arXiv:1611.05523v3 [gr-qc] 26 Dec 2017 ugigadigigpicpenl ietosa they the as of directions each null along principle ingoing focus with and streams rest outgoing accreting rotating a at of hole, interior the remain black in Similarly, CMB. who to the to observers respect familiar (CMB) freely-falling) world (comoving, Background preferred set the not selects Microwave and preferred in frames, preferred Cosmic a defines example, the define of For turn humans, set observers. in preferred which of a frames, provide Lorentz may environment physical without must instability method physical follow numerical instability. to huge. gauge the The enough grow to robust unstable metric be anisotropies. intrinsically the is of collapse of growth One that gradients is that challenges. Another numerical is severe challenge presents holes black Belinskii- spacelike and a stalls, to ensues. inflation [4] accretes, surface singular that do, collapse hole holes (BKL) black [2]. Khalatnikov-Lifshitz black rotating Penrose a real by in as that out blueshift indicate paper pointed infinite subsequent instability[3] a first in the inflationary presented horizon computations of Numerical The inner consequence discovered the nonlinear at [1]. instability the Israel inflationary holes is & the black Poisson rotating to of by rotating horizons subject realistic inner are The astronomically holes. of black structure interior ∗ [email protected] ai dabhn h rsn ae sta the that is paper present the behind idea basic A rotating realistic of structure interior the Calculating the of problem the is paper this for motivation The ASnmes 04.20.-q numbers: PACS s WEBB multivector-v of the structure. language ADM becomes mathematical the including into system approaches, translated the other are Gravity, then Some motion, hyperbolic. of pr strongly gra is equation the field own is magnetic its gravitational that the tensor iden If spatial 6 hypelectromagnetism. trace-free and motio initial a laws), the of define conservation on identities equations by arranged thereafter 24 guaranteed be = Lor are must i the 12 which integration + on constraints, 12 numerical class imposed comprise interv facilitate feat be equations may line to A Hamilton’s which freedoms the momenta. gauge vierbein, of Lorentz conjugate the allows 12 components it their spatial that and 12 parts, the antisymmetric are differential coordinates multivector-valued of language mathematical aitna praht h qain fgnrlrelativi general of equations the to approach Hamiltonian A oain aitna erdapoc onmrclrelat numerical to approach tetrad Hamiltonian covariant A .INTRODUCTION I. et srpyia lntr cecs .Clrd,Bou Colorado, U. Sciences, Planetary & Astrophysical Dept. IA o 4,U ooao ole,C 00,UAand USA 80309, CO Boulder, Colorado, U. 440, Box JILA, nrwJSHamilton S J Andrew Dtd uy1,2018) 18, July (Dated: fteCiodagba(emti ler)gnrtdby element generated an algebra) is (geometric vectors algebra multivector Clifford A keep the multivector- of To forms. of to language advantageous differential mathematical valued highly abstract is structure. the it use underlying transparent, structure d’indices”), “d´ebauches their the (what the mess obscures called a that into Preface] the degenerate [9, of to complexity Cartan tend notation the out index written of equations in Because gravitational the group, [8]. symmetry by relativity reviewed numerical are to ap- approaches tetrad Tetrad-based and the coordinate proach. manifest, and both distinct keeping symmetries Lorentz of strategy follows and alternative paper present first the gauge- The its Lorentz derivatives. and treatments coordinate metric with second Most the only namely work quantities, invariant relativity 6-dimensional transformations. numerical Lorentz the of of and group transformations), translations of group (coordinate 4-dimensional the are 7], [6, symmetries conditions gauge connections. For Lorentz geodesic geodesic the on on which interval. imposed built line in are the [5] congruences as theorems the singularity well on a example, freedoms as allow gauge connections impose to conditions, Lorentz to designed gauge freedom choosing is the in including freedoms paper of present range approach broader The the (equiva- interval). in connections line the proposed Lorentz of derivatives the on lently, on conditions gauge along CMB. directions frames the preferred null to define analogous The focus streams accreting the horizon. which inner the approach ikwk metric, Minkowski ue rdc wdepout.Terae sasmdto assumed is reader The product). (wedge product outer eea eaiiyi ag hoyo w local two of theory gauge a is relativity General imposing involve may frames preferred the Selecting le om,bign u hi underlying their out bringing forms, alued om.I h prah h gravitational the approach, the In forms. mtdt nidpnetfil satisfying field of independent field an magnetic to omoted the of analog vitational oecalnigpolm.Te40 The problems. challenging some n iis(eodcascntans.Te6 The constraints). class (second tities γ rufc fcntn ie u which but time, constant of ersurface ∗ SN EB n opQuantum Loop and WEBB, BSSN, , m yi rpsduigtepowerful the using proposed is ty l(h iren nldn their including vierbein) (the al r ftepooe omls is formalism proposed the of ure { ≡ ,1 osriteutos(first equations constraint 10 n, on than rather connections entz se,wihi nw obe to known is which ystem, dr O839 USA 80309, CO lder, γ 0 γ , m γ 1 · , γ γ n 2 , = γ 3 η } mn ihinrpoutthe product inner with n nantisymmetric an and , ivity 2 be familiar with both multivectors (see [10]) and forms conservation laws (§B), on Cartan’s Test (§D), and on [11, 12]. In this paper, local Lorentz transformations electromagnetism (§E), are deferred to Appendices. transform multivectors while keeping forms unchanged; while coordinate transformations transform forms while keeping multivectors unchanged. II. GRAVITATIONAL EQUATIONS Following Dirac [13], the system of Hamilton’s equations for a constrained Hamiltonian system, which The structure of the gravitational equations derived includes local gauge theories such as general relativity, in this section is quite similar to the structure of divides into three groups: (1) equations of motion, which the equations of electromagnetism, except that the are evolution equations that propagate the coordinates gravitational equations are more complicated. In and their conjugate momenta forward in time; (2) particular, the decomposition of Hamilton’s equations constraint equations (first class constraints, in Dirac’s into equations of motion, constraints, and identities terminology), which must be arranged to be satisfied in the two theories is quite similar. For reference, on the initial hypersurface of constant time, but Appendix E presents an exposition of electromagnetism which are guaranteed thereafter by conservation laws; that brings out the similarity of structure. and (3) identities (second class constraints, in Dirac’s terminology), which define redundant fields in terms of others. The constraints and identities share the property A. Line interval, connection, torsion, and curvature that they involve only spatial, not time, derivatives of the coordinates and momenta. The degrees of freedom of the gravitational field are The number of constraint equations equals the contained in the line interval e, or vierbein, a vector 1- dimensionality of the symmetry group of the gauge form with 4×4 = 16 degrees of freedom, and the Lorentz theory, which in the case of general relativity is 10, the connection Γ, a bivector 1-form with 6 × 4 = 24 degrees dimensionality of the Poincar´egroup. The constraint of freedom, equations of general relativity are commonly called k κ Hamiltonian and momentum constraints, which arise e ≡ ekκ γ dx , (1a) k l κ from symmetry under local translations (coordinate Γ ≡ Γklκ γ ∧ γ dx , (1b) transformations), and Gaussian constraints, which arise from symmetry under local Lorentz transformations. implicitly summed over distinct antisymmetric sequences of paired indices. The components Γ of the Lorentz In general relativity, there are altogether 40 gravita- klκ connections, defined by the rate of change of the tetrad tional Hamilton’s equations, including the 10 constraint vectors γ along directions dxκ relative to parallel equations. In the approach proposed in this paper, l transport, the remaining 30 Hamilton’s equations divide into 24 equations of motion and 6 identities. The 6 identities ∂γ Γ ≡ γ · l , (2) define a trace-free spatial tensor that is the gravitational klκ k ∂xκ equivalent of the magnetic field of electromagnetism. If the gravitational magnetic field is promoted to an are sometimes called Ricci rotation coefficients, or independent field satisfying its own equation of motion, (especially when expressed in a Newman-Penrose double- then the system becomes the Wahlquist-Estabrook- null tetrad basis) spin coefficients. Whereas the Buchman-Bardeen (WEBB) [14–16] system. vierbein coefficients ekκ are tensors with respect to both coordinate and Lorentz transformations, the Lorentz The version of general relativity considered in this connection coefficients Γ are coordinate tensors but paper is actually Einstein-Cartan theory, which is klκ not Lorentz tensors. the extension of general relativity to include torsion. It is convenient to use the notation ep to denote the Standard general relativity is recovered by setting torsion normalized p-volume element to zero; but from a fundamental perspective setting torsion to zero is, as Hehl [7, 17] has emphasized, p terms 1 1 artificial. Spin- 2 fields generate torsion: fermions have ep ≡ e ∧···∧ e , (3) spin angular-momentum that sources nonzero torsion. p! z }| { Torsion, whose effects are negligible under normal which is both a p-form and a grade-p multivector. The (accessible to humans) conditions, could potentially play factor 1/p! compensates for multiple counting of distinct a role under the extreme conditions attending inflation indices, so that ep has the correct normalization for and collapse inside black holes. a p-volume element. For example, e2 is the 2-volume The approach is presented in §II, and its integrability element, or area element, while e is the 1-volume element, assessed in §III. The approach is compared to a number or line interval. The scalar product of a multivector of other approaches, including ADM, BSSN, WEBB, K p a ≡ aK γ of grade p with the p-volume element e and Loop Quantum Gravity, in §IV. To keep the main p Λ is the p-form aΛ d x , text compact, some details of the notation (§A), along p p Λ with some extra material on the Bianchi identities and a · e = aΛ d x , (4) 3 implicitly summed over distinct antisymmetric sequences (standard) solution to the excess degrees of freedom is Λ of p coordinate indices. to integrate the Hilbert action by parts so that the The commutator of the (coordinate and Lorentz) gravitational variables become the line interval e and its covariant derivative defines the torsion vector 2-form S conjugate momentum π, the pseudovector 2-form defined and the Riemann curvature bivector 2-form R in terms by of exterior derivatives of the line interval and connection, π ≡−e ∧ Γ . (10)

1 Γ Specifically, the term −e2 ∧ dΓ in the Hilbert La- S ≡ de + 2 [ , e] , (5a) Γ 1 Γ Γ grangian (8) integrates by parts to R ≡ d + 4 [ , ] , (5b) − e2 ∧ dΓ = dϑ + de ∧(e ∧ Γ) . (11) which are Cartan’s equations of structure. Cartan’s equations (5) are definitions of torsion and curvature, not where ϑ is the expansion defined by equation (22). equations of motion. Equations of motion follow from The total exterior derivative term dϑ is the Gibbons- varying the action. Hawking-York [19, 20] boundary term. With the The exterior derivative d is coordinate covariant but boundary term discarded, the integration by parts brings not Lorentz covariant. The (coordinate and Lorentz) the Hilbert Lagrangian (8) to covariant exterior derivative of any multivector form a is ′ I L = L − dϑ = − e ∧ Γ ∧ de + 1 e2 ∧[Γ, Γ] g g κ 4 1 Da = da + [Γ, a] , (6) I
2 = π ∧ de + 1 [Γ, e] . (12) κ 4 the last term accounting for the change in the tetrad
vectors relative to parallel transport. Thus the right The modified Hilbert Lagrangian (12) is in super- hand side of the definition (5a) of torsion is the covariant Hamiltonian form with gravitational coordinates the line exterior derivative De of the line interval, which is a interval e, conjugate momenta π/κ, and the same super- (coordinate and Lorentz) tensor. The right hand side Hamiltonian (9) as before. of the definition (5b) of curvature does not have the The Lorentz connection Γ and conjugate momentum appearance of a covariant exterior derivative, but Γ is π are invertibly related, equation (10) inverting to not a Lorentz tensor, and the definition (5b) is precisely ∗∗ ∗∗ such as to make the Riemann curvature R a (coordinate Γ = − π⊤ + e ∧ ϑ⊤ , (13) and Lorentz) tensor. ∗∗ where denotes the double dual, equation (A13), ⊤ the transpose operation (A6), and ϑ is the expansion defined B. Gravitational action by equation (22). ′ Variation of the gravitational action Sg with the The Hilbert action is modified Hilbert Lagrangian (12) yields

′ I I Sg ≡ Lg , (7) δS = π ∧ δe + δπ ∧ S − Π ∧ δe , (14) Z g κ I κ Z Π where Lg is the Hilbert Lagrangian scalar 4-form where the curvature pseudovector 3-form is defined to be I 2 I 2 1 L ≡− e ∧ R = − e ∧ dΓ + [Γ, Γ] , (8) 1 1 g κ κ 4 Π ≡ e ∧ R − S ∧ Γ = dπ + [Γ, π] − e ∧[Γ, Γ] . (15)
2 4 where I is the pseudoscalar, equation (A3), and κ ≡ 8πG The e ∧ R term in Π equals the double dual of the vector is the normalized Newton’s gravitational constant. 1-form Einstein tensor G, The Hilbert Lagrangian (8) is in super-Hamiltonian ∗∗ (in the terminology of [18]) form Ip ∧ dq − Hg with G = e ∧ R . (16) gravitational coordinates the Lorentz connection q = Γ, conjugate momenta the area element p = −e2/κ, and For any given species of matter, the spin angular- super-Hamiltonian momentum Σ, a bivector 1-form, and the matter energy- momentum T , a vector 1-form, are conventionally defined I H = e2 ∧ 1 [Γ, Γ] . (9) by the variation of the matter action with respect to g κ 4 respectively the connection Γ and the line interval e [7]. In multivector-forms language, However, the area element e2, a bivector 2-form with 6 × 6 = 36 components, has excess degrees of freedom ∗∗ ∗∗ δSm = I Σ ∧ δΓ + δe ∧ T , (17) compared to the 16-component line interval e. A Z 4

∗∗ ∗∗ where Σ and T are the double duals of the spin angular- The term e · ϑ equals the transpose of the expansion, Σ momentum and energy-momentum T of the matter. ⊤ The spin angular-momentum is so called because it e · ϑ = ϑ . (25) vanishes for gauge fields such as electromagnetism, but is finite for spinor fields [7]. The exterior derivative of the expansion ϑ is the pseudoscalar 4-form dϑ satisfying, on substitution of Variation of the matter action Sm instead with respect to δπ and δe defines modified versions Σ˜ and T˜ of the Hamilton’s equations (21), spin angular-momentum and the energy-momentum, κ dϑ + 1 e2 ∧[Γ, Γ]= − e ∧ T˜ + Σ˜ ∧ π (26) 4 2 Σ˜ ˜ ∗∗
δSm = I − δπ ∧ + T ∧ δe . (18) 1 Σ˜ Z = −κ 2 T + ∧ π ,
The original and modified spin angular-momentum and ∗∗ ∗∗ energy-momenta of matter are invertibly related by where T ≡ e ∧ T is the double dual of the trace T of the energy-momentum tensor T . ∗∗ Σ = e ∧ Σ˜ , (19a) ∗∗ T = T˜ − Σ˜ ∧ Γ . (19b) D. 3+1 split

The combined variation of gravitational and matter The torsion equation (21a) comprises, apparently, 24 actions yields Hamilton’s equations for the torsion S and Π equations for the 16 components of the line interval e, curvature , while the Einstein equation (21b) comprises 16 equations S = κΣ˜ , (20a) for the 24 components of the conjugate momentum π. The apparent mismatch between the number of equations Π = κT˜ . (20b) and degrees of freedom is not a practical barrier to solving Hamilton’s equations (21). The 24 torsion equations can Equation (20b) is Einstein’s equation (with torsion). be reinterpreted as defining the 24 components of the More explicitly, Hamilton’s equations (20) are connection Γ (and hence π, equation (10)) in terms of the Σ˜ de + 1 [Γ, e]= κΣ˜ , (21a) spin angular-momentum and the exterior derivative of 2 the line interval e; and then the 16 Einstein equations are 1 Γ 1 Γ Γ ˜ dπ + 2 [ , π] − 4 e ∧[ , ]= κT . (21b) second-order differential equations for the line interval. This is the Lagrangian (second-order) approach. In most problems in general relativity, the spin Σ However, it is advantageous to pursue the Hamiltonian angular-momentum is taken to vanish, = 0, and thus (first-order) approach, since it opens the door to the torsion vanishes, S = 0, as indeed standard general powerful apparatus of canonical transformations, sheds relativity assumes. Spin angular-momentum is retained light on the fundamental structure of the gravitational here however for completeness. equations, and offers a possible pathway to quantization. The apparent mismatch between equations and degrees of freedom stems from two reasons. First, the 16- C. Expansion component line interval e and 24-component momentum π have excess degrees of freedom arising from the 4 The contraction of the momentum π defines the coordinate and 6 Lorentz gauge degrees of freedom. expansion ϑ, a pseudoscalar 3-form with 4 components, Second, a covariant description requires redundant fields that can be expressed entirely in terms of other fields. ϑ ≡ 1 e ∧ π = −e2 ∧ Γ . (22) 2 The simplest example of a redundant field is the magnetic The transpose of the double dual of the expansion, which field of electromagnetism. In electromagnetism, after a enters equation (13), is 3+1 split (see §E 3), the coordinates are the 3 spatial components A of the electromagnetic potential 1-form, ∗∗ ϑ⊤ =Γp γk . (23) and their conjugate momenta are the electric field, the 3 kp spatial components ∗F of the dual electromagnetic field. After a 3+1 space-time split, §II D, the tetrad time The remaining 3 components of the electromagnetic field, p component Γ0p is what is commonly called the trace comprising the magnetic field, the 3 spatial components of the extrinsic curvature, or three times the expansion, F , are redundant because they equal the spatial exterior justifying the nomenclature. derivative dA of the spatial components of the potential; The exterior derivative of the 3-volume e3 is the but the magnetic field nevertheless constitutes part of the pseudovector 4-form de3 satisfying, on substitution of 6-component 4-dimensional covariant electromagnetic Hamilton’s equation (20a), field. The standard approach to isolating excess degrees 3 1 Γ 3 3 2 Σ˜ de + 2 [ , e ] = de − e · ϑ = κe ∧ . (24) of freedom is, following Dirac [13], to perform a 3+1 5 space+time split of the equations, and to interpret only derivative here is the 1-form dt¯ ≡ (∂/∂t)dt. More those equations involving time derivatives as genuine explicitly, the equations of motion (30) are equations of motion. The remaining equations are either 1 Γ 1 Γ Σ˜ constraint equations (first class constraints), which are dt¯eα¯ + 2 [ t¯, eα¯] + dα¯et¯ + 2 [ α¯, et¯]= κ t¯ , (31a) equations that must be arranged to be satisfied in the 1 Γ 1 Γ 1 Γ Γ dt¯πα¯ + [ t¯, πα¯] + dα¯ πt¯ + [ α¯ , πt¯] − (e ∧[ , ])¯ initial conditions but which are thereafter guaranteed 2 2 4 t ˜ by conservation laws, or else identities (second class = κTt¯ . (31b) constraints), which define redundant fields in terms of others. Variation of the action with respect to the variations δe¯ and δπ¯ of the time components of the coordinates In splitting a multivector form a into time and space t t and momenta yields equations involving only spatial components, it is convenient to adopt a notation in which coordinate derivatives, no coordinate time derivatives. a¯ (subscripted t¯) represents all the coordinate time t t These purely spatial equations are not equations of parts of the form, while a (subscriptedα ¯) represents α¯ motion; rather they are either constraints, if their the remaining spatial coordinate components. The t¯ and continued satisfaction is guaranteed by a conservation α¯ subscripts should be interpreted as labels, not indices. law, or else identities. Thus a multivector p-form a splits as Variation of the action with respect to the 4- K p tA K p B component time component δet¯ of the line interval yields a = a¯ + aα¯ ≡ aKtA γ d x + aKB γ d x , (27) t the 4 Hamiltonian/momentum constraints, implicitly summed over distinct antisymmetric sequences Π ˜ K of Lorentz indices and A and B of spatial coordinate 4 Ham/mom: α¯ = κTα¯ , (32) indices. Note that only the coordinate components are or more explicitly split: the Lorentz components are not split into time and space parts. The time component of a product (geometric 1 Γ 1 Γ Γ ˜ 4 Ham/mom: dα¯ πα¯ + 2 [ α¯, πα¯]− 4 (e ∧[ , ])α¯ = κTα¯ . product of multivectors, exterior product of forms) of any (33) two multivector forms a and b satisfies The 4 components of et¯, which are commonly called (minus) the lapse and shift, equation (60), are arbitrarily (ab)t¯ = at¯bα¯ + aα¯bt¯ (28) adjustable by a coordinate transformation. Associated with this symmetry is a conservation law of energy- with no minus signs (minus signs from the antisymmetry momentum (Appendix B 3). The conservation law of form indices cancel minus signs from commuting dt guarantees that if the Hamiltonian/momentum con- through a spatial form). straints (32) are arranged to be satisfied on the initial Splitting the variation (14) of the gravitational action hypersurface of constant time t, then they will satisfied into time and space parts gives thereafter. It should be noted that the mere fact that et¯

tf tf can be treated as a gauge variable arbitrarily adjustable ′ I I by a coordinate transformation does not mean that e¯ δSg = (π ∧ δe)t¯ + (π ∧ δe)α¯ (29) t κ Iti κ I ti must be treated as a gauge variable. But regardless of I the choice of gauge, there are always 4 constraints whose + δπα¯ ∧ S¯ + δπ¯ ∧ Sα¯ − Πα¯ ∧ δe¯ − Π¯ ∧ δeα¯ . κ Z t t t t ongoing validity is guaranteed by conservation of energy- momentum. After the 3+1 split, the gravitational coordinates are the Variation of the action with respect to the 12- 12 spatial components eα¯ of the line interval, and their component time component δπt¯ of the conjugate momen- conjugate momenta are the 12 spatial components πα¯. tum yields 6 Gaussian constraints (first class constraints) The two surface integrals in equation (29) are first an and 6 identities (second class constraints), integral over time at the spatial boundary, and second a Σ˜ difference of integrals over spatial caps at the initial and 6 Gauss + 6 ids: Sα¯ = κ α¯ , (34) final times ti and tf . Variation of the combined gravitational and matter or more explicitly actions with respect to the variations δe and δπ of α¯ α¯ 6 Gauss + 6 ids: d e + 1 [Γ , e ]= κΣ˜ , (35) the spatial coordinates and momenta yields 12+12 = 24 α¯ α¯ 2 α¯ α¯ α¯ equations of motion involving time derivatives, The 6 Gaussian constraints in equations (35) are Σ˜ associated with symmetry under the 6 degrees of freedom 12 eqs of mot: St¯ = κ t¯ , (30a) of Lorentz transformations. There is some freedom in Π ˜ 12 eqs of mot: t¯ = κTt¯ . (30b) choosing gauge variables arbitrarily adjustable under a Lorentz transformation. One choice is to treat Γ Equations (30) are equations of motion in the sense that the 6-component time component t¯ of the Lorentz they determine the time derivatives dt¯eα¯ and dt¯πα¯ of connection bivector 1-form as an arbitrarily adjustable the gravitational coordinates and momenta. The time gauge variable. The Gaussian constraints then follow 6

Γ from varying the action with respect to δ t¯, and comprise connections Γκλµ are not the same as the Christoffel the subset of equations (34) obtained by contracting with symbols (of which there are 64, or 40 if torsion vanishes). the spatial line interval e, In accordance with its definition (5a), the spatial 1 torsion Sα¯ in equation (34) involves [Γ, e]α¯ , which Σ˜ 2 6 Gauss: (e ∧ S)α¯ = κ(e ∧ )α¯ . (36) comprises the 12 coordinate-frame Lorentz connections Γκ[βγ] antisymmetrized over the last two spatial indices Associated with the Lorentz symmetry is a law of βγ, conservation of angular momentum (Appendix B 2). This conservation law guarantees that if the Gaussian 1 Γ Γ κ 2 βγ 2 [ , e]α¯ = α¯ · eα¯ = −2Γκ[βγ] γ d x , (42) constraints (36) are arranged to be satisfied on the initial hypersurface of constant time, then the constraints the sum over βγ being over distinct pairs of spatial will continue to be satisfied thereafter. The Gaussian indices. The 9 all-spatial components Γαβγ of the constraints (36) may be rewritten coordinate-frame Lorentz connections are invertibly related to the 9 all-spatial components Γα[βγ] by 2 Σ˜ 6 Gauss: dα¯eα¯ − eα¯ · πα¯ = κ(e ∧ )α¯ , (37) Γαβγ =Γα[βγ] +Γβ[γα] − Γγ[αβ] . (43) which is a differential relation between the spatial coordinates and momenta eα¯ and πα¯, whose evolution The coordinate-frame spatial Lorentz connections Γαβγ is determined by their equations of motion (31). resolve into a 3-component trace and a 6-component The 6 identities in equations (35) comprise its trace- trace-free part, free spatial part. Projected into the coordinate frame, ∗∗ the 9 spatial components Sγαβ of the torsion are Γαβγ ≡ gγ[α π β] + Bαβγ . (44)

k Sγαβ = e γ Skαβ . (38) The trace part is the BSSN [21] momentum variable,

δ βγ ∗∗ The 3-component spatial trace of the spatial coordinate- Γαδ ≡ g Γαβγ = −π α , (45) frame torsion is where gβγ is the inverse of the spatial met- β βγ ∗∗ Sαβ ≡ g Sγαβ , (39) ric gαβ, and π α is the spatial double-dual (Ap- pendix A 6) of the all-spatial coordinate-frame momen- βγ α β γ 2 δǫ where g is the inverse of the spatial metric gαβ (which tum παβγδǫ γ ∧ γ ∧ γ d x . The trace-free part of the in general is not the same as the spatial components of coordinate-frame spatial Lorentz connection Γαβγ defines the coordinate-frame spacetime metric gκλ). The trace- the 6-component gravitational magnetic field Bαβγ , free spatial part of equations (35) defines the 6 identities, δ Bαβγ ≡ Γαβγ + gγ[αΓβ]δ . (46) δ ˜ ˜ δ 6 ids: Sγαβ + gγ[αSβ]δ = κ Σγαβ + gγ[αΣβ]δ . (40)
The gravitational magnetic field Bαβγ , equation (46), can be written alternatively as the spatial dual (Ap- E. Gravitational magnetic field pendix A6) of a symmetric 3 × 3 spatial matrix Bδγ ,

δ The 6 identities (40) can be interpreted as defining 6 Bαβγ = εαβ Bδγ . (47) redundant components of the spatial Lorentz connection The 12-component time component π¯ of the momen- Γα¯ in terms of spatial exterior derivatives dα¯eα¯ of the t spatial line interval. These 6 redundant components tum can be expressed as a linear combination of the 6- can be called the gravitational magnetic field, equa- component gravitational field, the 6-component gauge Γ tion (46), since the situation is analogous to that in field t¯, and the 12-component spatial momentum πα, electromagnetism, where the 3-component magnetic field with coefficients depending on the 16 components of the is redundant because it can be replaced by the spatial vierbein e. The calculation is somewhat intricate, and is exterior derivative of the spatial components of the relegated to Appendix C. electromagnetic potential, equation (E20). A practical numerical way to calculate πt¯, adopted Let Γκλµ denote the 24 components of the Lorentz by [3], is to use Singular Value Decomposition (SVD). connections projected into the coordinate frame, The 12 components of πt¯ are determined by 6 gauge conditions and the 12 constraints and identities (34). k l Γκλµ ≡ e κe λΓklµ . (41) This is 18 equations for 12 unknowns, including 6 constraints whose validity is in principle guaranteed The coordinate-frame Lorentz connections Γκλµ inherit by conservation of angular momentum. SVD finds antisymmetry in their first two indices κλ from the a solution to an overcomplete set of linear equations antisymmetry of the Lorentz connections Γklγ in their essentially by discarding redundant linear combinations first two indices kl. The 24 coordinate-frame Lorentz of the equations. 7

F. Expansion ADM formalism, the gravitational coordinates are the 6 components of the spatial metric gαβ, and their 6 After the 3+1 split, equation (24) is an equation of conjugate momenta define the 6 components of the 3 extrinsic curvature. Where then do the extra 6 degrees motion for the spatial 3-volume eα¯, of freedom in each of eα¯ and πα¯ come from? The answer 3 3 2 Σ˜ dt¯eα¯ + dα¯et¯ − et¯ · ϑα¯ − eα¯ · ϑt¯ = κ(e ∧ )t¯ . (48) is that the extra degrees of freedom are gauge degrees of freedom associated with Lorentz transformations. Equation (26) is an equation of motion for the spatial Recall that a motivating idea of this paper is that expansion ϑα¯ the environment defines preferred Lorentz frames (the 1 2 most familiar example being the Cosmic Microwave d¯ϑ + d ϑ¯ + e ∧[Γ, Γ] (49) t α¯ α¯ t 4 t¯ Background). The 12 components of the spatial line κ
interval e ≡ e γk dxα encode not only the spatial e T˜¯ e¯ T˜ Σ˜ ¯ π Σ˜ π¯ α¯ kα = − α¯ ∧ t + t ∧ α¯ + t ∧ α¯ + α¯ ∧ t . k 2   coordinate metric gαβ = e αekβ, but also the boost and Whereas the equation of motion (31b) for the spatial rotation of the vierbein with respect to the preferred Lorentz frame. momentum πα¯ is sourced only by the time component ˜ Tt¯ of the energy-momentum pseudovector 3-form, the equation of motion (49) for the spatial expansion ϑα¯ is ˜ ˜ H. Conventional Hamiltonian sourced by eα¯ ∧ Tt¯ + et¯ ∧ Tα¯, which depends in addition on the spatial component T˜α¯ of the energy-momentum. Thus direct numerical integration of the equation of After the 3+1 split, the alternative Hilbert La- motion (49) for the spatial expansion ϑα¯ will yield a grangian (12) can be written slightly different result from integrating the spatial line tf interval eα¯ and spatial momentum πα¯ and then inferring ′ I I ′ L = π ∧ e¯ + π ∧ d¯e − H , (51) the spatial expansion from its definition ϑ = 1 e ∧ π . g α¯ t α¯ t α¯ g α¯ 2 α¯ α¯ κ Iti κ Z A modified version of the equation of motion (49) may be obtained by subtracting from the right hand with conventional (not-super) Hamiltonian side an arbitrary factor n/2 times the wedge product Π ˜ et¯ ∧( − κT )α¯ of et¯ with the Hamiltonian/momentum ′ I H = (− π¯ ∧ S + Π ∧ e¯) . (52) constraints (it would also be possible to subtract an g κ t α¯ α¯ t Σ˜ arbitrary factor times (S − κ )α¯ ∧ πt¯, but that nicety will be glossed over here, since usually spin angular- The conventional Hamiltonian (52) is a sum of the gauge momentum and torsion are taken to vanish identically), and redundant components et¯ and πt¯ of the coordinates and momenta wedged with the components Πα¯ and Sα¯ of ϑ ϑ 1 e2 Γ Γ 1 e Π the curvature and torsion that are subject to constraints dt¯ α¯ + dα¯ t¯ − 4 ∧[ , ] t¯ + 2 n t¯ ∧ α¯ (50) and identities. κ ˜
˜ Σ˜ Σ˜ = − eα¯ ∧ Tt¯ + (1 − n)et¯ ∧ Tα¯ + t¯ ∧ πα¯ + α¯ ∧ πt¯ . 2   If the arbitrary factor is n = 0, the result recovers I. Generalization to other dimensions equation (49). If the arbitrary factor is n = 1, then the right hand side of equation (50) depends only on the The treatment in this section generalizes without time component T˜¯ of the energy-momentum, and the t obstacle to any number N ≥ 3 of spacetime dimensions. result of integrating the equation of motion for ϑ will α¯ The momentum π conjugate to the line interval e in N be equivalent to the result of integrating e and π and α¯ α¯ spacetime dimensions is, generalizing equation (10), the deducing ϑ therefrom. If the arbitrary factor is n = 2, α¯ pseudovector (N−2)-form the result is the Raychaudhuri equation, which is a key ingredient of singularity theorems. − − π ≡ (−)N 3eN 3 ∧ Γ . (53)

Π G. The 12 coordinates and 12 momenta The curvature pseudovector (N−1)-form is, generaliz- ing equations (15),

− − A perhaps surprising aspect of the equations of Π ≡ eN 3 ∧ R − eN 4 ∧ S ∧ Γ motion (30) is that there are 12 + 12 of them for the π 1 Γ π 1 eN−3 Γ Γ 12 gravitational coordinates eα¯ and their 12 conjugate = d + 2 [ , ] − 4 ∧[ , ] . (54) momenta πα¯. This is surprising because one is familiar 1 2 with the notion that the gravitational field has 6 degrees There are 2 N (N+1) Hamilton’s equations, comprising 1 of freedom (for example, the 2 scalar, 2 vector, and 2N(N−1) equations of motion, 2 N(N−1) Gaussian 2 tensor modes of cosmology; although only the 2 constraints, N Hamiltonian/momentum constraints, and 1 tensor degrees of freedom are propagating). In the 2 N(N−1)(N−3) identities. 8

III. STRONG HYPERBOLICITY AND THE WEBB FORMALISM TABLE I. Numbers of equations and variables Approach Eqs. mot. Constraints Identities Variables The ensemble of equations (21) constitute an Exterior This paper 24 10 6 40 Differential System (EDS). For such a system there is ADM 12 10 18 40 an algorithm, called Cartan’s Test [11, 12], to decide BSSN 15 10 15 40 whether the system is integrable. A system is integrable WEBB 30 16 0 46 if, given an appropriate set of initial conditions, a solution exists and is unique. Appendix D shows that LQG 18 7 0 25 the system (21) passes Cartan’s Test, and is therefore integrable. However, integrability does not guarantee good nu- system of tetrad-based equations proposed by Buchman merical behavior, if small errors in the initial con- & Bardeen [15] based on the work of Estabrook, Robinson ditions blow up exponentially. A condition that & Wahlquist [14]. [15] prove that the WEBB system is guarantees good numerical behavior is that the system strongly hyperbolic. be strongly hyperbolic [22–24]. Loosely speaking, [15] work with the all tetrad-frame Lorentz connec- µ strong hyperbolicity requires that perturbations to initial tions Γklm ≡ em Γklµ, which they decompose into conditions propagate as waves ∼ eiωt rather than growing various parts labeled with different letters. The 9- αt exponentially ∼ e . component all-spatial part Γabc of the tetrad-frame Strong hyperbolicity for a first-order system of partial Lorentz connection is labeled N. If the ADM gauge differential equations is defined as follows [24]. Let ui condition (57) is imposed, so that the tetrad-frame denote a set of variables satisfying the first-order system spatial hyperplane coincides with the coordinate-frame spatial hyperplane, then spatial tetrad-frame vectors Aa ∂ui α ∂uj and spatial coordinate-frame vectors Aα are related by + Aij α + ··· =0 , (55) a ∂t ∂x the spatial vierbein and its inverse, Aα = e αAa and A = e αA . The spatial tetrad-frame connection Γ where ··· does not involve derivatives of the variables. a a α abc then decomposes into a 3-component trace part (which The matrix Aα for each spatial coordinate direction ∗∗ ij [15] label n), identified here as the BSSN variable π , α is called the principal symbol of the system. The b and a 6-component trace-free part identified here as the system is called weakly hyperbolic if, for every direction gravitational magnetic field B , α, all the eigenvalues of the principal symbol are real. abc The system is called strongly hyperbolic if in addition, ∗∗ Γabc = εabdNcd = ηc[a π b] + Babc , (56) for every α, the eigenvectors of the principal symbol form a complete set, and the eigenvector matrix and which is the tetrad-frame version of equation (44). its inverse are uniformly bounded. A key advantage of The WEBB system is discussed further in §IV E. the BSSN formalism over the ADM formalism is that BSSN is strongly hyperbolic whereas ADM is only weakly hyperbolic [22, 23]. IV. COMPARISON TO ADM, BSSN, WEBB, The equations under consideration are the 24 equations AND LQG of motion (31) for the 24 variables ui = {ekα, πklmαβ } comprising the components of eα¯ and πα¯. The thing For comparison, this section recasts some other that complicates the analysis of the hyperbolicity of formalisms into the language of multivector-valued these equations is that the 18-component spatial Lorentz differential forms. connection Γα¯ depends not only on the 12-component The Arnowitt-Deser-Misner (ADM) formalism [25, 26], spatial momentum πα¯ but also on the 6-component provides the backbone for most modern implementations gravitational magnetic field, §II E, which itself depends of numerical relativity. The Baumgarte-Shapiro-Shibata- on spatial derivatives of the spatial coordinates eα¯ . The Nakamura (BSSN) formalism founded on the work of term dα¯ πt¯ in the Einstein equations (31b) then includes [21, 27] has gained popularity because it proves to have some second-order spatial derivatives of eα¯, while the better numerical stability than ADM when applied to 1 Γ 1 Γ Γ terms 2 [ α¯ , πt¯] and 4 et¯ ∧[ α¯ , α¯ ] are quadratic in first- problems such as the merger of two black holes [28]. order spatial derivatives of eα¯ . The Wahlquist-Estabrook-Buchman-Bardeen (WEBB) The difficulty can be overcome by promoting the [16] formalism is a system of tetrad-based equations gravitational magnetic field Bαβγ to a set of 6 inde- proposed by Buchman & Bardeen [15] based on the work pendent variables governed by their own equation of of Estabrook, Robinson & Wahlquist [14]. motion. The operation of promoting derivatives of Table I summarizes for each approach considered in variables to independent variables and enlarging the this paper the number of equations of motion (equations system of differential equations is called prolongation. governing the evolution of variables with time), the The system obtained by prolonging the gravitational number of constraints (equations that must be satisfied magnetic field proves to be the WEBB formalism [16], a in the initial conditions but are thereafter guaranteed), 9 the number of identities (equations defining redundant spatial line interval and spatial area element subject to variables in terms of others), and the total number of the ADM gauge condition (59) are invertibly related to variables. each other. Since the spatial line interval eα¯ and spatial area 2 element eα¯ subject to the ADM gauge condition (59) 2 A. ADM gauge condition are invertibly related, either ea¯α¯ or ea¯α¯ may be used as gravitational coordinates. The momenta conjugate to the 2 ADM and BSSN start by supposing that spacetime is spatial area element ea¯α¯ are the 3 × 3 = 9 components of Γ 0 a α sliced into spatial surfaces of constant time t. The unit the Lorentz connections − 0¯¯α ≡ Γa0α γ ∧ γ dx with normal to the spatial surfaces defines the tetrad time axis one Lorentz index the tetrad time index 0, also called γ0 at each point of spacetime. Usually ADM and BSSN the extrinsic curvature. In the literature the extrinsic work entirely in a coordinate frame, without reference to curvature is commonly denoted by the symbol K, tetrads; but there is no loss of generality to introduce a K ≡ Γ . (61) locally inertial tetrad γk and to project into a coordinate aα a0α frame at a later stage. The condition that the tetrad The extrinsic curvature is a spatial tensor, that is, it is a time axis be normal to spatial surfaces of constant time, tensor with respect to the 6-dimensional spatial subgroup γ0 · eα¯ = 0, imposes the 3 gauge conditions of the 10-dimensional Poincar´egroup consisting of spatial coordinate transformations (at fixed t) and spatial e0α =0 . (57) rotations of the spatial tetrad. If torsion vanishes, then the extrinsic curvature is symmetric (it equals its Physically, the ADM gauge condition (57) is equivalent transpose, equation (A8)). The momenta conjugate to to imposing that the 3-dimensional coordinate spatial hy- the spatial line interval e are the 9 components of perplane coincides with the 3-dimensional tetrad spatial a¯α¯ π¯ ≡ −e ∧ Γ¯ . In conventional notation, the 9 hyperplane (more formally, the hyperplane spanned by 0¯α a¯α¯ 0α ¯ ∗∗ components of the double dual π ¯ of the conjugate the 3 spatial coordinate 1-forms dxα is the same as the a¯t momenta π¯ are invertibly related to the 9 components hyperplane spanned by the 3 spatial tetrad directions 0¯α of the extrinsic curvature by γa). The ADM gauge condition (57) uses up the 3 degrees of freedom of Lorentz boosts. The ADM gauge ∗∗ π atα = Kaα − eaαK, (62) choice (57) is also a basic ingredient of Loop Quantum a Gravity, §IV F. where K ≡ Ka is the trace of the extrinsic curvature. To facilitate comparison to the approach proposed in the present paper, choose the 9 coordinates and B. Double 3+1 split 9 momenta in the ADM or BSSN formalisms to be 2 ea¯α¯ and π0¯¯α (one could equally well choose ea¯α¯ and Γ To elucidate the structure of the ADM formalism, it is − 0¯¯α). Variation of the combined gravitational and convenient to extend the coordinate 3+1 split (27) to a matter actions with respect to variations δπ0¯¯α and δea¯α¯ double 3+1 split of Lorentz as well as coordinate indices. yields 9 + 9 = 18 equations of motion for the spatial line Thus a multivector form a splits into 4 components interval ea¯α¯ and their conjugate momenta π0¯¯α, a0¯t¯, a0¯¯α, aa¯t¯, and aa¯α¯ that represent respectively Σ˜ the time-time, time-space, space-time, and space-space 9 eqs of mot: Sa¯t¯ = κ a¯t¯ , (63a) ¯¯ ¯ ¯ Π ˜ components of the multivector form (the 0t, 0¯α,a ¯t, 9 eqs of mot: 0¯t¯ = κT0¯t¯ . (63b) anda ¯α¯ subscripts should be interpreted as labels, not indices), Variation of the action with respect to δe0¯α¯ yields 3 equations of motion, a = a¯¯ + a¯ + a ¯ + a . (58) 0t 0¯α a¯t a¯α¯ Π ˜ 3 eqs of mot: a¯t¯ = κTa¯t¯ . (64) In this notation, the ADM gauge condition (57) is Note that the ADM gauge condition e0¯¯α = 0 is a gauge condition, fixed after equations of motion are derived, e0¯¯α =0 . (59) so it is correct to vary e0¯¯α in the action, leading to equation (64). Equation (64) is an equation of motion The coordinate time components e0¯t¯ and ea¯t¯ of the line interval are commonly called the lapse α and shift βa, in the sense that it involves a time derivative dt¯πa¯α¯ ; but πa¯α¯ is not one of the momenta π0¯¯α conjugate to the e0t = −α, eat = −βa . (60) line interval ea¯α¯ , so equation (64) has a different status from the 9 + 9 equations of motion (63). As described in The ADM gauge condition (59) reduces the number of §IV C and §IV D below, the treatment of equation (64) degrees of freedom of the spatial line interval eα¯ from 12 is what distinguishes the ADM and BSSN formalisms: 2 to 3 × 3 = 9, and of the spatial area element eα¯ from 18 ADM treats it as a constraint equation, while BSSN to the same number, 3 × 3 = 9. The 9 components of the retains it as an equation of motion. 10

Variation of the combined gravitational and matter the metric, and their conjugate momenta are, modulo action with respect to δπa¯α¯, δπa¯t¯, δπ0¯t¯, δea¯t¯ and δe0¯t¯ a trace term, the components Γα0β of the extrinsic Γ yields 19 equations involving only spatial derivatives, curvature − 0¯¯α projected into the coordinate frame γ namely 9 identities, 6 Gaussian constraints, and 4 (specifically, the conjugate momenta are Γα0β − gαβΓ0γ , Hamiltonian/momentum constraints, cf. equation (62)). For vanishing torsion, as ADM assumes, the extrinsic curvature Γα0β is symmetric and Σ˜ 3 ids: S0¯t¯ = κ 0¯t¯ , (65a) therefore also has 6 components. Thus in ADM there Σ˜ are 6 + 6 = 12 equations of motion. The remaining 3 Gauss: S0¯¯α = κ 0¯¯α , (65b) gravitational degrees of freedom are embodied in the 18 Σ˜ 3 Gauss + 6 ids: Sa¯α¯ = κ a¯α¯ , (65c) spatial components of the coordinate-frame connections Π ˜ (the Christoffel symbols, not Lorentz connections). The 3 mom: 0¯¯α = κT0¯¯α , (65d) 18 spatial coordinate connections are determined by Π ˜ 1 Ham: a¯α¯ = κTa¯α¯ . (65e) 18 identities relating them to spatial derivatives of the spatial metric. The 3 identities (65a) are not equations of motion (they In ADM, the 6 Gaussian constraints cease to appear involve no time derivatives), despite having a form label explicitly. Where have they gone? The answer t¯. Explicitly, is that they have disappeared into the 6-component 1 1 1 antisymmetric part of the Einstein equations, which 3 ids: dα¯e¯¯ + [Γa¯α¯ , e¯¯]+ [Γ¯¯, ea¯α¯]+ [Γ¯ , e¯ ] 0t 2 0t 2 0t 2 0¯α tα¯ vanishes identically if torsion vanishes, as ADM assumes. Σ˜ = κ 0¯t¯ . (66) Altogether, the ADM equations comprise 6 + 6 = 12 equations of motion, 18 identities, 3 momentum As earlier, equation (36), the 3 Gaussian constraints con- constraints, 1 Hamiltonian constraint, and 6 Gaussian tained in equations (65c) comprise the subset obtained by constraints absorbed into the antisymmetric part of the wedging with the spatial line interval ea¯α¯ , Einstein equations, a total of 40 Hamilton’s equations.

3 Gauss: (e ∧ S)a¯α¯ = κ(e ∧ Σ˜ )a¯α¯ . (67)

The combined set of 40 Hamilton’s equations (63)–(65) D. BSSN formalism are identical to the earlier equations (30), (32), and (34), but grouped differently. The different grouping results The BSSN formalism follows ADM for the most part, because ADM and BSSN treat e0¯¯α not as a dynamical except that it retains the 3 equations (64) as equations variable, but rather as a gauge variable set to zero. of motion. As a result, BSSN retains the 3 Gaussian constraints (67) as constraints. In all, the BSSN equations constitute 6 + 6 + 3 = C. ADM formalism 15 equations of motion, 15 identities, 3 momentum constraints, 1 Hamiltonian constraint, 3 Gaussian con- The ADM formalism treats the 3 equations (64) as straints, and 3 Gaussian constraints absorbed into the constraint equations, which is justified because they antisymmetric spatial part of the Einstein equations, a arise from variation of the action with respect to the total of 40 Hamilton’s equations. 3 components of e0¯¯α, which ADM treats as gauge variables. Actually, ADM simply discards equations (64) unceremoniously. If torsion vanishes, then Lorentz E. WEBB formalism gauge symmetry enforces that the Einstein tensor is symmetric, equation (B12), in which case the 3 The WEBB formalism [14, 15] is a tetrad system of equations (64) should equal the transpose of the 3 equations for numerical relativity similar in spirit to the momentum constraints (65d), and can be dropped. approach of the present paper. The WEBB formalism Because ADM uses up the gauge freedom under replaces the 6 identities (40) defining the gravitational Lorentz boosts on the vierbein, the gauge freedom can magnetic field Bαβγ in terms of spatial derivatives of the no longer be used on the connections (the 3 components spatial coordinates eα¯ by equations of motion, essentially Γ of 0¯t¯ can no longer be treated as gauge variables). Thus by replacing some of the equations for the spatial in ADM the 3 Gaussian constraints (67) cease to be torsion Sα¯ by their exterior derivatives, equation (68c), a constraints; rather, they become identities. process called prolongation. Promoting the gravitational Commonly, after the 3 gauge freedoms under Lorentz magnetic field to a set of independent variables governed boosts have been fixed by the ADM gauge choice (59), the by their own equation of motion has the virtue that the ADM equations are cast entirely in the coordinate frame. resulting system is strongly hyperbolic, as proved by [15]. This has the advantage that the remaining variables are Because the WEBB approach works entirely in the gauge-invariant under the remaining Lorentz symmetries, tetrad frame, the WEBB system of equations does not namely spatial rotations. After a 3+1 split, the gravita- translate directly into the language of multivector-valued tional coordinates are the 6 spatial components gαβ of differential forms used in the present paper. However, if 11 the ADM gauge condition (57) is imposed, so that the coordinates. The momentum conjugate to the spatial 2 3-dimensional tetrad-frame and coordinate-frame spatial area element ea¯α¯ is the 9-component extrinsic curvature Γ tangent planes coincide, then there is a direct translation. 0¯¯α. LQG keeps all of the 9+9 equations of motion (63), Subject to the ADM gauge condition, the WEBB system including their antisymmetric parts. has 30 equations of motion, Like ADM, LQG discards the 3 BSSN equations (64) as constraint equations enforced by the symmetry of Σ˜ 12 eqs of mot: St¯ = κ t¯ , (68a) the time-space components of the Einstein tensor, Π ˜ for vanishing torsion. LQG retains the 4 Hamilto- 9 eqs of mot: 0¯t¯ = κT0¯t¯ , (68b) nian/momentum constraints (65d) and (65e), and the 3 Σ˜ 9 eqs of mot: (dS)a¯t¯ = κ(d )a¯t¯ . (68c) Gaussian constraints (67). In the seminal paper that subsequently led to LQG, Equations (68a) govern the evolution of the 12 com- Ashtekar [33] noticed that bivectors in 4 spacetime ponents e of the line interval. Equations (68b) α¯ dimensions (and only in 4 dimensions) have a natural are equivalent to equation (39) of [15], and govern complex structure. The complex structure allows the 24- the evolution of the 9 components of the extrinsic component bivector 1-form Lorentz connections Γ to be curvature Γ¯ . The 9 equations (68c) are equivalent 0α¯ split into distinct 12-component complex right- and left- to equation (40) of [15], and constitute 3 equations handed chiral parts equivalent to the BSSN equations (64) plus 6 equations of motion for the gravitational magnetic field, as follows Γ± ≡ (1 ± γ )Γ , (70) from the decomposition (44) of the 9 all-spatial Lorentz 5 Γ connections a¯α¯ into the (dual of the) 3 all-spatial where γ5 ≡−iI is the chiral operator, which is minus the components πa¯α¯ of the conjugate momentum and the product of the quantum-mechanical imaginary i and the 6-component gravitational magnetic field Ba¯α¯ . pseudoscalar I of the geometric algebra, equation (A3). The 9 equations of motion (68c) must be supplemented The chiral Cartan’s equations (5b) are by the 9 initial conditions Sa¯α¯ = κΣ˜ a¯α¯, which comprise Γ 1 Γ Γ 3 Gaussian constraints and 6 equations relating the R± ≡ (1 ± γ5)R = d ± + 4 [ ±, ±] . (71) gravitational magnetic field to spatial derivatives of the spatial coordinates eα¯. These 6 equations are constraints That is, the right(left)-handed Riemann curvature de- since they must be arranged to be satisfied on the initial pends only on the right(left)-handed Lorentz connection. spatial hypersurface, but are thereafter guaranteed by Ashtekar pointed out that the imaginary part of the the Bianchi identity (B1a). I call them the B constraints. (right-handed, without loss of generality) chiral Hilbert Altogether the WEBB system has 16 constraints: Lagrangian, Σ˜ I 3 Gauss: S0¯α¯ = κ 0¯α¯ , (69a) L ≡− e2 ∧ R , (72) + κ + 3 Gauss + 6 B: Sa¯α¯ = κΣ˜ a¯α¯ , (69b) vanishes if torsion vanishes, and that the resulting 3 mom: Π¯ = κT˜¯ , (69c) 0α¯ 0α¯ Hamilton’s equations of motion are unchanged from Π ˜ 1 Ham: a¯α¯ = κTa¯α¯ . (69d) those of general relativity, if torsion vanishes. Ashtekar proposed as fundamental gravitational vari- In effect, WEBB replaces the 6 identities (40) by 6 ables the 9 components of the complex chiral spatial equations of motion and 6 constraints. Lorentz connection Γ+¯aα¯ , and the 9 components of the 2 spatial area element ea¯α¯. These 9 + 9 variables are called Ashtekar variables, F. Loop Quantum Gravity Γ 2 +¯aα¯ , ea¯α¯ Ashtekar variables . (73) Having come this far, it is useful to comment briefly on Loop Quantum Gravity (LQG; e.g. [29–32]). The The spatial line interval ea¯α¯ is the double dual of the 2 language of multivector-valued forms is perfectly suited area element ea¯α¯ in 3 spatial dimensions, to LQG. ∗∗ 2 Like ADM and BSSN, LQG starts by making the ea¯α¯ = e a¯α¯ , (74) ADM gauge choice (59). Unlike ADM and BSSN, LQG treats the remaining 3 Lorentz degrees of freedom a trick that works once again only in 4 spacetime under spatial rotations as having central importance. dimensions, hence 3 spatial dimensions. As remarked in the third paragraph of §IVB, the 9- A key point is that keeping only 9 of the 18 spatial component spatial line interval ea¯α¯ and 9-component components Γa¯α¯ of the Lorentz connection means that 9 2 spatial area element ea¯α¯ subject to the ADM gauge components are discarded, which allows the 9 identities condition (59) are invertibly related, so either ea¯α¯ or contained in equations (65a)–(65c) to be discarded. 2 Γ ea¯α¯ may be used as gravitational coordinates. LQG Likewise 3 of the 6 time components a¯t¯ of the Lorentz 2 adopts the spatial area element ea¯α¯ as the gravitational connection are kept, and 3 are discarded, allowing 12 the 3 Gaussian constraints (67) to be discarded. In Poincar´esymmetry group of general relativity is kept all, the Ashtekar equations comprise 9 + 9 equations manifest. The symmetry group is kept manifest by using of motion, zero identities, 4 Hamiltonian/momentum the powerful mathematical language of multivector- constraints, and 3 Gaussian constraints, a total of 25 valued differential forms. equations. The missing 15 equations, compared to the The most immediate practical advantage of the original set of 40 Hamilton’s equations, are the 3 BSSN approach is that it allows a wider range of gauge choices equations (64), which for vanishing torsion are enforced than the traditional ADM [38] or BSSN [28] approaches. by the antisymmetry of the momentum components of Specifically, the proposed approach allows gauge choices the Einstein equation, and the 9 identities and 3 Gaussian to be imposed not only on the line interval but also on the constraints determining the 12 discarded components of Lorentz connections. The broader range of gauge options the Lorentz connection of opposite chirality. could be advantageous in some challenging problems such The resulting simplification of the gravitational equa- as the interior structure of accreting, rotating black holes, tions of motion (there being zero identities), and the addressed in a subsequent paper [3]. polynomial form of quantities in the Ashtekar variables In the proposed approach, the 40 Hamilton’s equations 2 Γ ea¯α¯ and +¯aα¯, led to hopes that canonical quantization divide into 24 equations of motion and 6 identities, along could proceed. Unfortunately the device of complexifying with 10 constraint equations. The 6 identities define a 6- the Lorentz connection proved too high a price, and that component gravitational magnetic field analogous to the hope was not realized. magnetic field of electromagnetism. If the gravitational Subsequently it began to be realized, starting with [34], magnetic field is promoted to a set of independent that if gravity is treated as a gauge theory of spatial variables governed by their own equation of motion, Lorentz transformations, then a gauge rotation (a spatial then the system becomes the WEBB [14–16] system rotation) by 2π around a loop will lead to nontrivial with 30 equations of motion and 16 constraints, which quantum mechanical consequences. Thus Loop Quantum the authors of [15] have shown constitutes a strongly Gravity was born. hyperbolic system. Later versions of LQG are based on a real Lagrangian It has been remarked that Loop Quantum Gravity can [35] be cast elegantly in the language of multivector-valued forms.

I 2 LLQG ≡− e ∧ Rβ , (75) κ ACKNOWLEDGMENTS in which the imaginary i in equation (71) is replaced by a (usually) real parameter β called the Barbero-Immirzi This research was supported in part by FQXI mini- parameter [36, 37], grant FQXI-MGB-1626. I thank Gavin Polhemus for the suggestion to use multivector-valued forms, and Frank Estabrook for useful correspondence. I thank I ∗∗ R ≡ 1+ R . (76) β  β  Professor Fred Hehl for pointing out that the −e·T term in equation (B11) can be interpreted as the covariant exterior derivative of orbital angular momentum, §19(c) of [39]. V. SUMMARY

This paper has presented a Hamiltonian approach to REFERENCES numerical relativity in which the full 10-dimensional

[1] E. Poisson and W. Israel, “Inner-horizon Evgeny M. Lifshitz, “A general solution of the Einstein instability and mass inflation in black holes,” equations with a time singularity,” Advances in Physics Phys. Rev. Lett. 63, 1663–1666 (1989). 31, 639–667 (1982). [2] Roger Penrose, “Structure of space-time,” in Battelle [5] Jos´eM. M. Senovilla, “Singularity theorems and their Rencontres: 1967 lectures in mathematics and physics, consequences,” Gen. Rel. Grav. 30, 701–848 (1998). edited by C´ecile de Witt-Morette and John A. Wheeler [6] James M. Nester and Chiang-Mei Chen, (W. A. Benjamin, New York, 1968) pp. 121–235. “Gravity: a gauge theory perspective,” [3] Andrew J. S. Hamilton, “Mass inflation followed by Int. J. Mod. Phys. D25, 1645002 (2016), Belinskii-Khalatnikov-Lifshitz collapse inside accreting, arXiv:1604.05547 [gr-qc]. rotating black holes,” Phys. Rev. D 96, 084041 (2017), [7] Milutin Blagojevi´c and Friedrich W. Hehl, Gauge arXiv:1703.01921 [gr-qc]. Theories of Gravitation (Imperial College Press, 2013). [4] Vladimir A. Belinskii, Isaak M. Khalatnikov, and [8] James M. Bardeen, Olivier Sarbach, and Luisa T. 13

Buchman, “Tetrad formalism for numerical relativ- Phys. Rev. 116, 1322–1330 (1959). ity on conformally compactified constant mean cur- [26] R. Arnowitt, S. Deser, and C. W. Misner, “The dynamics vature hypersurfaces,” Phys. Rev. D 83, 104045 (2011), of general relativity,” in Gravitation: an introduction to arXiv:1101.5479 [gr-qc]. current research, edited by L. Witten (John Wiley & [9] Elie´ Cartan, Le¸cons sur la Gˆeom´etrie des Espace de Sons, 1963) pp. 227–265. Riemann, Vol. 21 (Gauthier-Villars, 1951). [27] Masaru Shibata and Takashi Nakamura, “Evolution of [10] Chris Doran and Anthony Lasenby, Geometric Algebra three-dimensional gravitational waves: Harmonic slicing for Physicists (Cambridge University Press, Cambridge, case,” Phys. Rev. D 52, 5428–5444 (1995). England, 2003). [28] Thomas W. Baumgarte and Stuart L. Shapiro, Numerical [11] Elie´ Cartan, Riemann Geometry in an Orthogonal Frame Relativity: Solving Einstein’s Equations on the Computer (World Scientific, 2001). (Cambridge University Press, 2010). [12] T. A. Ivey and J. M. Landsberg, Cartan for Beginners: [29] Thomas Thiemann, Modern canonical quantum general relativity Differential Geometry via Moving Frames and Exterior (Cambridge University Press, 2008) Differential Systems, 2nd Edition, Graduate Studies in arXiv:gr-qc/0110034. Mathematics, Vol. 175 (American Mathematical Society, [30] Carlo Rovelli, Quantum Gravity (Cambridge University Providence, 2016). Press, 2007). [13] Paul A. M. Dirac, Lectures on Quantum Mechanics [31] Carlo Rovelli, “Loop quantum gravity,” (Belfer Graduate School of Science, New York, 1964). Living Reviews in Relativity 11, 5 (2008). [14] Frank B. Estabrook, R. Steve Robinson, and [32] Martin Bojowald, “Loop quantum cosmology,” Hugo D. Wahlquist, “Hyperbolic equations Living Reviews in Relativity 11, 4 (2008). for vacuum gravity using special orthonormal [33] A. Ashtekar, “New Hamiltonian formulation of general frames,” Class. Quant. Grav. 14, 1237–1247 (1997), relativity,” Phys. Rev. D 36, 1587–1602 (1987). arXiv:gr-qc/9703072. [34] Ted Jacobson and Lee Smolin, “Nonperturbative quan- [15] L. T. Buchman and James M. Bardeen, “A hyperbolic tum geometries,” Nucl. Phys. B299, 295 (1988). tetrad formulation of the Einstein equations for [35] T. Thiemann, “Anomaly — free formulation numerical relativity,” Phys. Rev. D 67, 084017 (2003), of nonperturbative, four-dimensional Lorentzian [Erratum: Phys. Rev.D72,049903(2005)], quantum gravity,” Phys. Lett. B380, 257–264 (1996), arXiv:gr-qc/0301072. arXiv:gr-qc/9606088. [16] L. T. Buchman and J. M. Bardeen, “Schwarzschild [36] J. Fernando Barbero G., “Real Ashtekar tests of the WEBB tetrad formulation for variables for lorentzian signature space numerical relativity,” Phys. Rev. D 72, 124014 (2005), times,” Phys. Rev. D 51, 5507–5510 (1995), arXiv:gr-qc/0508111. arXiv:gr-qc/9410014. [17] Friedrich W. Hehl, “Gauge theory of gravity and [37] Giorgio Immirzi, “Real and complex spacetime,” (2012), arXiv:1204.3672 [gr-qc]. connections for canonical gravity,” [18] Charles W. Misner, Kip S. Thorne, and John A. Class. Quant. Grav. 14, L177–L181 (1997), Wheeler, Gravitation (W. H. Freeman and Co., 1973). arXiv:gr-qc/9612030. [19] James W. York, Jr., “Role of conformal three [38] Luis Lehner, “Numerical relativity: a review,” Class. geometry in the dynamics of gravitation,” Quant. Grav. 18, R25–86 (2001), arXiv:0106072 [gr-qc]. Phys. Rev. Lett. 28, 1082–1085 (1972). [39] E. M. Corson, Introduction to Tensors, Spinors, and [20] G. W. Gibbons and S. W. Hawking, “Action in- Relativistic Wave-Equations (Blackie & Son, London and tegrals and partition functions in quantum gravity,” Glasgow, 1953). Phys. Rev. D 15, 2752–2756 (1977). [40] Friedrich W. Hehl, J. Dermott McCrea, Eckehard W. [21] Thomas W. Baumgarte and Stuart L. Shapiro, “On Mielke, and Yuval Ne’eman, “Metric affine gauge the numerical integration of Einstein’s field equations,” theory of gravity: Field equations, Noether identities, Phys. Rev. D 59, 024007 (1998), arXiv:gr-qc/9810065. world spinors, and breaking of dilation invariance,” [22] Heinz O. Kreiss and Omar E. Ortiz, “Some mathe- Phys. Rept. 258, 1–171 (1995), arXiv:gr-qc/9402012. matical and numerical questions connected with first and second order time dependent systems of partial differential equations,” The conformal structure of space- time: Geometry, analysis, numerics, Lect. Notes Phys. 604, 359–370 (2002), arXiv:gr-qc/0106085. Appendix A: Notation [23] Gabriel Nagy, Omar E. Ortiz, and Oscar A. Reula, “Strongly hyperbolic second order Einstein’s 1. Indices evolution equations,” Phys. Rev. D 70, 044012 (2004), arXiv:gr-qc/0402123. [24] David Hilditch, “An introduction to well-posedness Latin indices k, l, ... refer to Lorentz frames. Greek and free-evolution,” Proceedings, Spring School indices κ, λ, ... refer to coordinate frames. Early on Numerical Relativity and High Energy Physics latin a, b, ... and greek α, β, ... indices refer to spatial (NR/HEP2): Lisbon, Portugal, March 11-14, coordinates only. Capital latin K, L, ... and greek Λ, Π, ... 2013, Int. J. Mod. Phys. A28, 1340015 (2013), indices refer to distinct antisymmetric sequences of arXiv:1309.2012 [gr-qc]. [25] R. Arnowitt, S. Deser, and C. W. Misner, “Dynamical indices. Implicit summation is over distinct antisymmet- structure and definition of energy in general relativity,” ric sequences of indices, since this removes ubiquitous factorial factors that would otherwise appear. 14

2. Multivectors Commutators [a, b] of multivector forms are defined in the usual way, Multivectors [10] are elements of the Clifford algebra, or geometric algebra, generated by basis elements (axes) [a, b] ≡ ab − ba . (A5) γ equipped with an inner product k In accordance with the convention of this paper, the products on the right hand side of equation (A5) γk · γl = ηkl , (A1) are geometric products of multivectors, and exterior and an antisymmetric outer product γk ∧ γl = −γl ∧ γk. products of forms. The commutator of a bivector form In this paper there are 4 spacetime dimensions, hence 4 with a multivector form of grade n is a multivector form axes γk ≡ {γ0, γ1, γ2, γ3}, and their inner product ηkl of the same grade n. is Minkowski. The geometric algebra is identical to the Clifford algebra of Dirac γ-matrices, hence the notation. 4. Transpose of a multivector form The metric generalizes to a multivector metric ηKL. If the indices are K = k1...kn and L = l1...ln, then the ⊤ multivector metric ηKL is The transpose a of a multivector form a of grade n and form index p is defined to be the multivector form, [n/2] ηKL ≡ γK · γL = (−) ηk1l1 ··· ηknln , (A2) of grade p and form index n, with multivector and form indices transposed by the vierbein, which has an extra sign (−)[n/2]. Multivector indices are ⊤ lowered and raised with the multivector metric and its ⊤ k l p µν.. a = akl..µν.. γ ∧ γ ∧ .. d x inverse. A similar convention applies to forms: there is k l µ ν
m n n κλ.. an extra sign factor of (−)[p/2] when p indices of a form = e κe λ ·· em en ·· akl..µν.. γ ∧ γ ∧ · · d x m n n κλ.. are lowered or raised. = aκλ..mn.. γ ∧ γ ∧ · · d x , (A6) The pseudoscalar I is the highest grade element of the geometric algebra, implicitly summed over distinct antisymmetric sequences kl.., mn.., κλ.., µν.. of indices. Transposing a multivector I ≡ γ0 ∧ γ1 ∧ γ2 ∧ γ3 . (A3) form twice leaves it unchanged,

⊤ ⊤ The square of the pseudoscalar in 4 spacetime dimensions (a ) = a . (A7) is −1, and in that sense behaves like an imaginary number, The transpose of a symmetric tensor a, one satisfying l l akλ ≡ akle λ = alke λ ≡ aλk, is itself, I2 = −1 . (A4) ⊤ k λ ⊤ k λ k λ a = (akλ γ dx ) = aλk γ dx = akλ γ dx = a . (A8) 3. Multiplication As a particular example, the line interval is symmetric, e⊤ = e, because the tetrad metric is symmetric. In this paper, the product of 2 multivector forms a and b is the geometric product of the multivectors, and the exterior product of the forms. The wedge symbol is not 5. Hodge duals used to denote the exterior product of forms, since that notation would conflict with the use of the same symbol Hodge dual operations are defined for both multivec- for the outer product of multivectors. Rather, all form tors and forms. The Hodge dual of a multivector a of products are by default exterior products. grade n is Ia, which is a multivector of grade N−n in The wedge symbol ∧ denotes the highest grade element N spacetime dimensions. The form dual of a p-form p Λ of the geometric product of multivector forms. Thus a = aΛ d x with multivector coefficients aΛ is the the wedge product a ∧ b of multivector forms a and b multivector (N−p)-form ∗a given by of grades m and n is the grade m + n component of their ∗ ∗ N−p Π Λ N−p Π geometric product. a = aΠ d x = εΠ aΛ d x , (A9) The dot symbol · denotes the lowest grade element of the geometric product of multivector forms (except that where εΠΛ is the totally antisymmetric pseudoscalar the dot product of a scalar, a multivector of grade 0, coordinate tensor, and the implicit summation is over with a multivector is zero). Thus the dot product a · b of distinct antisymmetric sequences Λ and Π of respectively multivector forms a and b of nonzero grades m and n is p and N−p coordinate indices. The antisymmetric tensor the grade |m − n| component of their geometric product. is normalized to ε0...(N−1) = 1 in the tetrad frame, K In accordance with the convention of this paper, a · b is which ensures that the pseudoscalar satisfies I = ε γK a dot product of multivectors, but an exterior product of implicitly summed over the single distinct antisymmetric forms. sequence K = 0...(N−1) of N indices. The definition 15 of the form dual is arranged such that the operations of The Riemann tensor R is itself determined by its taking the dual and the transpose commute, so that definition (5b) in terms of the Lorentz connection Γ and Γ ∗ ⊤ ⊤ its exterior derivative d . a = (I(a )) . (A10) The contracted Bianchi identities, obtained by wedging The double dual of a multivector form a, both a the Bianchi identities (B1) with the line interval e, are multivector dual and a form dual, is denoted with a ∗∗ d(e ∧ S)+ 1 [Γ, e ∧ S] − 1 [e2, R]=0 , (B3a) double-asterisk overscript , 2 2 1 Γ ∗∗ d(e ∧ R)+ [ , e ∧ R] − S ∧ R =0 . (B3b) a ≡ I( ∗a)= ∗(Ia) . (A11) 2 The term 1 [e2, R] in the contracted torsion Bianchi The double dual of a double dual is the identity, 2 identity (B3a) is a bivector 4-form whose 6 components ∗∗ ∗∗ comprise the antisymmetric part of the Ricci tensor, a = a . (A12) 1 e2 R e e R e e R The double-dual transpose of a multivector form can be − 2 [ , ]= ∧( · )= · ( ∧ ) k l 4 κλµν written = −ekκelλR[µν] γ ∧ γ d x . (B4) ∗∗⊤ ⊤ a = I(Ia) . (A13) The contracted Bianchi identities (B3a) and (B3b) enforce conservation respectively of total angular mo- mentum and of total energy-momentum. 6. Coordinate-frame spatial dual Combining the contracted Bianchi identity (B3b) with the torsion Bianchi identity (B1a) yields the Equations (44), (45), (47), and (C5) involve a pseudovector 4-form identity for the curvature Π defined coordinate-frame spatial dual. The dual involves by equation (15), applying the totally antisymmetric coordinate-frame Π 1 Γ Π 1 Γ 1 Γ Γ spatial tensor εαβγ, which is defined by d + 2 [ , ] − 2 [e, R] ∧ + 4 S ∧[ , ]=0 . (B5) klmn εαβγ ≡ εtαβγ = ektelαemβenγε = |e|[αβγ] , (A14) 2. Conservation of angular momentum where |e| is the determinant of the 4 × 4 spacetime vierbein matrix. To ensure that the dual of a dual is Invariance of the action of a matter species under the identity, the indices on εαβγ must be lowered and kl Lorentz transformations implies the law (B11) of con- raised with the spatial coordinate metric gαβ ≡ ekαelβη and its inverse gαβ (which is not the same as the spatial servation of the angular momentum of that matter components of the coordinate-frame spacetime metric species. If the spin angular-momentum of a matter gκλ). The inverse totally antisymmetric coordinate- species vanishes, then the conservation law implies that frame spatial tensor εαβγ is the energy-momentum tensor of that matter species is symmetric, equation (B12). The energy-momentum − εαβγ = |g| 1|e|[αβγ] , (A15) tensor is generically not symmetric for a matter species with non-vanishing spin angular-momentum. The where |g| is the determinant of the 3 × 3 spatial metric actions of gauge fields such as electromagnetism are matrix. independent of the Lorentz connection, so gauge fields carry no spin angular-momentum. Fermionic fields on the other hand do depend on the Lorentz connection, Appendix B: Bianchi identities and conservation laws and carry spin angular-momentum. Under an infinitesimal Lorentz transformation gen- k l erated by the infinitesimal bivector ǫ = ǫkl γ ∧ γ , 1. Bianchi identities any multivector form a whose multivector components transform like a tensor varies as The Jacobi identity applied to the triple product of the 1 (coordinate and Lorentz) covariant derivative implies the δa = 2 [ǫ, a] . (B6) Bianchi identities In particular, since the vierbein ekκ is a tetrad vector, 1 Γ 1 dS + 2 [ , S]+ 2 [e, R]=0 , (B1a) the variation of the line interval e under an infinitesimal 1 Γ Lorentz transformation ǫ is dR + 2 [ , R]=0 . (B1b) 1 δe = 1 [ǫ, e] . (B7) The 2 [e, R] term in the torsion Bianchi identity (B1a) 2 is a vector 3-form whose 16 components comprise the Γ antisymmetric part of the 36-component Riemann tensor, The components of the Lorentz connection do not constitute a tetrad tensor, so its variation under a 1 n 3 κλµ 2 [e, R]= e · R = R[κλµ]n γ d x . (B2) Lorentz transformation is not given by equation (B6). 16

Rather, the variation of the Lorentz connection follows also for multivector-valued forms, since multivectors are from a difference of tetrad-frame tensors, scalars under coordinate transformations. However, the magic formula encounters the difficulty that the Lie 1 Γ δDΓa − DΓδa = 2 [δ , a] , (B8) derivative of a multivector-valued form is not a tetrad tensor. Consequently neither Lǫe nor LǫΓ is a tetrad where DΓ denotes the Lorentz-covariant exterior deriva- tensor. However, as pointed out at the beginning of §5.2.1 tive of [40], the Lagrangian is a Lorentz scalar, so in varying 1 the Lagrangian 4-form L, the exterior derivative d can DΓa ≡ da + [Γ, a] . (B9) 2 be replaced by the Lorentz-covariant exterior derivative The variation δΓ of the Lorentz connection under an DΓ defined by equation (B9), infinitesimal Lorentz transformation generated by ǫ is LǫL = LΓǫL ≡ ǫ .(DΓ L) + DΓ(ǫ . L) . (B15) then The advantage of this replacement is that the Lorentz- δΓ = − dǫ + 1 [Γ, ǫ] . (B10) 2 covariant Lie derivatives of the line interval and Lorentz
Inserting the variations (B7) and (B10) of the line connection are then (coordinate and tetrad) tensors, and interval and Lorentz connection into the variation (17) the resulting law of conservation of energy-momentum of the matter action, integrating by parts, and dropping is manifestly tensorial, as it should be. The Lorentz- e the surface term, yields the law of conservation of angular covariant variation δ of the line interval under an momentum, infinitesimal coordinate transformation (B13) generated by ǫ is ∗∗ ∗∗ ∗∗ 1 dΣ + [Γ, Σ] − e · T =0 . (B11) 1 2 Γǫ Γ δe = −L e = − d(ǫ . e) − 2 [ , ǫ . e] − ǫ . S . (B16) If the conservation law (B11) is summed over all matter Γ ∗∗ The Lorentz connection is a coordinate tensor but not species, and Σ is eliminated using the relation (19a) and a tetrad tensor, so the formula for the Lorentz-covariant the equation of motion (20a), then the contracted torsion Lie derivative does not apply directly to the Lorentz Bianchi identity (B3a) is recovered. connection. Rather, the Lorentz-covariant variation δΓ If the spin angular-momentum of a matter species of the Lorentz connection follows from equation (B8), vanishes, then the conservation law reduces to implying

∗∗ 1 Γ Γǫ Γ Γ Γǫ e · T =0 , (B12) 2 [δ , a]= − L D a + D L a , (B17) which translates into the statement that the energy- which leads to momentum tensor of the matter species is symmetric, ⊤ δΓ ≡ −LΓǫΓ = −ǫ . R . (B18) T = T . Inserting the variations (B16) and (B18) of the line in- terval e and Lorentz connection Γ into the variation (17) 3. Conservation of energy-momentum of the matter action, integrating by parts, and dropping the surface term, yields the law of conservation of energy- Invariance of the action of a matter species under momentum, coordinate transformations implies the law (B19) of ∗∗ ∗∗ ∗∗ ∗∗ conservation of energy-momentum of that matter species. (ǫ · e) ∧ dT + 1 [Γ, T ] − (ǫ . S) ∧ T − Σ ∧(ǫ . R)=0 . κ 2 Any infinitesimal 1-form ǫ ≡ ǫκ dx generates an
(B19) infinitesimal coordinate transformation I don’t know a way to recast equation (B19) in κ κ κ multivector-forms notation with the arbitrary 1-form ǫ x → x + ǫ . (B13) factored out, but in components equation (B19) reduces to The variation of any quantity a with respect to an infinitesimal coordinate transformation ǫ is, by construc- m mn 1 λmn D˚ Tκm + T Kmnκ + Σ Rλκmn =0 . (B20) tion of the Lie derivative, minus its Lie derivative, δa = 2 ǫ κ −L a, with respect to the vector ǫ . The Lie derivative of Here Kmnκ are the components of the contortion bivector a form a with respect to a 1-form ǫ is given by “Cartan’s 1-form K defined by Γ ≡ ˚Γ + K, where ˚Γ (with a magic formula,” ˚ overscript) is the torsion-free (Levi-Civita) Lorentz connection. If the energy-momentum conservation Lǫa = ǫ .(da) + d(ǫ . a) , (B14) law (B19) is summed over all matter components, and the ∗∗ total spin angular-momentum Σ and energy-momentum where . denotes the form inner product (the form dot ∗∗ . is written slightly larger than the multivector dot · to T eliminated in favour of torsion S and curvature distinguish the two). Cartan’s magic formula (B14) holds R using Hamilton’s equations (20), then the law of 17 conservation of total energy-momentum recovers the analogously to the inversion formula (13) for Γ in terms contracted Bianchi identity (B3b). of π and its contraction ϑ. The double duals on the right If the spin angular-momentum of a matter species van- hand side of equation (C5) are to be interpreted as double ishes, then the energy-momentum conservation law (B19) duals with respect to the coordinate spatial metric gαβ for that species reduces to and its inverse gαβ (see Appendix A 6). ∗∗ ∗∗ 1 ˚Γ dT + 2 [ , T ]=0 , (B21) Appendix D: Cartan’s Test for integrability with ˚Γ the torsion-free Lorentz connection. The equations of motion (30) along with the con- straints and identities (32) and (34), form an Exterior Appendix C: The time component of momentum Differential System (EDS). For such a system there is an algorithm, called Cartan’s Test [11, 12], to decide In the numerical scheme, the equations of motion (31) whether the system is integrable. A system is integrable determine the evolution of the 12 spatial coordinates eα¯ if, given an appropriate set of initial conditions, a solution and 12 spatial momenta πα¯. The 4 time components exists and is unique. et¯ of the coordinates are gauge variables, the lapse and Cartan’s algorithm starts by collecting the 4 coordi- κ shift. The 12 time components πt¯ of the momenta can be nates x and variables, here the 16+ 24 components ekκ computed by first inferring the spatial Lorentz connection and Γklκ of the line interval and Lorentz connection, into Γ α¯ , as elaborated below, and then deducing πt¯ from the a single 44-dimensional manifold, definition (10), κ {x ,ekκ, Γklκ} . (D1) Γ Γ πt¯ = − et¯ ∧ α¯ − eα¯ ∧ t¯ . (C1) The tangent space of this manifold is spanned by the 44 differentials The 18 components Γklα of the spatial Lorentz κ connection Γα¯ can be computed from their coordinate- {dx , dekκ, dΓklκ} . (D2) frame components Γκλα, equation (41). The 18 For brevity and ease of algebraic manipulation, it is coordinate frame components Γκλα split into 9 all-spatial advantageous to treat the variables and their differentials components Γβγα and 9 time-space components Γtβα. k k l as multivectors eκ ≡ ekκγ and Γκ ≡ Γklκγ ∧ γ , and The 9 all-spatial components Γαβγ are given by likewise de ≡ de γk and dΓ ≡ dΓ γk ∧ γl. Cartan π κ kκ κ klκ equation (44) in terms of the spatial momentum α¯ and interprets the 44 differentials (D2) as 1-forms satisfying the gravitational magnetic field Bαβγ . The 6-component an exterior calculus, that is, the exterior derivatives of gravitational magnetic field Bαβγ itself, §IIE, can be the 1-forms are zero, and their exterior products are calculated either from spatial derivatives of the spatial antisymmetric. The equations of motion (21) are recast e coordinates α¯ , or in the WEBB formalism from its as relations between the 1-forms, equation of motion. κ 1 Γ 2 κλ Σ˜ 2 κλ The 9 components Γtβγ follow from the spatial deκdx + 2 [ κ, eλ]d x = κ κλd x , (D3a) momentum πα¯ and from the all-spatial components Γαβγ 2 κλ 1 1 3 κλµ dπκλd x + [Γκ, πλµ] − eκ ∧[Γλ, Γµ] d x given by equation (44). The definition (10) of the 2 4 ˜ 3 κλµ
trivector 2-form π implies that = κTκλµd x , (D3b) with implicit summation over distinct antisymmetric π γt ∧ γα ∧ γβ d2xγδ tαβγδ sequences of indices. Equation (D3a) is a vector of 4 t α β 2 γδ = −(2gtγΓαβδ +4gβγΓtαδ) γ ∧ γ ∧ γ d x , (C2) relations between 2-forms, while equation (D3b) is a pseudovector of 4 relations between 3-forms. As usual in where gκλ is the coordinate-frame spacetime metric. The this paper, all products of forms are to be understood as summation in equation (C2) is over distinct sets of pairs exterior products, the wedge symbol ∧ being reserved to αβ and γδ of spatial indices. Define µt to be the 9- indicate an outer product of multivectors. After the EDS component spatial bivector 2-form equations (D3) have been solved, yielding solutions eκ(x) and Γ (x) for the variables e and Γ as functions of µ ≡ (π +2g Γ ) γα ∧ γβ d2xγδ . (C3) κ kκ klκ t tαβγδ tγ αβδ the spacetime coordinates xκ, then the differentials of Further, define the 1-component spatial trivector 3-form the variables can be replaced by an expansion in partial derivatives, and the EDS equations (D3) then reproduce νt be the contraction of µt with the spatial metric g ≡ α β e κ gαβ γ dx , the equations of motion (21). For example, d κdx in equation (D3a) becomes equal to the usual exterior 1 νt ≡ 2 g ∧ µt . (C4) derivative de, ∂e ∂e ∂e Then equation (C2) inverts to de dxλ = λ dxκdxλ = λ − κ d2xκλ = de , λ ∂xκ ∂xκ ∂xλ  α β ∗∗ ∗∗ Γtαβ γ dx = − µt + g ∧ ν t , (C5) (D4) 18 where the sum in the second expression is over both κ and submanifold at a point suffice to guarantee polynomial λ, while the sum in the third expression is over distinct solutions over an infinitesimal open neighborhood of the antisymmetric pairs κλ of indices. point. To proceed, project equations (D3a) and (D3b) re- Cartan’s approach to constructing a solution of the spectively along arbitrary coordinate directions dxκ and EDS is iterative. First choose a generic (what generic arbitrary coordinate planes d2xκλ, yielding a “Pfaffian means is addressed below) coordinate direction, call it system” of 16 + 24 = 40 1-form equations, dx1, and project the equations of motion (D5) along that direction. The torsion equation (D5a) projected along 1 Γ λ Σ˜ λ 1 deκ + 2 [ [κ, eλ]]dx = κ [κλ]dx , (D5a) the 1-form dx yields a vector of 4 1-forms, 1 Γ 1 Γ Γ µ dπκλ + 2 [ [κ, πλµ]] − 4 e[κ ∧[ λ, µ]] dx 1 Γ λ Σ˜ λ de1 + [ [1, eλ]]dx = κ 1λdx . (D7) ˜ µ
2 = κT[κλµ]dx . (D5b) The 4 equations (D7) define a 4-dimensional subspace of Cartan’s Test states that the system (D5) is integrable the tangent space to the putative integral submanifold. if it is “involutive” and “torsion-free” (this is a different The Einstein equation (D5b) projected along the 1- meaning of torsion). Involutive means that the number form dx1 yields nothing, since the equations involve of independent equations (here 40) equals the number of antisymmetrized pairs κλ of indices, and the pair 11 is variables (here 40). Torsion-free means that the exterior not antisymmetric. The number 4 of equations added derivatives of equations (D5) vanish on the equations of at the first iteration is called the first Cartan character, motion. The exterior derivatives of equations (D5a) are c1 = 4. the 16+24 = 40 2-form equations Next, choose a second generic coordinate direction, call it dx2, and project the equations of motion (D5) 1 Γ 1 Γ λ Σ˜ λ 1 2 2 [d [κ, eλ]] − 2 [ [κ, deλ]] dx = κd [κλ]dx , (D6a) along the directions dx and dx . The projection of the 1 Γ 1 Γ
1 Γ Γ torsion equations (D5a) yields 4 more equations similar [d [κ, πλµ]] − [ [κ, dπλµ]] − de[κ ∧[ λ, µ]] 2 2 4 to (D7) but with index 2 in place of 1. The projection of 1 Γ Γ µ ˜ µ + 2 de[κ ∧[ [λ, d µ]] dx = κdT[κλµ]dx . (D6b) the Einstein equations (D5b) yields a vector of 4 1-form
equations But equations (D6) are guaranteed by the Bianchi identities (B1a) and (B5) for S and Π. That is, the 1 Γ 1 Γ Γ µ dπ12 + 2 [ [1, π2µ]] − 4 e[1 ∧[ 2, µ]] dx Bianchi identities (B1a) and (B5) express dS and dΠ as = κT˜ dxµ .
(D8) linear combinations of S, Π, and the antisymmetric part [12µ] [e, R] of the Riemann tensor R. When the equations The second iteration thus adds 8 1-form equations to the Σ˜ Π ˜ of motion S = κ and = κT are substituted into 4 1-form equations of the first iteration, bringing the total the Bianchi identities, the conclusion is that dS = number of 1-form directions lying in the tangent space of Σ˜ Π ˜ κd and d = κdT , which are none other than the integral submanifold to 4 + 8 = 12. The number equations (D6). Note that the Bianchi identity (B1a) 8 of equations added at the second iteration defines the Σ˜ with S → κ essentially defines the contribution second Cartan character, c2 = 8. to the antisymmetric Riemann tensor [e, R] from any Finally, choose a third generic coordinate direction, component of spin angular-momentum Σ˜ . The exterior call it dx3, and project the equations of motion (D5) derivative equations (D6) thus hold on the equations of along the three directions dx1, dx2, and dx3. The motion, and are therefore torsion-free in Cartan’s sense. projection yields 4 more torsion equations similar Since the system (D5) is both involutive and torsion-free, to (D7), and 8 more Einstein equations similar to (D8), it is integrable. bringing the total number of 1-form directions lying in The solution to the Cauchy problem comprises an “in- the tangent space of the integral submanifold to 4 + 8 + tegral submanifold,” a submanifold of the 44-dimensional 12 = 24. The number 12 of equations added at the third manifold (D1) of coordinates and variables (below, the iteration defines the third Cartan character, c3 = 12. submanifold will be shown to be 24-dimensional). For The three iterations define 3 successive Cauchy an integrable system, the integral submanifold passing problems. The first problem is to solve the 4 equations through any point of the parent manifold is unique. at the first iteration along a 2-dimensional coordinate The tangent space to the integral submanifold has the submanifold (42-dimensional submanifold of the full 44- defining property that all the 1-forms of the Pfaffian dimensional manifold (D1)) where the last 2 coordinates system (D5) vanish in the tangent space. Cartan shows x3 and x4 are fixed (so dx3 = dx4 = 0). The 4 equations that the Cauchy problem reduces to a local problem in govern the evolution of the 4 components e1 along the an infinitesimal neighborhood of a point on the integral direction x2. The solution involves 4 initial conditions, submanifold. Cartan solves the Cauchy problem by the values of the components of e1 along an initial line showing that there exist solutions for the variables as of x1 at fixed x2. The second problem is to solve the 12 finite-order polynomials in the coordinates xκ with origin equations at the second iteration along a 3-dimensional shifted to the point in question. Cartan’s solution shows coordinate submanifold where the last coordinate x4 is that conditions on the tangent space to the integral fixed (so dx4 = 0). The solution involves 12 initial 19 conditions, the values of the components of e1, e2, and of the variables eα and παβ on the initial spatial 1 2 3 π12 along an initial surface of x and x at fixed x . hypersurface of constant time t. But in addition there The third problem is to solve the 24 equations at the are 4 + 12 = 16 differential conditions (constraints and third iteration along the 4-dimensional submanifold of identities) on the variables eα and παβ over the initial coordinates. The solution involves 24 initial conditions, hypersurface, a total of 40 initial conditions. the values of eα and παβ with α, β running over 1, 2, 3, along an initial 3-dimensional hypersurface of x1, x2, and x3 at fixed x4. The solution at the third iteration fills Appendix E: Electromagnetism out all 4 spacetime dimensions, so no further iteration is needeed. The tangent space of the integral submanifold The purpose of this Appendix is to show that the at the third iteration has 4 + 8 + 12 = 24 dimensions, so structure of the equations of electromagnetism is quite the integral submanifold has dimension 24. The Cartan similar to the structure of the gravitational equations characters are derived in the text.

{c1,c2,c3} = {4, 8, 12} , (D9) 1. Electromagnetic potential and field summing to the dimension 24 of the integral submanifold. The number of initial conditions is 4+12+24 = 40, which Electromagnetism is a U(1) gauge theory. The coor- is as it should be for a system of 40 equations governing ν 40 variables. In terms of Cartan characters, the number dinates of electromagnetism are a 1-form A ≡ Aν dx which arises as the connection of an electromagnetic of initial conditions is 3c1 +2c2 + c3 = 40. The initial conditions comprise 4 functions of a single variable x1 gauge-covariant exterior derivative, 2 3 4 along a line at fixed x , x , x , plus 12 functions of 2 D ≡ d+ ieA , (E1) variables x1, x2 along a surface at fixed x3, x4, plus 24 A functions of 3 variables x1, x2, x3 along a hypersurface which acts on fields ϕ of dimensionless charge e. Under at fixed x4. an electromagnetic U(1) gauge transformation of the field The above construction required that the coordinate ϕ by some scalar θ, directions dx1, dx2, and dx3 be chosen “generically.” − Not all coordinate 1-forms are generic. For example, ϕ → e ieθϕ , (E2) the 1-form dφ of the azimuthal coordinate in spherical polar coordinates is ill-defined at the poles. As another the electromagnetic gauge-covariant derivative of ϕ is example, if the coordinate 1-form is chosen to be null, required to transform as then the vierbein matrix e will have zero and infinite − kκ D ϕ → e ieθD ϕ . (E3) eigenvalues (albeit finite determinant). Physically, if A A the coordinate 1–form is null, then the locally inertial This requires that the electromagnetic potential A vary k frame defined by the tetrad γ is moving through the under a gauge transformation as coordinates at the speed of light. For example, this happens at the horizons of stationary black holes with A → A + dθ . (E4) respect to stationary coordinates. These are called coordinate singularities. The solution is to choose generic The commutator of the electromagnetic gauge- coordinate directions that are not null. Of course there covariant derivative defines the electromagnetic field 2 µν are also genuine singularities, such as those inside black 2-form F ≡ Fµν d x (implicit sum over distinct holes, where the locally inertial description of spacetime antisymmetric pairs µν), breaks down irretrievably, and beyond which coordinates F ≡ dA , (E5) cannot be continued. In numerical relativity, one wishes to choose a which is invariant under electromagnetic gauge transfor- coordinate system that is everywhere well-behaved, the 4 mations. In components, equation (E5) is coordinate 1-forms dxκ being nowhere null. This means choosing a coordinate system with 3 spatial coordinates 2 µν ∂Aν ∂Aµ 2 µν α F d x = − d x . (E6) x and 1 time coordinate t, such that the spatial 1- µν  ∂xµ ∂xν  forms dxα are everywhere spacelike, and the time 1-form dt is everywhere timelike. A coordinate system is well- behaved over a region of spacetime if and only if the 4×4 2. Electromagnetic action matrix of vierbein coefficients ekκ is positive definite (all eigenvalues of the vierbein matrix are strictly positive) The electromagnetic Lagrangian 4-form, expressed in over that region. terms of the electromagnetic coordinates A and their 4 Thus in practice the fourth coordinate x of Cartan’s derivatives, is construction is chosen to be a timelike coordinate t. 1 ∗ There are 24 initial conditions comprising the values Le = 2 (dA) ∧ dA . (E7) 20

The electromagnetic Lagrangian (E7) is invariant under Taking the exterior derivative of the second Hamilton electromagnetic gauge transformations, as it must be. equation (E14b) yields The units here are Heaviside (SI units with ε0 = µ0 = 1); in Gaussian units the right hand side of equation (E7) dF =0 , (E16) would have an extra factor of 1/(4π). The momenta conjugate to the coordinates A are which comprises Maxwell’s source-free equations. δL e = ∗dA = ∗F , (E8) δdA 3. 3+1 split ∗ which is the 2-form dual F of the electromagnetic Splitting the variation (E12) of the combined electro- field F . Recast in super-Hamiltonian form, the magnetic and matter action into time and space parts electromagnetic Lagrangian (E7) is yields ∗ Le = F ∧ dA − He , (E9) tf ∗ ∗ δ(Se + Sm)= F ∧ δA (E17) with coordinates A, momenta F , and super- I ti Hamiltonian tf ∗ ∗ ∗ ∗ + − (d F − j)t¯ ∧ δAα¯ − (d F − j)α¯ ∧ δAt¯ 1 ∗ He = F ∧ F . (E10) Zti 2 ∗ ∗ + δ Fα¯ ∧(dA − F )t¯ + δ Ft¯ ∧(dA − F )α¯ . The variation of the electromagnetic action with ∗ respect to the coordinates A and momenta F is After the 3+1 split, the electromagnetic coordinates are the 3 spatial components Aα¯ of the electromagnetic ∗ ∗ ∗ δS = F ∧ dδA + δ F ∧ dA − δ F ∧ F . (E11) potential 1-form, and their conjugate momenta are the 3 e ∗ Z spatial components Fα¯ of the dual electromagnetic field, Integrating the ∗F ∧ dδA term in equation (E11) by which 3 components comprise the electric field. Variation A ∗F parts brings the variation of the electromagnetic action of the action with respect to the variations δ α¯ and δ α¯ to of the 3 coordinates and their 3 conjugate momenta yields 3 + 3 equations of motion that give time derivatives of ∗ ∗ ∗ the coordinates and momenta, δSe = F ∧ δA + − d F ∧ δA + δ F ∧(dA − F ) . I Z ∗ ∗ ∗ ∗ (E12) 3 eqs of mot: (d F )t¯ ≡ dt¯ Fα¯ + dα¯ Ft¯ = jt¯ , The variation of the matter action with respect to the (E18a) ∗ electromagnetic coordinates A defines the 3-form dual j 3 eqs of mot: (dA)t¯ ≡ dt¯Aα¯ + dα¯ At¯ = Ft¯ . (E18b) of the matter electric current 1-form j, ∗ Note that Fα¯ is dual to Ft¯, and those components ∗ ∗ comprise the electric field; while F is dual to F¯, and δSm = j ∧ δA . (E13) α¯ t Z those components comprise the magnetic field. The time component A¯ of the electromagnetic Requiring that the variation δ(S + S ) of the t e m potential can be regarded as a gauge field, arbitrarily combined electromagnetic and matter actions vanish adjustable by an electromagnetic gauge transforma- with respect to arbitrary variations δA and δ ∗F of the tion (E4). Variation of the action with respect to electromagnetic coordinates and momenta, subject to variations δA¯ of the gauge variable imply the constraint the condition that δA vanishes on the boundary, yields t equation Hamilton’s equations, ∗ ∗ ∗ ∗ 1 constraint: (d F ) = j . (E19) d F = j , (E14a) α¯ α¯ dA = F . (E14b) The constraint equation (E19) has the property that, provided that it is satisfied in the initial conditions, it is The first Hamilton equation (E14a) is a 4-component 3- guaranteed thereafter by conservation of electric current. form comprising Maxwell’s source-full equations. The Conservation of electric current is a consequence of the second Hamilton equation (E14b) is a 6-component U(1) symmetry of electromagnetism. 2-form that enforces the relation (E5) between the Variation of the action with respect to the variation electromagnetic field F and the electromagnetic potential ∗ δ Ft¯ of the 3 components of the magnetic field implies A. the 3 identities Taking the exterior derivative of the first Hamilton 2 equation (E14a) yields, since d = 0, the electric current 3 ids: (dA)α¯ = Fα¯ . (E20) conservation law The identities (E20) express the 3 components of the ∗ d j =0 . (E15) magnetic field Fα¯ as the spatial exterior derivative of 21 the spatial potential Aα¯ . Equation (E20) is not an it is possible to work with the 3-component magnetic equation of motion since it does not involve any time field Fα¯ ≡ (dA)α¯ , in which case the variables are the ∗ derivatives; nor is it a constraint equation, since its 6 components of the electric and magnetic fields Fα¯ continued satisfaction is not enforced by any conservation and Fα¯. This is the traditional (century-old) approach. law. Rather, the identities (E20) mean that the magnetic The electric and magnetic fields are electromagnetic field Fα¯ is a redundant field that can be eliminated gauge-invariant, but not coordinate gauge-invariant (the by replacing it with (dA)α¯ . The spatial potential Aα¯ electric and magnetic fields form a coordinate tensor, is itself determined by its equation of motion (E18b). not a set of coordinate scalars). In general relativity, Although the magnetic field is redundant, it is required the analogous situation is that the Hamiltonian variables for a covariant description of the 4-dimensional 2-form after a 3+1 split are the 12-component spatial vierbein electromagnetic field. Moreover, the magnetic field ea¯α¯ and its conjugate momentum the 12-component ∗ (or rather its dual Ft¯) appears in the equation of spatial momentum πa¯α¯ . In place of these it is motion (E18a) for the electric field. possible, if torsion vanishes, to work with Lorentz In all, the 10 Hamilton’s equations of electromagnetism gauge-invariant variables, the 6-component spatial metric comprise 3+3 = 6 equations of motion, 1 constraint, and gαβ, its 6-component conjugate momentum the extrinsic 3 identities. curvature Kαβ, and the 18-component spatial coordinate In electromagnetism, the Hamiltonian variables after connections Γαβγ (Christoffel symbols). This is the ADM a 3+1 split are the 3-component spatial potential Aα¯ formalism. Thus the ADM formalism can be thought and its conjugate momentum the 3-component spatial of as a general relativistic analog of the traditional ∗ electric field Fα¯ . In place of the spatial potential Aα¯ electromagnetic formalism.