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a new theoretical concept which , once formulated , naturally emerges in many related contexts , deserves further study . even more so , if it makes us view well - established theories in a novel way , and meaningfully points beyond standard theory . area metrics , we argue in this paper , are such an emerging notion in fundamental physics . an area metric may be defined as a fourth rank tensor field which allows to assign a measure to two - dimensional tangent areas , in close analogy to the way a metric assigns a measure to tangent vectors . in more than three dimensions , area metric geometry is a true generalization of metric geometry ; although every metric induces an area metric , not every area metric comes from an underlying metric . the mathematical constructions , and physical conclusions , of the present paper are then based on a single principle :    _ spacetime is an area metric manifold . _    we will be concerned with justifying this rather bold idea by a detailed construction of the geometry of area metric manifolds , followed by providing an appropriate theory of gravity , which finally culminates in an application of our ideas to cosmology . in the highly symmetric cosmological area metric spacetimes , we can compare our results easily to those of einstein gravity . we obtain the interesting result that the simplest type of area metric cosmology , namely a universe filled with non - interacting string matter , may be solved exactly and is able to explain the observed  @xcite very small late - time acceleration of our universe , see the figure on page , without introducing any notion of dark energy , nor by invoking fine - tuning arguments . it may come as a surprise , but standard physical theory itself predicts the departure from metric to true area metric manifolds . more precisely , the quantization of classical theories based on metric geometry generates , in a number of interesting cases , area metric geometries : back - reacting photons in quantum electrodynamics effectively propagate in an area metric background @xcite ; the massless states of quantum string theory give rise to the neveu - schwarz two - form potential and dilaton besides the graviton , producing a generalized geometry which may be neatly absorbed into an area metric @xcite ; the low energy action for d - branes @xcite is a true area metric volume integral @xcite ; canonical quantization of gravity  la ashtekar  @xcite naturally leads to an area operator  @xcite , such that the classical limit of the underlying spin network structure is also likely a generic area metric manifold , rather than a metric one . the emerging picture is that area metric manifolds are generalized geometries . in the case of string theory , one may even reverse the argument by observing that the geometry of an area metric background forces one to consider strings rather than point particles  @xcite . in the present paper this plays a role in our discussion of fluids in area cosmology ; fluids on area metric spacetime can not consist of particles , but must feature strings as the minimal mechanical objects , which leads us to develop the notion of a string fluid . generalized geometries begin to play an increasingly important role also in mainstream string theory , despite the fact that one initial starting point for its formulation is a metric target space manifold . non - geometric backgrounds in string theory , meaning backgrounds that do not admit a metric geometry , for instance emerge in flux compactifications on which one acts with t - dualities or in mirror symmetry @xcite . they also appear in compactifications with duality twists , which in some cases have been shown to be equivalent to asymmetric orbifolds  @xcite . generalized geometries built to understand these situations have been originally proposed by hitchin @xcite and , the t - fold idea , by hull @xcite . these have found a number of applications  @xcite , one recent example discusses the stabilization of all moduli through fluxes in a specific non - geometric background  @xcite . maybe the most striking example of a classical theory , where area metrics play a natural role , is gauge theory in general , and maxwell electrodynamics in particular . given that electrodynamics is the historical birth place of the concept of a spacetime metric , this is certainly noteworthy . it is known that electrodynamics may be formulated on any @xmath0-dimensional smooth manifold without even introducing the concept of a metric @xcite . the idea of this so - called pre - metric approach goes back to a paper by peres @xcite , and is based on the observation that charges , and in certain situations , magnetic flux lines can be counted , which is nicely explained in @xcite . hence one may define the notions of a field strength two - form @xmath1 and an electromagnetic induction @xmath2-form @xmath3 . the equations of vacuum electrodynamics are then given by @xmath4 and @xmath5 . the induction two - tensor  @xmath6 dual to  @xmath3 must be related to the field strength @xmath1 by some constitutive relation in order to close the system of equations . while peres originally took this to be a definition of the metric , hehl et al . @xcite generalized this to an arbitrary linear relation @xmath7 described by a tensor @xmath8 of fourth rank , and investigated which conditions imply light propagation along a lorentzian lightcone . such general relations between the field strength and the induction are known from the description of electrodynamics in continuous media ; the lensing of light rays in some materials can not be described by geodesics in a metric background . it is therefore not too daring to suspect that gravitational lensing may be equally rich , which amounts to the assumption that spacetime is an area metric manifold  our central assumption in this paper .    of course , neither the most extensive list of known phenomena , nor the most suggestive hints for generalizations alone will be able to justify our proposal to consider spacetime an area metric manifold . however , the area metric gravity theory developed in this paper seems a particularly worthwile testbed for our hypothesis ; this is not least due to the fact that we do not introduce any new parameter into the theory . in this sense , it is not a deformation of einstein - hilbert gravity ( as every alternative action based on metric geometry necessarily is ) , but rather an extension in which the metric ricci scalar is replaced by its area metric analogue . the prediction of an accelerated expansion of our universe at late times , without any additional assumptions or scales being put in by hand , should count as a promising indication in favour of the idea of area metric spacetime . this paper is divided into two largely self - contained parts . part one ( sections 17 ) develops the foundations of area metric geometry in detail . its practical results , however , are concisely summarized in the first section of part two ( sections 8 - 14 ) where the area metric version of einstein - hilbert gravity is formulated in general , and then applied to area metric cosmology . a more detailed outline of the individual sections of the paper is given at the beginning of each of the two parts . we conclude the paper with a discussion of our results and point out future directions . appendix  [ conventions ] lists our conventions , while appendices  [ appvary ] and  [ qsimplification ] respectively derive the general equations of motion of four - dimensional area metric gravity , and those simplified for the almost metric case . this first purely mathematical part of the paper may be skipped at first reading . its practically relevant results are concisely summarized at the beginning of part two , so that the reader mainly interested in the application of area metric geometry to gravity and cosmology may fast - forward to the second part , and come back later to the in - depth treatment of area metric geometry presented here . the precise definition of area metrics , densities and volume forms is given in section  [ manifolds ] , before the non - linear structure of the space of oriented areas is briefly discussed in section  [ sec_areabundles ] . the ensuing construction of area metric geometry is canonical in the sense that it does not rely on additional structure beyond area metric data @xcite . a central issue , namely the extraction of some effective metric from an area metric , is resolved in three steps : identification of the fresnel tensor associated with an area metric in section  [ gauge ] , construction of a family of pre - metrics from the fresnel tensor in section  [ normal ] , and finally selection of a unique , non - degenerate member of that family in section  [ sec_effmetric ] . the aside on area metric symmetries in section [ sec_killing ] prepares our discussion of cosmology later on . from a practical point of view , the most important result of this first part of the paper is the construction of area metric curvature tensors which are downward compatible to their metric counterparts , in section  [ curvature ] . knowing how to measure lengths and angles , one knows how to measure areas . more precisely , if @xmath9 is a metric manifold , one may define the tensor @xmath10 that measures the squared area of a parallelogram spanned by vectors @xmath11 as @xmath12 . we will call @xmath13 the area metric induced from the metric @xmath14 .    the basic idea of area metric geometry consists in promoting area metrics to a structure in their own right , independent of whether there is some underlying metric or not . to achieve this generalization , we simply introduce the area metric by keeping some of the salient algebraic properties of the metric - induced area metrics ( [ cg ] ) . formally , we define an area metric manifold @xmath15 as a smooth @xmath0-dimensional manifold @xmath16 equipped with a fourth - rank covariant tensor field @xmath17 that satisfies the following symmetry and invertibility properties at each point of the manifold :    1 . @xmath18 for all vector fields @xmath19 in @xmath20 , 2 . @xmath21 for all vector fields @xmath19 in @xmath20 , 3 . @xmath22 , defined as @xmath23 by continuation , is invertible . here @xmath24 denotes the space of all contravariant antisymmetric tensors of rank two ; for our conventions concerning components of @xmath25 tensors see appendix [ conventions ] . we will see at the end of section [ gauge ] that in three dimensions every area metric is metric - induced ; from four dimensions onwards , however , there exist area metrics that can not be induced from any metric . in addition to the symmetries _ ( i ) _ and _ ( ii ) _ which we included in our definition , a metric - induced area metric ( [ cg ] ) features a third symmetry , namely cyclicity : @xmath26 we emphasize that we do not impose cyclicity as a property of generic area metrics . in fact , we will see shortly that non - cyclicity plays a central role in two related problems : in the extraction of metric information from an area metric and in the construction of area metric compatible connections . the price to pay for non - cylicity is that the area metric tensor @xmath17 is algebraically reducible . more precisely , any area metric decomposes uniquely into a cyclic area metric and a four - form , which are both irreducible . it will turn out to be advantageous for technical reasons to consider such a decomposition for the inverse area metric , @xmath27}\,.\ ] ] in this context note that the cyclic components @xmath28 and the four - form components @xmath1 of the inverse area metric generically mix under inversion ; in other words , the cyclic part of the inverse area metric is not simply the inverse of the cyclic part of the area metric .    an area metric @xmath17 naturally gives rise to the scalar density @xmath29 of weight @xmath30 , where the capitalized determinant @xmath31 is understood to be taken over the square matrix of dimension @xmath32 representing the map @xmath33 . the correct transformation behaviour under diffeomorphisms is easily seen by direct calculation , noting that for any square matrix @xmath34 of dimension @xmath0 we have @xmath35}{}_{b_2 } = ( \det t^a{}_b)^{d-1}\,,\ ] ] where the determinant @xmath36 denotes the standard determinant . hence any area metric manifold is naturally equipped with a volume form @xmath37 , whose components in some basis are given by @xmath38 where @xmath39 is the levi - civita tensor density normalized such that @xmath40 . now that we have given a precise definition of area metric manifolds , we should mention some relations to the mathematical literature . the idea to base geometry on some measure of area goes back to work by cartan @xcite , and has been generalized under the name of ` areal spaces ' , see @xcite and references therein , to geometries based on reparametrization - invariant integrals , i.e. , to any given volume measure . although these geometries are more general than our physically motivated area measure , this does not mean that we will simply reproduce , or specialize , known mathematics in this paper . in fact , it is precisely our tensorial approach that allows us , in a novel way , to construct an effective metric and connections on the embedding bundle of areas . moreover the physical motivation behind our construction leads to a successful application to gravity theory . at first sight , it seems that an area metric assigns a measure to parallelograms . but actually , an area metric @xmath17 assigns equal area measure to any two co - planar parallelograms @xmath11 and @xmath41 in @xmath42 that are related by an @xmath43 basis transformation . to see this , let @xmath44 and @xmath45 with @xmath46 ; then @xmath47 the equivalence class of all parallelograms that are @xmath48-related to some representative parallelogram @xmath11 is algebraically neatly realized as the wedge product @xmath49 , since @xmath50 the quotient space of all parallelograms by this @xmath48 identification will be denoted by @xmath51 , and its elements are the oriented areas over @xmath52 . similarly , we denote the bundle of oriented areas over @xmath16 by @xmath53 . whereas parallelograms constitute a vector space @xmath54 , the oriented areas @xmath51 at some point @xmath55 can not carry a vector space structure . to see this note the following : while @xmath49 clearly is an element of the vector space @xmath24 , a generic vector @xmath56 decomposes into a finite sum of such wedge - products , rather than being a single wedge - product . a useful necessary and sufficient criterion for @xmath57 to be a simple wedge product , and hence an oriented area , is @xmath58 in which case @xmath57 is called simple . thus the space of oriented areas @xmath51 is a subset of the vector space @xmath24 defined by the vanishing of the four - tensor @xmath59 . since this condition is quadratic in @xmath57 , the set @xmath51 is recognized as an affine variety in @xmath24 . hence the bundle @xmath53 fails to be a vector bundle . while this does not prevent the construction of a connection , from some connection on the underlying principal bundle , it is not possible to define a covariant derivative on @xmath53 . but it is possible to define a covariant derivative on the vector bundle @xmath25 , into which @xmath53 is embedded , and we will do so in section [ curvature ] .    when discussing strings and string fluids in section [ sec_sfluids ] , we will also need to address integrability issues , i.e. , under which circumstances a distribution of oriented areas is tangent to some underlying two - surface . the physical postulate put forward in this paper is that physical spacetime is an area metric manifold @xmath15 , rather than a metric manifold . this immediately prompts the question of whether a generic area metric may give rise to some effective metric on the manifold  @xmath16 . the latter will play a significant role in the construction of an area metric curvature scalar which is downward compatible to the metric ricci scalar , in section [ curvature ] .    with the aim of constructing an effective metric in mind , it turns out to be extraordinarily instructive to probe the geometric structure of an area metric manifold by abelian gauge theory . in particular this allows us to study wave propagation , which in the geometric - optical limit gives insight into the geometry of rays . in the metric - induced case , the ray surfaces reduce to the familiar null cones . in our more general setting , the fresnel tensor describes the geometry of rays , and its derivation presents the first step towards the extraction of metric information from the underlying area metric manifold , which will be completed in sections [ normal ] and [ sec_effmetric ] . consider the following action for a one - form potential @xmath60 with field strength @xmath61 on an area metric manifold of dimension @xmath62 : @xmath63 the equations of motion , derived by variation with respect to @xmath60 , are simply given by the bianchi identity @xmath4 and by @xmath5 in terms of the dual electromagnetic induction  @xmath3 , which is a @xmath2-form on @xmath16 with components @xmath64    to obtain geometric information from these equations , we study the geometric - optical limit . in this limit one considers waves as propagating discontinuities in the derivatives of the fields @xmath1 and of the electromagnetic induction @xmath2-form @xmath3 along a wavefront surface described by the level lines of some scalar function @xmath65 on @xmath16 . the rays of the wavefront are then given by the gradient @xmath66 . extending an insightful argument of hehl , obukhov and rubilar @xcite , where a detailed derivation may be found , to arbitrary dimension @xmath0 , we find that the wavefront gradients must obey the fresnel equation @xmath67 where @xmath68 is the totally symmetric tensor density of weight @xmath69 given by @xmath70 with @xmath28 being the cyclic part of the inverse area metric @xmath71 as in ( [ c+f ] ) . as explained in appendix [ conventions ] we use the convention that the summation over numbered _ anti_-symmetric indices is ordered , i.e. , @xmath72 and similarly for the @xmath73 . it is a remarkable fact that , in the geometric - optical limit , the propagation of wave fronts is determined entirely in terms of the cyclic part of the inverse area metric , due to ( [ dense ] ) . it will be convenient to cast the information encoded in the density @xmath68 into the form of a tensor @xmath74 . in order to not introduce non - cyclic information into this tensor , we are led to de - densitize @xmath68 according to @xmath75 note that the determinant factor presents the required scalar density of weight @xmath76 because  @xmath28 is a contravariant tensor . the tensor @xmath74 describing the ray surfaces will be called the fresnel tensor induced from the area metric @xmath17 .    in order to illustrate the geometric role of the fresnel tensor @xmath74 in a familiar setting , consider the action ( [ edaction ] ) on a lorentzian manifold @xmath9 , on which , for the time being , the area measure is simply induced from the metric @xmath14 by virtue of ( [ cg ] ) . a more subtle discussion of metric - induced measures will follow in section [ sec_effmetric ] . in this case the action reduces to the standard form @xmath77 . moreover , a direct calculation shows that the fresnel tensor simplifies to @xmath78 so that the fresnel equation ( [ fresnel ] ) takes the factorized form @xmath79 this result is of course equivalent to the standard null condition for light rays propagating on a metric manifold . the insight to be drawn from this comparison is the fact that , a priori , it is not a metric that describes the propagation of rays ( not even in the metric case ) , but rather the fresnel tensor @xmath74 ( which however happens to factorize neatly to the form ( [ factoredfresnel ] ) in the case of a metric background ) . hence , it is useful to make a conceptual difference between the background structure ( metric or area metric ) employed to define one - form dynamics on the one hand , and the structure that describes that theory in the geometric - optical limit ( the fresnel tensor ) on the other hand , although we are used to think of these structures as synonymous in the metric case . there is an important corollary from the above findings , of which we will make essential use in section [ sec_killing ] . for an area metric in three dimensions , the ( totally symmetric ) fresnel tensor @xmath80 is of second rank , and with some amount of algebra , one checks that @xmath80 is invertible . furthermore , @xmath81 induces the area metric we started with , by expression ( [ cg ] ) . hence , in three dimensions , every area metric is induced from some metric . this is no longer the case in more than three dimensions . as a second step towards the extraction of an effective metric from the area metric , we identify in this section a family @xmath82 of pre - metrics on  @xmath16 which are induced by an area metric via the fresnel tensor . if the area metric data distinguish a non - degenerate member of that family ( in a way we will explain below ) , the area metric manifold will be called normal . in other words , normal area metrics allow for the extraction of a metric . probing an area metric manifold of arbitrary dimension by abelian gauge theory , we found that wave propagation in the geometric - optical limit is described by the fresnel tensor ( [ fresnel ] ) , not by a metric . we may use the totally symmetric fresnel tensor to define a symmetric bilinear form on a subbundle of the cotangent bundle @xmath83 ( with bundle projection @xmath84 ) by the following construction , which in similar form finds application in finslerian geometry @xcite . we choose local coordinates @xmath85 on @xmath86 which induce local coordinates @xmath87 on @xmath88 , by virtue of @xmath89 . then we define the characteristic function @xmath90 on that portion @xmath91 of the cotangent bundle where @xmath92 takes real values . thus @xmath93 is a level set of the function on the total bundle space defined by the imaginary part of @xmath92 , and hence defines a subbundle of @xmath83 , with @xmath94 . for any @xmath95 , the function @xmath96 is homogeneous of degree two in @xmath97 , i.e. , @xmath98 . from the characteristic function we may now define a symmetric tensor field @xmath99 by differentiating  @xmath92 twice with respect to the fibre coordinate @xmath100 this is indeed a tensor under diffeomorphisms of @xmath16 since they do not depend on the fibre coordinates @xmath97 . it is quickly verified that @xmath101 so that the bilinear form @xmath92 only depends on the direction of  @xmath97 in the fibre , not on its ` length ' . this scaling behaviour of @xmath99 is consistent with the scaling of the characteristic function @xmath92 . a simple and fruitful way of looking at this construction is to think of an area metric as giving rise to a family @xmath82 of symmetric contravariant tensors on @xmath16 with components @xmath102 parametrized by sections @xmath103 . viewing the characteristic function @xmath92 as a generalized norm squared on cotangent vectors @xmath97 , the symmetric covariant tensors @xmath82 present linearizations of this norm around the chosen section @xmath104 . we emphasize again the fact that only the cyclic part @xmath28 of the inverse area metric @xmath71 contributed to the fresnel tensor @xmath74 , and hence to the construction of the symmetric tensors @xmath105 .    an important class of area metric manifolds are those for which one may construct a particular section @xmath106 alone from the data of the area metric tensor @xmath17 , such that  @xmath107 is non - degenerate on all of @xmath16 . then @xmath108 will be called a normal area metric manifold . we will see that normality of an area metric is an inevitable requirement for the construction of area metric curvature tensors that are downward compatible to their metric counterparts . but it is not a big restriction . we will show in the next section that one - forms @xmath104 can always be constructed from the area metric , albeit not always uniquely . also , @xmath109 is an open condition so that normality is the rule not the exception . in particular , we will show that area metrics induced from riemannian metrics are always normal . for lorentzian metrics in even dimensions , we find that the construction of a normal area metric interestingly requires that the lorentzian metric should describe a stably causal spacetime . the signature of a normal area metric manifold @xmath108 is defined as the signature @xmath110 of the metric @xmath111 constructed from @xmath17 and @xmath112 . with this definition , we may define the totally antisymmetric tensor density @xmath113 to be normalized such that @xmath114 . this in turn induces the contravariant tensor @xmath115 with components @xmath116 it is necessary to distinguish the tensors @xmath117 and @xmath115 , because their components are in general not related by lowering and raising of indices with the area metric and inverse area metric , respectively . with these definitions , we obtain for the contraction over @xmath118 indices the useful identity @xmath119}_{\,\,\,b_1 \dots b_{d - k}}\,.\ ] ] we now complete the construction of an effective metric from a generic area metric manifold @xmath15 without specifying additional data . from our findings in sections  [ gauge ] and  [ normal ] , we know that the extraction of a metric from area metric data amounts to the construction of a section @xmath104 of the subbundle @xmath91 , as defined by reality of the characteristic function  ( [ character ] ) , such that  @xmath120 is normal . then the inverse of the tensor  @xmath82 as defined in  ( [ hsigma1 ] ) provides the desired metric . in order to achieve downward compatibility with metric geometry , we further require that for metric - induced area metrics the inducing metric  @xmath14 is recovered up to a sign ( more can not be expected since the metric - induced area metrics are quadratic expressions ) . this requirement will lead to a modification of the induction formula  ( [ cg ] ) for lorentzian metrics . the fresnel tensor for a metric - induced area metric ( [ cg ] ) takes the simple form ( [ metricfresnel ] ) . then the characteristic function ( [ character ] ) is given by @xmath121 in both cases this is real on the entire cotangent bundle , so that @xmath122 . now , however , appears a crucial difference between metric manifolds of different signatures . first consider the case of a riemannian metric @xmath14 where @xmath123 independent of the dimension . in this case all members of the family @xmath105 of metrics coincide , and are given , for an arbitrary section @xmath104 , by @xmath124 this means that any choice of section @xmath125 recovers the inducing riemannian metric @xmath14 from the area metric @xmath126 , including the zero section that does not even require area metric data . thus @xmath127 is normal if @xmath14 is a riemannian metric . the same conclusion holds if @xmath14 is an odd - dimensional lorentzian metric . but now let @xmath14 be an even - dimensional lorentzian metric . then the family @xmath82 of metrics is given by @xmath128 in contrast to the riemannian case , it is now not possible to choose the zero - section in order to recover the inducing lorentzian metric . this means we must obtain a non - trivial section from the area metric in order to be downwards compatible to the metric - induced case . otherwise the construction would fail for the even - dimensional lorentzian case . we first observe that the required section @xmath104 can not be constructed alone from the cyclic part @xmath28 of the inverse area metric . this is because the only one - forms obtainable from @xmath28 are of the form @xmath129 where @xmath130 and @xmath131 are a finite set of non - negative integers and @xmath132 and @xmath133 . now for an area metric that is metric - induced according to  ( [ cg ] ) , we have @xmath134 and @xmath135 , so that both @xmath28 and its inverse are cylic ( which , we recall , does not hold for generic area metrics ) . then it is easily verified that all of the one - forms  ( [ candidateone ] ) vanish . thus we are compelled to conclude that information from the area metric beyond the cyclic part  @xmath28 is needed for the definition of a non - trivial section  @xmath104 . we hence turn to the four - form part of the inverse area metric as a potential carrier of this information . this fits nicely together with the observation that  @xmath136 is defined entirely in terms of the cyclic part of the inverse area metric . so the four - form part will exclusively contribute to the construction of the section  @xmath104 which replaces  @xmath97 , while the cyclic part exclusively contributes to the definition of the bilinear symmetric tensor  @xmath136 . for an area metric manifold of dimension @xmath137 , we may schematically decompose the inverse area metric as @xmath138 so that the information in the four - form part is succinctly encoded in a @xmath139-form @xmath140 . now four - dimensional area metric manifolds are distinguished since , in this dimension , the construction of a one - form from the four - form part of @xmath71 is unique . the only possible choice is @xmath141 any rescaling of this choice , even by a function , is without effect because of relation ( [ deg0 ] ) . due to the symmetries of @xmath71 and @xmath115 , it is also not possible to construct further non - vanishing one - forms from @xmath142 . already in five dimensions , there are several possibilities , @xmath143 in even higher dimensions , the number of possibilities further proliferates . so in four dimensions only may one speak of _ the _ normal area metric manifold @xmath120 constructed from an area metric manifold . ( at least , unless one could formulate further meaningful criteria to construct , also in higher dimensions , a natural section @xmath112 to single out the effective metric from the family @xmath82 of contravariant tensors . ) from now on , we will therefore restrict attention to the four - dimensional case , which of course also is the one of immediate physical interest . a four - dimensional normal area metric manifold will henceforth be denoted @xmath144 where @xmath145 is the unique metric constructed from the area metric  @xmath17 . generically , the metric @xmath146 of course only encodes part of the information contained in the area metric . other than in dimension three , the area measure @xmath147 re - induced from the effective metric generically does not agree with the measure determined by  @xmath17 . the above constructions play out nicely for area metric manifolds induced by stably causal spacetimes @xmath9 in four dimensions . for the latter , there always exists a global time function @xmath148 on @xmath16 such that @xmath142 is everywhere @xmath14-timelike @xcite . hence after a choice of global time function @xmath148 has been made , the latter may be encoded directly into the totally antisymmetric part of the inverse area metric : @xmath149 due to our mainly plus signature convention , the inducing lorentzian metric may be recovered , using ( [ zzz ] ) and ( [ lorentzh ] ) , as @xmath150 . so downward compatibility is maintained . from now on , we will denote the inverse area metric ( [ stablycausallift ] ) induced by some lorentzian metric @xmath14 and a suitable scalar field @xmath148 always by @xmath151 . inverting this one finds the relation @xmath152 this is the closest a normal area metric may come to a lorentzian metric , and presents an important class of area spacetimes which can be easily compared to standard metric spacetime , see section [ solar ] . in the following section , we will see that cosmological symmetries , for instance , enforce this special , almost metric , form of an area metric . at this point it is useful to study isometries of area metric manifolds . this will provide us with more evidence for the natural role non - cyclic area metrics play and prepare our discussion of area metric cosmology . a diffeomorphism @xmath153 is called an area metric isometry if it preserves the area metric in the sense that for all smooth vector fields @xmath154 on @xmath16 and all @xmath55 , we have @xmath155 where @xmath156 denotes the push - forward with respect to @xmath92 . as in the metric case , it is useful to consider the generators of isometries , i.e. , killing vector fields @xmath157 . clearly , @xmath157 is a killing vector field for an area metric manifold @xmath15 if @xmath158 . this condition implies that the killing vectors of a given area metric manifold , together with the standard commutator , constitute a lie algebra . it is reassuring to verify that for a metric - induced area metric manifold , @xmath157 is a killing vector of @xmath159 if and only if @xmath157 is a killing vector of @xmath9 . this is indeed the case : the implication @xmath160 is evident . conversely , assume @xmath161 . in particular , for any pair of @xmath14-orthogonal vectors @xmath162 , we have @xmath163 now consider a @xmath14-orthonormal basis @xmath164 , @xmath165 of @xmath166 . for any three distinct vectors @xmath167 in such a basis , the relation above implies    @xmath168    this set of equations is homogeneous and non - degenerate with respect to @xmath169 , @xmath170 , and @xmath171 , so that it admits as the unique solution @xmath172 . to complete the proof , note that we then also have @xmath173 so that @xmath174 . it follows that @xmath175 for all @xmath176 ; in other words , @xmath177 . it is also worthwhile to note that , in the case @xmath178 , the theorem easily extends to @xmath179 . equipped with the definition of killing vectors , we can now discuss physically relevant applications . in particular , we will be able to construct the analogue of a friedmann - lematre - robertson - walker ( flrw ) cosmology for an area metric background . we will discover that a cosmological area metric in @xmath180 is precisely of the form ( [ stablycausallift ] ) , with the cyclic part being induced by a standard flrw metric , and with the scalar field providing an additional degree of freedom in comparison to metric geometry . first recall that , given a lie group action @xmath181 on some smooth manifold @xmath16 , it is useful to define the following notions @xcite . the orbit of some point @xmath55 is the submanifold @xmath182 . the isotropy group @xmath183 at some point @xmath55 is that subgroup of @xmath17 whose action leaves @xmath97 invariant . a submanifold @xmath184 of dimension @xmath185 is said to be _ homogeneous _ if there is a transitive action of some lie group @xmath17 on @xmath184 , i.e. if @xmath186 for all @xmath187 . it is said to be _ spherically symmetric _ around a point @xmath97 if the isotropy group of @xmath97 is @xmath188 , and the relative orbit of any other point @xmath189 around @xmath97 is topologically equivalent to an @xmath190-sphere .    in complete analogy with the standard , rigorous definition of a ( spatially ) homogeneous and isotropic manifold in lorentzian geometry , we can now define area metric cosmology , i.e. , flrw area metric manifolds , as follows : @xmath15 is a @xmath0-dimensional flrw area metric manifold if and only if it can be suitably sliced in @xmath191-dimensional hypersurfaces , which are homogeneous and spherically symmetric around each point . additionally , the hypersurfaces must be spacelike , in the sense that the restriction of the area metric to them must be positive definite . in the four - dimensional case , on which we will focus , the general form of the area metric compatible with these requirements can be easily determined . first of all , we note that the area metric on the slices must be metric - induced , since the slices are three - dimensional ( compare the corollary at the end of section [ gauge ] ) . the inducing metric @xmath192 must describe three - dimensional maximally symmetric riemannian manifolds with line element @xmath193 for normalized curvature @xmath194 , in the usual system of coordinates @xmath195 . for the restriction of the area metric to the slices of constant @xmath196 we then obtain the expression @xmath197 for some function @xmath198 . moreover , on each of the spacelike slices , @xmath199 behaves as a three - dimensional metric with respect to the entire group of isometries . therefore , introducing some function @xmath200 , we have @xmath201 finally , explicitly solving killing conditions for the remaining components of the area metric , one finds for the remaining non - vanishing components the form @xmath202 for some function @xmath203 . employing a suitable rescaling of the time coordinate and redefining the free functions @xmath204 , it is easy to verify that an flrw area metric takes the form @xmath205 , see ( [ stablycausallift ] ) , where @xmath206 is the standard flrw metric , so that @xmath207 and @xmath208 only depends on time . inverting @xmath205 and redefining the scalar field , also the inverse area metric decomposes into the form @xmath209    thus in comparison with the standard metric case , flrw area metric manifolds in @xmath180 feature a time - dependent scalar degree of freedom @xmath210 additional to those of the metric , which happens to be encoded in the totally antisymmetric part of the ( inverse ) area metric . if non - vanishing , the section @xmath211 will be @xmath212-timelike , and thus render the flrw area metric normal , as discussed at the end of the previous section . consequences of this feature for the dynamics of such cosmological models will be discussed in sections  [ sec_cosmology ] and  [ sec_exact ] . our aim in this section is the construction of an area metric compatible connection and area metric curvature , which are downward compatible to their metric counterparts . as discussed in @xcite , we principally need a connection on the non - vector bundle of area spaces @xmath53 embedded in  @xmath25 . in order to keep technicalities to a minimum , we will determine this connection in terms of a covariant derivative @xmath213 on the vector bundle @xmath25 . consider a four - dimensional normal area metric manifold @xmath214 , where  @xmath146 is the unique metric constructed from @xmath17 , as explained in section [ sec_effmetric ] . the metric @xmath146 immediately gives rise to the torsion - free levi - civita connection @xmath215 , which of course lifts to the @xmath25-bundle in the standard way . while @xmath215 provides a covariant derivative on @xmath25 and is downward compatible , it is not of direct relevance for the area metric manifold @xmath15 . on the one hand , it does not contain all the information contained in the area metric , which follows from a simple counting of the number of independent components of the effective metric that defines @xmath215 . on the other hand , @xmath215 does not obey the important geometric property that the area metric should be covariantly constant : @xmath216 in general . but covariant constancy of the area metric ensures , for instance , that the simplicity condition  ( [ simple ] ) of a section  @xmath57 of  @xmath25 is not violated by parallel transport ; in other words , areas are preserved under parallel transport  @xcite . but most importantly , we require area metric compatibility because it allows us to determine a unique connection as follows . first note that an arbitrary connection @xmath213 on @xmath25 , including the area metric compatible one we are seeking , differs at most by some tensor @xmath157 from the lift of the levi - civita connection , i.e. , @xmath217 or , in coordinates , @xmath218 . in order to uniquely determine an area metric - compatible connection @xmath213 , we need to constrain the class of @xmath25-connections we are looking for . to this end note that for an arbitrary connection on the bundle @xmath25 over a normal area metric manifold @xmath144 , we may define the tensor @xmath219 for any vector @xmath220 and sections @xmath221 of @xmath25 . we call @xmath222 the relative torsion of the @xmath25-connection  @xmath213 with respect to @xmath144 . the virtue of having such a tensor is that one may formulate a consistent condition on a generic @xmath25 connection by requiring @xmath222 to vanish identically . this requirement corresponds to the symmetry condition @xmath223 on the components of the tensor @xmath157 defined in ( [ xdef ] ) , where indices have been lowered using the area metric @xmath17 . vanishing relative torsion and area metric compatibility together , @xmath224 then uniquely determine the tensor @xmath157 : using the first requirement @xmath225 gives @xmath226 while the second requirement @xmath227 allows us to equate this to @xmath228 . then @xmath157 must be of the form @xmath229    this result can also be obtained in the coordinate - free language of pre - connections introduced in @xcite . then the relative torsion - free , area metric compatible @xmath25-connection is given by @xmath230,\cdot)\,,\ ] ] with the symmetric and antisymmetric pre - connections    @xmath231 & = & g(\nabla_z^{lc}\omega,\sigma ) - g(\omega,\nabla_z^{lc}\sigma)\,,\end{aligned}\ ] ]    respectively . here the symmetric pre - connection is determined by @xmath225 alone , while the antisymmetric pre - connection then is determined by @xmath227 . note that while in  @xcite a metric  @xmath14 was assumed to be given as data in addition to an area metric in order to construct a connection , the significant advance of this paper consists in having constructed the metric  @xmath146 in a unique manner from the area metric data alone . hence we have now achieved the construction of a true area metric connection , without additional data . the identification of this unique connection @xmath213 associated with a normal four - dimensional area metric manifold @xmath15 now allows for the definition of the area metric curvature in standard fashion @xmath232}\omega\,,\ ] ] whose components can be calculated to be @xmath233}{}_{b_2]ij}+\left(\nabla^{lc}_i x^{a_1a_2}{}_{b_1b_2j}+x^{a_1a_2}{}_{f_1f_2i}x^{f_1f_2}{}_{b_1b_2j}-(i\leftrightarrow j)\right),\ ] ] where @xmath234 is the curvature of the levi - civita connection @xmath215 obtained from @xmath146 . natural contraction over indices 1 and 2 with indices 3 and 5 yields the area metric ricci tensor @xmath235 another second rank tensor is defined by @xmath236 } = \mathcal{r}_g{}^{pq}{}_{pqij}\,.\ ] ] due to its antisymmetry , however , there is no contraction to a non - zero scalar of first order in  @xmath184 ; the lowest order scalar one may build from @xmath237 is @xmath238 . this invariant is therefore of no interest to area metric gravity , as its inclusion in an action would drive the derivative order of the equations of motion automatically beyond two . so the unique area metric curvature scalar of linear order in @xmath239 , is the area metric ricci scalar @xmath240    in the case of an area metric @xmath13 induced from a riemannian metric we have  @xmath241 from  ( [ xx ] ) . hence it is immediately clear that the area metric ricci tensor and ricci scalar reduce to their metric counterparts while the tensor @xmath184 vanishes . the lorentzian case is less obvious ; recall that inducing a normal area metric from a lorentzian metric requires the addition of a scalar @xmath148 whose gradient @xmath142 is a globally non - null one - form field ; according to  ( [ stablycausallift ] ) the area metric is  @xmath242 . it may now be checked , however , that also in this case all of the above curvature tensors reduce to their metric counterparts , and that  @xmath184 vanishes identically . this proves the required downward compatibility to metric geometry . we will use the curvature invariant  @xmath243 in the following second part of the paper for the construction of an area metric gravity theory . the discussion of area metric gravity in this second part is self - contained , since the relevant results of area metric geometry developed in part one are concisely summarized in the following section [ synopsis ] . the general formalism of area metric gravity and its coupling to matter is developed in section [ gravtheory ] , followed by a comparison of almost metric vacuum solutions to standard general relativity in section [ solar ] . fluids on area metric spacetimes are necessarily string fluids , as is explained in sections [ pfluids ] and [ sec_sfluids ] . building on these preliminaries , we apply area metric geometry to cosmology in section [ sec_cosmology ] , and find the exact solution for a homogeneous and isotropic area metric universe filled with string dust in section [ sec_exact ] . this leads to the prediction that area metric cosmology may explain , without dark energy or fine - tuning , the small late - time acceleration of our universe . in this brief section , we give a practical guide to the construction of area metric curvature in _ four _ dimensions . for derivations and details of the construction in coordinate - free fashion , see part one of this paper . a lorentzian area metric on a smooth four - dimensional manifold  @xmath16 is a covariant tensor @xmath244[cd ] } = g_{[cd][ab]}\,,\ ] ] whose inverse is defined as the contravariant tensor @xmath245 satisfying @xmath246}_d\,.\ ] ] considering ordered pairs of indices @xmath247 , one may write an area metric in four dimensions in petrov notation , i.e. , as a @xmath248 matrix @xmath249 with @xmath162 taking values in the set @xmath250 , [ 02 ] , [ 03 ] , [ 12 ] , [ 13 ] , [ 23]\}$ ] . defining @xmath251 as the determinant taken over the matrix  @xmath249 , one may define covariant and contravariant area metric volume tensors @xmath252 where the totally antisymmetric tensor densities @xmath39 are normalized so that @xmath253 and @xmath254 , respectively . the inverse area metric uniquely decomposes as @xmath255 i.e. , into a cyclic part @xmath256}=0 $ ] and a totally antisymmetric part , which in four dimensions amounts to the specification of a function  @xmath148 on  @xmath16 . the definition of  @xmath257 is that of  @xmath37 , with  @xmath28 replaced by @xmath17 . a normal area metric allows to extract a unique effective metric @xmath258 from the area metric data . in case the area metric is almost metric , i.e. , @xmath259 for some metric @xmath14 , the effective metric @xmath146 recovers the inducing metric @xmath14 up to a sign , which validates the present construction independent of its deeper geometric meaning discussed in part one of this paper . finally , the area metric curvature tensor is given by @xmath260}{}_{[b_1b_2][ij ] } = 4\delta^{[a_1}_{[b_1}r^{a_2]}{}_{b_2]ij}+\big(\nabla^{lc}_i x^{a_1a_2}{}_{b_1b_2j}+\frac{1}{2}x^{a_1a_2}{}_{pqi}x^{pq}{}_{b_1b_2j}-(i\leftrightarrow j)\big),\ ] ] where @xmath234 and @xmath215 are the riemann tensor and the levi - civita connection of the effective metric @xmath146 , and the non - metricity tensor @xmath157 is defined by @xmath261}{}_{[b_1b_2]f}\,.\ ] ] the area metric curvature tensor , as well as the associated area metric ricci tensor @xmath262 and area metric ricci scalar @xmath263 , reduce to their metric counterparts for almost metric area metrics . these correspondences ensure in particular that area metric geometry is downward compatible to metric geometry , which is therefore contained as a special case . area metric geometry allows us to devise a gravity theory different from einstein s without modifying the form of the einstein - hilbert action ; the latter is just re - interpreted as an action for an area metric manifold , with all metric quantities being replaced by their area metric counterparts . we restrict our study of the area metric version of the einstein - hilbert action to four dimensions , as only there the standard metric version enjoys its truly special status due to lovelock s theorem @xcite . thus we adopt the following action for four - dimensional normal area metric spacetimes @xmath15 : @xmath264 from which the equations of motion are derived by variation with respect to the ( inverse ) area metric @xmath17 . the gravitational constant @xmath265 is not yet determined at this stage . the quantity @xmath266 represents the matter lagrangian scalar density , which in many cases is simply @xmath267 in terms of a scalar lagrangian @xmath268 . the diffeomorphism invariance of the action immediately implies an area metric bianchi identity for the gravitational part of the action , and the conservation law for the area metric energy - momentum tensor @xmath34 . in order to study the latter , consider a diffeomorphism on  @xmath16 generated by a vector field @xmath269 . the induced variation of the inverse area metric may be expressed covariantly as a lie derivative , @xmath270pcd}+2 \nabla^{lc}_p \xi^{[c}g^{d]pab}\,,\ ] ] where the second equality uses the levi - civita connection of the effective metric . requiring invariance of the matter action under this variation by setting @xmath271 , i.e. , @xmath272 , and performing some partial integration , we are led to the conservation equation @xmath273}_i\right)=0\,.\ ] ] for the fourth rank tensor @xmath274 we call this tensor @xmath34 the generalized energy - momentum tensor of matter on an area metric manifold . since it is derived by variation with respect to the inverse area metric , it has the symmetries of the area metric . in particular it may contain a totally antisymmetric part in the decomposition under the local frame group . computationally performing the variation of the gravitational part of the action is rather involved ; it is given in full detail in appendix [ appvary ] . from the reducibility of the inverse area metric @xmath17 into a cyclic and totally antisymmetric part , however , it is clear that in four dimensions the equations of motion for generic area metrics can be separated accordingly . so if the tensor @xmath275 denotes the variation of the gravitational part of the action  ( [ totalaction ] ) with respect to the inverse area metric @xmath71 , i.e. , @xmath276 then , using equation ( [ varg ] ) , we may decompose the total variation into those terms arising from the gravitational part @xmath277,\ ] ] and into those from the matter part @xmath278.\ ] ] taking care to impose the symmetries of @xmath279 on the expressions in brackets that are contracted with it , we may thus define the cyclic contributions @xmath280 and @xmath281 , and the scalar contributions @xmath282 and @xmath283 of the gravitational tensor @xmath284 and the energy - momentum tensor @xmath34 to the equations of motion . in terms of these quantities , whose explicit form is derived in appendix [ appvary ] , the full equations of motion for area metric gravity take the form    [ generalform ] @xmath285    of course , these field equations are far too complex to be exactly solved in general , and one has to content oneself with studying exact solutions for highly symmetric spacetimes . for area metric cosmology we will start this program below .    as an example for the general theory outlined above , consider the theory of electrodynamics defined by the action ( [ edaction ] ) . variation of this action with respect to the inverse area metric yields the generalized energy momentum tensor @xmath286 as an aside we note here that @xmath180 is special because it is only in this dimension that we have @xmath287 for arbitrary electromagnetic fields . we may now substitute the expression for @xmath34 into the conservation equation above . the coupling of the area metric background and the gauge theory is then only consistent if this equation is satisfied . this is in fact to be expected since the energy momentum tensor was derived from a diffeomorphism invariant action . in order to see explicitly that this is indeed the case , we rewrite the resulting conservation condition for electrodynamics in the form @xmath288}+\frac{1}{2}f_{di}\left(\nabla^{tm}_c+\frac{1}{2(d-1)}x_c\right)\left(g^{cdab}f_{ab}\right ) = 0\,.\ ] ] all covariant derivatives in this expression may be replaced with partial derivatives by using the definition of the dual electromagnetic induction @xmath2-form , see ( [ induction ] ) . in this way we obtain @xmath289}+\frac{1}{2}(-1)^d f_{i_d i}\omega_g^{i_1\dots i_d}\partial_{[i_1}h_{i_2\dots i_{d-1}]}=0\,.\ ] ] so the equations of motion of area metric electrodynamics , namely the bianchi identity @xmath4 , and @xmath5 imply energy conservation , rendering the matter coupling consistent . a question of immediate interest is of course how standard einstein gravity fits into the more general area metric framework , given its huge phenomenological success especially regarding observations within our solar system . more precisely , we would like to know under which circumstances there exist solutions of  ( [ generalform ] ) where the inverse area metric is of almost metric form @xmath290 , see  ( [ stablycausallift ] ) , which is the closest one may get to the standard lorentzian case . ( recall from section [ sec_effmetric ] that the field  @xmath148 is needed in order to render the area metric induced from a lorentzian metric normal . ) interestingly , area metrics with cosmological symmetries are of almost metric form , as we found in section [ sec_killing ] . thus the results of the present section will be useful in our discussion of area metric cosmology in the following sections  [ pfluids][sec_exact ] . the main aim here , however , is to study the conditions under which area metric gravity reduces to einstein gravity .    in doing so , it is important to withstand the temptation to discuss the reduction of ( [ totalaction ] ) to the almost metric case at the level of the action . this can not be meaningful a priori , because variation with respect to @xmath17 sweeps out more variations than variations with respect to a metric and a scalar field can do @xcite . if one simply inserted the ansatz @xmath291 into the total action , this would reduce the gravitational part to its standard metric analogue , up to a @xmath148-dependent conformal factor . but we will now show that this does not correspond to what generically happens at the level of the full equations of motion including matter . so we insert the almost metric form @xmath291 into the full equations of motion , which , as shown in technical detail in appendix [ qsimplification ] , neatly factorizes the cyclic contribution  @xmath280 of the gravitational variation as @xmath292b ] } + s_{[c[a}g_{b]d]}\,,\end{aligned}\ ] ] for some symmetric tensor @xmath293 . if the cyclic contribution @xmath281 of the energy - momentum tensor to the equations of motion is generated in like fashion from some symmetric second rank tensor , then the field equations are equivalent to    [ vacuumgg ] @xmath294    where we define @xmath295 , @xmath296 and @xmath297 . when we discuss cosmological solutions below , we will have to make sure that @xmath298 does not leave its allowed range @xmath299 we will see that this can be understood as a consistency constraint on the initial conditions . it is thus clear that for there to be an almost metric solution at all , the cyclic part of the energy - momentum tensor at hand needs to factorize in the same fashion as ( [ factorized ] ) .    in vacuo , this is of course trivially the case . it is also reassuring that the vacuum system with @xmath300 is causally well - behaved : in order to see this , we will now perform suitable field redefinitions , and thus reveal that this system is conformally equivalent to einstein - hilbert gravity minimally coupled to a massless scalar field . first , using the trace of the first equation we may replace the system by a simpler one :    @xmath301    now a conformal transformation of the metric and a simple redefinition of the scalar field according to @xmath302 shows that the vacuum equations of motion for an area metric of the almost metric form @xmath291 are conformally equivalent to the system @xmath303 these are precisely einstein s equations for @xmath304 coupled to a massless scalar field @xmath305 , and thus we know that also the original , conformally related theory is causal . we emphasize again that the above equations are only valid for almost metric area metrics in vacuo . this very special case , however , allows for a most important conclusion : any vacuum solution @xmath9 of einstein - hilbert gravity is a vacuum solution of area metric gravity ( setting either @xmath306 , or  @xmath307 with appropriate conditions on the derivatives , see ( [ vacuumgg ] ) ) .    upon the inclusion of matter however , it is no longer true that metric solutions of einstein - hilbert gravity lift to solutions of area metric gravity . for instance , the area metric energy - momentum tensor ( [ edt ] ) for electrodynamics does not take the factorized form ( [ factorized ] ) for the almost metric ansatz @xmath291 , simply because the first term does not even contain a metric . thus , not only can electrodynamics live on a non - metric area metric spacetime , but also in general electrodynamics backreacts in such a way as to generate a non - metric area metric spacetime ! this can only be avoided by restricting the admissible geometries ( and thus the variation ) to a standard metric one . in other words , while standard einstein - hilbert solutions are of course stationary points of the einstein - hilbert action , they may fail to be stationary within the wider spectrum of area metric manifolds , in the presence of matter whose energy momentum tensor does not factorize . thus it is the fact that we vary with respect to the area metric which causes the potential departure from einstein - hilbert gravity in the presence of matter , while the vacuum solutions of standard general relativity are also vacuum solutions of area metric gravity . in general relativity perfect fluids present an effective device to discuss the gravitational effects of fundamental matter averaged over large scales . especially in order to obtain calculationally manageable cosmological models , the description of matter as an ubiquitous fluid is necessary . in this section we will introduce fluid matter on area metric spacetimes as the most general form of energy momentum consistent with area metric cosmology . this discussion paves our way towards a first comparison between area metric cosmology and einstein cosmology . the general form of the energy momentum four - tensor is restricted by the killing symmetries of a given area metric spacetime @xmath15 . consider a diffeomorphism of @xmath16 generated by a vector field @xmath269 , and the resulting change @xmath308 of the variation of the gravitational action with respect to @xmath71 . there are two ways to express this quantity . the first simply is the lie derivative of the four - tensor @xmath284 along @xmath269 , i.e. , @xmath309 . the second uses the fact that @xmath284 is completely determined by the area metric @xmath17 ; schematically it follows that @xmath310 . hence if @xmath157 is a killing symmetry of @xmath17 , i.e. , if @xmath311 , then we conclude @xmath312 the general form of the area metric gravity equations of motion , including matter , is @xmath313 the lie derivative of the left hand side along any killing symmetry @xmath157 of the area metric @xmath17 vanishes , hence consistency requires that the energy momentum tensor should also satisfy @xmath314 for each of the background s killing symmetries . in other words , consistent energy momentum tensors inherit the symmetries of the background . this conclusion is analogous to the one used in standard general relativity to restrict the class of admitted sources in a spacetime with specified symmetry properties . we may repeat our discussion of section [ sec_killing ] to impose the symmetries of an flrw area metric cosmology on the four - tensor @xmath34 . recall that @xmath71 is defined in terms of the standard flrw metric @xmath14 and a time - dependent scalar @xmath148 , and that we have a distinguished time variable @xmath315 . one then finds the components of @xmath34 to be proportional to those of the area metric up to time - dependent functions . we choose the following parametrization ,    [ fluidenergy ] @xmath316    where the three functions @xmath317 and @xmath184 describe the local macroscopic properties of the fluid . below , we will match these functions with the usual notions of density and pressure for a fluid . but first note that like in the metric case , the fluid energy momentum tensor is not obtained from some lagrangian , and so the conservation law ( [ conservation ] ) is not automatically ensured . we have to impose the energy - momentum conservation condition on the tensor @xmath34 , which , using the hubble function @xmath318 , leads to the equation of motion of the fluid , @xmath319    before we turn to our main application of fluid matter , which will be to area metric cosmology , we will introduce in the next section the notion of string fluids which allows us to give a covariant interpretation to fluid matter on area metric spacetimes . area metric spacetimes @xmath15 do not provide a measure of length which would be required to formulate dynamics for the worldlines of point particles . but they do provide a measure of area , and so one may regard string worldsheets as the minimal dynamical , and hence fundamental , objects on area metric manifolds @xcite . it is therefore reasonable to expect that any collection of matter fields averaged over large scales should have an approximate description in terms of a fluid made out of strings ; this is in complete analogy to standard general relativity where perfect ( point particle ) fluids are used for this purpose . our aim in this section is twofold : first we will formulate a simple energy momentum four - tensor for a string fluid on an arbitrary area metric spacetime , and give a geometric interpretation for the consistency conditions arising from the energy conservation equation  ( [ conservation ] ) . second , we will rewrite the general energy momentum tensor consistent with area metric cosmology covariantly in terms of a string fluid ; any collection of averaged matter fields in cosmology can thus be understood as a continuous distribution of string worldsheets . we recall a few basic facts about classical strings on @xmath15 from @xcite . string worldsheets are tangent surfaces to an integrable distribution of areas @xmath320 in @xmath53 . thus they satisfy the simplicity condtion  ( [ simple ] ) and the frobenius integrability criterion : the commutator of @xmath321 and @xmath322 lies within the plane spanned by @xmath323 . in components this is equivalent to @xmath324}=0\,.\ ] ] string dynamics follow from the stationarity of the worldsheet area integral , i.e. , from @xmath325 . the equality follows from the simplicity of @xmath57 , if we explicitly decompose the area metric as @xmath326 into the cyclic algebraic curvature tensor @xmath327 and a four form @xmath328 in analogy to ( [ c+f ] ) . it has been shown in @xcite that the string equation of motion takes a very simple form after fixing the worldsheet reparametrization invariance to constant squared area @xmath329 . the stationarity , or minimal surface , condition then simply becomes @xmath330(\omega\wedge z)=0\ ] ] for any vector @xmath220 . only the cyclic part of the area metric contributes to this equation . the energy momentum four - tensor of a string fluid should be expressible in terms of the area metric and the area distribution @xmath57 . we now write down the simplest two terms that obey the required symmetries of an area metric . one of these will then be shown to represent fluids composed of non - interacting strings ; the other term is an example for a simple interaction . define @xmath331 in terms of two functions @xmath332 and @xmath333 . in order to demonstrate the geometric meaning of these terms we analyse the energy momentum conservation condition . as argued in the preceding section this condition is not automatically satisfied , but nevertheless has to be imposed for consistent matter coupling . we substitute ( [ sfluid ] ) into ( [ conservation ] ) ; the resulting equation can be rewritten as follows : @xmath334ab}\omega^{ab}\right ) -4 g_{diab}\omega^{ab}\left(\nabla^{lc}_c+\frac{1}{6}x_c\right)\left((\tilde\rho+\tilde p ) \omega^{cd}\right)+8\partial_i\tilde p\\\ ] ] observe that the totally antisymmetric part of the area metric in the first term does not contribute . this is a consequence of the simplicity ( [ simple ] ) and the integrability ( [ integrability ] ) of @xmath57 . hence we may replace @xmath17 in the first term by its cyclic part @xmath327 . due to the antisymmetrization in this term we may also replace the covariant derivatives by partials . let us discuss the case @xmath335 first . the first term then is recognizable as the stationarity condition ( [ stationarity ] ) : the local string worlsheets in the string fluid are stationary , as is classical string motion . the vanishing of the remaining second term in the string fluid conservation equation is implied by a generalized continuity equation @xmath336 where the right hand side represents the divergence of a densitized current . in form - language the continuity equation is equivalent to @xmath337=0 $ ] for the two - form @xmath338 . the geometrical picture for this equation is as follows . choose any compact four - volume  @xmath339 with boundary @xmath340 , and any hypersurface defined by @xmath341 with gradient @xmath342 . the projection on the hypersurface of the flow of the string fluid through @xmath343 is @xmath344\wedge d\psi\,.\ ] ] this vanishes if the volume @xmath339 lies entirely within the domain of the string fluid ; then its inflow and outflow through the boundary @xmath340 are precisely equal , independent of the choice of projection on @xmath342 . thus the string fluid energy momentum tensor ( [ sfluid ] ) is amenable to a simple analysis for  @xmath335 : energy momentum conservation is valid provided that the string fluid obeys both the stationarity equation and a generalized current conservation , or continuity , equation . while the worldsheet minimal surfaces of this situation correspond to zero mean curvature form @xmath345 , switching on a non - constant pressure @xmath346 means prescribing the local mean curvature by the projection of the gradient field @xmath347 into the surfaces non - zero  @xmath333 ( even a constant one ) also deforms the continuity equation . the term proportional to  @xmath333 hence represents a certain class of interactions between the strings that form the string fluid . finally , we remark that constant @xmath348 ( with @xmath349 ) generates precisely the same terms that one would obtain by adding an area metric cosmological constant to the gravitational action @xmath350 , in perfect analogy with the metric case . we now approach our second aim , and part , of this section : the covariant rewriting of the most general energy momentum tensor consistent with area metric cosmology , see the preceding section , in terms of a string fluid . recall that area metric cosmologies are of the almost metric type , featuring the flrw metric @xmath14 and an additional time - dependent scalar @xmath148 . string fluid matter in cosmology is tightly constrained ; consistency with isotropy forbids a one - component string fluid , since the local string worldsheets then would distinguish a preferred spatial direction . we therefore must resort to a string fluid with three components @xmath351 , with three @xmath14-spacelike vectors  @xmath352 that may isotropically fill out the spatial sections in a four - dimensional cosmology . due to the distinguished cosmological time variable , @xmath315 , the fluid energy momentum tensor ( [ fluidenergy ] ) has components proportional to the @xmath353 , @xmath354 and @xmath355 projections of the area metric , respectively . these can be defined covariantly using the three - component string fluid .    using the decomposition ( [ gdec ] ) we first define a projection ( which is an endomorphism on  @xmath25 ) to the purely spatial indices  @xmath356 $ ] by writing @xmath357}_{cd}+q^{ab}{}_{cd}\,,\qquad q^{ab}{}_{cd}=\sum_i \omega^{ab}_i g^c_{cdi_1i_2}\omega^{i_1i_2}_i\,.\ ] ] to guarantee the right - action of this operator is a projection we need to satisfy @xmath358 , which is easily achieved by the requirement @xmath359 which relates the three components of the string fluid . for later convenience we solve this by considering @xmath360 and @xmath361 vielbeins of the three - dimensional metric @xmath362 : then we have the useful relation @xmath363 . the second projection we need is the one to the mixed indices @xmath364 $ ] . it is simply given by the right - action of @xmath365 for which of course @xmath366 . there is one further way to covariantize the purely spatial components  @xmath367 of the area metric . consider the operator @xmath368 since the @xmath369 all contain the preferred time vector @xmath370 , we have @xmath371 , and hence @xmath372 . so @xmath373 is not a projection ; it rather maps tensors with mixed indices @xmath364 $ ] to tensors with purely spatial indices . in terms of the three - component string fluid we hence rewrite the fluid energy momentum as a sum of projections of the area metric to its different components , weighted each by time - dependent functions : @xmath374.\end{aligned}\ ] ] we do not display two further non - vanishing terms involving @xmath375 and @xmath376 since they simply result in shifts of the functions @xmath377 and @xmath378 . making use of the schematic relation @xmath379=-g^4(\cdot,\cdot)$ ] which follows from the fact that we have chosen the @xmath380 as vielbeins of @xmath362 , we may rewrite the energy momentum tensor as @xmath381 note that we have four functions @xmath382 here , although three functions @xmath383 as in ( [ fluidenergy ] ) should be sufficient . without loss of generality we can in fact fix @xmath384 . if we do so , and calculate the components @xmath385 , @xmath386 and @xmath387 , comparison to ( [ fluidenergy ] ) gives the invertible relation @xmath388    to conclude , the three - component string fluid tensor with @xmath384 is a covariant way to rewrite the general source tensor for all types of matter averaged over large scales in an area metric cosmology as @xmath389 the energy momentum conservation equation follows from ( [ fleq ] ) and ( [ invrel ] ) , using for later convenience the redefinition @xmath390 , and the hubble function , as @xmath391    a special case of the general expression ( [ gensfluid ] ) is the string fluid representing the sum of three components of non - interacting strings , see ( [ sfluid ] ) , which we will call string dust . from our previous discussion we read off the characteristic relations @xmath392 we will use string dust in our discussion of the late universe , where interactions presumably are neglibible . as a first application of the dynamical area metric manifold theory developed so far , we derive in this section the equations of area metric cosmology , with the aim to compare our theory to einstein s general relativity .    from our discussion of killing symmetries we know that cosmological area metrics are of the almost metric type , fully characterized by just two fields : the flrw metric @xmath14 and a time - dependent scalar field @xmath148 . thus we can use the simplified equations of motion  ( [ vacuumgg ] ) discussed in section [ solar ] . these are of the general form @xmath393 we perform a time space split by writing @xmath315 , where greek indices denote spatial coordinates . the characteristic property of the flrw metric , which obeys @xmath394 , also implies @xmath395 . employing the definition @xmath396 , and similarly for the spatial components of the cosmological ricci tensor , @xmath397 , we explicitly obtain the following expressions for the non - vanishing components @xmath398 and @xmath399 :    [ eqgrav ] @xmath400    where @xmath318 is used . the expression for  @xmath282 is given by equation ( [ phiva ] ) . we now consider the matter contribution to the gravitational equations of motion . as argued above the most general form of matter in area metric cosmology is given by a string fluid energy momentum tensor of the form ( [ gensfluid ] ) . from the timelike normalizations @xmath401 we immediately find the string fluid contribution to the scalar equation which simply reads @xmath402 the four - tensor contribution @xmath403 is more involved . since @xmath404 here , this projection to an algebraic curvature tensor is @xmath405 where the indices on @xmath369 are lowered with the metric @xmath14 . importantly there are no non - vanishing components @xmath406 . this is guaranteed by our construction and again shows consistency with the area metric cosmology background . we proceed to calculate the non - vanishing components of @xmath281 , which again uses the convenient form of the @xmath407 as vielbeins of @xmath362 . the results are @xmath408 it is easy to see that @xmath281 is induced by a symmetric two - tensor in the same way as is  @xmath280 in  ( [ factorized ] ) . we now extract the components of @xmath296 ; writing @xmath409 this gives @xmath410 according to ( [ eqform ] ) , we finally combine our results for the gravitational contributions ( [ eqgrav ] ) and our results for the matter contributions ( [ eqmatt ] ) to the general equations of area metric cosmology coupled to any type of matter . the latter is described in a parametrization by a three - component string fluid . we find the simple @xmath148-equation @xmath411 calculating the trace of @xmath412 and using ( [ req ] ) , we deduce a standard scalar field equation for @xmath298 , @xmath413 with potential @xmath414    finally , using both ( [ req ] ) and ( [ phieq ] ) to simplify ( [ eqform ] ) , we explicitly obtain the time - time and spatial components of the ricci tensor    [ graveqs ] @xmath415    we may now easily compare our equations to those of einstein gravity , i.e. , to @xmath416 , coupled to a perfect fluid with standard energy momentum @xmath417 . performing the time space split these equations are @xmath418 thus our string fluid gives rise to a perfect fluid with equation of state parameter @xmath419    before we analyze this effective behaviour further we have to perform a final consistency check . since the string fluid energy momentum four - tensor was not derived , by variation with respect to the inverse area metric , from an action , we have to ensure that the energy momentum conservation equation ( [ constring ] ) is satisfied . only then is the coupling of the string fluid to the gravitational equations consistent . to see why this is true , we start from the time - component of the contracted bianchi identity @xmath420 which is the standard fluid equation of motion @xmath421 . in our string fluid variables we thus have @xmath422 we expand the derivative of the term @xmath423 ; then we replace @xmath424 by using the identity @xmath425 which holds for any flrw cosmology , and @xmath426 by employing the equation of motion of @xmath298 . up to a sign , this precisely yields the energy momentum conservation condition ( [ constring ] ) . our cosmological three - component string fluid thus is consistent without any further restrictions on the functions @xmath427 and @xmath377 . let us turn to a closer inspection of the effective equation of state parameter @xmath428 . we obtain the following results which are generic for any string fluid , and thus for any type of matter in area metric cosmology ( in string fluid parametrization ) . in the limit @xmath429 we obtain the parameter @xmath430 of an effective radiation fluid . note that this limit is exactly realized by the vacuum which , in particular , obeys @xmath431 . thus the vacuum cosmology in area metric geometry is equivalent to einstein cosmology filled with a radiation fluid .    in the limit @xmath432 we obtain the parameter @xmath433 describing a universe with zero acceleration @xmath434 . consider now the condition @xmath435 for an accelerating universe ; it is satisfied if either @xmath436 or @xmath437 . for values of @xmath438 close to the simple pole in @xmath428 at @xmath439 , we may obtain any value of @xmath428 , both positive or negative . so string fluids in principle should be able to describe any physical universe . moreover , the diverging effective equation of state parameter @xmath440 close to the pole is a temptation to speculate on a possible explanation for the inflationary phase of the universe . whether this can be realized , however , requires a careful analysis and solutions of the equations .    in the next section we will explicitly analyze the important case of the late universe , which is characterized by non - interacting matter . we will show that area metric cosmology provides a natural explanation for the observed acceleration of our universe . in this section we exactly solve the equations of area metric cosmology filled with non - interacting matter , which we describe by string dust . this is the simplest scenario to which we can apply our theory . moreover , it should describe the late universe , where matter on average has spread out so much that interactions are no longer important . the defining relations for string dust have been discussed in ( [ sdust ] ) . we employ these and collect the complete set of equations of motion . the scalar equation of motion follows from  ( [ potphi ] ) as @xmath441 we also have the gravitational equations ( [ graveqs ] ) which , in the string dust case , read    [ gequ ] @xmath442    in the non - vacuum case where @xmath443 , an equivalent system is obtained by replacing one of the three equations above by the simpler continuity equation which follows from ( [ constring ] ) , i.e. , @xmath444 and we will , in fact , replace the scalar equation of motion .    dividing the first of the gravitational equations ( [ gequ ] ) by @xmath3 we deduce @xmath445 for constant @xmath446 . so @xmath298 is determined by the scale factor . the continuity equation is easily integrated , and yields @xmath332 in terms of the scale factor and a constant @xmath447 . explicitly we find    @xmath448    the remaining equation we have to solve in order to obtain the string dust solution follows from the second gravitational equation in ( [ gequ ] ) : @xmath449 we are interested in the effective equation of state parameter for the perfect fluid that is created by area metric cosmology in addition to einstein cosmology . hence we rewrite the defining relations for the parameters @xmath450 and @xmath438 in ( [ statepar ] ) using the above equations . this yields @xmath451 for @xmath452 . hence the definition of the parameter @xmath428 results in @xmath453 we conclude that once we have a solution of ( [ aeq ] ) for the scale factor @xmath454 , the whole system of equations is integrated , and provides @xmath332 , @xmath298 and @xmath428 . equation ( [ aeq ] ) is exactly solved by the scale factor @xmath455 for integration constants @xmath456 and @xmath457 . the parameter @xmath269 is defined as above . for the sake of completeness , we will now discuss three possible cases @xmath458 , @xmath459 and @xmath460 , assuming @xmath461 ( the discussion for @xmath462 is very similar but does not contain consistent @xmath463 cosmologies ) . however , it is evident that the only relevant case for our model of late time cosmology is @xmath460 , since @xmath458 requires fine - tuning , and @xmath459 evolves towards a contracting universe , against the assumed predominance of the string dust component .    [ [ i - case - xi0 . ] ] ( i ) case @xmath458 . + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    this case with @xmath464 can only be realized by positively curved @xmath465 cosmologies ; this fixes @xmath447 which is not generic . we also need positive @xmath466 and @xmath467 for a real solution . we find @xmath468 so that the effective equation of state parameter follows from ( [ parst ] ) as @xmath469 it is not difficult to show that @xmath470 for all @xmath471 . in the limit @xmath472 we find @xmath473 . consistency requires @xmath474 , see the discussion for ( [ constraint ] ) . since @xmath475 one can show consistency for @xmath476 and large @xmath477 . so this case of a string dust filled area metric cosmology describes an open , eternally decelerating , universe with initial singularity , for which the acceleration tends to zero for late times , see figure [ figure1 ] . [ [ ii - case - xi0 . ] ] ( ii ) case @xmath459 . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    the solution for the scale factor holds for positive @xmath466 and is the upper half of an ellipsoid , defined for @xmath478 . we deduce @xmath479 , but this leads to an inconsistency . for late times , i.e. , for @xmath480 , the derivative @xmath481 diverges , and hence @xmath298 so that @xmath474 can not be satisfied . therefore this closed universe is not a valid solution .    [ [ iii - case - xi0 . ] ] ( iii ) case @xmath460 . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    this case will turn out to be the most interesting . it has @xmath482 , and so it can be realized by negatively curved @xmath483 and flat @xmath463 cosmologies without further restrictions . it can also be realized by @xmath465 cosmologies , but this requires a matter density @xmath484 . again @xmath479 so that the effective equation of state parameter takes the form @xmath485 but here , for @xmath460 , we have to discuss two subcases depending on the sign of the integration constant @xmath456 . if @xmath466 then the solution is defined for @xmath486 . if @xmath487 then the solution is defined for all @xmath196 . for both signs of @xmath456 we obtain the late time limit of @xmath473 . we can also check consistency for late times @xmath488 . if @xmath489 this limit ensures @xmath490 . the solution with @xmath466 corresponds to an open , eternally decelerating , universe with initial singularity , for which the acceleration tends to zero for late times . the solution for @xmath487 describes an open , eternally _ accelerating _ , universe without singularities that passes a minimal radius @xmath491 and has a late - time acceleration tending to zero , see figure [ figure1 ] . it is worthwhile to note that the sign of @xmath456 is , in principle , deducible from evaluation of the hubble parameter and of the density @xmath332 at present time , since @xmath492 . in the case of a flat universe , for example , acceleration ( @xmath487 ) is obtained if @xmath493 ( note , however , that the precise value of @xmath265 has still to be fixed , e.g. , from comparison with solar system experiments . )    _ the solid curves in this sketch show the area metric cosmologies filled with string dust . from bottom to top these are the cases @xmath458 , @xmath460 with @xmath466 , and @xmath459 with @xmath487 . the dashed curve depicts the inconsistent case @xmath459 . the top curve is an important new result from area geometry ( and can not be obtained as a dust - filled einstein cosmology ) : late - time acceleration that tends to zero for @xmath472._,width=288 ]    we conclude that non - interacting string dust matter in area metric cosmology is equivalent to einstein cosmology filled with a perfect fluid , but , importantly , it provides a more interesting phenomenology than does an einstein universe filled with standard pressure - free dust . first , area metric cosmology predicts an open late universe , since the closed cosmology is inconsistent . second , and most strikingly , area metric cosmology may naturally explain , for all @xmath463 and @xmath494 cosmologies , the late - time acceleration of the universe ! this explanation neither invokes fundamentally unexplained concepts such as a cosmological constant or other forms of dark energy , nor does it invoke fine - tuning since the acceleration automatically tends to small values at late time ; these results become consequences of area geometry applied to the very simplest scenario , cosmology . this paper is based on the hypothesis that spacetime is an area metric manifold , which presents a true generalization of metric geometry in dimensions greater than three . by constructing a rigid extension of general relativity to area metric geometry , which can explain the acceleration of the late universe without additional assumptions , we have shown that this idea can be taken to surprisingly interesting conclusions . the physically immediately relevant case of a four - dimensional area metric manifold is distinguished , since we have shown that the area metric geometry on three - dimensional submanifolds is equivalent to some metric geometry . this reconciles the idea of an area metric spacetime with the phenomenological fact that we can measure lengths and angles in our individual spatial sections . paying tribute to the phenomenal success of the metric description of spacetime in general relativity , we have constructed curvature invariants that are downward compatible to their metric counterparts . crucial for this achievement was the extraction of an effective spacetime metric from the fundamental area metric data . the availability of an area metric curvature scalar , which reduces to the metric ricci tensor precisely for a metric - induced area metric , in particular allows us to read the einstein - hilbert action as dynamics for an area metric . remarkably , the equations of motion , derived by variation with respect to the area metric , are of second differential order . this circumvention of lovelock s theorem @xcite ( which for metric manifolds asserts that standard general relativity is the only geometric gravity theory with second order equations ) underlines one remarkable aspect of this modification of general relativity . another distinguished feature is that the area metric extension of einstein gravity is rigid , in the sense that it does not use an undetermined deformation ( length ) scale to add derivative or curvature corrections to the einstein - hilbert action , see e.g. @xcite , nor does it simply add additional fields propagating on a given background metric , as do scalar tensor theories , see e.g. @xcite . we saw that the theory is downward compatible to general relativity in vacuo : every vacuum solution of einstein s equations also solves the vacuum equations of area metric gravity . upon the inclusion of matter , however , solutions of einstein - hilbert gravity no longer induce solutions of area metric gravity . a case in point is the area metric einstein - maxwell theory , which is only consistent for true area metric backgrounds . in other words , not only can electromagnetic fields propagate on an area metric background , but their backreaction requires a deviation from a purely metric spacetime . the simplest departure from standard metric theory occurs for area metrics of almost metric type , which are determined by a metric and a single scalar . we have shown that the vacuum equations of area metric gravity for this situation are causally well - behaved ; conformal transformation and field redefinition demonstrates their equivalence to einstein gravity coupled to a free scalar field . it should be noted , however , that the equations arising from area metric gravity have to be interpreted without these transformations , since they are not available in the generic , not almost metric , case . the scalar field in the almost metric setting therefore must not be regarded as a field on a metric background , but is itself part of the area metric geometry . the consequences of this mechanism become most apparent in cosmology , which is of the almost metric type , where the scalar in fact behaves as an effective radiation fluid , unlike a standard scalar field . an area metric structure of spacetime implies that strings , not point particles , are the minimal mechanical objects one may discuss @xcite . an important consequence of this fact is that the notion of a fluid in cosmology has to be built on an integrable distribution of local worldsheets , rather than on a velocity field for worldlines . we have discussed string fluids arising in this way , and shown that they covariantly describe any collection of matter fields consistent with area metric cosmology . so string fluids can be seen as matter averaged over large scales . we find a remarkable correspondence between , on the one hand , area metric cosmology coupled to the most general consistent string fluid , and , on the other hand , einstein cosmology coupled to a standard perfect fluid . the energy density and pressure of this effective perfect fluid are given in terms of four functions : the scalar and three parameters of the string fluid . this implies an enormous freedom in the effective equation of state parameter which may , in principle , take any real value .    for the late universe one may assume that matter has spread out so much that interactions are no longer important . we have derived exact solutions for area metric cosmology filled with string dust , i.e. , with non - interacting string fluids . while they show some similarity to the solutions of a dust - filled einstein universe , a remarkable feature emerges that can not be realized in general relativity : independent of the spatial curvature , area metric cosmology explains the observed small late - time acceleration of our universe @xcite . this effect is a consequence of area geometry ; it does not invoke a cosmological constant nor additional quintessence fields , see e.g. @xcite . neither does this result depend on fine - tuning , since one of the nice features is that the acceleration automatically tends to zero with time . so the idea of spacetime as an area metric manifold can be taken very far indeed , has a common conceptual basis in the structure of string and gauge theory , and provides a very encouraging first cosmological prediction , too . but , as can be expected from any novel theoretical framework , a number of open ends remains to be addressed . the most pressing one from a phenomenological point of view is whether area metric gravity passes solar system tests . this question is certainly solvable , but complicated by the fact that a spherically symmetric area metric is generically not almost metric , but contains one further scalar degree of freedom . while we have shown that every vacuum solution of einstein gravity , i.e. , in particular the schwarzschild solution , induces a vacuum solution of area metric gravity , it is not clear whether this vacuum solution consistently extends to a solution with matter  having learnt that the inclusion of matter may require deviations from metric geometry . in the metric case life is simpler because of birkhoff s theorem . another such physical issue is the discussion of fermions , which might be introduced by their dirac algebra with respect to the effective metric .    on the mathematical side , a precise understanding of the degrees of freedom united in the area metric is needed . this includes in particular the connection to string theory , where true area metrics emerge as combinations of a metric and a two - form potential . this problem is likely related to the question of possible relations between area metric geometry and generalized geometries emerging in string theory , such as generalized complex geometry  @xcite or t - folds  @xcite . if one ventures to consider an area metric spacetime structure as a geometrization of the string idea , which we feel is structurally quite natural , one ultimately has to find out how to quantize strings on an area metric background ; given that the trace anomaly in quantized string theory is determined essentially by the two - dimensional worldsheet curvature @xcite and that the area metric curvature vanishes in two - dimensions , one may for instance speculate that a successful string quantization based on area metric geometry modifies the standard no - ghost theorems @xcite . a detailed investigation of these issues is of course needed , and remains the subject of future work . in this appendix we lay down our conventions for the coordinate representations of objects and operations . let @xmath16 be a @xmath0-dimensional smooth manifold and @xmath495 a basis of @xmath20 . this basis canonically induces the basis @xmath496 of @xmath497 which is of dimension @xmath498 . any section @xmath57 of @xmath497 can be expanded in this basis in the following form , @xmath499 which defines the components of @xmath57 . note that we adhere to the convention that sums over numbered indices @xmath500 , _ only _ on totally _ anti_-symmetric objects , are ordered sums over @xmath501 . as an immediate consequence of this convention we find the components of exterior products of sections @xmath502 and @xmath503 : @xmath504}\,.\ ] ] a basis @xmath505 of @xmath83 induces the canonical basis @xmath506 of @xmath507 which is dual to ( [ basis1 ] ) if @xmath505 is dual to @xmath495 , in which case @xmath508}_{b_k]}\,,\ ] ] where the right hand side are precisely the components of the identity on @xmath497 . we also define contraction symbols for @xmath509 and @xmath510 :    @xmath511    for k - forms @xmath512 we define the exterior derivative such that @xmath513}\epsilon^{a_1}\wedge\dots\wedge\epsilon^{a_{k+1}}\,.\ ] ] this appendix contains full technical detail of the derivation of the equations of motion of area metric gravity in four dimensions . these are obtained from variation , with respect to the inverse area metric , of the action @xmath350 proposed in ( [ totalaction ] ) in section [ gravtheory ] . this action contains , through the area metric curvature tensor ( [ areacurv ] ) , covariant derivatives of the @xmath157-tensors , which we remove by partial integration . for this purpose note that covariant derivatives involving the tangent bundle connection @xmath215 integrated over the area manifold are no longer surface terms , since the area metric is not covariantly constant under this connection . instead the following formula holds for partial integration : @xmath514 where @xmath515 . using this result we obtain the four - dimensional action in the alternative form @xmath516 which is classically equivalent to the original action on manifolds without boundary . as a shorthand we have used the definition @xmath517 in contracting the @xmath157-tensors , where deltas with multiple indices simply denote products of single deltas , e.g. @xmath518 . we proceed to vary the action ( [ varyac ] ) with respect to the area metric , which is determined by its decomposition into the irreducible algebraic curvature tensor @xmath28 and the scalar @xmath148 as in ( [ scheme ] ) . the calculation is not difficult , but rather long , and will be presented in several steps . we begin with a variation with respect to the effective metric @xmath146 , which of course still depends both on @xmath28 , via the fresnel tensor , and on @xmath148 via the chosen natural section @xmath142 of the cotangent bundle . the dependence on the effective metric in all covariant derivatives @xmath215 , also in the tensors @xmath157 , has to be taken account . variation of the levi - civita connection and of the corresponding riemann curvature tensor with respect to a given metric are standard . we explicitly provide these variations with respect to the inverse metric . then , for connection and curvature    [ h1 ] @xmath519    where we use another notational shorthand , @xmath520 the same remark as above applies for deltas with multiple indices . the variation of the @xmath157-tensor with respect to the effective metric follows as @xmath521}_{t|j]}\right)\nabla^{lc}_x\delta g_g^{\alpha\beta}\,.\label{h3}\ ] ] employing these relations and performing all necessary partial integrations one hence obtains the expression @xmath522}_{t|j ] } \right)\right ) \nonumber\\ & & { } - \frac{1}{6}\left(\delta^{xw}_{\alpha\beta}-g_g^{xw}g_{g\,\alpha\beta}\right)\left(\nabla^{lc}_xx_w+\frac{1}{6}x_xx_w\right).\end{aligned}\ ] ]    our second step is the variation of the action with respect to all explicitly occurring area metrics ; these appear in the volume form , and also in the @xmath157-tensors . explicitly we have    @xmath523    application of these two identities finally yields the expression @xmath524    in order to obtain the full variation of the action , we last but not least have to consider the dependence of the effective metric and of the area metric itself on the irreducible parts  @xmath28 and  @xmath148 of the area metric , and then to combine all terms . for the area metric we find @xmath525 the expressions for the effective metric are more elaborate . in our mainly plus signature convention these are given by @xmath526 and @xmath527 where the characteristic function @xmath92 evaluated on the section @xmath142 , see ( [ character ] ) , appears . we also use the abbreviations @xmath528 , @xmath529 and @xmath530 . the final result for the combined terms resulting from the total variation of the action @xmath531 may now be written in the simple form @xmath532 note that in reaching this result another partial integration has to be performed on the terms arising from the variation @xmath533 . the final result for @xmath282 is then given by the expression @xmath534 the almost final result for @xmath280 is given by the following expression @xmath535 which , however , does not yet have the correct symmetry structure : @xmath536 to obtain the correct @xmath280 note that this term must have the symmetries of an algebraic curvature tensor since these symmetries are imposed by its contraction with @xmath537 in equation ( [ result ] ) . to correct procedure thus is to project onto this irreducible representation by removing the totally antisymmetric part : @xmath538 the positive sign between both terms is a consequence of the lorentzian signature . finally , we arrive at the vacuum gravity equations for four - dimensional area metric manifolds . vanishing variation @xmath539 with respect to the inverse area metric implies the following two equations , corresponding to its cyclic and totally antisymmetric parts , respectively : @xmath540 in section [ solar ] we have discussed almost metric area metrics @xmath291 , which are determined by a metric and scalar field , as those closest to standard lorentzian manifolds . for this rather simple case occurs a significant simplification of the four - dimensional equations of motion that have been derived in detail in the preceding section . we will now present this simplification in some more detail as a basis for our discussion of area metric cosmology in the main text .    essentially because of the simple form ( [ metricfresnel ] ) of the fresnel tensor in the almost metric case , we find that the inner bracket in the @xmath148-variation , see @xmath282 in ( [ vphi ] ) , is identically zero , and hence its derivative is . hence @xmath541 in vacuo this expression must vanish by equation ( [ vac ] ) , which simply requires that @xmath542 should be an algebraic curvature tensor .    a similarly drastic simplification occurs for the variation with respect to @xmath28 . we expand all terms that multiply @xmath543 in equation ( [ vcalmost ] ) , while at the same time imposing the algebraic curvature tensor symmetries on the indices @xmath355 . this results in @xmath544\beta}+\frac{1}{2}\delta^{(ab)}_{\alpha[\gamma}g_{g\,\delta]\beta}\,.\ ] ] the simplified form of the @xmath28-equation then follows as @xmath545\beta}+\frac{1}{2}v_{g_g\,[\alpha[\gamma}g_{g\,\delta]\beta]}\nonumber\\ & & { } + \frac{1}{12}\phi k^\phi g_{g\,\alpha[\gamma}g_{g\,\delta]\beta}+v_{g\,\alpha\beta\gamma\delta}+\frac{1}{24}k^\phi\omega_{\alpha\beta\gamma\delta}\,.\end{aligned}\ ] ] the antisymmetrization in the second term is supposed to act only on the index pairs @xmath356 $ ] and @xmath546 $ ] . this equation indeed has the symmetries of an algebraic curvature tensor ; this is ensured by the addition of the last term which removes the totally antisymmetric part of  @xmath542 . the inverse @xmath547 of @xmath151 has already been displayed in ( [ invlift ] ) . this expression immediately enables us to calculate the @xmath157-tensor as @xmath548}_{cd}\right).\ ] ] we now have all the information needed to further simplify the above expressions for @xmath282 and @xmath280 in the almost metric case . for the @xmath148-variation we simply find @xmath549 the @xmath28-variation @xmath280 which is a rank four tensor turns out to conform to a very particular structure . it is in fact induced from a symmetric rank two tensor @xmath293 as shown in equation  ( [ factorized ] ) , where @xmath550 our main application of these expressions in this paper is to area metric cosmology . + d.  n.  spergel _ et al . _ [ wmap collaboration ] , astrophys . j.  suppl . * 148 * ( 2003 ) 175 [ arxiv : astro - 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area metric manifolds emerge as effective classical backgrounds in quantum string theory and quantum gauge theory , and present a true generalization of metric geometry . here , we consider area metric manifolds in their own right , and develop in detail the foundations of area metric differential geometry . based on the construction of an area metric curvature scalar , which reduces in the metric - induced case to the ricci scalar , 
 we re - interpret the einstein - hilbert action as dynamics for an area metric spacetime . 
 in contrast to modifications of general relativity based on metric geometry , no continuous deformation scale needs to be introduced ; the extension to area geometry is purely structural and thus rigid . 
 we present an intriguing prediction of area metric gravity : without dark energy or fine - tuning , the late universe exhibits a small acceleration .