aa r X i v : . [ g r- q c ] A ug Equiaffine braneworld
Fan Zhang
Gravitational Wave and Cosmology Laboratory, Department of Astronomy, Beijing Normal University, Beijing 100875, China (Dated: August 21, 2020)Higher dimensional theories, wherein our four dimensional universe is immersed into a bulk am-bient, have received much attention recently, and the direction of investigation had, as far as we candiscern, all followed the ordinary Euclidean hypersurface theory’s isometric immersion recipe, withthe spacetime metric being induced by an ambient parent. We note in this paper that the indef-inite signature of the Lorentzian metric perhaps hints at the lesser known equiaffine hypersurfacetheory as being a possibly more natural, i.e., less customized beyond minimal mathematical formal-ism, description of our universe’s extrinsic geometry. In this alternative, the ambient is deprivedof a metric, and the spacetime metric becomes conformal to the second fundamental form of theordinary theory, therefore is automatically indefinite for hyperbolic shapes. Herein, we advocateinvestigations in this direction by identifying some potential physical benefits to enlisting the helpof equiaffine differential geometry.
I. INTRODUCTION
Within the mathematical literature, there are a numberof different theories that were constructed to describe theextrinsic geometry of a hypersurface immersed into a bulkambient. While the intrinsic geometry of our spacetimeis verifiable best described using the pseudo-Riemanniangeometry, we still have (the freedom) to choose its extrin-sic counterpart, should we envision a braneworld scenarioand thus a bulk ambient that physically exists. The vari-ous extrinsic theories differ mainly on how much machineryis made available to us. More infrastructure would facil-itate more specialization, resulting in simpler expressionsand easier equations to solve. Specifically, when the am-bient is equipped with a metric, we can define angles andnorms for the vector bases in reference frames, thus con-centrate only on orthonormal frames and work with smallerprincipal bundles.Our natural bias towards convenience, coupled with thefact that the intrinsic side General Relativity is a metrictheory, perhaps led to the universal assumption that theambient should also be equipped with a metric, and that ourspacetime is isometrically immersed into it. There is how-ever the complication that the intrinsic metric is of an indef-inite Lorentzian signature. Since a positive-definite ambientmetric would not be able to induce a Lorentzian one on thebrane, the ambient metric has to be indefinite as well. Suchpseudo-metrics are rather pathological. Besides the generalcounter-intuitiveness of the hyperbolic trigonometry [1, 2],the fact that their metric balls are noncompact complicatesthe search for compatible topologies into a rather messybusiness (see e.g., [3]). Such problems render the pseudo-metrics arguably quite awkward to use.In other words, if Nature had intended to grant us theconvenience of an ambient metric, it is only doing so half- Codimension one is the simplest case and appears to be sufficientfor yielding the desired physical implications discussed in this paper,we note nonetheless that generalizations to higher codimensions ispossible. We also assume flat ambient throughout this paper. In-deed, the integrability conditions we consider later are equivalent todemanding that the ambient curvature vanishes. However, general-izations to ambients of constant sectional curvatures should also bepossible. heartedly. It is then, at the very least prudent, to alsoconsider the possibility that there is no such intention, andour intrinsic Lorentzian metric is not induced by an ambientmetric, but instead bespeaks a different concept in somenon-metric extrinsic theory. Such theories do exist, andone is called the equiaffine differential geometry, wherein asymmetric bilinear form called equiaffine metric is laid ontothe immersed hypersurface, but which is in fact conformalto the second fundamental form of the immersion, in themore familiar ordinary Euclidean terminology.In this short note, we outline this proposal that our rela-tivistic Lorentzian metric might profitably be interpreted asan equiaffine metric (the relevant concepts are introduced inSec. II). Specifically, we argue in Sec. III, that the Einstein’sequations can possibly then be regarded as the equiaffineGauss equations in disguise, with the dark energy arisingnaturally through a mean curvature term. We also specu-late on some potential global cosmological implications ofthe equiaffine braneworld scenario in Sec. IV. II. CONCEPTS
Roughly speaking, equiaffine differential geometry con-siders concepts that are invariant under the special lineartransformations and translations, which unfortunately donot include orthogonality as determined by an ambient met-ric. Therefore, the equiaffine differential geometry has tobe constructed without summoning help from any ambientmetric (thus the name “affine”). Such more rudimentarymathematics, with less circumstance-specific appendages,are particularly likely to infiltrate physics, since they facil-itate more useful physical laws that are applicable to moreexperimental settings and engineering situations . In other Or special affine or unimodular or sometimes simply affine; we willsubstitute out these alternative terms from literature to avoid clut-tering nomenclature. Also, in deference to more established con-vention, we will refer to theories equipped with an ambient metricas being “ordinary Euclidean”, even though we allow indefinite am-bient metrics in the context of our physical discussion. This is a physical representation of the Erlanger programme’s ratio-nale for developing non-Euclidean geometries. By loosening the re-strictions on the transformations, so less of the geometric constructs words, the bottom-up experiments-driven organic growthof physical models may go against the top-down theorybuilders’ wishes (and perhaps at times overly optimistic as-sumptions) to have more tools in the box.An immediate consequence of the depletion of the tool-box is that we cannot use the usual procedure to obtain thefirst fundamental form. Fortunately though, we notice thatthe second fundamental form Π ab can still be defined andeven be appropriately scaled to become equiaffine invariant[4]. It can thus possibly reincarnate into some sort of re-placement intrinsic metric. Heuristically, since the eigenvec-tors of Π ab are conventionally regarded as being mutuallyorthogonal, and the eigenvalues can be used as referencescales, the principal structure of Π ab contains the neces-sary ingredients for the construction of an intrinsic metricgravity theory. But there is another problem. To obtain asecond fundamental form, one has to have a normal first,but we cannot define a normal in the usual way by demand-ing it be orthogonal to the tangents of the hypersurface. Analternative procedure has to be developed, which we brieflysummarize in Appendix A (see e.g.,[5] for further details).Those steps are sufficient to pick out a unique bilinearform Π ab associated with a unique equiaffine normal ν α ,which is then used as a metric called the equiaffine first fun-damental form [6] or equiaffine metric [7], but obviously notnecessarily positive-definite. Despite this equiaffine metricnot being a measure of distances in the traditional sense,the equiaffine theory nevertheless, by construction and inparticular due to the efforts of Blaschke and colleagues, en-joys substantial computational similarity with its Euclideancounterpart (e.g., the conditions in Appendix A are de-signed to mimic properties seen in the Euclidean theory).This lack of operational distinction might have contributedto the conspicuous lack of explicit references to equiaffinegeometry from the physical studies, whereby the theoriesare reverse-engineered to, at least initially, be phenomeno-logical descriptions of experiments, and thus one has directaccess only to the mechanics, and not the underlying geo-metric significances that might have set the two differentialgeometries apart. A potential opportunity for equiaffine ge-ometry to break away from anonymity though, arises whenboth intrinsic and extrinsic curvatures make simultaneousappearances, as in the Gauss equations (1) below, to makehiding the double life of the equiaffine metric untenable,outing it as a curvature in actuality. We will exploit theconsequences of this revelation in Sec. III, to model thedark energy or cosmological constant. We highlight thisparticular facet of cosmology, precisely because it offers ahandle for us to assess the potential relevance of equiaffinegeometry in a physical context.Besides the equiaffine metric, there is another importantquantity in equiaffine geometry, the totally symmetric cu-bic form F abc (see Eq. A5 in particular), which is also calledthe equiaffine second fundamental form [6]. Just as Π ab describes the next order warping of the tangential or os-culating hyperplane that produces a better approximating stay invariant, we isolate and concentrate on only those propertiesthat are the most robust, the computations concerning whom si-multaneously apply to larger collections of geometric objects. osculating quadric, F abc describes how the quadric can becrafted further into an even more snug-fitting osculatingcubic ( F abc is related to the third order Taylor expansioncoefficients in the local graph representation of the hyper-surface, and Π ab to the Hessian [6, 8]). In other words, con-cepts in the equiaffine theory are shifted up one order (ormore, see Footnote 6 below) as compared to the Euclideantheory.This hike in orders implies in particular, that in theequiaffine context, the local Minkowski approximation to aspacetime is not the flat osculating hyperplane, as would bethe case with isometric immersion. It is instead the osculat-ing quadric, with (equiaffine) “flatness” now meaning thatthe third order adjustments F abc vanish. It is more specifi-cally a paraboloid or improper affine hypersphere to boot,meaning its equiaffine normals are parallel to each other,and it is thus also “flat” in the sense that its shape oper-ator (representing fourth order adjustments, see Footnote6 below), given by Eq. A3, vanishes. Such simplicity thatequiaffine geometry endows upon the Minkowski space isone of the reasons we favour equiaffine over its centroaffineor Euclidean-affine (see Appendix A) siblings within thefamily of alternative affine geometries. Because we believethat when physical laws were drafted, the authors tried to,consciously or not, save as much ink as possible when itcomes to describing the empty stage, and thus we enjoybetter odds at finding General Relativity in the equiaffinecorner of the overarching general affine ballpark. III. EQUATIONS
The Gauss formula and Weingarten equation (togetherconstitute the structure equations) utilized in Appendix Abreak ambient derivatives of tangential and normal vectorfields, respectively, into tangential and normal parts, andinform us about the bending of the hypersurface throughthe cross-mixings, quantified by Π ab and the shape operator S ab . Given an arbitrary prescription of Π ab and S ab though,there is no automatic guarantee that the desired warping isachievable inside a flat ambient that is only one dimensionhigher.Taking cues from the ordinary Euclidean theory, oneneeds to take further derivatives, and obtain the Gaussand Codazzi equations as the integrability conditions forthe structure equations. The existence part of the funda-mental theorem for hypersurfaces then ensures that givena first and a second fundamental form that are compliantwith these conditions, we are guaranteed a successful im-mersion. With the isometric immersion of spacetime in thattheory though, the physical equations of motion have noth-ing to do with the Gauss and Codazzi equations, thus whenGeneral Relativity provides the intrinsic metric, no inte-grability is guaranteed a priori , and one has yet to ensurethat there exists a second fundamental form, that solvesboth the Gauss and Codazzi equations. In other words, westill need to find a way to warp the hypersurface, in such away that the resulting stretching and squeezing reproducesthe prescribed intrinsic distance changes. This is a highlynontrivial task (see e.g., [9–19]), requiring for general space-times a ten dimensional flat pseudo-Riemannian ambient.In the equiaffine theory, the fundamental forms are re-placed by the equiaffine metric Π ab and the cubic form F abc (in place of S ab , see Footnote 6), stipulated to satisfya similarly named set of integrability equations [4, 5, 20].But when it comes to braneworld physics, instead of simplytranscribing the Euclidean case, we make the observationthat if the ambient does physically exist, then the integra-bility conditions must somehow already be asserting them-selves, in disguise, as physical laws. Or else our theorieswould be yielding copious illegitimate solutions that can-not be realized in Nature because they prevent our branefrom fitting into the ambient, which doesn’t appear to bethe situation. Specifically, we conjecture to identify Ein-stein’s equations with the equiaffine Gauss equations, andlink up matter contents with the extrinsic curvature, sothat their presence warps spacetime, in a quite literal ex-trinsic geometric sense. This way, the metric and matterfields as solutions to the Einstein’s equations and matterequations of motion serve up Π ab and F abc that are, by con-struction, best positioned to satisfy the integrability condi-tions (note in particular, the Minkowski setup of constant Π ab = diag( − , , ,
1) and F abc = 0 satisfies these condi-tions trivially).Besides immersibility, there are other advantages to iden-tifying Einstein’s equations with the equiaffine Gauss equa-tions, whose once-traced form for a four dimensional hyper-surface can be turned into Π R ab − H Π ab = F acd F bdc − F abc ; c , (1)after substituting in the once-traced first equiaffine Codazziequations S ab = H Π ab − F abc ; c , (2)where H ≡ S aa / We caution that, by a naive counting of indices, the equiaffine inte-grability equations are overdetermined, so in principal, some patho-logical solutions we obtain using a subset of the equations (e.g.,the intrinsic metric can be obtained using the Ricci part of theequiaffine Gauss equations, plus gauge conditions, without referringto the Weyl half) could possibly violate the remaining ones. Theinternal consistency between these equations may prevent this, asin the case of electromagnetism and General Relativity (Einstein’sequations and Bianchi identities) – after a 3 + 1 split, some equa-tions may become constraints that are preserved by the evolutionequations and thus trivialize. However, we haven’t managed to findtheorems clarifying the extent to which this is guaranteed for theequiaffine integrability conditions, thus the circumspect statement. Note that while the integrability conditions take care of local im-mersibility, there may be additional global embeddability conditionsthat prevent self-intersections. They could possibly manifest as inte-gral versions of energy conditions, with the mostly attractive natureof gravity being the consequence of it being much easier to embed anoverall (local infringements can be compensated by nearby regions)more (as compared to the ambient) positively curved hypersurface(cf. [21] vs. [22]) – it is difficult to fit a stretchy hypersurface into acomparatively more crumpled up ambient without having to fold itback on itself. The first equiaffine Codazzi equation can be seen as a constraintequation relating the Weingarten form (equals shape operator withone index lowered by Π ab , according to the Ricci equation) S ab andthe first derivatives of F abc . In this sense, S ab can be seen as anauxiliary variable defined to break a second order equation for F abc down into two sets of first order equiaffine Codazzi equations. semi-column denotes covariant derivative using the Levi-Civita connection of Π ab . Note also that the apolarity con-dition (A6) enforces that the divergence term in Eq. (2)is traceless. Using the equiaffine Theorema Egregium (ob-tained by taking another contraction over Eq. 1), this ex-pression can be processed further into one containing theEinstein tensor (sign conventions match those of [23]) Π G ab + 3 H Π ab = F acd F bdc − F abc ; c − F def F def Π ab . (3)In principal, H is a function of Π ab and F abc , but it hasa special status for physical membranes that have surfacetension in them. Being the divergence of the equiaffine nor-mal (see Eq. A3), H measures how buckled our hypersur-face is, and if it varies rapidly, we would end up with a verybumpy surface. The local high frequency bumpiness con-tributes nothing towards satisfying the global constraints ofany variational isoperimetric problem (cf., the fixed bound-ary of a Plateau problem for a soap bubble supported by awire frame, or the fixed enclosed volume for a free-floatingbubble), but increases the surface area. So whatever con-straints our universe has to conform to , it is quite likelythat variations in H is suppressed (cf. the smoothing prop-erties of mean curvature and in particular, surface tensionflows), allowing it to masquerade as the cosmological con-stant. Such an entry, proportional to Π ab , takes an appear-ance in Eq. (3) because Π ab really describes the extrinsicshape of the hypersurface. A mean curvature rescaled sec-ond fundamental form is in fact also present in the ordinaryEuclidean theory’s version of the Gauss equations, but sinceover there, it is the first fundamental form that serves as theintrinsic metric instead, such a term cannot then readily beidentified with the cosmological constant contribution.Finally, the matter stress-energy tensor must correspondto the F abc terms on the right hand side of Eq. (3). Thesecond equiaffine Codazzi equations, governing the secondderivatives of F abc , should then be compatible with theirequations of motion. As a toy model for exploring possibil-ities of how this might happen, consider a situation whereour region of interest, near some type of particles, happensto be approximated by a neighbourhood of an affine hy-persphere (these are quite diverse and can become rathercomplicated, see e.g., [25] for some visual illustrations), thenwe have S ab = H Π ab , or equivalently F abc ; c = 0 , (4) The ensuing qualitative argument applies to either the equiaffine orthe Euclidean normal, leading to similarities, e.g., within equiaffinegeometry, a minimal hypersurface is still characterized by H = 0,and hyperspheres (whose equiaffine normal directions meet at a sin-gle ambient point, which can be at infinity) are still of constant H . Global constraints often restrict admissible topology, and scalarsconstructed out of S ab (i.e., functions of the coefficients in its char-acteristic polynomial, including H ) provide densities that integrateinto topological invariants, so one would expect H to be inverselyrelated to the overall sizes of the universe. Indeed, as noted bye.g. [24], inverse squareroot of the cosmological constant is on theorder of 10Gly, roughly matching the age of the universe in natu-ral units, which is to be expected if we are not fine-tuned to residein a special era in the history of the universe, so our distance tolandmark events like the big bang is generic. according to Eq. (2), with H being a constant. It is easy tocheck that the second equiaffine Codazzi equations S ab ; c − S ac ; b = F abd S cd − F acd S bd (5)are satisfied (both sides vanish). In other words, Eq. (4),together with the constant H condition, can be regarded asa sufficient replacement to Eq. (5). Furthermore, Eq. (4)also takes out the middle term on the right hand side ofEq. (3). The resulting expression and Eq. (4) bear cursoryresemblances in form to the stress-energy tensor and thesourceless equations of motion for the Yang-Mills fields, soa formal mapping fitting (some sectors of) the StandardModel into F abc might not be prohibitively difficult to con-trive. However, it would be much more satisfying if wecould unearth the underlying equiaffine extrinsic geometricsignificances, if there is indeed any, of the Standard Modelentities, an arduous task that we will relegate to future stud-ies. We highlight here one particular difficulty, that indexsymmetry already restricts the number of independent de-grees of freedom in F abc to at most 20, yet there are 118 inthe Standard Model. So unless we augment the equiaffinefreedoms by e.g., increasing the codimensions, the afore-mentioned mapping would be highly non-injective (multi-ple particle configurations warp spacetime the same way)and thus non-invertible, so one perhaps shouldn’t hope torecover all necessary insights by examining the equiaffinerendition of gravity alone. IV. DISCUSSION AND CONCLUSION
So far, we have confined ourselves to more local consid-erations. On the other hand, the equiaffine braneworld sce-nario has obvious global cosmological implications. For ex-ample, it is difficult to embed a compact hypersurface thatis everywhere non-convex [6] (hard to constrict the hyper-surface while not being able to find any support planes),so it is not unreasonable to consider situations where someparts of our universe are strongly convex, i.e., where theequiaffine metric is positive/negative-definite, because e.g.,as a part of the isoperimetric consideration, our brane uni-verse may need to enclose a fixed ambient volume. Anysmooth change in metric signature will then inevitably leadus to a transition boundary, that can be identified withthe big bang, where at least one of the eigenvalues of Π ab vanishes, so it becomes degenerate, and the machinery ofequiaffine geometry breaks down (at the very least Π ab di-verges). In other words, the big bang appears singular onlybecause of the limitations of the particular mathematicalinfrastructure we implicitly adopt. Such is a rather desir-able situation, since there is then no fundamental obstruc-tion preventing us from improving our modelling and beable to impose initial conditions for our Lorentzian side ofthe universe. In particular, there is no need to censor thebig bang, justifying excluding it from the cosmic censorshipconjectures. In fact, even without major remodelling ef-forts, such degenerate locales can often be handled to someextent with finesse, e.g., the Frenet frame can be general-ized to smoothly extend across an inflection point, wherethe normal of a curve is not defined [26]. Similarly in thespacetime case, a degenerate boundary may be handled viathe matching of limits of regular quantities obtained on ei-ther side (see e.g., [27, 28]). In fact, in the ordinary Euclidean sense, the big bangwould be flatter than elsewhere (cf. the low initial gravita-tional entropy issues [29]), in the sense that Π ab is conformalto the second fundamental form of the ordinary Euclideantheory, thus at least one of the Euclidean principal cur-vatures vanishes there. It being a highly warped place isinstead in the equiaffine sense, meaning it is quite far frombeing a quadric, viz. some components of F abc are large.This is necessary for the metric signature switch to happen,because the defining quadratic polynomial ϕ for which anyquadric is the zero set of, would have its Hessian being con-formal to Π ab (see Appendix A, and Ref. [6] Vol. 3, Chap. 3for a 2-D illustration). The Hessian of a quadratic polyno-mial is constant, so the big bang region would have to shunall quadric shapes in order to host a convexity change. Al-ternatively stated, the ambient-induced connection ˜ ∇ a hasdifficulty preserving Π bc via parallel transportation wheneven its qualitative fundamentals like rank and signaturechange, thus F abc is large according to Eq. A4.A byproduct of this feature is that the Pick invariant J ≡ F abc F abc /
12 in Eq. (3) would likely end up being ap-preciable near the big bang, plausibly demanding the pres-ence of a significant inflaton potential style component inthe matter stress-energy tensor. In other words, inflationand the big bang come hand in hand, unless chance cancel-lations occur during index contractions to suppress J evenwhile F abc is component-wise (as measured by e.g., the L orL ∞ norm) not small. This can only happen when Π ab has anindefinite signature though, and we could simply approachthe big bang from the convex side and evoke continuity ar-guments instead, which incidentally reveals that J shouldget quite large on that side of the big bang already, and ourLorentzian side of the universe would be born directly intoongoing inflation, without any potentially problematic [30]delay.Due to the foundational role the Lorentzian metric playsin modern physics, there would be much to explore in wayof consequences of our proposal that it be an equiaffinemetric, we can but skim only the most obvious ones here.Our discussions are also frustratingly broad-stroked in na-ture, because we are pulling a bottom block from a massiveJenga and the whole thing has to be rebuilt before we canget to an altitude where definitive and precise experimentaltests can be proposed. Even on the mathematical front,the knowledge of equiaffine hypersurfaces (especially thosewith indefinite Π ab ) is somewhat limited, due in part to theadded difficulties resulting from the raised order of oscu-lating approximates and thus of the differential equations.Nevertheless, we think the glimpse of possibilities is suf-ficiently interesting that we wish to share with the widercommunity, to solicit interest for further forays. Acknowledgments
This work is supported by the National Natural ScienceFoundation of China grants 11503003 and 11633001, the In-terdiscipline Research Funds of Beijing Normal University,and the Strategic Priority Research Program of the ChineseAcademy of Sciences Grant No. XDB23000000. [1] G. S. Birman and K. Nomizu, The American MathematicalMonthly , 543 (1984), ISSN 00029890, 19300972, URL .[2] E. Kasner, Nature (London) , 434 (1921).[3] R. S. Hoekzema (Essay as a part of the 2010-2011 lectureon Quantum Fields in curved spacetimes by W.G. Unruh,2011).[4] W. Blaschke, Vorlesungen ¨uber Differentialgeometrie II:Affine Differentialgeometrie (Springer, 1923).[5] U. Simon, A. Schwenk-Schellschmidt, and H. Viesel,
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Equiaffine differential geometry studies invariants underactive global equiaffine transformations of the ambient (spe-cializing to the dimension we are interested in)ASL(5 , R ) ≡ SL(5 , R ) ⋉ R , (A1)i.e., special linear transformations and translations insidethe ambient affine space for locations, accompanied by onlythe special linear normal subgroup actions on the tensors re-siding over the tangent and cotangent vector spaces. Quan-tities and concepts that, when computed and assessed forthe post-transformation hypersurface, match the transfor-mation of the original counterparts, are considered invari-ant. Equiaffine transformations preserve only parallelism,partition ratio of three points, as well as parallelepiped vol-ume and orientation. They thus deform more strongly thanrigid rotations of the ordinary Euclidean hypersurface the-ories, altering in particular orthogonality relations as de-fined by any ambient metric, which then cannot be usedto pick out an equiaffine invariant normal to the hypersur-face. More precisely, let ˆ ι α = L βα ι β + b α be an ASL(5 , R )transformation on the immersion function ι α of a hyper-surface (Greek indices run over five dimensions, and Latinfour), with L βα ∈ SL(5 , R ), then ˆ µ α (ˆ ι q ) = L βα µ β ( ι q ), where µ α (ˆ µ α ) denotes the normal in the sense of being orthogo-nal to the immersed hypersurface traced out by ι (ˆ ι ), and q denotes an arbitrary point on the abstract hypersurface(domain of the immersion functions). An alternative ap-proach is therefore needed to arrive at an unique equiaffineinvariant normal.One begins with a generic transverse vector field ν α . Toget to Π ab associated with it, one computes firstly a bilinearform by taking the derivative of a conormal. A conormal n α is an ambient one form or covector, which is ∝ ϕ ,α if the hy-persurface is defined as a level surface of ϕ in a local neigh-bourhood. The orientation of n α is fixed, but we cannotmap it into a normal direction since the natural duality be-tween vectors and covectors is precisely the ambient metricthat we don’t have. Nevertheless, its directional certaintyis sufficient to ensure its contraction (no metrics involved,just covectors acting on vectors as linear functions) with anytangent vector to always vanish, so an ambient derivativealong any tangential direction on such a contraction yieldszero. On the other hand, application of the Leibniz’s rulesplits such a derivative into two terms, one with a derivativeon the n α half of the contraction which is just that bilinearform we prepared, while the other has the derivative actingon the tangent vector half, with the component along ν α ofthe outcome given by the second fundamental form Π ab weseek (i.e., this Π ab is defined as the coefficient in the Gaussformula). Equating the two terms to zero then allows us toextract Π ab from the bilinear form, simply through dividingit by − n α ν α (which is nonvanishing since ν α is required tobe transverse).There are various freedoms not yet fixed in this proce-dure: while the orientation of n α is certain, we cannot nor-malize its scaling due to the lack of an ambient metric;both orientation and scaling of ν α are also free. In orderto further pin down a unique equiaffine invariant ν α witha unique corresponding Π ab , we then impose the followingconditions to gradually narrow down the choices:1. An obvious step one can take to partially remove thearbitrariness is by synchronizing n α and ν α with thecondition n α ν α = 1 , (A2)which, for a fixed n α , can be seen as a normalizationfor ν α , but is unfortunately not sufficient to determine ν α completely, since there isn’t a preferred orientationfor it. We thus want to supplement and enhance thisnormalization condition. We do so by taking inspi-rations from the Euclidean theory, where fixing theamplitude of a normal has the consequence that onlytangential components are present in the Weingartenequation governing the derivative of said normal alongthe hypersurface. We impose the same condition on ν α , and call the transverse fields satisfying this con-dition relative normals. This strategy works, yieldinga unique ν α to any given n α , and allows for S ab ≡ − ν a,b (A3)to serve as the shape operator, in a fashion analogousto the ordinary Euclidean theory.Furthermore, it is equivalent [31] to demanding thatthe hypersurface volume form ˜ ω induced by ν α shouldbe parallelly transported by the intrinsic covariantderivative ˜ ∇ a , induced from the flat ambient connec-tion by borrowing the expression for the Gauss for-mula from the ordinary Euclidean theory. The vol-ume ˜ ω is defined with determinants (top differentialforms), by eliminating, through contracting with ν α ,a conormal contribution from within the ambient vol-ume form. Eq. (A2) ensures that this conormal is infact n α .2. There is only then the scaling freedom in n α , or equiv-alently a conformal freedom in Π ab [5], that still needsto be fixed. Because ˜ ω is obtained by factoring out n α from the ambient volume form, a condition on ˜ ω would be quite effective. For the equiaffine case (thereare other conditions leading to sibling theories, likecentroaffine or Euclidean-affine, whose unique nor-mals are invariant under different subsets of the gen-eral linear plus translation transformations), we im-pose the apolarity condition, requiring ˜ ω to agreewith the intrinsic pseudo-Riemannian volume form Π ω ∝ p | det( Π ab ) | . This provides some assurancesthat Π ω , even though defined through an indefinite Π ab that can output negative distances, would not endup straying too far from how one expects a volume tobehave. There is a unique relative normal satisfyingthis condition, called the equiaffine (or Blaschke) nor-mal, given by ν α = Π (cid:3) ι α /
4. Roughly, it finds a cen-tral direction around which the hypersurface locally looks as symmetric as possible – see [4, 32] for morerigorous descriptions. As equiaffine transformationssheer the hypersurface shape, the central directiontilts concomitantly, allowing this particular relativenormal to be an equiaffine invariant.Note, this is generally a different fixing than the Eu-clidean normal, wherever an ambient metric is pro-vided, so one has to be careful about borrowing in-tuitions from the Euclidean theory. Nevertheless, theEuclidean normal is also a legitimate relative normal,and thus the Π ab s corresponding to the Euclideanand equiaffine normals are conformally related, andmany important properties, such as rank and signa-ture (modulo interchanging the positive and negativeslots), are shared between them. However, it is thefirst fundamental form and not the second that’s usedas the intrinsic metric for the ordinary Euclidean dif-ferential geometry, so that theory is different fromthe Euclidean normal-fixed version of affine geome-try (viz. Euclidean-affine), and its intrinsic pseudo-Riemannian (recall Footnote 2) counterpart theory isnot conformally related to the intrinsic partner of theequiaffine theory.To implement the apolarity condition, we need tobring in the totally symmetric cubic form F abc thatevaluates the breakdown of metricity, when thatambient-induced covariant derivative ˜ ∇ a of the lastenumeration point is used on the candidate equiaffinemetric Π ab , i.e., F abc = −
12 ˜ ∇ a Π bc . (A4)Raising indices using Π ab defined through Π ab Π bc = δ ac from here on and throughout the paper, we ob-tain F abc that measures the difference between theambient-induced connection (associated with ˜ ∇ a ),formally written in component form as ˜Γ cab , and theChristoffel symbol Π Γ cab of Π ab , viz., F abc ≡ ˜Γ cab − Π Γ cab . (A5)The apolarity condition is then imposed as the alge-braic relation0 = Π bc F abc = F abb = ˜Γ bab − Π Γ bab = (ln ˜ ω ) ,a − (ln Π ω ) ,a = (cid:18) ln ˜ ω Π ω (cid:19) ,a , (A6)or the ability to propagate an Π ω = ˜ ωω