc. The curl of 2 value. — The “rough Laplacian” does not have prop- the velocity field with the amplitude vc is proportional to erties (1) and (2), if the Ricci tensor∇ is not zero. — The the angular velocity of matter relative to the axes of gy- de Rham-Hodge Laplacian is also easier to compute than roscopes at the positions of the cosmic sources. But this the “rough Laplacian”, because no Christoffel symbols has the severe drawback that vc is not measurable without resp. connection coefficients are needed to compute it. prior knowledge of the time-evolution of gyroscope axes The difference between the de Rham-Hodge Laplacian all over the cosmos, which should be the output of solving ~ 1 ~ 1 ~ and the “rough Laplacian” is given by the Weitzenb¨ock the momentum constraint: Ωgyro = 2 Bg = 2 curlAg formula, i.e. by curvature terms. The Weitzenb¨ock for- (g) − − with Ai = g0i. Bardeen’s version of the momentum mula for vector fields, is derived in Sec. 3 of the Ap- constraint (and the version advocated by Biˇc´ak et al) pendix. For a FRW universe with K = 1 and with the needs an absolute element in the input: the local non- curvature radius a 1, the Weitzenb¨ock± formula gives c ≡ rotating frames all over the cosmos. This violates the ∆A~ = ( 2 2K)A.~ ∇ − crucial starting demand: No absolute element in the in- The de Rham-Hodge Laplacian applies to totally an- put. The requirement “the theory should not contain an tisymmetric tensor fields. For totally symmetric ten- absolute element”, discussed in [35], is closely connected sor fields, e.g. for cosmological tensor perturbations, to Mach’s starting point: No absolute space. the appropriate Laplacian is the Lichnerowicz Laplacian Einstein commented about the energy-momentum ten- [33, 34]. For vector fields the Laplacians of Lichnerowicz sor Tµν in a coordinate basis: “If you have a tensor Tµν and of de Rham-Hodge coincide. and not a metric ... the statement that matter by itself determines the metric is meaningless” [36]. This objec- tion of Einstein directly applies to the approach advo- F. No absolute element in the input: cated by Biˇc´ak et al [16, 19]. On the other hand we have The exponential cutoff at the H-dot radius shown in Sec. IX of [7] that this objection of Einstein does not apply to the LONB components of the energy- The crucial difference between our momentum con- momentum tensor. straint, Eq. (5), and the corresponding equation of Bardeen, Eq. (4.12), and the equation advocated by 2 G. The solution of the momentum constraint Biˇc´ak et al [16, 19] is our µ A~g term, which causes the Yukawa cutoff in Eq. (7) below, and which is absent in the solution of the equation advocated by by Biˇc´ak et al. The solution of the momentum constraint Eq. (5) for 2 Our µ A~g term is due to the difference of the sources. cosmological gravitomagnetism on a background of open Our guiding principle and starting point: “No Ab- FRW universes is given by identical expressions for K =0 and K = 1, solute Element in the Input”. The source, the matter − input, must be measurable without prior knowledge of g = A(g). The latter is the output of solving the momen- B~ (P )= 4G d(vol )[~n J~ (Q)]Y (r ), (7) 0i i g − N Z Q P Q × ε µ P Q tum constraint. From Eq. (4) we see that B~ = curlA g g d 1 directly gives the local nonrotating frame all over the Yµ(r)= − [ exp( µr)] = Yukawa force, (8) universe, which should not be in the input, because it dr R − would be an absolute element in the input. — We write where rP Q is the measured geodesic (radial) distance the momentum constraint in terms of the LONB compo- from the field point P (gyroscope) to the source point Q, nents T 0ˆ, which are measurable without prior knowledge ˆi and (2πRP Q) is the measured circumference of the great of A~g, i.e. without an absolute element in the input. circle through Q with the center P. Hence r = aχ, where For Mach’s principle, the relevant factor in the source a is the scale factor, and χ is the radial coordinate (co- moving geodesic distance from the origin). — The struc- is the angular velocity of cosmic matter, Ω~ matter, mea- sured from our position and relative to geodesics (on Σ ) ture of Eq. (7) is analogous to Amp`ere’s force for ordinary t 2 which start in the directions of our chosen fiducial axes. stationary magnetism, however the 1/r of Amp`ere’s In the local ortho-normal bases, the transverse part of force is replaced by the Yukawa force of Eq. (8). — In 3 the energy current T 0ˆ = J (ε) is directly proportional the curved space H the geodesic distance rP Q appears ˆi ˆi in the exponential cutoff and in the derivative of Eq. (8), to Ωmatter (relative to our fiducial axes), which can be ˆi while RP Q appears in the denominator. (g) measured without prior knowledge of g0i = Ai . The Although the structure of Eq. (7) is analogous to the measured transverse matter velocity is directly propor- solution of Amp`ere’s law for ordinary stationary mag- tional to Bardeen’s gauge-invariant velocity amplitude netism, there is a fundamental difference to Amp`ere’s vs, which he does not use in his momentum constraint. law, since Eq. (7) is valid for time-dependent situations, In contrast, Bardeen’s Eq. (4.12) (and the equation ad- i.e. gravitomagneto-dynamics. vocated by Biˇc´ak et al) is written in the coordinate basis The vector structure of Eq. (7) needs explanation, since 0 3 with the source T i , and this is proportional to Bardeen’s there is no global parallelism on a hyperbolic 3-space H . 6

The vectors J~ε(Q) and ~nP Q are in the tangent space at From Eqs. (4, 7) we obtain the source point Q, and ~nP Q is the unit vector pointing ∞ along the geodesic from P to Q. We parallel transport ~ ~ equiv ~ equiv Ωgyro = < Ω matter > dr Ω matter(r) W (r),(9) [~nP Q J~ε(Q)] along the geodesic from Q to P, where the ≡ Z0 contributions× from all sources are added. — We use the 1 W (r) = 16π G (ρ + p) R3 Y (r) (10) arrow-notation of Thorne et al [22] for vectors in tangent 3 N µ spaces to Riemannian 3-spaces from a (3 + 1)-split. This emphasizes the structure analogous to Amp`ere’s law. for perturbations of open FRW universes. Yµ(r) is the Yukawa force given in Eq. (8). The crucial equation for establishing exact (as opposed to partial) dragging of local inertial axis directions is the H. Frame invariance, no boundary conditions condition that the weight function W (r) must be normal- ized to unity,

There is a second fundamental difference between the ∞ solution Eq. (7) for cosmological gravitomagnetism and dr W (r)=1, (11) the corresponding solutions in other theories, Amp`ere’s Z0 magnetism, electromagnetism in Minkowski space, and as it must be for a proper averaging weight function in general relativity for the solar system. The solutions any problem. Our weight function with its Yukawa force, in all these other theories contain an absolute ele- Eq. (10), fulfills Eq. (11) as shown in Sec. V C. ment: Amp`ere’s law and the laws of electromagnetism ~ in Minkowski space do not hold in frames which are ro- Eqs. (9, 11) state our most important result: Ωgyro is tating relative to inertial frames, unless one introduces equal to the weighted average (with proper normalization ~ fictitious forces. to unity) of Ωmatter(r) for all types of matter-energy and Einstein’s equations by themselves are form-invariant. for all types of energy-current distributions. The time- But for general relativity of the solar system, when work- evolution of inertial axes exactly follows the weighted av- ing in rotating frames relative to the asymptotic nonro- erage of cosmic matter motion. Hence the evolution of in- tating frame (the absolute element), one must impose ertial axes is fully determined (not merely influenced) by boundary conditions on the solutions in the asymptotic the cosmic energy currents. There is exact dragging (not Minkowski space. The boundary conditions encode the merely partial dragging) of inertial axes by the weighted absolute element, they explicitely encode fictitious forces average of cosmic energy currents, as postulated by Ernst (which do not have sources within the solar system), Mach [4, 5]. Mach had asked: as explained in Sec. XI of [7]. Hence the solution of “What share has every mass in the determi- Einstein’s equations for the solar system is not form- nation of direction ... in the law of inertia? invariant when going to a frame which is rotating relative No definite answer can be given by our expe- to asymptotic inertial frames. riences.” In contrast, the solution for cosmological General Rel- ativity, Eq. (7), remains valid as it stands, i.e. the solu- Exact dragging of inertial axes by arbitrary cosmologi- tion is form-invariant, if one goes to a frame which is in cal energy currents in linear perturbation theory has been globally rigid rotation relative to the previous reference demonstrated for the first time in our papers [6, 7] for frame: The form of the solution is frame-independent, K = 0 and in the present paper for K = 1. because both sides of Eq. (7) change by the same term. ± Therefore the solution Eq. (7) contains no absolute ele- ment. — No boundary conditions at spatial infinity are J. Alternative, inequivalent postulates proposed by needed for regular solutions in Eq. (7). authors after Mach

Our presentation of Mach’s principle closely agrees with the one of Misner, Thorne, and Wheeler [37] and I. Exact dragging of inertial axes Weinberg [38]. However, many alternative and inequiv- alent postulates have been proposed under the heading The solution Eq. (7) can be rewritten to show exact “Mach’s principle” by many authors after E. Mach. An dragging of inertial axes explicitly. For reasons of sym- incomplete list of 10 inequivalent versions of Mach’s prin- metry under rotations and space reflection relative to a ciple has been given by Bondi and Samuel [1], see also gyroscope at the origin, a general velocity field of mat- [8]. ter for rP Q fixed can contribute to the gyroscope’s pre- In the list of Bondi and Samuel the version “Mach 9” cession only through its term with (ℓ = 1, P = +) in (“The theory contains no absolute elements”) is the pos- the spherical harmonic decomposition. This term is a tulate of the so-called “relativists”: Huyghens, Bishop toroidal vorticity field, and it is equivalent to a rigid ro- Berkeley, Leibniz, Mach, and others. It was the starting ~ equiv tation with the angular velocity Ω matter(r), see Sec. V A. point for Mach. See our “Test 3” in Sec. VH. — The 7 version “Mach 10” (“Overall rigid rotations of a system the radial Green functions for the operator (∆ µ2) act- 3 3 − 3 are unobservable”) is a necessary condition, formulated ing on toroidal A~g-fields with ℓ =1on(E ,S ,H ). The by Mach in his earliest writings [5] quoted in Sec. I A. final result is the gravitomagnetic B~g-field at the position See our “Test 4” in Sec. V H. Globally rigid rotations of the gyroscope, i.e. the precession of the gyroscope, are discussed in Sec. V G. expressed by the energy currents in the universe, and In contrast the versions “Mach 1, 2, 6, 7” are not true the equation which expresses exact dragging of inertial in cosmological General Relativity, as already stated in axes. We also show that Mach’s principle remains valid [1]. — The version “Mach 3” (“local inertial frames in- for perturbations of a universe with arbitrarily small to- fluenced ...”) is irrelevant, because with only partial tal energy inside the Hubble radius, i.e. arbitrarily close dragging one violates Mach’s postulates “no absolute el- to the Milne universe, and for a universe arbitrarily close ement” in the input and “frame-invariance” in the solu- to a de Sitter universe. — We derive the gravitomag- tions (see the quote of Mach in Sec. I.A). — “Mach 4” netic fields for the zero modes of the operator (∆+4K), (“The Universe is spatially closed”) is irrelevant, because i.e. for time-dependent globally rigid rotations of coordi- we show that exact dragging also holds in open uni- nates and FIDO axes in a FRW universe. — In Sec. VH verses. — “Mach 5” (“The total angular momentum of we give Six Fundamental Tests for Mach’s Principle, all the universe is zero”) is not true for that angular momen- intimately related to Mach’s original formulation and all tum which can be defined and is observable without an explicitely fulfilled by our solutions. absolute element in the input (i.e. without prior knowl- The Appendix gives the tools for the de Rham - Hodge edge of the local non-rotating frames), see Sec. IF and the Laplacian on vector fields in Riemannian 3-spaces, specif- end of Sec. II. — “Mach 8” (“ρ/ρcrit is of order unity”) is ically the explicit equations connecting the calculus of irrelevant, because we show in Secs. VD, VF that exact differential forms with the elementary notation with curl dragging also holds for ρ/ρ 1 and ρ/ρ 1. crit ≫ crit ≪ etc. These tools are essential for cosmological gravito- With respect to this list of 10 versions of Mach’s prin- magnetism. ciple we conclude: The criticisms “There are many for- mulations of Mach’s principle” and “It is not clear, what Mach meant” do not apply. II. THE MOMENTUM CONSTRAINT

K. Outline The momentum constraint written in local orthonor- mal tetrad components is given in Eq. (5). It can be de- rived via two different general methods: (i) One can use In Sec. II we derive the momentum constraint for (1) the local orthonormal tetrad (LONB) method of Cartan that momentum density which can be measured as an from beginning to end. This method has been presented input without prior knowledge of the solution of the mo- in Secs. V and VI of [7], where the connection coeffi- mentum constraint (the output), and for (2) the Lapla- cients for vorticity perturbations are given for FRW back- cian ∆ of de Rham-Hodge as opposed to the “rough 2 grounds with K = (0, 1), and G0ˆˆi is derived for K =0. Laplacian” . We have used this method± to derive Eq. (5) also for ∇ ~ In Sec. III we show that our Bg field on a Minkowski K = 1, but we shall not present details again. (ii) Al- background uniquely determines all nonzero components ternatively± one can, in a first step, take the detour via the of the Riemann tensor. standard coordinate-component method with Christoffel To derive the Green function for the momentum con- symbols to obtain the momentum constraint in a coor- straint for the curved 3-spaces S3 and H3, we need the so- 0 0 dinate basis with T i as the source. R i has been de- lution in source-free regions, i.e. we need vorticity eigen- rived in Bardeen [21], Eq. (A2b), and in Kodama and fields of the Laplacian. Sazaki [28], Eq. (D.14b). In a second step one applies the In a first step, in Sec. IVA, we derive scalar eigenfunc- basis transformation from the coordinate basis to the lo- 3 3 tions of the Laplacian on S and H . Our new result: We cal orthonormal basis. — A modification of method (ii) is give the radial functions of the scalar harmonics in a sim- quicker: The basis transformation at the end of method 3 3 3 ple, identical form for all three geometries (E ,S ,H ). (ii) can be replaced by taking Bardeen’s momentum con- In Secs. IVB - IVD we give the expansion of vorticity straint and moving the difference (T 0 T 0ˆ), which is i ˆi fields (toroidal and poloidal) in terms of vector spheri- meaningful at the level of linear perturbation− theory, to ~ ± cal harmonics Xℓm and Yℓm~eχ. We show that the radial the left-hand side of the momentum constraint. eigenfunctions of the Laplacian for toroidal vector fields 0 The momentum constraint for R i of Bardeen and of are the spherical Bessel, Neumann, and Hankel functions Kodama and Sazaki reads in our notation generalized to S3 and H3 by the simple replacements r (χ resp R) given in Sec. I G. ( 2 +2K/a2)Ai = 16πG (ρ + p )vi , (12) ⇒ ∇ c g N 0 0 c In Sec. V we show that only toroidal A~g-fields with 2 ℓ = 1 are relevant for the precession of gyroscopes at the where ac denotes the curvature scale, and refers to the i∇ i origin and that the precession of a gyroscope cannot be physical scale. The 3-vector index i in Ag and vc is low- caused by scalar or tensor perturbations. We then derive ered and raised by the 3-metric. — The zero modes of the 8

2 operator (∆+4K/ac) correspond to time-dependent rigid on Bardeen’s right-hand side we use the “second” Fried- rotations of the coordinates as demonstrated in Sec. V G. mann equation, For non-zero modes of (∆+4K/ac), our gravitomagnetic vector potential Ai , which is equal to our shift vector field K g (dH/dt) = 4πG (ρ + p). (18) i 2 N β , is directly proportional to Bardeen’s gauge-invariant − ac − amplitude Ψ (a combination of shift amplitude and shear amplitude), Combining the first and second steps gives the new op- erator on the left-hand side (geometric output side) of 2 2 Ai (x, t)= βi(x, t)= Ψ(t)Q(1)i(x), (13) the momentum constraint: [∆ + (2 + 2 4)K/ac µ ]= g − [∆ µ2], i.e. the three curvature terms− cancel. The− new − for a given wave number and polarization. For zero term on the right-hand side (measured input side) is our ~ modes there is no shear, and a nonvanishing gauge- energy current Jǫ. This gives Eq. (5). invariant amplitude does not exist. — Similarly for From Eqs. (17, 13) we obtain vi = dxi/dt = vi Ai . s c − g Bardeen’s gauge-invariant velocity amplitude vc Therefore the difference between the momentum den- i i sities (ρ + p)vc, used by Bardeen, and (ρ + p)vs, used i (1)i vc(x, t)= vc(t)Q (x). (14) by us, is analogous to the important difference between i the canonical momentum pi = ∂L/∂x˙ and the kinetic To convert Bardeen’s momentum constraint Eq. (12) momentum ki = m dxi/dt in Lagrangian mechanics for to our momentum constraint, the first step is replacing point particles of charge q in an electromagnetic field, 2 i the “rough Laplacian” = i by “the” Laplacian i i i ∆ (de Rham-Hodge Laplacian)∇ ∇ via∇ the Weitzenb¨ock for- k = p qA , (19) i − i i mula. With the Ricci tensor of the 3-space for FRW the (ρ + p) vs = (ρ + p)[vc Ag], (20) Weitzenb¨ock formula gives Eq. (A.19), − where the 3-indices are raised and lowered by the 3- 2 ~ 2 ~ (∆ )A = (2K/ac)A. (15) metric. The kinetic (resp. canonical) momentum is (resp. − ∇ − i i is not) measurable without prior knowledge of A and Ag. Hence Bardeen’s momentum constraint written in terms The kinetic momentum density is equal to the LONB of the de Rham-Hodge Laplacian is components T 0ˆ = (ρ + p)dx /dt, while the canonical ˆi ˆi momentum is equal to the coordinate-basis components 2 i i (3) (∆+4K/ac)Ag = 16πGN(ρ0 + p0)vc. (16) 0 j j T i = (ρ + p) gij [dx /dt + Ag]. To prove the postulate formulated by Mach (exact The second step is replacing the gauge-invariant mat- dragging of inertial axis directions) the kinetic momen- ter velocity amplitude vc, used by Bardeen, by his other tum, resp the kinetic angular momentum of matter is gauge-invariant amplitude vs, used by us. In our gauge, i the appropriate input. The canonical angular momentum which is shear-free, dx /dt is equal to Bardeen’s gauge- 0 i (e.g. around the z-axis: T φ), cannot be used, because its invariant vs. measurement resp. definition needs an absolute element The replacement of vc by vs is necessary, because the in the input. — More details on kinetic versus canonical field with the amplitude vc, used by Bardeen, is directly momentum and angular momentum in Sec. VIII of [7]. related to the angular velocity of matter relative to spin A cosmological term in Einstein’s equations is diagonal axes of local gyroscopes at the sources throughout the in a LONB, hence a cosmological constant cannot appear universe. This is not a measurable input without prior in the momentum constraint in LONB components. If a knowledge of the solution of Einstein’s equations, i.e. of cosmological constant Λ is absorbed in ρ and p, one has g , A~ , and B~ . — In contrast, the field with the am- 0i g g (pΛ/ρΛ)= 1, and evidently Λ cannot make a contribu- plitude vs, used by us, is directly related to the angular − tion to J~ε = (ρ + p)~v. velocity of matter measured relative to geodesics (on Σt) which start at our position and go in the directions of the chosen spatial axes along our world line (resp geodesics to asymptotic quasars in an open asymptotically unper- III. RIEMANN TENSOR UNIQUELY GIVEN BY OUR ~ FIELD turbed FRW universe). The input needed in the context Bg of Mach’s principle is the field with the amplitude vs. — The difference between Bardeen’s two gauge-invariant ve- In this section we show that our choice of FIDOs and ~ ~ locity amplitudes is hence our (Eg, Bg) fields are singled out uniquely. We specialize to gravitomagnetism on a Minkowski back- (vc vs)= Ψ. (17) ground, since the material in this section will not be − − needed later, although it is very important: Our specific The Ψ term is a geometric output, hence it must be choice of FIDOs is singled out by the property that only moved to the left-hand side, the geometric side (output with our choice of FIDOs the (E~g, B~g)-fields uniquely de- side), of the momentum constraint. For the prefactors termine the Riemann tensor and the gravito-electric and 9

R R αβγδ gravito-magnetic tidal tensors ( ˆiˆj , ˆiˆj ) for vector per- The curvature invariant is RαβγδR = E B 1 · ≡ turbations, 8 [ ˆiˆj ˆiˆj 4 ˆiˆj ˆiˆj ], and the curvature pseudoinvariant ⋆ERE R− B1 B ρσ αβγδ g is ( 2 εαβρσR γδ)R = 4 ˆiˆj ˆiˆj . These in- Rˆˆˆˆ = ˆˆ = E , (21) · ≡ − E B − i0j0 Eij (ˆi|ˆj) variants are not useful for first-order perturbation theory, ⋆ g because local Lorentz transformations of and are 2( R)ˆi0ˆˆj0ˆ εˆipˆqˆ Rpˆqˆˆj0ˆ = ˆiˆj = B ˆ ˆ , (22) ˆiˆj ˆiˆj − ≡ B (i|j) second-order in perturbations. E B 1 R δ R = (curlB~ ) . (23) kˆ0ˆ ≡ rˆsˆ kˆrˆ0ˆˆs 2 g kˆ The connection in the opposite direction, finding our B~ g field from the Ricci tensor: Any vector field in the The purely spatial Riemann tensor is zero for vector per- 3-vector sector, i.e. with zero divergence, is the sum of a turbations, i.e. the intrinsic geometry of each slice Σt field determined by its curl plus a harmonic field, which remains unperturbed, as explained at the beginning of in Euclidean 3-space is a homogeneous field, because we Sec. I B. — A vertical bar denotes the covariant derivative require that the B~ -fields are not singular at spatial in- in 3-space, and a round bracket around two indices de- g ~ notes symmetrization. — The equation for R is Eq. (35) finity. Therefore the Ricci tensor determines our Bg-field kˆ0ˆ ~ of our companion paper [7]. up to a homogeneous Bg-field, which is equivalent to The first equalities in Eqs. (21, 22) directly follow a time-dependent spatially rigid rotation of all FIDOs from the operational definition of the Riemann tensor: according to Eq. (4). — Our solutions of the equations of cosmological gravitomagnetism on FRW backgrounds (1) Rˆi0ˆˆj0ˆ gives the relative acceleration of neighboring with K = (0, 1), e.g. Eq. (7), are form-invariant un- free-falling, quasistatic particles (geodesic deviation), i.e. ± the gravito-electric tidal field . The tidal 3-tensors der time-dependent spatially rigid rotations, see Sec. I H. Eˆiˆj ( , ) must be traceless by definition. The 3-trace The exact dragging of inertial axes, Eq. (9), is indepen- Eˆiˆj Bˆiˆj dent of the choice of FIDOs. of Rˆi0ˆˆj0ˆ is a 3-scalar, therefore it is zero in the vec- ⋆ tor sector. — (2) The Hodge-star dual ( R)ˆi0ˆˆj0ˆ Before Einstein’s General Relativity, gravitomagnetic 1 1 − ≡ and gravitoelectric fields were discussed by Oliver Heav- 2 εˆi0ˆˆpqˆRpˆqˆˆj0ˆ 2 εˆipˆqˆRpˆqˆˆj0ˆ gives the relative precession of− neighboring≡ quasistatic gyroscopes’ spins. The 3-trace iside [42] based on a gravitational-electromagnetic anal- ogy. With his equations Heaviside could have derived of (εˆipˆqˆRpˆqˆˆj0ˆ) is zero because of the cyclic identity of the Riemann tensor, Rabcd +Racdb +Radbc 0. Replacing the the partial dragging of inertial axes by a rotating star ≡ (Lense-Thirring effect), although a factor 4 would have relative precession by the gravito-magnetic tidal field ˆiˆj 1 B been missing. gives a factor ( 2 ) according to Eq. (4) for any choice of FIDOs. − The second equalities in Eqs. (21-23) are only true for our (E~g, B~g)-fields, defined via our specific choice of FI- DOs. Our FIDOs are singled out by the property that all 13 nonzero Riemann components are uniquely determined IV. EIGENFUNCTIONS OF THE LAPLACIAN by spatial derivatives of our 2 vector fields (E~g, B~ g). If FOR SCALAR FIELDS AND VORTICITY FIELDS IN (E3,S3,H3) one also uses time-derivatives of the B~g field and uses curlE~ = ∂ B~ from Eq. (29) in [7], one concludes that g − t g our B~ g field by itself determines all 13 nonzero compo- Cosmological vorticity fields are of a different type nents of the Riemann tensor: Our B~ g field gives an ef- from vorticity fields typical in fluid dynamics and con- ficient and physically transparent way to represent the densed matter physics with their vortex lines (line singu- entire information in the Riemann tensor. larities) and fields often idealized with cylindrical symme- The Weyl tensor C (defined by having no non- try. In cosmology we consider vorticity fields ( vector αβγδ ≡ trivial 4-contractions) and its Hodge dual ⋆C, have been fields with vanishing divergence) which are regular, and used in the literature [39, 40, 41] to define the electric and we expand them in a spherical basis. These same fields magnetic tidal 3-tensors by contracting twice with a time- also occur in the multipole expansion of electromagnetic like unit vector. However, our tidal tensors have a differ- radiation fields [43]. In this section we generalize these ent normalization, because tidal tensors must be directly fields to the curved 3-spaces S3 and H3. — In the first connected to relative acceleration and relative precession paragraph of Sec. IV B we explain, why the methods used (and hence to relative gravito-magnetic fields). The nor- in textbooks and in the literature for CMB-analyses are malization of our tidal tensors in terms of the Weyl tensor not adapted for our purpose. is given by C = 1 and ⋆C = 1 . The fac- ˆi0ˆˆj0ˆ 2 Eˆiˆj ˆi0ˆˆj0ˆ − 2 Bˆiˆj Our aim is to solve the momentum constraint with its 1 1 2 tor 2 in the first equation comes from Cˆi0ˆˆj0ˆ = 2 Rˆi0ˆˆj0ˆ operator (∆ µ ), i.e. the inhomogeneous Helmholtz 1 − valid for vector perturbations. The factor ( 2 ) in the equation. As a preparation for the Green function of second equation comes from the conversion from− the rel- (∆ µ2) we derive the eigenfunctions of the Laplacian ative precession (given by the Riemann tensor) to the for (−E3,S3,H3) inside resp. outside a shell with sources. relative gravito-magnetic field. First we do this for scalar fields. 10

A. Eigenfunctions of the Laplacian for scalar fields The solutions regular at the origin are in (E3,S3,H3) (K) ℓ 1 d ℓ sin qχ ˜qℓ (χ)= R ( ) ( ), (30) Our new results in this subsection: We give the radial −R qdχ qR functions of the scalar harmonics in a simple, identical which we call generalized spherical Bessel functions. form for all three geometries (E3,S3,H3), e.g. Eq. (30), a Writing the radial equation and the solutions in this form which is familiar from the spherical Bessel functions form gives a unified notation for all three geometries for elementary wave physics in Euclidean 3-space. This is (E3,S3,H3). Compared to spherical Bessel functions (fa- in contrast to the complicated derivations and results in miliar from wave physics in Euclidean 3-space) the only the literature. — In Subsec. IV D we shall show that the changes are the replacements of the independent variable, radial functions are also identical for scalar and toroidal r χ, in the radial oscillations and in the derivative, vorticity harmonics, if and only if one uses those vector → ~ − and the replacement of the denominators and prefactors, spherical harmonics, Xℓm, which have point-wise norm r R(χ). — From Eq. (30) we obtain the generalized and LONB components independent of R for fixed (θ, φ). → spherical Bessel functions ˜(K)(χ) for ℓ = (0, 1), Since the momentum constraint Eq. (5) is an ellip- qℓ tic equation (no partial time-derivatives), we consider, (K) 1 in this and all following sections, a FRW universe with ˜q,ℓ=0(χ) = sin (qχ), (31) K = 1ata given time with units of length chosen such qR ± 1 d 1 that the curvature radius is ac = a = 1 at the given time. ˜(K) (χ) = [ sin(qχ)]. (32) We also include the case of FRW with K = 0 (Euclidean q,ℓ=1 − q2 dχ R 3-space E3) at a given time with the scale factor a =1, The generalized spherical Neumann functions are ob- a = 1 (24) tained from Eqs. (30, 31, 32) by the replacements sin(qχ) cos(qχ) and cos(qχ) + sin(qχ). The gen- ds2 = dχ2 + R2 (χ)[dθ2 + sin2 θ dφ2] (25) →− → K eralized spherical Hankel functions are given by h˜(1,2) = RK (χ) = χ, sin χ, sinh χ ,K = 0, +1, 1 . (26) ˜ { } { − } j in.˜ — Relevant for Mach’s principle are the spheri- cal± Bessel and Hankel functions for ℓ = 1 generalized to The following relations will be useful, K = 1. They are given by Eq. (32) and ± R′′ = KR, R′ 2 =1 KR2, (27) − − ˜(1,2)(K) 1 d 1 hq,ℓ=1 (χ)= i 2 [ exp( iqχ)]. (33) ′ ± q dχ R ± where = d/dχ, and R is short for RK (χ). For scalar fields the angular eigenfunctions of the The eigenfunctions of the Laplacian will be used in Laplacian ∆ are spherical harmonics Yℓm(θ, φ) with the context of Green functions with sources at χ = χs, eigenvalues ℓ(ℓ + 1)/R2. — The radial eigenfunctions hence they will be used only for 0 χ<χ resp. for − s of the Laplacian are the generalization to K = 1 of χ>χ . Therefore there are no geometric≤ restrictions on ± s the spherical Bessel functions jℓ(qr), which are the ra- the eigenvalues in our context. dial eigenfunctions in Euclidean 3-space. These general- The simplicity of the elementary functions (32, 33), ized spherical Bessel functions, the hyperspherical Bessel their identical form for all three geometries, the familiar (K) functions, will be denoted in this paper by ˜qℓ (χ) to em- appearance from elementary wave physics in Euclidean phasize the close correspondence to ˜(K=0)(χ) j (qχ). 3-space, and the straightforward derivation are in con- qℓ ≡ ℓ trast to the complicated derivations for the hyperspher- We shall always denote the eigenvalues of the de Rham- ν Hodge Laplacian ∆ by ( k2). For ∆ acting on scalar ical Bessel functions Φℓ (χ) via the associated Legendre − −1/2−ℓ eigenfunctions and on vector or tensor eigenfunctions in functions P−1/2+β(cos y) discussed extensively in [44] and the scalar sector we define q by used by [45].

scalar sector: q2 = k2 + K. (28) B. Vector spherical harmonics It will turn out that q is the wave number in the radial oscillations sin(qχ) in Eqs. (30, 31, 32). In this subsection we introduce the 3-dimensional vec- The radial part of the Laplace operator acting on a tor spherical harmonics (X~ ± , ~e Y ) and the Regge- −2 d 2 d ℓm χ ℓm scalar function f(χ) is ∆f = R dχ (R dχ f). Using ± ˜ 2 2 Wheeler harmonics (˜xℓm, dχYℓm), and we obtain their −1 d −1 d R 2 R = K we obtain ∆f = R 2 (Rf)+ Kf. curls and divergences in Eqs. (44) - (49), which are valid dχ − dχ Hence the radial eigenfunctions for scalar harmonics sat- and form-identical not only for the 3-spaces (S3,H3, E3), isfy but for any spherically symmetric Riemannian 3-space. Our curl- and div-equations for this general case are very 1 d2 ℓ(ℓ + 1) much simpler than the equations in the literature for the (R ˜(K)) = [ q2 + ]˜(K). (29) R dχ2 qℓ − R2 qℓ special case of Euclidean 3-space. — The general and 11 simple equations (44) - (49) are all that’s needed to con- where we have also listed the third basis field needed for + − struct vorticity fields (divergenceless) and to explicitly 3 dimensions, ~eχYℓm. The triple (X~ , X~ , ~eχ) is point- write down the de Rham-Hodge Laplacian on vorticity wise orthogonal and has positive orientation. X~ + and fields, ∆V~ = curl curl V,~ for any Riemannian 3-space X~ − have the same norm point-wise. − with spherical symmetry. — Textbooks usually present ℓ + The parity is P = ( 1) for X~ and ~eχYℓm, while the vector spherical harmonics of [46, 47], which are − ℓm P = ( 1)ℓ+1 for X~ − . not adapted to cosmological perturbation theory, because − ℓm ~ ± they mix the vector and the scalar sector, see the end of All three basis fields Xℓm, ~eχYℓm are eigenfunctions { } 2 this subsection. — The spin-s spherical harmonics used of the total angular momentum operators J ,Jz with eigenvalues ℓ(ℓ + 1),m . This follows because{ exterior} in CMB analyses [31] are not adapted to our problem, { } because they mix parities, i.e. the toroidal and poloidal differentiation d,˜ taking the Hodge dual ( ), and mul- ∗ sectors of vector spherical harmonics, see the end of this tiplication with ~eχ commute with rotations of the total subsection and Sec. IVC. system. ~ ± Vector spherical harmonics Xℓm(θ, φ) form a basis for The normalization and orthogonality relation on the vector fields tangent to the 2-sphere. In the notation 2-sphere at any R is of differential forms, the vector spherical harmonics of ′ ′ Regge and Wheeler [48], which we denote by a lower case ~ (p) ~ (p ) ~ (p) ~ (p ) ± (Xℓm , Xℓ′m′ ) = dΩ < Xℓm , Xℓ′m′ > x˜ℓm, are given by Z = ℓ(ℓ + 1)δℓℓ′ δmm′ δpp′ . (43) x˜+ dY , x˜− (2) dY , (34) ℓm ≡ ℓm ℓm ≡ − ∗ ℓm This relation is crucial for the projection of a general i.e. the gradient of Y , and its Hodge dual on the 2- ℓm velocity field on the term with (ℓ =1, P = +). The global sphere, (2) . In component notation Eq. (34) gives ∗ (Hilbert space) scalar product on the 2-sphere is denoted + by (V,~ W~ ), and < V,~ W~ > denotes the point-wise scalar (xℓm)α = ∂α Yℓm, (35) product g(V~ ∗, W~ ), where the superscript ∗ stands for the (x− ) = ε gβγ ∂ Y , (36) ℓm α − αβ γ ℓm complex conjugate. where α = (θ, φ), ε = (2)gε , ε +1. In the Computations of curl and div and results are much sim- αβ αˆβˆ θˆφˆ ≡ pler for Regge-Wheeler harmonics ~x ± than for the ’phys- combination ε gνλ = ε λpthe radius R of the 2-sphere ℓm µν µ ical’ harmonics X~ ± .— Also the basis fields [~e R−2Y ] φ −1 θ ℓm χ ℓm drops out, εθ = (sin θ) , εφ = (sin θ). are computationally much simpler than the basis fields The vector harmonics of Regge− and Wheeler have co- [~eχYℓm], because the divergence of the former vanishes variant components (1-form components) which are in- apart from a δ-function at the origin, which turns out to dependent of the radial coordinate χ, be irrelevant in our context, ± ∂χ(xℓm)α =0, α = (θ, φ). (37) + ℓ(ℓ + 1) div ~xℓm = Yℓm (44) In contrast, the ’physical’ vector spherical harmonics − R2 ~ ± curl ~x + = 0 (45) are denoted by the upper case Xℓm. They are used in ℓm the literature on classical electrodynamics [43] and called − div ~xℓm = 0 (46) pure-spin vector harmonics by Thorne [49]. They are de- − ℓ(ℓ + 1) fined with an R-factor which makes the radial covariant curl ~xℓm = 2 Yℓm~eχ (47) derivative vanish, − R ~eχ div [Yℓm ]=0 (48) X~ ± R ~x± , (38) R2 ℓm ≡ ℓm ± ~eχ 1 − X~ = 0. (39) curl [Yℓm ] = ~x . (49) ∇χ ℓm R2 − R2 ℓm This is equivalent to requiring that the point-wise norm ~ ± ~ ±∗ ~ ± ~ ± If we had computed the divergence and curl of Xℓm and g(Xℓm , Xℓm) and the LONB components of Xℓm are in- dependent of R for fixed (θ, φ). Yℓm~eχ, there would have been twice as many terms, and We rewrite the 2-dimensional Hodge dual of Eq. (34) the terms would have been more complicated. (2) ~ ~ Computations of curl V~ = ~ V~ and div V~ = ~ V,~ as Yℓm = ~eχ Yℓm. The generator of rotations, ∇× ∇ · ∗ ∇ − × ∇ where ~ is the covariant derivative, are made very simple the angular momentum operator of wave mechanics in ∇ units of ¯h, is L~ = i(R~ ), where R~ R(χ)~e . Hence with the calculus of differential forms, where no Christof- − × ∇ ≡ χ fel symbols are needed, see Eqs. (A.2, A.5). + The spin-s spherical harmonics of Newman and Pen- X~ = R~ Yℓm (40) ℓm ∇ rose [49, 50, 51], which are widely used for cosmic mi- X~ − = iLY~ = (R~ ~ )Y = ~e X~ + (41) ℓm ℓm × ∇ ℓm χ × ℓm crowave anisotropies [31], are not adapted to our prob- ~eχYℓm, (42) lem, because they mix parities, i.e. they mix toroidal and 12 poloidal vorticity fields (see Sec. IV C), while the drag- gyroscope at the origin. Therefore poloidal eigenfunc- ging of inertial axis-directions is caused by energy-flows tions of the Laplacian are not considered in this paper, with J P =1+, i.e. in the toroidal sector (see Sec. V A). but they will be needed and presented in a forthcoming In most textbooks vector spherical harmonics are con- paper, “The other half of Mach’s principle: Frame drag- structed by coupling a definite orbital angular momentum ging for linear acceleration” [52]. with spin-1 basis vectors to obtain a definite total angular In this subsection we derive the following important momentum via Clebsch-Gordan coefficients [46, 47, 49]. and simple result: These basis states are not adapted to cosmological per- 1. In the eigenfunctions of the Laplacian for toroidal turbation theory, because they mix the poloidal vorticity vorticity fields on (S3,H3, E3): The physical vector sector and the scalar sector, see the following subsection. ~ − spherical harmonics Xℓm are multiplied by radial functions which are identical to those for scalar har- C. Toroidal and poloidal vorticity fields monics, i.e. the generalized spherical Bessel func- (K) tions ˜qℓ (χ) of Eq. (30) resp. the corresponding Neumann and Hankel functions. For 3-dimensional vorticity fields the 3-divergence − must be zero. The vector spherical harmonics ~xℓm for 2. For toroidal vorticity harmonics we have the rela- m = 0 are vector fields purely along the direction of tion q = k: − ~eφ and independent of φ. Therefore the ~x are named ± ℓm 2 (K) ~ − ’toroidal’, and evidently their divergence is automatically (∆ + q )[˜qℓ (χ) Xℓm(θ, φ)] =0. (53) zero. The toroidal harmonics with m = 0 are generated 6 In all sectors q denotes the wave number in the from those with m = 0 by rotations, and their divergence radial oscillations sin(qχ), and ( k2) denotes the is again zero, as also seen from Eq. (46).— Multiplica- − tor eigenvalue of the de Rham-Hodge Laplacian. Note tion with a radial function g (χ) gives the 3-dimensional that the relation for scalar harmonics is q2 = toroidal vorticity fields, k2 + K, Eq. (28). ~ tor tor − To prove this result, all that’s needed is the expres- Vℓm (χ,θ,φ)= gℓ (χ) ~xℓm(θ, φ). (50) sion for the de Rham-Hodge Laplacian for vorticity fields, The 3-divergence is not zero for the vector spherical ∆= curl curlA,~ and the simple Eqs. (44) - (49). — For + + − harmonics ~xℓm by themselves, div ~xℓm = 0. To obtain the rest of this subsection we drop the subscripts (ℓ,m). ~ 6 ~ tor tor − vector fields V with zero 3-divergence we must add a We start with a toroidal vorticity field A = gA ~x and 2 part perpendicular to S , ~eχF (χ,θ,φ), where F can evaluate its curl. Using the vector identity curl (g~v) = be expanded in scalar spherical harmonics Yℓm. Hence g curl~v + (grad g) ~v and Eq. (47) gives the 3-dimensional poloidal vorticity fields can be written × ~ tor tor − curl A = curl (gA ~x ) ~ pol (pol, tg) + (pol, rad) −2 d Vℓm (χ,θ,φ)= gℓ ~xℓm + gℓ [YℓmR ~eχ]. = ( gtor)~x + ℓ(ℓ + 1)gtor[Y R−2~e ]. (54) (51) − dχ A − A χ Both terms have P = ( 1)ℓ. For m = 0 this − In a second step we take the curl a second time, which equation gives vorticity fields V~ with only radial and ~ tor + gives minus the de Rham-Hodge Laplacian of A , θ components, since ~xℓ,m=0 points along the meridians. Hence− the name ’poloidal’ for these vorticity fields. ∆A~ tor = curl curl A~ tor − The condition for vanishing divergence of the poloidal d2 ℓ(ℓ + 1) = [ gtor gtor]~x −, (55) vorticity fields is evaluated using the vector identity dχ2 A − R2 A div (gV~ )= g div V~ + V~ grad g and Eqs. (44, 48), · where we have used Eqs. (45, 49). Up to here all for- d mulae in Secs. IV B - IV D are valid for any Riemannian g(pol,rad) = ℓ(ℓ + 1) g(pol,tg). (52) dχ ℓ ℓ 3-space with spherical symmetry. We now specialize to (S3,H3, E3). We rewrite the toroidal vorticity fields in ~ − − Again, the equations in this subsection are valid for any terms of the physical vector harmonics Xℓm = R ~xℓm Riemannian 3-space with spherical symmetry. and their associated radial functions Gtor R−1gtor, ≡ hence gtor~x− = GtorX~ −. Comparing Eq. (55) rewritten in terms of Gtor with Eq. (29) gives the main result of D. The radial functions for vorticity harmonics this section, Eq. (53). In contrast to [16] we have given a simple, unified treat- Relevant for rotational frame-dragging are energy cur- ment of all three cases, K = (0, 1), with form-identical rents and vector potentials in the toroidal sector as derivations and results. — Ref.± [16] discussed the solu- demonstrated in Sec. V A. Hence we now focus on the tions of the homogeneous momentum constraint (which toroidal eigenfunctions of the Laplacian. — Poloidal en- are eigenfunctions of the Laplacian) without referring to ergy currents cannot contribute to the precession of a the Laplacian. 13

V. MACH’S PRINCIPLE but this can only produce source-fields J~ε in the natural parity sequence, 0+, 1−, 2+, etc, which cannot contribute A. Symmetries and Mach’s principle to the precession. — For tensor perturbations (gravita- tional waves) all linear perturbations are given by a trace- We treat all linear perturbation fields on all types less, divergenceless 3-tensor, from which one cannot form ~ of FRW backgrounds, K = ( 1, 0), and all energy- an axial vector field to generate a Bg field at the origin. momentum-stress tensors, i.e. all± types of matter (also The proof of theorem (2) uses the following facts: For a dark energy and a cosmological constant), not necessarily general energy-momentum-stress tensor (not necessarily of the perfect-fluid form, and for totally general field con- of the perfect-fluid type) the local center-of-mass veloc- ˆ ε 0ˆ −1 0 figurations of energy currents J T . This is in striking ity ~v is given by vkˆ (ρ0 + p0) T ˆ in linear pertur- kˆ ≡ kˆ ≡ k contrast to the previous literature, which only treated the bation theory, where ρ0 and p0 refer to the unperturbed artificial situation of spherical shells of matter rotating FRW background. — For fixed χ, a toroidal velocity field ~ − rigidly around one given axis. — However the mathemat- Xℓ=1,m is a flow equivalent to a rigid rotation of a shell ics of totally general perturbations can be reduced to the of matter. ~ mathematics of the special case of spherical shells of mat- The gravitomagnetic vector potentials Ag relevant for ter in rigid rotation around a given gyroscope with the the gyroscope’s precession, must also be toroidal with help of two theorems, which are based on the symmetries ℓ = 1. The gravitomagnetic field B~ g = curlA~g, which relevant for Mach’s principle, rotation and parity: causes the gyroscope’s precession, must be poloidal with ℓ =1. 1. The precession of a gyroscope (relative to given local LONB axes) cannot be caused by scalar perturba- tions nor, in linear perturbation theory, by tensor B. Green functions for (∆ − µ2) on vorticity fields perturbations. — In the vector sector the preces- in open FRW universes sion of a gyroscope can be caused only by energy- current fields with J P = 1+ relative to the given We derive the Green function for the gravitational vec- gyroscope’s position, i.e. of the toroidal type and tor potential A~g for K = ( 1, 0) generated by an energy- with ℓ =1. current source which is toroidal,− has ℓ =1, m =0, and is concentrated at the geodesic distance χ relative to the 2. On every mathematical spherical shell centered on s gyroscope considered. Such a source corresponds to a the gyroscope considered: The field component (in shell of cosmological matter in rigid rotation around the the harmonic decomposition of the general energy- z-axis, current field) which is toroidal and has ℓ = 1 (rel- φ ative to the gyroscope) is given by an equivalent v =Ωs δ(χ χs). (57) rigid rotation with an equivalent angular velocity of − ~ matter The Green function for arbitrary ℓ is not needed to matter Ω equiv (χs). The equivalent angular veloc- ity of matter is given by the global scalar product of compute the dragging of inertial axes at the origin. The ~ ~ − derivation for general ℓ is totally analogous to the case the Jε-field with the toroidal fields Xℓ=1,m on the shell with radius χ : ℓ = 1 treated here, and it gives the general solution of the momentum constraint for the toroidal sector. As noted at the beginning of Sec. IV A, all equations dΩ < X~ − ∗ , J~ (χ,θ,φ) > (J )− (χ) Z ℓ=1,m ε ≡ ε ℓ=1,m in Secs. IV and V involve the perturbation quantities at matter one time, and at the chosen time we set the scale factor = 16π/3 (ρ0 + p0) R(χ) [Ωm(χ)]equiv , (56) a equal to one. −p Inside and outside the rotating shell, A~ is a toroidal where < ... , ... > denotes the point-wise inner vector eigenfield of (∆ µ2) according to the momentum product, and dΩ is the element of solid angle, while constraint Eq. (5). The− wave number q in the radial Ω denotes spherical-basis components of the an- m oscillations is q = iµ according to Eq. (53). We choose gular velocity. The transformation between the ± ~ the positive sign. spherical and the Cartesian components of Ω is Inside the shell the radial function G(χ)= f(χ) which given by Ω = Ω , and Ω = ( Ω m=0 z m=±1 ∓ x − multiplies X~ − according to Eq. (53) is the generalized iΩ )/√2. ℓ=1,m y spherical Bessel function for K = ( 1, 0), Eq. (32), ± The proof of theorems (1) uses the following facts: The d 1 precession of a gyroscope relative to given LONB axes, iµ2˜j(K) (χ)= − [ sinh(µχ)]. (58) q=iµ, ℓ=1 dχ R d(Sˆi)/dt, which equals the torque on the gyroscope, is an axial vector, J P = 1+. The precession of a gyroscope is Note that any regular toroidal vector field must vanish at caused locally by the gravitomagnetic field B~ g(0), which the origin. is also an axial vector. — For scalar perturbations all Outside the shell the radial function for an open uni- fields are derived from scalar fields (via differentiation), verse, K = ( 1, 0), which is regular at spatial infinity − 14 for q = iµ, is the generalized Hankel function of the first and all other components are zero. We multiply kind, Eq. (33), exponentially decreasing for χ , disc[f (pol,tg)] with (X+ ) to obtain disc[B ], which → ∞ B ℓ=1,m=0 θˆ θˆ d 1 we insert in Eq. (62) to determine N, iµ2 h˜(1)(K) (χ) = − [ exp( µχ)] (59) − q=iµ,ℓ=1 dχ R − 3 K −1 = Y (χ) = Yukawa force. (60) N = (16πGN)(ρ + p)Rs Ωsµ(1 + 2 ) 4π/3. (66) µ − µ p Note that the Yukawa potential is (1/R) exp( µχ), while − This completes the computation of the Green function, the Yukawa force is minus the gradient (the χ-derivative) Eq. (61), in an open universe, K = ( 1, 0), for the vec- of the potential. — For a closed universe this solution is − tor potential generated by the toroidal current J~ with not acceptable, because the 1/R factor makes it singular ε ℓ = 1, m = 0 at χ , specifically by the velocity field of at χ = π, the antipodal point to the origin. This will be s Eq. (57). treated in subsection V D. ~ The vector potential A~ tor is tangential to the shell, The gravitomagnetic field B at the origin, the posi- ~ and it must be continuous across the shell, because the tion of the gyroscope, is obtained from Eq. (54). Bg(0) ~ component of B~ normal to the shell must be continuous, written for arbitrary orientation of Ωs is otherwise divB~ = 0 on the shell. In an open universe, 6 2 3 K = ( 1, 0), we obtain B~ (0) = 2Ω~ gyro(0) = Ω~ s [16πGN(ρ + p)]R Yµ(χs), − − − 3 s (67) f tor(χ,χ )= N ˜j(χ ) h˜(1)(χ ), (61) A s < > valid in an open universe, K = ( 1, 0). This equation ~ − where we have dropped the subscripts q = iµ, ℓ = 1 shows, how Ωs, the angular velocity of the rotating shell, { } and the superscript (K = 1, 0) for better readability. determines Ω~ gyro, the angular velocity of precession of a − The arguments χ< resp. χ> denote the smaller resp. gyroscope at the center of the rotating shell. 2 larger of the two arguments (χ,χs), and χs refers to the In Eq. (67) we denote the prefactor byµ ˜ , radius of the shell. 2 The normalization factor N is determined by the µ˜ 16πGN(ρ + p) (68) 0ˆ 2 ≡ strength of the source J~ǫ on the shell: Einstein’s G µ 4(dH/dt). (69) φˆ ≡ − equation, Eq. (5), determines the discontinuity of the The connection betweenµ ˜ and µ is given by the (dH/dt) tangential component Bˆ, θ equation, which follows by subtracting the first from the sing ε second Friedmann equation, (curl B~ ) = δ(χ χs) disc[Bˆ]= 16πGN J φˆ − θ − φˆ = δ(χ χs)16πGN(ρ + p)ΩsRs sin θ. (62) K − − (dH/dt) 2 = 4πGN(ρ + p) − ac − The discontinuity of Bθˆ is determined by the discontinu- (pol,tg) µ 2 µ˜ 2 ity of its radial function [f ]. For an open universe, ac 1 ( ) + K = ( ) . (70) B ≡ ⇒ 2 2 K = ( 1, 0), Eqs. (52, 54, 61) give − Three length scales occur in our problem: the H-dot ra- (pol,tg) d tor dius (µ/2)−1, the (ρ + p)-radius (˜µ/2)−1, and the curva- disc [fB ]= disc[ fA ] − dχ ture radius ac 1. d ≡ = N disc[ ˜j(χ ) h˜(1)(χ ) ] − dχ{ < > } s C. Exact dragging of inertial axes for = N W [ ˜j, h˜(1) ] , (63) − s perturbations of an open FRW universe where we denote the Wronskian by W [f,g] (fg′ f ′g) with = d . Our Wronskian multiplied with≡ R2 is− inde- The main result of this paper: (1) The equations for ′ dχ pendent of χ, as can be seen from Eq. (29). Therefore we “Dragging of Inertial Axes”, Eqs. (9, 10), are obtained (K) directly from Eq. (67) by integrating over all shell radii. evaluate R2W in the limit χ 0. We use ˜j (χ) → q=iµ,ℓ=1 → (2) The equation, which shows that there is exact drag- (iχµ/3)(1+K/µ2)andn ˜(K) (χ) (µχ)−2 for χ 1, q=iµ,ℓ=1 → ≪ ging (as opposed to partial dragging) of local inertial and we obtain axis directions, is the statement that the weight function 1 K W (χ) has its normalization equal to unity, Eq. (11). W [ ˜j(K), h˜(1)(K) ] ℓ=1 = (1 + ) (64) q,ℓ q,ℓ q=iµ µR2 µ2 We now give the proof that the normalization of the weight function is indeed equal to 1 for the case of an valid for K = ( 1, 0) and for all R. We note that for open universe, K = ( 1, 0). First one does a partial inte- ± − (ℓ =1,m = 0) gration of (d/dχ) in the Yukawa force. This gives Wtot = µ˜2 dχRR ′ exp( µχ). A partial integration of the factor + − − 2 2 −1 (X )ˆ = (X ) ˆ = 3/(4π) sin θ, (65) exp(R µχ) gives W =µ ˜ dχ(1 2KR )µ exp( µχ), ℓ=1,m=0 θ ℓ=1,m=0 φ − tot p − R − − 15 where we have used Eq. (27). If instead of the last step, gravitomagnetodynamics is an elliptic equation (it has one performs a partial integration of the factor (RR′), no partial time-derivatives). These facts have already 2 2 one obtains Wtot =µ ˜ dχ(R /2)µ exp( µχ). Adding been recognized by [11, 13, 14] among others. — (2) The µ2 times the first expressionR plus (4K) times− the second fact that the tensor sector (gravitational waves) in lin- 2 2 expression one obtains (4K + µ )Wtot =µ ˜ , and using ear perturbation theory (and of course the scalar sector) (K=−1,0) ~ Eq. (70) gives Wtot =1. cannot produce a gravitomagnetic field Bg nor a torque Our answer to Mach’s question “what share?” is given on a gyroscope has been demonstrated in Sec. VA. by (1) the radial weight function W (χ) with its Yukawa- The vector structure of Eqs. (72, 73, 75) needs explana- force cutoff, and (2) the angular weight function X~ − ∗ tion, since there is no global parallelism on a hyperbolic ℓ=1,m 3 of Eq. (56) in the projection of the general velocity field 3-space H . This is explained at the end of Sec. I G. on the toroidal vorticity sector with ℓ =1. The differences between Eq. (75) and Amp`ere’s law in The kinetic angular momentum L~ per dχ is directly integral form are: obtained from Ω~ using the moment of inertia per matter 1. the replacement of the current of electric charge J dχ of a fluid shell, which is (8π/3)(ρ + p)R4, q by the energy current Jε, ~ dL 8π 4 = (ρ + p) R Ω~ matter(χ). (71) 2. the factor ( GN), as in the transition from dχ 3 Coulomb’s law− to Newton’s law, ~ Evidently the kinetic angular momentum L is determined 3. the additional factor 4, which occurs in the transi- by angular velocity measurements without prior knowl- tion from Amp`ere’s law of ordinary magnetism to edge of g0φ, which is the output of solving Einstein’s equa- gravitomagnetism, tions. The gravito-magnetic moment per dχ, d~µg/dχ, is de- 4. the replacement of the 1/r2 force in Amp`ere’s law fined in the same way as in ordinary magnetism, except by the Yukawa force Yµ(χ) for cosmological gravit- that the current of charge is replaced by the current of omagnetism, energy J~ε : 5. the need to distinguish between R (in the denomi- ~ ∞ d~µg 1 dL 1 2 nator of the Yukawa force) and χ (in the exponent = = (4πR ) dΩ[R~ J~ε], (72) dχ 2 dχ 2 Z0 × and in the derivative) for FRW backgrounds with 4π curved 3-space, K = 1, dΩ[~n J~ ] = dΩ < X~ − ∗ J~ >,(73) ± Z × ε m − 3 Z ℓ=1,m · ε 6. the remarkable fact that the law (75) for cosmo- where ~n is the radial unit vector at the source point, and logical gravitomagnetism is valid for general non- R~ = R ~n. The last two equations show that the gravit- stationary situations, while Amp`ere’s law is valid omagnetic moment and the angular momentum involve only for stationary currents. a projection of the energy-current field on the sector of toroidal vorticity fields with ℓ =1. What distance interval χ of sources dominates drag- ging of local inertial frames at P ? — For a spatially flat Ω~ is determined by an integral over the density of gyro universe, the weight function W (χ) grows linearly from kinetic angular momentum: From Eqs. (9, 71) χ = 0 to χ µ−1 =µ ˜−1, and it decays exponentially for χ µ−1≈. Hence the sources Q around χ = µ−1 1 dL~ ≫ P Q B~ g(P )= 2Ω~ gyro = 4GN d(volQ) Yµ(χ). dominate the frame-dragging at P. — For a spatially hy- − − Z {R d(vol)} −1 perbolic universe, K = 1, and forµ ˜ ac 1 the (74) situation is approximately− the same as for≫ K =≡ 0, since ~ Expressing the kinetic angular momentum L by the the frame dragging is caused by sources at distances much ~ energy current Jε we obtain the fundamental law for cos- smaller than the curvature radius. — But for K = 1 − mological gravitomagnetism in integral form for an open and for χ much larger than the curvature radius ac 1, FRW universe, (K = 1, 0) : ≡ − the effective exponential cutoff factor in the weight func- tion W (χ) is exp[ (µ 2)χ]. From (ρ + p) > 0 and from ~ ~ Bg(P )= 2Ωgyro(P )= Eqs. (69, 70) with− K −= 1, it follows thatµ ˜ > 0 and − − 2 ~ µ > 2. Forµ ˜ << 1 we have (µ 2) µ˜ /4, the cutoff = 4GN d(volQ)[~nP Q Jε(Q)]Yµ(χP Q). (75) 2 − ≈ − Z × factor becomes exp( µ˜ χ/4), and we have three regimes for W (χ) : For 0 −χ << 1 the weight function W (χ) ≤ −2 Again: Ω~ gyro is given by the sources J~ε at the same time. grows linearly, for 1 << χ << µ˜ the weight function is This follows directly from: (1) The facts that dragging constant, and for χ >> µ˜−2 it decays exponentially. On effects are given directly by the momentum constraint, a logarithmic scale the sources near χ =µ ˜−2 dominate and that the momentum constraint in the vorticity sec- frame dragging forµ ˜ << 1. — For all open universes tor, Eq. (5), for the general time-dependent context of the dragging of inertial frames is dominated by sources 16 at a distance equal or larger than the (ρ + p)-radius: Lo- = N¯ W [ ˜j(χ), ˜j(π χ)]s. (77) cal inertial frames are rigidly “in the grip of the distant − universe”. To evaluate the Wronskian W we use the method given Form-invariance of the solution Eq. (75) under trans- before Eq. (64). Specifically, (R2W ) is independent of formation to a rigidly rotating frame: If the reference χ, and we evaluate this expression for χ 0. We use ˜(K=+1) 2 ˜(K=+1)→ FIDO at the position of the gyroscope changes his lo- jq=iµ,ℓ=1(χ) (i/3)χµ(1+1/µ ) and jq=iµ,ℓ=1(π χ) ∗ 2 2 → − → cal spatial axes to new ones, ~eˆ , with angular velocity i/(χ µ ) sinh(µπ) for χ 1, and we obtain i ≪ Ω~ ∗(0) relative to the old ones, Eq. (75) remains valid − ˜(K) ˜(K) K=+1 as it stands, because both sides of Eq. (75) change by the W [ jq,ℓ (χ), jq,ℓ (π χ)] q=iµ, ℓ=1 ~ − same term. On the right-hand-side (~n Jε) changes by 1 1 ⋆ × [ (ρ+p)Ω (0) R sin θ ~e ], where we have put the z-axis of = 2 (1 + 2 ) sinh(µπ). (78) − θˆ µR µ the spherical coordinates in the direction of Ω~ ⋆. The in- − tegration over the solid angle gives (8π/3)(ρ+p)R Ω~ ⋆(0). Hence the normalization N¯ is The integration over χ is the same as in Eq. (11). The re- N¯ = N sinh−1(µπ), (79) sulting change on the right-hand side is [ 2Ω~ ⋆(0)], which − is equal to the change on the left-hand side. where N N (K=0,−1) is given by Eq. (66). To obtain definite values for measurements of fields J~ ≡ ε The gravitomagnetic field B~ at the origin generated and B~ , one must make an arbitrary choice of a definite g by a rotating shell Ω(χ)= δ(χ χ )Ω on a FRW back- state of rotation of FIDO-axis directions along one world s s ground with K = +1 is given− by taking the result (67) line, as explained at the end of Sec. I B. But for any such for K = (0, 1) and making the replacement choice: The field of measured energy currents J~ε all by − ~ −1 itself completely determines the gravitomagnetic field Bg exp( µχ) sinh (µπ) sinh[µ(π χ)]. (80) − ⇒ − at all points, i.e. the precession Ω~ gyro of gyroscopes at all points: No absolute element is needed in the input, as For an arbitrary velocity field it directly follows with this required in the starting point of Mach [4, 5]. replacement from the result of Eq. (9) that Ω~ gyro is equal For cosmological gravitomagnetism and for given to the weighted average of Ω~ matter, ~ sources Jε, measured via angular velocities, distances, π and (ρ + p): The solution Eq. (75) for the momentum Ω~ gyro = dχ Ω~ matter(χ) W (χ), constraint is unique. There exists no ambiguity of adding Z0 a solution of the homogeneous equation. No regular so- 1 W (χ) = µ˜2R3 sinh−1(µπ) lution exists for the homogeneous momentum constraint 3 2 in cosmological gravitomagnetism, (∆ µ )A~g = 0. — d 1 − − sinh[µ(π χ)] . (81) Therefore for regular solutions no boundary conditions dχ {R − } are needed at spatial infinity for open universes. See also Sect. XI of [7]. Again, the averaging weight function is normalized to π unity, Wtot = 0 dχW (χ) = 1. The proof uses the same steps as theR ones given to prove unit normaliza- D. Exact dragging of inertial axes for tion of W (χ) for open universes in the second paragraph perturbations of a closed FRW universe of Sec. VC. Hence for perturbations of a closed FRW universe the evolution of inertial axes exactly follows the To find the Green function for A~g in a closed universe weighted average of cosmic matter motion, there is exact and in the toroidal sector, we note that a toroidal vector dragging of inertial axes by cosmic energy currents, as ~ tor field Vℓ=1,m which is a regular vector field at χ = 0 and stated in Mach’s postulate. at χ = π must have a point-wise norm which vanishes The fundamental law for gravitomagnetism in inte- linearly at these two points. Therefore the radial solution grated form follows with the replacement (80) from tor tor ~ − Eq. (75), GA = fA , which multiplies Xℓ=1,m, must also vanish linearly at χ = 0 and at χ = π. Hence the radial part of the Green function can be written as B~ (P ) = 4G sinh−1(µπ) d(vol ) (~n J~ ) − N Z Q P Q × ε tor ¯ ˜ ˜ fA (χ,χs)= N j(χ<) j(π χ>). (76) d 1 − − [sinh[µ(π χ)] . (82) We have again dropped the subscripts (q = iµ, ℓ = 1) dχ {R − } and the superscript (K = +1) for better readability. The solutions (81, 82) are again frame-invariant, i.e. The discontinuity in the derivative of the radial part form-invariant when going to a reference frame which is of the Green function, the analog of Eq. (63), is in globally rigid rotation relative to the previous reference d frame. The proof is identical to the one given for open disc [ f tor ] dχ A universes in Sec. V C. 17

Einstein, Wheeler, and others, recently e.g. Biˇc´ak et behaviour of this body?” — If the density of galaxies al [16], have held the view that from the Machian point was e.g. 10−9 per Hubble volume in an otherwise empty of view closed universes are preferable. In contrast our universe, a particle could be a distance of 103 Hubble results show that Mach’s hypothesis, exact dragging of radii away from all other matter. All the same, if galax- inertial axes, holds for linear perturbations of all FRW ies in the unperturbed universe were homogeneously and universes, open or closed. isotropically distributed, we have shown that for a uni- verse with linear vorticity perturbations Mach’s principle still holds. E. Exact dragging of inertial axes for The Milne universe is the limit ρ/ρcrit 0 (for p/ρ perturbations of Einstein’s static, closed universe fixed) of a FRW universe. The first Friedmann→ equa- 2 2 tion gives H + K/ac 0, hence we have a hyperbolic 2 → For a static, closed universe µ 4(dH/dt)=0, and universe with a curvature radius ac equal to the Hubble (˜µ/2)2 = K = +1. Taking the limit≡−µ 0 of the result radius H−1. The Milne universe is equivalent to a part of for a FRW universe with K = +1, Eq.→ (82), one obtains Minkowski space-time, the forward light-cone originating the fundamental law for gravitomagnetism for Einstein’s at the space-time point where the Hubble expansion (of static, closed universe, test particles at rest in the Milne universe) started. How can exact dragging of inertial axes by cosmic en- B~ (P )= ergy flows work for ρ/ρ 1, when the total energy crit ≪ d 1 within a Hubble volume is arbitrarily small? According 4G d(vol ) (~n J~ ) − [ (1 χ/π)].(83) − N Z Q P Q × ε dχ R − to the discussion in Sec. V C, the weight function becomes 2 2 2 W (χ) (˜µ /4) exp( µ˜ χ/4) forµ ˜ 16πGN(ρ + p) ≈ 2 − ≡ ≪ Again Ω~ gyro is equal to the weighted average of Ω~ matter, 1. The prefactor (˜µ /4) is arbitrarily close to zero, but the integral for Wtot would be linearly divergent without the π cutoff at χ = 4˜µ−2. Therefore the integral without ~ ~ cutoff Ωgyro = dχ Ωmatter(χ) W (χ), the prefactor gives (4/µ˜2), and this is arbitrarily large. Z0 4 d 1 The product of these two factors gives Wtot = 1, exact W (χ) = R3 − [ (1 χ/π)], (84) dragging of inertial axes, as demonstrated in Sec. V C. 3 dχ R − The de Sitter universe is the limit (p/ρ) ( 1) of ց − where W = π dχ W (χ)=1. Eq. (84) states that also FRW universes with K = 0, 1. These three cases cor- tot 0 respond to three different slicings± of one de Sitter space- for perturbationsR of Einstein’s static closed FRW uni- verse the evolution of inertial axes exactly follows the time geometry or part of it. weighted average of cosmic matter motion, there is exact How does Mach’s principle work arbitrarily close to the de Sitter limit, i.e. for [(p/ρ) + 1] 1 and ρ/ρ finite dragging of inertial axes by cosmic energy currents, as ≪ crit stated in Mach’s postulate. — The solution (83) is again and fixed, when the energy currents within the Hubble volume can be made arbitrarily small for any observer? form-invariant when going to a reference frame which is 2 2 in globally rigid rotation relative to the previous refer- For K = 0 we have µ =µ ˜ = 16πG(ρ + p). Hence the prefactor in W (χ) is of order µ2 1. On the other hand ence frame. ≪ the integral for Wtot would be quadratically divergent Einstein’s static, closed universe (3-sphere) has been −1 important in the work of Ozsv`ath and Sch¨ucking [53], without the cutoff at χcutoff = µ . Hence the integral without the prefactor is of order 1/µ2 1. The two who added a Bianchi IX amplitude (the lowest mode of ≫ tensor perturbations) to Einstein’s universe, and then in- effects cancel, and Wtot = 1 as stated in Eq. (11). vestigated Mach’s principle. Unfortunately Ozsv`ath and For K = 1 the discussion of the de Sitter limit, (ρ + p) 0, is exactly− the same as the discussion given above Sch¨ucking considered the vanishing of the local vorticity → relative to a gyroscope to be a test of Mach’s princi- for the Milne limt. ple. This is in contradiction to our conclusion that the For K = +1 and (ρ + p) ρ, i.e.µ ˜ 1, one obtains µ 2i(1 µ˜2/8), and W≪(χ) = 8(3π)≪−1 sin4 χ, i.e.µ ˜2 weighted cosmic average of the energy currents in Eq. (84) → ± − π directly determines the time-evolution of gyroscope axes. drops out. This gives 0 W (χ)dχ =1. Our result agrees with Mach’s hypothesis that some (to R him unknown) cosmic average of matter motion would determine the time-evolution of gyroscope axes. G. Globally rigid rotations: Zero modes of (∆+4K)

We have seen in Sec. I B that for vorticity perturba- F. Mach’s principle for the limits to the Milne and tions the fixed-time slices Σt are uniquely determined, the de Sitter universe and that each slice has the unperturbed 3-geometry of E3,S3,H3 . For the unperturbed 3-geometries we have Many physicists (but definitely not Mach) have asked {chosen unperturbed} 3-coordinates: spherical coordinates again and again: “If you could remove a body far away for E3,S3,H3 , resp. Cartesian coordinates for E3. from all sources of gravity, what would be the inertial This{ does not yet} fix the coordinatization (gauge) com- 18 pletely, because there still exist residual gauge transfor- 1. Exact dragging of local inertial axes by a weighted mations by time-dependent globally rigid rotations of the average of cosmological energy currents. The spherical coordinates for 3-space. In this subsection we weight function must be normalized to unity. show that these residual gauge transformations are zero modes of (∆+4K), where ac =1. 2. “Totally determined”: The time-evolution of local Since this paper on Mach’s principle focusses on ro- inertial axes (relative to any given set of axes) is tational motion of fiducial axes, not on linear accelera- totally determined by cosmological energy currents. tion of frames, we apply a change of the spatially global (rigid) reference frame such that the 3-coordinate veloc- 3. No absolute element as an input. The time- ity dxi/dt of the reference FIDO at the spatial origin, evolution of local inertial axes must be an output χ = 0, is left unchanged. The angular velocity of the determined by cosmic energy currents, not an in- new reference-FIDO axes at the spatial origin relative to put. the old ones is denoted by [ Ω~ ⋆(χ = 0)]. In the new coordinates the− matter 3-velocity field is 4. “Solutions Frame-Invariant”: The solution for ⋆ Ω~ gyro in terms of cosmological energy currents must changed by the additional field ~vmatter, which is equal to ~⋆ be form-invariant under transformations to glob- the negative of the additional shift vector field β and also ally rotating frames, i.e. frame-invariant. equal to the negative of the additional gravitomagnetic ~⋆ vector potential Ag, 5. No boundary conditions at spatial infinity in open cosmologies are needed to totally determine local ⋆ ⋆ ⋆ ~v = A~ =Ω (0) ~eφ. (85) non-rotating frames. matter − g The momentum constraint, Eq. (5), for the additional 6. No reference-frame condition along one world line ⋆ ~ ⋆ 2 fields ~vmatter = Ag , with the definition ofµ ˜ for the is needed to totally determine local non-rotating coefficient in the− source term, Eq. (68), and with the frames: In open universes such a reference-frame relation (˜µ2 µ2)=4K, Eq. (70), gives condition along a world line at spatial infinity is − equivalent to boundary conditions. In closed uni- ~ ⋆ (∆+4K)Ag =0. (86) verses such reference-frame conditions along one world line play the analogous role to boundary con- ~ ⋆ i.e. the additional field Ag arising from a globally rigid ditions for open universes. rotation is a zero mode of the operator (∆+4K). The curl of A~⋆ gives the additional gravitomagnetic Tests (1)-(4) represent different aspects of the postulate g as formulated by Mach. — Test (3) is the postulate of the ~ ⋆ field Bg , “relativists” (Huyghens, Bishop Berkeley, Leibniz, Mach, and others) that only relative motion of physical objects B⋆ = 2Ω⋆(0) cos θ χˆ − is relevant. This was the starting point of Mach. — Test dR B⋆ = +2Ω⋆(0) sin θ (4) is a necessary condition, formulated by Mach in his θˆ dχ earliest writings [5] quoted in Sec. I A. — Test (1) is an ex- ⋆ B ˆ = 0. (87) plicit formulation of Mach’s Principle, given in Sec. I A, φ while Tests (2)-(6) are less explicit about the implemen- For K = 0 (Euclidean 3-space) the additional B~ ⋆-field tation. — Tests (1) and (2) are unambiguous for linear g perturbation theory. In the nonlinear case, there is now ~ ⋆ and hence the Ωgyro-field are homogeneous fields. But agreement that gravitational waves make contributions. for K = 1 the additional B~ ⋆-field is not a homogeneous Hence Einstein’s formulation of Mach’s principle [54], ± g field, its magnitude varies as a function of χ. “the (metric tensor) field gµν is entirely determined ... ⋆ ~ ⋆ ~ ⋆ The additional fields ~vmatter, Ag , Bg are equal to the by the energy tensor of matter” is untenable. But the fields, which arise in an unperturbed FRW universe, if the definition of “energy currents” ( = “momentum densi- coordinate system is in rigid rotation relative to FRW ties”) and “angular momentum densities” of the gravi- coordinates, and if the FIDOs and their LONBs are tational field has been controversial. Frame-dragging ef- adapted to the rotating coordinates. — But note that fects by “the angular momentum of gravitational waves” the constraint for a given vanishing source field J~ 0 has been investigated in special, non-cosmological mod- ε ≡ has the unique solution A~g 0. els in [55, 56]. — Tests (3) and (4) are meaningful in the ≡ presence of gravitational waves, even if the concept of en- ergy currents of the gravitational field is not available in H. Six fundamental tests a precise way. — Tests (5) and (6) are intimately related. for the principle formulated by Mach The kinetic energy currents, i.e. the LONB compo- nents T 0ˆ can be measured without prior knowledge of the kˆ The following six tests cover different aspects, inti- local non-rotating frames. Hence no absolute element is mately related, of the principle formulated by Mach: needed in the input for our approach. — In contrast the 19 canonical energy currents, i.e. the coordinate-basis com- on cosmological perturbation theory do not treat the top- 0 ponents T k cannot be measured without the prior knowl- ics of the de Rham-Hodge Laplacian and the Weitzenb¨ock edge of the local non-rotating frames all over the universe. formula. The latter gives the difference (∆ 2)A,~ which 0 −∇ Hence using coordinate-basis components T k means that is needed when comparing with the literature on cos- one needs an absolute element as an input, e.g. in the mological vector perturbations. In books about differ- approach emphasized and advocated by [13, 16, 19]. ential geometry these topics are treated in [23, 24]. — The six tests listed above are not fulfilled explicitly But, as far as we know, no book on General Relativ- (manifestly), if one uses the momentum constraint for ity or differential geometry and no paper on cosmologi- 0 the coordinate-basis components T k, as done and advo- cal perturbations gives the explicit equations which con- cated e.g. in [13, 16, 19]. These papers give no proof nect the notation of differential forms with the elemen- that any one of the six tests is satisfied. tary notation for vector calculus in Riemannian 3-spaces: Einstein’s theory of General Relativity applied to the (curl~a)µ = (⋆d a˜)µ, Eq. (A.2), and for vorticity fields solar system fails tests (1)-(5). Hence it is not a “theory (∆~a)µ = (curl curl~a)µ = (⋆d⋆d a˜)µ, Eqs. (A.10, − − of relativity” in the sense of the relativists before Ein- and A.11). Ref. [23] does not give such equations, and stein, as has been recognized for a long time. Ref. [58] instead gives a “rough, symbolic identifica- In this paper we have shown that the six fundamental tion”, which cannot be used literally in the equation tests listed above are manifestly (explicitly) fulfilled for ∆~a = curl curl~a, see the paragraph following the iden- − linear perturbations of FRW universes in Cosmological tity (A.2). General Relativity. Our aim in this Appendix is to give the derivations of the tools and results needed in cosmological gravitomag- netism. We present this for cosmologists with little or no experience in the calculus of differential forms. Acknowledgments The identities for vector calculus in Riemann 3-spaces at the level of first covariant derivatives are indistin- We like to thank T. Frankel, J. Fr¨ohlich, G.M. Graf, guishable from identities for vector calculus∇ in Euclidean J. Hartle, and N. Straumann for discussions, and 3-space, given e.g. in [57], because curvature effects can- D.R. Brill and R.M. Wald for hospitality and discussions. not appear at the level of first derivatives.— At the level of second covariant derivatives , the standard identi- ties of vector calculus in Euclidean∇ 3-space, e.g. in [57], APPENDIX: THE DE RHAM-HODGE remain true in Riemannian 3-space, if and only if one LAPLACIAN FOR VECTOR FIELDS IN uses the de Rham-Hodge Laplacian ∆ as opposed to the RIEMANNIAN 3-SPACES: CONNECTING THE ‘rough Laplacian’ 2. CALCULUS OF DIFFERENTIAL FORMS AND The reasons which∇ single out the de Rham-Hodge THE CURL NOTATION Laplacian in the vector sector on curved 3-space, and which give its conceptual motivation in physics: The material in this appendix (in addition to Sec. I E) 1. If for a vector field all types of sources (div and is essential for cosmological gravitomagnetism, because: curl) are zero, then the de Rham-Hodge Laplacian (1) For vector perturbations the de Rham-Hodge Lapla- of this vector field is also zero, i.e. the vector field cian (“the” Laplacian in the mathematics literature) is harmonic; Sec. I E. has many crucial advantages (listed below) over 2, the ∇ “rough Laplacian”, which, unfortunately, has been used 2. The de Rham-Hodge Laplacian commutes with in the entire literature on cosmology. The calculus of dif- curl, div, grad; Sec. I E. ferential forms is needed for the de Rham-Hodge Lapla- cian. In addition, this calculus makes computations sim- 3. The identities of vector calculus in Euclidean 3- ple, since no Christoffel symbols are required for compu- space [57] and the important first identity of Green tations. Unfortunately this calculus has not been used in generalized to vorticity fields, Eq. (A.9), remain the literature on cosmological vector perturbations. true in Riemannian 3-spaces, if and only if one uses (2) On the other hand the elementary notation for vec- the de Rham-Hodge Laplacian. tor calculus in Riemannian 3-spaces with curl and div 4. For electromagnetism in curved space-time the is indispensable for physical insight in gravitomagnetism, equivalence principle forbids curvature terms, if i.e. for seeing directly that the mathematical structure of and only if the de Rham-Hodge Laplacian resp the momentum constraint for cosmological gravitomag- d’Alembertian is used; Sec. I E. netism on FRW backgrounds with K = ( 1, 0) is ex- ± plicitly form-identical (without curvature terms) to the 5. The action principle for Amp`ere magnetism in Rie- familiar structure of Amp`ere magnetism in Euclidean 3- mannian 3-space, Eq. (A.6), which is bilinear in space (except for the Yukawa cutoff at the H-dot radius). first derivatives, produces the Amp`ere equation (3) Unfortunately almost all books on differential ge- with the de Rham-Hodge Laplacian and without ometry and on General Relativity, and all research papers curvature terms, Eq. (A.10). 20

6. The momentum constraint for cosmological grav- LONB components of the metric tensor, gˆiˆj = δij ). — itomagnetism has identical mathematical structure But in the calculus of differential forms the Hodge dual to Amp`ere magnetism (except for the Yukawa cut- involves the totally antisymmetric ε-tensor in a coordi- λµν −1 off), if and only if the de Rham-Hodge Laplacian is nate basis, ελµν = ελˆµˆνˆ √g and ε = ελˆµˆνˆ (√g) . The used; Eq. (5). volume of a parallelepiped spanned by a triple of vectors is v(X,Y,Z) = ε XλY µZν, hence the volume 3-form 7. The Hodge decomposition for closed Riemannian 3- λµν v˜ has components vλµν = ελµν . — For the Hodge dual spaces, specialized to 1-forms, states that any vec- which takes a p-form into a (3 p)-form, one can either tor field can be uniquely decomposed into a sum of first raise the indices of the original− p-form and then act a gradient field plus a curl field plus a harmonic λµν with ελµν , or equivalently one can first act with ε and field. This theorem is valid, if and only if the then lower the indices to obtain the final (3 p)-form. de Rham-Hodge Laplacian is used to define har- − The operation curl takes a vector field ~a into another monic fields. vector field of opposite parity, ( curl~a ) = ( ~ ~a ), where 8. No Christoffel symbols resp connection coeffiecients is the covariant derivative. The formula∇× analogous to are needed to compute the de Rham-Hodge Lapla- Eq.∇ (A.1) gives, cian; Eqs. (A.2, A.5, A.14, A.15). (curl~a) = (~ ~a) = g εκµν ∂ a = (⋆ d ˜a) . (A.2) Every one of these properties does not hold for the “rough λ ∇× λ λκ µ ν λ Laplacian” 2. ∇ The last and the next-to-last expressions in Equa- tions (A.2) give the simplest method to compute a curl. 1. Differential forms and curl, grad, div But the first and second expressions are crucial to make the precise connection with the notation universally used One can represent a vector field ~v in Riemannian 3- in electrodynamics [43] via an identity in the form of space, a geometric object, by its Local Ortho-Normal an explicit equation. — The first step on the right-hand Basis (LONB) components, denoted by a hat over the side of Eq. (A.2), the exterior derivative (antisymmet- index, v , or by its contravariant components in a coor- ric derivative) d of a 1-forma ˜ gives the 2-form da˜ with ℓˆ components (da˜) ∂ a ∂ a . Because the calcu- dinate basis vλ, or by its associated 1-formv, ˜ resp. by µν µ ν ν µ lus of forms is tied to≡ (unspecified)− coordinate bases and its 1-form components (covariant components) v . The λ to totally antisymmetric covariant tensors, the covariant calculus of differential forms is fundamentally tied to the derivative , after antisymmetrization, can be replaced representation in some unspecified coordinate basis and µ by the ordinary∇ partial derivative ∂ . This is the first with covariant components. µ great simplification, and it takes place, because the con- For the vector product (~a ~b ) the two notations are × nection coefficients in the coordinate basis, the Christof- connected by the equation ρ fel symbols Γ µν , are symmetric in the second and third κµν indices. — Compared to other identities given for curl in (~a ~b )λ = gλκε aµbν = ( ⋆ [˜a ˜b ] )λ, (A.1) × ∧ the literature [24], the identity (curl~a)λ (⋆da˜)λ has the important advantage that it directly transcribes≡ the com- wherea ˜ and ˜b are the 1-forms (covariant vectors) as- putational prescription of elementary vector calculus for sociated with ~a and ~b, i.e. (˜a)µ = (~a)µ = aµ. A Cartesian coordinates, (curl~a)i = εijk∂j ak, to Rieman- tilde ˜ designates p-forms (totally antisymmetric covari- nian 3-spaces (or to curvilinear coordinates in Euclidean ant tensors of rank p). — The first step is the wedge 3-space). — Some authors, e.g. [58], make a ’dictionary’ product (exterior product, antisymmetric product) of A 1 ∧ with the ’rough, symbolic identification’ curl dα , two 1-forms, which gives a 2-formc ˜ with components where the 1-form α1 is the covariant expression⇔ for the ˜ cµν =[˜a b ]µν aµbν aν bµ. — The second step is tak- A ∧ ≡ − vector . This identification cannot be used literally in ing the Hodge dual (Hodge star operator) of the resulting the operation (curl curl A), because the Hodge star op- 2-form. All our bases will have right-handed orientation. erator is missing in front of dα1 in their rough, symbolic The primary definition for the Hodge dual is given in identification. The operation (curl curl) shows that the any LONB: One contracts with the totally antisymmet- output of the operation curl cannot be the 2-form dα1, ric ε-tensor, the Levi-Civita tensor, in a LONB, where the output must be a vector (represented e.g. by a 1- ε +1. Explicitely: the Hodge dual of of a p-formc ˜ 1ˆ2ˆ3ˆ ≡ form) as in our identity (A.2), which directly produces is given by (⋆c˜)ˆ ˆ = (1/p!) εˆ ˆ ˆ ˆ cˆ ˆ , i1,..,in−p i1,..,in−p,j1,..,jp j1,..,jp the identity (curl curl~a )λ = (⋆d⋆d a˜)λ. where n = 3 is the dimension of our Riemann space. Ex- The operation grad takes a scalar field φ into ~ φ. ample: h˜(1) = ⋆ ˜b(2) written in LONB components gives ∇ hˆ = bˆˆ. It follows that ⋆⋆ = 1 for p-forms in a Rieman- 1 23 (grad φ) = (~ φ) = ∂ φ = (dφ) . (A.3) nian 3-space. — The volume element spanned by a LONB λ ∇ λ λ λ with postive orientation is equal to +1: v(~e1ˆ, ~e2ˆ, ~e3ˆ) = ε(~e1ˆ, ~e2ˆ, ~e3ˆ)= ε1ˆ2ˆ3ˆ = +1. The LONB components of the The inner product (scalar product) of two vectors can ε-tensor are invariant under proper rotations (as are the be rewritten as an antisymmetric product by first going 21 to the dual of ˜b, a partial integration.— To make the surface terms disap- pear, we must either have a compact Riemann space, or, λ 1 λ µνκ in a non-compact Riemann space, one of the two forms, ~a ~b = a bλ = ελµν [ a (ε bκ)]= ⋆ [˜a ( ⋆ ˜b )], · 2 ∧ the variation, must have compact support.— For partial (A.4) 1 βγδ δ integration in curved space one needs the tool of differ- where we have used 2 εαβγε = δα. The wedge product ential forms. The adjoint operator to d in the sense of of a 1-forma ˜ with a 2-formc ˜ is given by [˜a c˜] = ∧ λµν the global scalar product for compactly supported forms aλcµν + aµcνλ + aν cλµ = [˜c a˜]λµν . Similarly the inner is denoted by d∗. Hence (β p, dα p−1) (d∗β p, α p−1), product of two p-forms α, β is∧ given by <α,β >= ⋆(α ≡ ∧ where the superscript p refers to p-forms. Expressing the ⋆β). point-wise inner product via a wedge product, Eq. (A.4), The divergence, div~a = ~ ~a, can be written as an an- and using the explicit coordinate-basis expression, partial ∇· tisymmetric derivative in analogy with Eq. (A.4). This integration gives is the second great simplification, because the antisym- d∗ αp = ( 1)p ⋆d⋆ αp. (A.7) metrization again eliminates the need for Christoffel sym- − bols. It provides the easiest way to obtain the explicit The superscript ∗ denotes the adjoint, the non- expression for ( ~ ~a ) at the end of the following sequence superscripted ⋆ denotes the Hodge dual.— The codiffer- of equations, ∇· ential operator δ in this paper is defined with the opposite sign from the adjoint, div ~a = ~ ~a = ⋆d⋆ a˜ δ d∗ δa˜ = div~a, (A.8) αβγ ∇ · δ µ = ε ∂α(εβγδa )=(1/√g) ∂µ (√ga ). (A.5) ≡− ⇒ wherea ˜ is a 1-form. Some books define δ with the op- posite sign from our sign, δ d∗.— The exterior deriva- Some well-known textbooks on General Relativity need ≡ more than a dozen steps to derive the well-known last ex- tive d takes a p-form into a (p + 1)-form, the codifferen- tial operator δ takes a p-form into a (p 1)-form, and pression in Eq. (A.5), because they work with Christoffel − symbols. — But note that the expression (⋆d⋆a˜) is much dd =0, δδ =0. more useful, since it gives e.g. div ~x± of Eqs. (44, 46) in Green’s first identity generalized to vector fields with a trivial way. zero divergence on Riemannian 3-spaces (partial integra- tion) reads

dv (curl ~α) (curl β~)= dv ~α ∆β~ (A.9) 2. The Laplacian on vector fields in curved 3-spaces Z · − Z · in the absence of surface terms. This identity is form- The Laplacian ∆ acting on vector fields resp 1-forms identical with the identity in Euclidean 3-spaces, and it in curved space is called de Rham Laplacian in [32], and can be used to give the primary operational definition of [58] writes: “This Laplacian was defined first by Kodaira the Laplace operator (de Rham-Hodge Laplacian) on vec- and independently by Bidal and de Rham.” The his- tor fields with zero divergence in Riemannian 3-spaces. — tory including references is given in [59], footnote on Equivalently, in our physics situation the primary con- p. 125. Since other authors use the name Hodge in cept of the Laplacian is given by starting from the energy this connection, and in order to be clearly understood, functional Eq. (A.6): The factor multiplying the varia- we shall call it the de Rham-Hodge Laplacian. The de tion of A~ in the integrand of Eq. (A.6) gives Amp`ere’s Rham-Hodge Laplacian ∆ must be distinguished from 2 α law for A,~ i.e. ∆A~ in the vorticity sector, the “rough Laplacian” α [25], [58]. Only when ∇ ≡ ∇ ∇ 2 applied to a scalar φ field, one has (∆ )φ = 0. — ∆A~ = curl curlA~ = 4πJ~q (A.10) For the primary conceptual definition of− the∇ Laplacian − − (∆A~) = (⋆d⋆d a˜) . (A.11) acting on vector fields, we start from an action integral, µ − µ because an action integral only involves first derivatives, Equation (A.11) with the operator (⋆d⋆d ) gives the hence it is independent of curvature effects. For Amp`ere simplest computational method in Riemannian 3-spaces magnetostatics in static curved 3-space, the action inte- (and for curvilinear coordinates in Euclidean 3-space). gral (times 1) reduces to the energy functional, On the other hand, Eq. (A.10) with the operator (curl − curl) in Riemannian 3-spaces is needed to make the pre- E[A~]= dv [(curlA~)2 4πA~ J~ ], (A.6) cise connection using the notation universally adopted in Z − · electrodynamics in Minkowski space in a (3+1)-split [43]. In the general case of vector fields with div ~a = 0 and where dv = d3x √g is the invariant volume element. E[A~] curl ~a = 0 the requirements for the Laplace operator6 ∆ is given by the global scalar product of two vector fields, of de Rham6 and Hodge acting on vector fields are: (.. , ..)= dv .. , .. , where .. , .. is the point-wise scalar product. R h i h i (1) 2nd order elliptic differential operator, To carry out the variation of the energy functional self-adjoint, (A.12) E[A~] with respect to variations of A,~ one has to perform (2) curl ~a =0 and div ~a =0 ∆~a =0. (A.13) ⇒ 22

2 Reversing the arrow in the statement (A.13) would be in- rived by first using (curl curl A)α = ( A)α + β − ∇ correct for open Riemannian 3-spaces (open universes) in (grad divA)α + [( β α α β)A] . For the last term contrast to statements in some textbooks. This is evident one uses the Ricci∇ identity,∇ − ∇ ∇ from simple counter-examples: For a FRW background with K = 0 a homogeneous B~ -field gives ∆A~ = 0, but α α β [( µ ν ν µ)A] = R βµν A , (A.16) curl A~ = 0. For a FRW background with K = 1 the ∇ ∇ − ∇ ∇ 6 ~ − analogous field is a poloidal vorticity field B with ℓ = 1 and the contracted Ricci identity, and q = 0, for which again ∆A~ = 0, but curl A~ = 0. — From Eqs. (A.12, A.13) it follows that 6 [( )A]µ = R Aβ, (A.17) ∇µ∇ν − ∇ν ∇µ νβ ∆~a = curl curl~a + grad div~a. (A.14) − α ∆˜a = ( ⋆d⋆d + d⋆d⋆)˜a where R is the Riemann tensor and Rνβ the Ricci − βµν = (δd + dδ)˜a tensor. The sign conventions for the Riemann and Ricci tensors are those of Misner, Thorne, and Wheeler [20]. = (d + d∗)2 a.˜ (A.15) − This gives the difference between the true Laplacian ∆ (de Rham-Hodge Laplacian) and the “rough Laplacian” Note that (dd∗) is self-adjoint, and (d+d∗)2 = (dd∗ +d∗d) α , i.e. the Weitzenb¨ock formula, is self-adjoint and positive semi-definite. ∇ ∇α 2 µν The “rough Laplacian” = g µ ν does not satisfy ∇ ∇ ∇ ˜ 2 ˜ β the requirement curl ~a = 0 and div ~a =0 ∆~a = 0 in (∆A A)α = R Aβ. (A.18) − ∇ α curved space, and{ it does not satisfy Green’s} ⇒ first identity Eq. (A.9). For a FRW universe with K = 1 and a curvature scale ± β 2 β ac, the Ricci tensor of 3-space is Rα = (2K/ac) δα , and the Weitzenb¨ock formula gives 3. The Weitzenb¨ock Formula (∆ 2)A˜ = (2K/a2)A.˜ (A.19) The Weitzenb¨ock formula [23, 60] for vector fields − ∇ − c [24] and in 3-dimensional Riemannian space is de-

[1] H. Bondi and J. Samuel, Phys. Lett. A 228, 121 (1997), (1984). gr-gc/9607009. [10] D.R. Brill and J.M. Cohen, Phys. Rev. 143, 1011 (1966). [2] http://einstein.stanford.edu [11] L. Lindblom and D.R. Brill, Phys. Rev. D 10, 3151 [3] I. Ciufolini and E.C. Pavlis, Nature 431, 958 (2004). (1974). [4] E. Mach, Die Mechanik in Ihrer Entwicklung: Historisch- [12] C. Klein, Class. Quantum Grav. 10, 1619 (1993), and Kritisch Dargestellt (Brockhaus, Leipzig, 1. Auflage 1883; references therein. 7., verbesserte und vermehrte Auflage 1912). English [13] D. Lynden-Bell, J. Katz, and J. Biˇc´ak, Mon. Not. R. translation by T.J. McCormack, The Science of Mechan- Astron. Soc. 272, 150 (1995); 277, 1600(E) (1995), and ics: A Critical and Historical Account of Its Develop- references therein. ment (Open Court Publishing, La Salle, Ill., 6th Ameri- [14] J. Katz, D. Lynden-Bell, and J. Biˇc´ak, Class. Quantum can ed., 1960, based on the 9th German edition of 1933). Grav. 15, 3177 (1998). See Chap. 2, Sec. 6, Subsec. 5. [15] T. Doleˇzel, J. Biˇc´ak, and N. Deruelle, Class. Quantum [5] E. Mach, Die Geschichte und die Wurzel des Satzes von Grav. 17, 2719 (2000). der Erhaltung der Arbeit (Calve, Prague, 1872; 2nd ed. [16] J. Biˇc´ak, D. Lynden-Bell, and J. Katz, Phys. Rev. D 69, Johann Ambrosius Barth, Leipzig, 1879); History and 064011 (2004). Roots of the Principle of the Conservation of Energy [17] I. Ciufolini and J.A. Wheeler, Gravitation and Inertia (Open Court, Chicago, 1911). See Note No. 1; pp. 77- (Princeton University Press, Princeton, NJ, 1995). 80 are quoted in [8] on p. 107. [18] H. Pfister, Gen. Relat. Gravit. 39, 1735 (2007). [6] C. Schmid, Cosmological Vorticity Perturbations, Grav- [19] J. Biˇc´ak, J. Katz, and D. Lynden-Bell, Phys. Rev. D 76, itomagnetism, and Mach’s Principle, Proc. COSMO-01, 063501 (2007). Rovaniemi, Finland (2001), gr-qc/0201095 [20] C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravita- [7] C. Schmid, Phys. Rev. D 74, 044031 (2006), tion (Freeman, San Francisco, 1973). gr-qc/0508066. [21] J.M. Bardeen, Phys. Rev. D 22, 1882 (1980). [8] J.B. Barbour and H. Pfister, editors, Mach’s Principle: [22] K.S. Thorne, R.H. Price, and D.A. MacDonald, editors, From Newton’s Bucket to Quantum Gravity (Birkh¨auser, Black Holes, The Membrane Paradigm (Yale University Boston, 1995). Index of Different Formulations of Mach’s Press, New Haven, 1986). Principle on p. 530. [23] J. Jost, Riemannian Geometry and Geometric Analysis, [9] H. Thirring, Phys. Zeitschr. 19, 33 (1918), 22, 29(E) 4th ed. (Springer, Berlin, 2005), Secs. 2.1, 3.3. (1921), English translation in Gen. Rel. Grav. 16, 712 [24] T. Frankel, The Geometry of Physics, 2nd ed. (Cam- 23

bridge Univ. Press, Cambridge, 2004), Secs. 14.1, 14.2. ogy, Part I, The Electrician, 31, 281-282 (1893), Part II, [25] M. Berger, A Panoramic View of Riemannian Geometry The Electrician, 31, 359 (1893). (Springer, Berlin, 2003), Sec 15.6. [43] J.D. Jackson, Classical Electrodynamics (Wiley, New [26] G. de Rham, Differentiable Manifolds (Springer, Berlin, York, 3rd ed., 1999, Sec. 9.7; 2nd ed., 1975, Sec. 16.2). 1984), Secs. 24 - 26. French original: Vari´et´etes differen- [44] L.F. Abbott and R.K. Schaefer, Ap.J. 308, 546 (1986). tiables (Hermann, Paris, 1955). [45] Hu et al [31], Appendix B. [27] K. Tomita, Prog. Theor. Phys. 68, 310 (1982). [46] M.E. Rose, Multipole Fields (Wiley, New York, 1955). [28] H. Kodama and M. Sasaki, Progr. Theoret. Phys. Suppl. [47] A.R. Edmonds, Angular Momentum in Quantum Me- 78, 1 (1984). chanics (Princeton Univ. Press, Princeton, 1957). [29] R. Durrer and N. Straumann, Helvetica Physica Acta, [48] T. Regge and J.A. Wheeler, Phys. Rev. 108, 1063 (1957). 61, 1027 (1988). [49] K.S. Thorne, Rev. Mod. Phys. 52, 299 (1980). [30] N. Straumann, Ann. Phys. (Leipzig), 15, 701 (2006), [50] E.T. Newman and R. Penrose, J. Math. Phys. 7, 863 hep-ph/0505249. (1966). [31] W. Hu, U. Seljak, M. White, and M. Zaldarriaga, Phys. [51] J.N. Goldberg et al., J. Math. Phys. 8, 2155 (1967). Rev. D 57, 3290 (1998). [52] C. Schmid, The other half of Mach’s principle: Frame [32] Misner, Thorne, and Wheeler [20], Sec. 22.4. dragging for linear acceleration, unpublished. [33] A. Lichnerowicz, Publ. Math. Inst. Hautes Et. Sci 10 [53] I. Ozsv`ath, and E. Sch¨ucking, Nature (London) 193, (1961), 293 - 344, see p. 314. 1168 (1962), and Ann. Phys. (N.Y.) 55, 166 (1969). [34] A. Lichnerowicz, in Relativity, Groups, and Topology, [54] A. Einstein, Ann. Phys. (Leipzig) 360, 241 (1918). edited by C. DeWitt and B. DeWitt (Gordon and Breach, [55] J. Biˇc´ak, J. Katz, and D. Lynden-Bell, Class. Quantum New York, 1964), see Eqs. (3.3, 3.4). Grav. 25, 165017 (2008). [35] J. Ehlers in [8], p. 458. [56] D. Lynden-Bell, J. Biˇc´ak, and J. Katz, Class. Quantum [36] A. Einstein, quoted by J. Ehlers in [8], p. 93. Grav. 25, 165018 (2008). [37] Misner, Thorne, and Wheeler [20], Sec. 21.12, pp. 548-49. [57] J.D. Jackson [43], vector formulas inside front cover (top [38] S. Weinberg, Gravitation and Cosmology (Wiley, New half of page). York, 1972). [58] T. Frankel [24], p. 94. [39] Thorne et al [22], Secs. V.A.2 and V.A.3. [59] G. de Rham, Vari´et´es Differentiables (Hermann, Paris, [40] M. Bruni, P.K.S. Dunsby, and G.F.R. Ellis, Ap.J. 395, 1955), see footnote on p. 125. 34 (1992). [60] R. Weitzenb¨ock, Invariantentheorie (Noordhoff, Gronin- [41] E. Bertschinger and A.J.S. Hamilton, Ap.J. 435, 1 gen, 1923). (1994). [42] O. Heaviside, A Gravitational and Electromagnetic Anal-