einstein s equations admit black hole solutions , and gravitational collapse in general relativity inevitably leads to the formation of these objects , as shown by the celebrated singularity theorems  .observational evidence for the existence of black holes is strong as well , both in the `` stellar - mass '' range (  ) , and in the `` supermassive '' range (  ) . in the stellar - mass range ,black - hole candidates in x - ray binaries are too heavy to be neutron stars or quark stars with reasonable equations of state  ; in the supermassive range , observations of stars around the black - hole candidate in the center of the milky way ( sgra ) show that this object is too compact and massive to be a dark cluster of low - luminosity bodies  or a fermion star  .also , near - infrared observations of sgra seem to provide circumstantial evidence that this object has an event horizon ( because if it had a surface , it would emit radiation , _ e.g. _thermally , and be more luminous than observed )  , thus disfavoring more exotic horizon - less alternatives such as boson stars or gravastars .similar observations also exist in low - mass x - ray binaries for stellar - mass black - hole candidates , which do not show the type - i bursts that would be expected from the interaction of the accreting gas with their surface , if they had one  .moreover , the current paradigms for quasars and active galactic nuclei , as well as for the formation of galaxies , also require that supermassive black holes be present in the center of almost every galaxy .in fact , stellar and gas dynamical evidence  and observations of the optical , ultraviolet ( uv ) and x - ray spectrum of active galactic nuclei  do indeed corroborate this assumption .the defining characteristic of a black hole is , as mentioned above , the presence of an event horizon . that is a causal boundary that separates ( or defines ) the interior and exterior of a black hole , and prevents any signal originating from the interior from reaching the exterior .the reason why spacetimes that posses an event horizon can exist in general relativity is rooted in the causal structure of the theory .the latter is inherited from special relativity , and confines signals , irrespective of their nature , to propagate within future - directed light cones .although event horizons depend in general on the global structure of the spacetime , in static , spherically symmetric spacetimes ( and in stationary , slowly rotating ones , which we will also consider here ) they are usually discussed in terms of the structure of light cones in a well - behaved coordinate system , i.e. they correspond to the location where these light cones `` tilt '' and cease allowing signals to propagate `` outwards '' .it is , therefore , clear that local lorentz invariance , being the symmetry underpinning special and general relativity , is central to the existence of black holes .it is then natural to wonder what happens to horizons and black holes if lorentz invariance is not an exact symmetry of nature .intuitively , the answer to this question seems to depend on the way local lorentz symmetry is violated , and on how this affects the causal structure of the theory .consider for instance a theory with a preferred frame , but where all excitations have linear dispersion relations .lorentz symmetry is then broken by the existence of a preferred frame , and certain modes may perhaps propagate faster than light in that frame , but all propagating modes have a finite speed. therefore , there are still `` propagation cones '' in such a lorentz - violating theory , and the causal structure remains qualitatively similar to general relativity .the modification that one may expect at the level of black holes is the presence of multiple horizons , i.e. modes propagating at different speeds would have different causal boundaries .however , there is no particular reason why dispersion relations should be linear once lorentz symmetry has been given up .consider instead dispersion relations [ mdsr ] ^2k^2+k^4 + , where is the frequency , the wave - number , a constant with appropriate dimensions , and the dispersion relation can potentially include higher powers of .perturbations with sufficiently large can then travel arbitrarily fast in the preferred frame. therefore , there are no propagation cones , and the causal structure is drastically different from special and general relativity , as the existence of a preferred frame and infinite - speed propagation imply the existence of hypersurfaces of simultaneity .the very notion of a black hole does not seem to fit in this framework .it is also worth mentioning that if one were to start with a theory with a higher - order dispersion relation , and then truncate it to order as a result of some low - momentum approximation , then the picture would drastically change into the one depicted above for a linear dispersion relation .so , one could say that excitations with sufficiently low momenta experience the existence of an event horizons , but more energetic ones would be able to escape them .alternatively , one could say that the existence of horizons is a low - energy artefact .this conclusion is obviously based on highly heuristic arguments and lacks a firm mathematical basis .it does , however , act as the motivation and starting point for a rigorous exploration of what happens to the notion of a black hole in gravitational theories that exhibit violations of lorentz invariance .this is indeed the purpose of this paper .we stress that this is not a purely academic question .as discussed above , the astrophysical evidence for the existence of black holes is rather convincing , and observations allow us to determine the characteristics of objects with ever - increasing accuracy .hence , black holes can be powerful laboratories for gravity , and in particular they can be used to constrain violations of local lorentz symmetry in the gravity sector .such constraints would be complementary to the very stringent constraints already existing for lorentz - symmetry violations in the matter sector  .quantifying the potential violations of symmetries that are considered as fundamental is an important goal in itself , but it requires a theoretical framework providing predictions to be confronted with observations .einstein - aether theory ( - theory ) has been quite successful at playing this role when it comes to lorentz - invariance violations in gravity . - theory is essentially general relativity coupled to a vector field ( the aether ) , which is constrained to have unit norm and be timelike .being of unit length , the aether can not vanish , and thus breaks boost invariance by defining ( locally ) a preferred frame .recently , additional motivation for lorentz - violating gravity theories has been put forward : hoava has suggested that giving up lorentz symmetry can lead to considerably better uv properties , and it appears possible to put together a power - counting renormalizable lorentz - violating theory of gravity  .this could be a pay - off significant enough to entertain the idea that lorentz symmetry might indeed be approximate , and not exact , in the gravity sector.  we will give a brief overview of both - theory and hoava gravity in section [ framework ] .in connection with what we discussed above , it is worth pointing out that dispersion relations are linear in the former , and higher - order in the latter .moreover , the low - energy limit of hoava gravity , which is a truncation of the theory at second order in derivatives , has been shown to be dynamically equivalent to - theory , provided that the aether is forced to be hypersurface - orthogonal at the level of the action .together , the two theories form an excellent theoretical framework for rigorously exploring the fate of black holes once local lorentz symmetry is violated . in section [ sphbh ]we indeed investigate static , spherically symmetric , asymptotically flat black holes in the two theories , following the lines of ref . , while in section [ slowly_rotating ] we move away from spherical symmetry and focus on slowly - rotating black holes ( partially following refs . but providing independent derivations of the results of those papers , and clarifying their physical interpretation ) .section [ cons ] contains our conclusions .  in this paper, we use a metric signature and units where .the action for - theory is [ s ] s_ =  ( -r -m^ _ _ u^_u^ ) d^4x where is the ricci scalar , the determinant of the metric , and m^ _ = c_1 g^g_+c_2^_^ _ + c_3 ^_^_+c_4 u^u^g _ , with being the covariant derivative associated to and being dimensionless coupling constants .also , is constrained to be a unit timelike vector , i.e. g_u^u^=1 .this can be explicitly imposed by adding the lagrange multiplier term to the action , or it can be taken into account implicitly during the variation .this constraint is a key feature of - theory : by virtue of it , the aether can never vanish , and always points to some timelike direction .so , in the locally flat coordinate system where becomes the minkowski metric , the aether breaks boost invariance , leading to violations of lorentz symmetry and defining a preferred frame .consider now a restricted version of - theory where the aether is always hypersurface - orthogonal .then , one can express the aether in terms of a scalar as [ ho ] u_= , where the unit constraint has already been taken into account . because the aether must be timelike, defines a time coordinate , and one can choose to foliate spacetime into constant- hypersurfaces . in this adapted foliation , one has [ preferred foliation ] u_=_^t ( g^tt)^-1/2=n_^t , where is the lapse function , and the action takes the form [ sbpsh ] s_h.o. = dt d^3x n ( k_ijk^ij - k^2 + ^(3)r + a_ia^i ) , where is the extrinsic curvature of the constant- hypersurfaces , is the induced spatial metric , its ricci curvature , and a_i=_i is the acceleration of the aether flow .we have also introduced new parameters which are given in terms of those appearing in action ( [ s ] ) as [ hlpar ] = = , = , = , where .only three coupling constants appear in action ( [ sbpsh ] ) , as opposed to the four in action ( [ s ] ) .this is because for a hypersurface - orthogonal aether field , any one of the , or terms can be expressed in terms of the other two without loss of generality .action ( [ sbpsh ] ) is what is referred to as the low - energy ( or infrared ) limit of hoava gravity . in its complete versionthe action of hoava gravity is [ sbpshfull ] s_hl= dt d^3x n(l_2+l_4+l_6 ) , where [ l2 ] l_2 = k_ijk^ij - k^2 + ^(3)r + a_ia^i , is a new mass scale , and and are respectively of fourth and sixth order in the spatial derivatives , but contain no derivatives with respect to .the presence of sixth - order terms in the spatial derivatives is crucial for power - counting renormalizability . a more precise prescription to construct and follows from the symmetries of the low - energy part of the action .action ( [ sbpsh ] ) is written in a preferred foliation of constant hypersurfaces , and the existence of this foliation is essential for the addition of the and terms .the action is invariant under diffeomorphisms that preserve this foliation , and , but not under arbitrary diffeomorphisms .therefore , and should contain ( respectively ) all forth and sixth - order terms that are compatible with this symmetry . in practice , this means that and can contain any 3-dimensional scalar constructed with , and their ( spatial ) derivatives .we have deliberately chosen to introduce hoava gravity through its relation with - theory for various reasons ( see also ref .first , the relation between the two theories will be extensively used in what follows .second , this choice shows that hoava gravity can be written in a covariant manner .third , it makes it straightforward to argue that the theory has an extra scalar degree of freedom ( since the transverse mode of the aether disappears due to the hypersurface - orthogonality condition ) , which becomes apparent and defines the preferred foliation .note that different versions of hoava gravity exist with extra symmetries , or different field content , but we will not consider them here ( see ref . for a brief review ) .the presence of terms of fourth and sixth order in the spatial derivatives leads to higher - order dispersion relations for the spin-2 graviton and the scalar degree of freedom , such as those discussed in the introduction . in - theory instead , all degrees of freedom have linear dispersion relations .the same property is shared by the low - energy limit of hoava gravity , but is an artefact of the truncation .thus , in combination , the two theories do indeed act as an excellent framework to address the issues raised in the introduction .the precise form of and will _ not _ be crucial for what follows .finding black hole solutions when those terms are included is very challenging from a computational and numerical point of view , so here we will rather focus on - theory and the low - energy part of hoava gravity .in particular , we will look for spacetimes with a metric horizon .this is a null hypersurface of the covariant 4-dimensional metric ( which couples minimally to the matter fields , as required by the weak equivalence principle ) , and acts as causal horizon for those fields . besides the metric horizon, these solutions are expected to possess different causal horizons for the propagating modes that reside in the gravity sector ( i.e. the spin-2 and spin-0 modes of hoava gravity , and the spin-2 , spin-1 and spin-0 modes of - theory ) , because these modes will generically travel at superluminal or subluminal speeds .these horizons will be null surfaces of the effective metrics where is the speed of the spin- mode ( see ref . for a detailed discussion ) . we will then proceed and discuss the effect the higher - order derivative terms and on the causal structure of these spacetimes , and discuss whether they really deserve to be called `` black holes '' .our analysis hinges on the assumption that is a sufficiently large mass scale , so that the presence of the and terms does not really affect the spacetime geometry in regions of interest for our analysis .this will be discussed in detail in the next section . is indeed bound from below from tests of lorentz symmetry . in particular , purely gravitational experiments impose the mild bound ev .test of lorentz symmetry in the matter sector can lead to constraints that are many orders of magnitude higher for , but which depend strongly on the details of the percolations of lorentz - symmetry breaking from the gravity to the matter sector . is also , perhaps more surprisingly , bound from above ( gev ) , from the requirement that the theory behave perturbatively at all scales , so that the power - counting renormalizability arguments put forth in ref . actually apply .we conclude this brief overview of - theory , hoava gravity and their mutual relation with a brief discussion of their field equations . varying the action ( [ s ] ) with respect to the metric and the aether , in the absence of matter fields and taking into account the unit constraint , yields [ gfe ] e _ & & - ( g _ - t^_)=0 , + [ aeeq ] _ & & ( _ j^-c_4_^u^ ) ( g_-u_u_)=0 where is the einstein tensor , [ tae ] t^_&=&_(j^_(u_)-j^_(u_)-j_()u^ ) + & & + c_1 + & & + u_u_+c_4 _ _ - l_ g _ , is the aether stress - energy tensor , and j^_=m^ _ _u^ _ = u^_u_. taking instead the variation with respect to and one obtains , again in the absence of matter fields and taking into account the unit constraint , e_+_(u_)=0,[einsteinae ] + _=0[aeeq2 ] .these equations are clearly equivalent to the system  and , and either set of equations represents the full field equations of - theory .( we will use both forms of the field equations in what follows ) .  imposing the constraint before the variation , and then varying the action ( [ s ] ) with respect to and yields [ hl1 ] & & e_+_(u_)=0 , + [ hleq ] & & _ ( ) = 0 , keeping the same notation as above .these are the equations of the low - energy limit of hoava gravity in covariant form .  based on the above , it is straightforward to see that solutions of - theory for which the aether is hypersurface - orthogonal will be solutions of the low - energy limit of hoava gravity as well , but the converse is not necessarily true .we also note that in ref . it was shown that in the low - energy limit of hoava gravity , equation  is actually implied by equation  thanks to the bianchi identity .this is something that we will use in section [ slowly_rotating ] .   the parameters define a 4-dimensional parameter space for - theory .similarly , , and define a 3-dimensional parameter space for low - energy hoava gravity . in this paper, we will only focus on the regions of these parameter spaces where the theories are dynamically well - behaved and observationally viable .the viable regions ( in light blue ) of the parameter space of - theory ( left ) and hoava gravity ( right ) .general relativity corresponds to the origin in both cases . for - theorythe viable region extends upwards in the vertical direction , and for hoava gravity both upwards and downwards in the vertical direction ., title="fig:",width=226 ]  the viable regions ( in light blue ) of the parameter space of - theory ( left ) and hoava gravity ( right ) .general relativity corresponds to the origin in both cases . for - theorythe viable region extends upwards in the vertical direction , and for hoava gravity both upwards and downwards in the vertical direction ., title="fig:",width=226 ]  particularly strong constraints follow from the requirement that the two theories reproduce the weak - field predictions of general relativity in the presence of matter ( i.e. when one adds matter fields minimally coupled to the metric ) .one can indeed perform a post - newtonian expansion , as in general relativity , and cast the results into the usual parametrized post - newtonian ( ppn ) framework  . in both theories ,all ppn parameters turn out to be identical to those of general relativity except two  , and .these parameters measure preferred - frame effects , and are experimentally constrained to very small values , and , in agreement with the predictions of general relativity , where . requiringthen that and vanish exactly , we obtain c_2=,c_4=- , for - theory and = 2 - 2 for hoava gravity , and we are therefore left with a two - dimensional parameter space for both theories . requiring vanishing preferred frame parameters might seem a strong requirement .however , since and are dimensionless , if one allows and to take values that are nonzero but satisfy the observational constraints , then the predictions of both theories will be close to those obtained for to within an accuracy , and it is therefore convenient to decrease the dimension of the parameter space by setting and exactly to zero .it is sensible to ask that all of the propagating modes of the two theories have positive energy and be free of gradient instabilities in flat space .additionally , there should be no modes propagating subluminally .if this were the case , matter could decay into these modes via a erenkov - like process , and this would lead to observable effects .these requirements impose further constraints onto the parameters of the two theories , identifying only specific regions of the aforementioned 2-dimensional surface as viable .details can be found in ref . . herewe limit ourselves to show a plot of the allowed part of the parameter space in figure [ parspace ] .the following redefinitions of parameters allow for a better visualisation : c_=c_1c_3 ; = /-1 , = ( -1)/ , we stress that there are further constraints on lorentz - violating gravity theories that could further restrict the viable parameter space .binary - pulsar observations have long been known to carry the potential for constraining the couplings to regions near the origin in the and planes  , but calculations have become sufficiently accurate for imposing precise constraints only very recently  .similarly , cosmological observations of the cosmic microwave background and the matter power spectrum constrain the couplings to values of order unity or less  .therefore , we focus here on the regions of the parameter spaces where and , since larger values of the couplings are unlikely to be compatible with these observations .in this section , we briefly review the derivation and main physical properties of spherically symmetric , static and asymptotically flat black holes in both low - energy hoava gravity ( referred to simply as hoava gravity from now on ) and - theory . for a more detailed discussion , see ref . , where these solutions were derived .   first , let us note that spherically symmetric , static and asymptotically flat black holes are actually exactly the same in the two theories  .this fact is not obvious .a spherically symmetric , static aether vector field is always hypersurface - orthogonal and therefore , as discussed at the end of section [ aeandhl ] , such solutions of - theory will also be solutions of hoava gravity , but the converse is not generically true .however , it can be shown to hold for spherically symmetric , static and asymptotically flat solutions . inserting the spherically symmetric and static anstze and ( with satisfying the unit constraint ) into equation , one straightforwardly obtains that identically , while implies .so , in order to show that any solution of hoava gravity is also a solution of - theory , it is sufficient to prove that equation  implies .this can be seen by integrating equation  ( which contains only and derivatives because of the spherically symmetric static ansatz ) between two arbitrary radii and .gauss s divergence theorem then gives one can then note that asymptotic flatness requires , and , and thus , so sending one obtains at the ( arbitrary ) radius .therefore , the static , spherically symmetric and asymptotically flat black - hole solutions are indeed the same in the two theories , and this allows us to focus on - theory only ( because the solutions that we will find will also be all of the solutions of hoava gravity ) . in order to actually find these black - hole solutions , it is convenient to use ingoing eddington - finkelstein coordinates , which are regular at the horizon . in these coordinates ,the line element and the aether field take the form [ efmetric ] ds^2=f(r)dv^2 - 2 b(r)dv dr - r^2d^2 , + [ efaether]=a(r ) _ v-_r .imposing asymptotic flatness , and have the following behavior  \frac{1}{\chi}\nn\\&&\label{asya}\end{aligned}\ ] ] where and are constants specifying the solution , and is a gauge parameter that can be set to 1 by rescaling the coordinate . is a measure of the `` characteristic size '' of the solution as measured at infinity , i.e. one can define a `` gravitational radius '' .this is in turn proportional to the total mass measured by a distant observer via , where is the gravitational constant measured by newtonian experiments ( related to the `` bare '' gravitational constant appearing in the action by  ] ; , , , are complicated expressions involving the couplings as well as , , and their derivatives , and the coefficients and are functions of radius only ( involving again the couplings , as well as , , and their derivatives ) .again , it is easy to show that , , , must evaluate to zero once the background spherically symmetric static solution is used .in fact , by transforming that solution under a coordinate change ( with being a constant ) , one obtains and . replacing in equations  andwe obtain that and must evaluate to zero . likewise , transforming the spherical solution under [ being a constant ] , we obtain and . replacing this in equation  andwe conclude that and must also evaluate to zero ., , , evaluate to zero using the solutions of ref . . ]  combining equations  and to eliminate the terms depending on the -derivatives , one obtains with \nonumber\\&&-\left(c_1+c_3 - 1\right ) s_1 ^ 2 a(r ) \left[4 b(r)-r b'(r)\right]\big\}\,,\end{aligned}\ ] ] where  ] .  the equations that we need to solve are , therefore and while this system of equations may be integrated numerically starting from the metric horizon of the background solution and imposing asymptotic flatness at infinity using a shooting method ( c.f .the analysis in ref . ) , a thorough study of these solutions , their geometry and physical observables in the viable region of the theory s parameters space is beyond the scope of this paper , and will be presented elsewhere  . here , we limit ourselves to using these equations to show that slowly rotating - theory black holes are necessarily different than hoava gravity ones , because they present no global preferred - time slicing , and therefore no universal horizon in the sense defined previously .  to show this ,let us first note that hypersurface - orthogonality of the aether implies vanishing vorticity vector . using this and the requirement that the aether be stationary and axisymmetric ( hence ) ,one obtains that it must be , where is a constant ( of order , due to the slow - rotation assumption ) . as noted in ref . , a coordinate change can then be shown to transform to zero , without affecting the metric ansatz . therefore ,if one imposes hypersurface - orthogonality for the aether , it is not restrictive to set in equations . one can then combine these equations into a first - order equation for , namely =0\ , .\end{aligned}\ ] ] because the combination can be checked to be non - zero using the background solution of ref . defined in equation ( 21 ) of ref . , which was shown to be non - zero there . ], it follows that constant .however , as noted in ref . , this is simply the background static spherically symmetric solution under a coordinate change , so no slowly rotating black - hole solutions exist in - theory if the aether is required to be hypersurface - orthogonal .therefore , no universal horizons seem to exist in - theory s slowly rotating black holes , at least in the sense defined in section [ uhor ] , because no global preferred - time foliation exists in the first place .   as discussed in section [ framework ] , the einstein equations for hoava gravity coincide with the einstein equations for - theory , provided that those are derived by varying and keeping fixed and that the aether is hypersurface - orthogonal  ( c.f .equations and ) .as we also stressed , the bianchi identity implies  that the hoava gravity einstein equations also include equation ( which follows from a variation of the action with respect to ) , so one does not need to impose equation explicitly when solving equations .  the only two independent fields equations at order are then given by the and components of the einstein equations .the former is given by equation , and proceeding as in - theory , it implies .the latter is then given by equation , which because of the hypersurface orthogonality condition ( which as discussed above implies ) becomes \right\}=0\ ] ] ( where we have used the fact , proved in section [ spin_ae ] , that and are zero once the spherically symmetric static solution is used to evaluate them ) .the solution to this equation then reads where is the location of the metric horizon , while and are integration constants .in particular , because asymptotic flatness imposes far from the black hole , from a comparison with the kerr solution of general relativity , it is clear that plays the role of the black - hole spin .also , it should be noted that can be eliminated with a coordinate transformation .  clearly , because the aether vector field does not present any terms at order , the preferred - time foliation is also unaffected at that order , and the slowly rotating solutions have the same universal horizon as the static spherically symmetric black holes reviewed in section [ sphbh ] .we have reviewed spherically symmmetric static black - hole solutions  in - theory and in the infrared limit of hoava gravity . in order to focus on the black holes that are expected to exist in the real universe ( e.g. ones forming from gravitational collapse ), we impose that the geometry of our solutions be regular everywhere apart from the central singularity  . because the matter fields ( which couple minimally to the metric ) and the propagating modes of the gravity sector have different propagation speeds in these theories , these black - hole solutions present multiple horizons , i.e. a `` metric '' horizon for the matter fields , as well as spin-0 , spin-1 and spin-2 horizons for the various modes that reside in the gravity sector .we have compared the geometry of these black holes ( outside the metric horizon ) to that of the schwarzschild solution of general relativity , and we have discussed the prospects of constraining the differences with future electromagnetic and gravitational - wave probes .lorentz - violating gravity theories are generically expected to have propagating modes that can travel arbitrarily fast in the uv limit .this would make the very concept of event horizon ( and therefore of a black hole ) meaningless , because modes with sufficiently high energies would be allowed to escape the multiple horizons mentioned above and probe the central singularity . however , the examples we studied here demonstrate that , when there is a preferred foliation , a new kind of horizon can exist , and it acts as a causal boundary for all modes , even instantaneous ones .the nonlinear stability of universal horizon certainly requires further investigation .we have also investigated whether such universal horizons exist for slowly rotating black holes , and we have shown that they do indeed exist for hoava gravity , while they do not seem to be present in - theory  .the reason for that is that slowly rotating black holes in - theory do not posses a preferred foliation .we are indebted to ted jacobson for countless stimulating conversations and insightful suggestions .section [ sphbh ] was largely based on results obtain in ref . , of which he is a coauthor .we also thank ian vega for checking the algebra of the equations of section [ slowly_rotating ] in the context of a related project . e.b .acknowledges support from the european union s seventh framework programme ( fp7/people-2011-cig ) through the marie curie career integration grant galformbhs pcig11-ga-2012 - 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Abstract: lorentz - symmetry and the notion of light cones play a central role in the definition of horizons and the existence of black holes . current observations provide strong indications that astrophysical black holes do exist in nature . here we explore what happens to the notion of a black hole in gravity theories where local lorentz symmetry is violated , and discuss the relevant astrophysical implications . einstein - aether theory and hoava gravity are used as the theoretical background for addressing this question . we review earlier results about static , spherically symmetric black holes , which demonstrate that in lorentz - violating theories there can be a new type of horizon and , hence , a new notion of black hole . we also present both known and new results on slowly rotating black holes in these theories , which provide insights on how generic these new horizons are . finally , we discuss the differences between black holes in lorentz - violating theories and in general relativity , and assess to what extent they can be probed with present and future observations .


introductory - level electricity and magnetism ( e&m ) has been investigated since the early 80 s ( see references in ) . yet , research at the upper - division level is relatively new .recently , the university of colorado developed the colorado upper - division electrostatics ( cue ) diagnostic. one of the primary purposes of the cue is to `` serve as a comparative instrument to assess upper - division e&m courses . ''  the cue is designed in a pre / post format .the optional 20-minute pre - test contains 7 out of the 17 post cue questions which junior - level students might reasonably be expected to be able to solve based on their introductory course experience .the post - test is designed to be given at the end of the first upper - division semester in a single 50-minute lecture .the cue contains open - ended questions where students do not actually solve the given problems . instead , the instructions for the first half of the post - test are as follows :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ for each of the following , give a brief outline of the easiest method that you would use to solve the problem .methods used in this class include but are not limited to : direct integration , ampere s law , superposition , gauss law , method of images , separation of variables , and multipole expansion ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  , with an off - center spherical cavity carved out of it .find e ( or v ) at point p , a distance 4r from the sphere.'' ]  .grading rubric for cue question 5 .reproduced from the cue assessment grading rubric. [ cols= " < , < , < " , ]   the first thing to note is that of the relevant responses , only 3% of cu students used a potential approach ( 9% of correct answers ) , in contrast to 11% of osu students ( 33% of correct answers ) .even more striking is the difference in the explicit use of the word `` superposition . ''of all relevant answers ( combining a , b , c , d ) , of cu students explicitly used the term `` superposition , '' compared to the same percentage of osu students who did * not * use this term .this pattern is also evident in the correct responses ( a and b only ) . of all correct answers( combining a and b ) , of osu students * did not * explicitly use the term `` superposition , '' compared to 82% of cu students who * did * use this term .  in order to look more closely at the issue of_ what _ is being superposed , we also did a comparison without considering the use of the word `` superposition '' or distinguishing between electric field and potential approaches .these results are presented in fig .[ fig : q5data - all ] , which groups all correct categories ( a and b ) and all incorrect or ambiguous categories ( c and d ) .the overall results are comparable for both universities .it was surprising to us that in both schools only of all students took a clearly correct ( electric or potential field ) approach to this problem ( of relevant responses ) .if we look only at relevant answers , in almost of cases students were either unclear about what they wanted to add / superimpose or were clearly talking about adding charges .in most e&m i courses , students begin with the electric field approach , and only after they master this concept do they move on to the potential approach . as a consequence ,the electric field approach is an intuitive and natural way for those students to tackle problems .this might explain why so few cu students tried to solve this problem using potentials , compared to osu students , who are introduced to potential before electric field .this is a question we are currently exploring.  regarding the difference in explicit use of the word `` superposition '' , the cu course materials, which include lecture notes , clicker questions , tutorials , _ etc ._ , seem to strongly emphasize the term `` superposition . ''for example , in both the instructor and student manuals for the first tutorial , one can find the following note :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ a common hw and exam problem asks for the e - field caused by a charged disc .hint : a disc is the sum of many rings . did someone say `` superposition '' ? _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  this emphasis is not similarly apparent in the paradigms materials.  additionally , as one might expect at the institution that developed the cue , one can see the impact of this test in the reformed course materials , such as clicker questions that are similar to questions on the cue , in whole or in part .this interaction between the development of the course materials and the development of the assessment is not unexpected , but it is important to consider when extending the assessment beyond the original institution .two of the stated goals of the cue are to `` serve as a comparative instrument to assess upper - division e&m courses '' and to help `` education researchers to learn more about specific areas where upper - level students are struggling.'' using the question on the superposition principle as an example , we identify two ways in which the cue is not fulfilling these goals as well as it could using the current rubric .the emphasis in the rubric on the terminology of `` superposition '' and lack of inclusion of a potential approach represent a potential bias toward the traditional subject order and emphasis that obscures interesting differences between approaches .even more striking is the lack of information the current rubric provides about an important area where students seem to be struggling , independent of approach .both osu and cu have made major reforms designed to help students understand superposition , though in different ways . yet , despite these reforms , more than 40% at each school do not recognize the need for superposition on this problem and of those that do , almost 70% do not clearly identify what they are superposing .the categorization we propose here allows one to identify how a teaching scheme may impact responses and even more importantly , illuminates an important issue about student understanding of superposition that is obscured in the original rubric .we suggest that a similar reexamination of the rest of the cue might provide additional examples where insight on student understanding would broaden the impact of the cue as a tool for assessing upper - division e&m courses .the university of colorado is currently working on a multiple - choice version of the cue .this provides an excellent opportunity for just such a re - analysis .a recent interview with two osu undergraduates suggests that the issues raised in this paper are still of concern with the multiple - choice version of the cue .supported in part by nsf due 1023120 .we would like to thank steve pollock , rachel pepper and bethany wilcox for conversations about the design and grading of the cue and for sharing some of the cu test data .

Abstract: while introductory electricity and magnetism ( e&m ) has been investigated for decades , research at the upper - division is relatively new . the university of colorado has developed the colorado upper - division electrostatics ( cue ) diagnostic to test students understanding of the content of the first semester of an upper - division e&m course . while the questions on the cue cover many learning goals in an appropriate manner , we believe the rubric for the cue is particularly aligned to the topics and methods of teaching at the university of colorado . we suggest that changes to the rubric would allow for better assessment of a wider range of teaching schemes . as an example , we highlight one problem from the cue involving the superposition principle . using data from both oregon state university and the university of colorado , we discuss the limitations of the current rubric , compare results using a different analysis scheme , and discuss the implications for assessing students understanding .   address = department of physics , oregon state university , corvallis , or 97331  address = department of physics , oregon state university , corvallis , or 97331  address = department of physics , oregon state university , corvallis , or 97331


quantum information processing relies on the capacity of sustaining coherence as well as the entanglement among different parties . however , many systems proposed for implementing such protocols are naturally subjected to local and unavoidable dissipation such as ion traps , cavity qed systems , and atomic ensembles  . needless to say ,this usually works against their quantum efficiency through the mechanism of decoherence .different strategies have been designed to partially protect or restore this coherence , such as entanglement distillation  , quantum repeaters  or feedback  , even though all these proposals present combinations of different costs like the increase of the amount of resources to create the necessary redundancy or the need to perform multi - qubit , _ i.e. _ non - local , operations .  in this paperwe show a fully deterministic and _ local _ scheme that counteracts the unavoidable action of dissipation .it relies on three basic elements : the capacity to monitor local environments detecting whether one or none excitation has leaked into each local reservoir ; the possibility to rapidly feedback the excitation into the lossy qubit ; and the use of three - level systems to encode qubits .note that even though the used encoding implies some sort of redundancy , the scheme is much less demanding than the usual quantum error correction codes for dissipative systems  because it adds only one local extra level to each part and does not require any extra entanglement or global operation .we discuss applications of this idea to improve the efficiency of quantum repeater protocols and to produce longer lasting coherence and/or entanglement for quantum information storage , teleportation or swapping .we also show that it can represent a physical implementation of the optimum singlet conversion protocol  .let us consider a global system composed of internal subsystems that are weakly coupled to their own local reservoirs .these couplings should respect typical markov and born approximations and one should be able to read information from each environment regarding the emission of single excitations from the respective subsystem in a time scale much shorter than the subsystem decay rate and still much larger than the correlation times of its respective reservoir .we then show that , within the framework of the well known quantum trajectories technique  , the monitoring of the local environments plus the classical communication of the obtained results ( hence , the first element of the scheme ) is already enough to enhance the quantum communication efficiency .however , in the common case of qubits encoded in two - level systems , only the so - called no - jump trajectory , obtained when no environment is disturbed , is of use .in fact , the scheme starts to fail as soon as any single qubit decays , since its particular state is projected to the low energy level and factored out of the rest of the system . in order to solve this problem, we introduce the two remaining elements to show that encoding the qubits in three - level systems , combined with feedback on individual systems powers up the scheme to the limiting point of completely suppressing the action of the local environments only through local operations .to start with , let us take a look at how environment - monitoring can enhance a simple quantum teleportation scheme for which alice and bob need to share a pair of maximally entangled qubits .consider a initial bell state given by where we have considered as the local decay rates , and are the lowering operators for the qubits of alice and bob respectively , and the interaction picture is implied .as time goes by , entanglement is lost , which can be evidenced by the entanglement of formation of state  as shown in fig .[ fig1 ] . due to this process ,the further alice and bob take to perform the teleportation , the largest the amount of resources needed , since they would need more and more copies in order to distill a maximally entangled pair .let us now assume that alice and bob can monitor the environments of their respective qubits , _i.e. _ they can continuously detect if their qubits have lost one excitation .it can involve a direct measurement of the reservoir , for example , by detecting an emitted photon , or an indirect probing of the loss of excitation through an auxiliary system  . under these conditions ,the time evolution of the system is no longer described by the master equation itself but rather by its so - called quantum - jump unraveling , in which the no - jump operator ( corresponding to no detection in the monitored environment ) is given by  ] , where gives the probability density for the trajectory where a jump happens at ( with ) .the trajectory - averaged entanglement is then given by .furthermore , we can choose to maximize the one - jump trajectory and to maximize the entanglement of the state when the atom reaches bob , hence , creating an even longer lasting entanglement .[ superficie2 ] compares the protocols as a function of and showing the optimum range in which each strategy is better and also that for all times and initial states , encoding qubits in qutrits is more efficient in the presence of decay and environmental monitoring .[ superficie2]d shows entanglement for different temperatures of the reservoir .typical microwave cavity qed experiments correspond to the intermediate curve  .note that at such low temperatures , thermal excitations play a minor role in the proposed scheme .   for ;b. ( upper right ) one - jump contribution for ; c. ; d. with for different temperatures with the mean environment excitation number given by . decay time as in the experimental setup in .,title="fig:",width=321 ] +the schemes proposed before can be readily incorporated in some previously known communication protocols .one of the main ideas of transmitting quantum information in 1-d networks relies on quantum repeaters . in this case , one wants to transmit entanglement through large distances and uses intermediate stations to recover it from time to time through entanglement distillation .once some degree of entanglement is recovered in an intermediate station , this entanglement is teleported to the next station .naturally , since using qutrits allows to keep entanglement for longer times between each station , there is a reduction in the total amount of necessary resources .  in higher dimensional networks, it was recently shown that the geometry of the graph defining the quantum network plays an important role in the problem of long - distance communication .entanglement percolation ideas were first presented for pure - state based networks .the main building block for these strategies is the possibility of performing the optimal singlet conversion . as we discussed before ,this task can be achieved by reservoir monitoring .this fact in turn shows a strategy of realizing classical entanglement percolation in noisy lattices .to conclude , we have shown that for systems undergoing dissipation , encoding qubits in qutrits preserves coherence and entanglement for much longer times if the excitations given to the local reservoirs can be detected by external observers .thus , systems with their environments under continuous measurement are not only suitable but advantageous for some quantum information protocols .moreover , the monitoring scheme is further improved if the parties are able to locally feedback the recovered information into the system .this leads naturally to a reduction in resources for protocols such as quantum repeaters , teleportation , swapping and error correction , as well as an improvement in the coherence time of quantum memories .we have also shown a way of using the information leakage to the environment to perform the optimal singlet conversion , the central task in classical entanglement percolation .finally , note that experiments already observe quantum jumps in different systems , e.g , harmonic oscillators ( microwave cavity fields )  , and in single ions  , which means that the scheme here proposed is clearly within nowadays technology .the authors thank a. acn and a.r.r .carvalho for enlightening discussions .support from brazilian agencies cnpq and fapemig and the european project qap is warmly acknowledged .this work is part of brazilian national institute of science and technology on quantum information .

Abstract: we study how to protect quantum information in quantum systems subjected to local dissipation . we show that combining the use of three - level systems , environment monitoring , and local feedback can fully and deterministically protect any available quantum information , including entanglement initially shared by different parties . these results can represent a gain in resources and/or distances in quantum communication protocols such as quantum repeaters and teleportation as well as time for quantum memories . finally , we show that monitoring local environments physically implements the optimum singlet conversion protocol , essential for classical entanglement percolation .


scale - invariant random spatial networks ( sirsns ) are a class of random networks defined as a route between each pair of points , with three types of properties .first , invariance properties , second , guarantees on mean lengths of routes in the euclidean metric and third guarantees on the mean length of intersection of a suitably truncated version of the network with a given compact set .it turns out that these conditions are enough to deeply constrain the network .for example , all sirsns have singly - infinite paths for which any subset is included in a route , but no such paths are doubly - infinite .the only known example is the binary hierarchy model , in two dimensions .it consists of minimum - time paths on a dyadic grid where speed depends on the two - valuation .invariance is obtained by a randomization construction .the latter feature is somewhat unsatisfying : the model itself is invariant , but realisations exhibit long - range dependence : observation of a small region gives much information on the network everywhere .a more `` natural '' candidate for a sirsn is therefore the _ poisson line process model_. intuitively , lines are thrown uniformly at random in , and marked with random speed limits .slower lines are dense in .then the route between two points is the minimum - time path made of segments of these lines . remarkably , even in dimension , when random lines almost surely do not intersect , such paths exist .the whole construction is invariant .the aim of this paper is to show that the poisson line process is indeed a sirsn for all .historically , introduced the notion of a sirsn , and proved a number of their properties , including those mentioned in the first paragraphs of this . a verbal description .the motivation was twofold .first , had proved that it was possible to build a road network connecting a prescribed set of points that both had routes almost as short as the segments between each pair of points , that is , the corresponding euclidean geodesics , and total road length almost as short as the steiner tree , that is , the shortest possible connecting network .however , the network was less efficient at small scales .thanks to their invariance properties , sirsns have the same efficiency at all scales .it turns out that there is a trade - off : for a sirsn , there is a lower bound on the expected total length of the network , which is decreasing in the expected route length between two points .the second motivation was to give a mathematical abstraction of road networks and maps , in particular online maps as they are used today .namely , we may change viewpoint , zoom in or out , and the appearance changes little , as smaller roads are shown and hidden .sirsns are then defined as statistically invariant under translation , rotation and change of scale .moreover , we are less interested in the roads than in the routes : how do we drive from to ? sirsns are thus defined by giving routes only , namely unique routes connecting pairs of points .it turns out that a notion of `` main roads '' at any scale can be built from the network of routes itself . to wit, the network of main roads at scale would be the network of routes deprived of balls of radius around their endpoints .it is finite in every compact .similarly , have shown that a number of statistics of real road networks do not depend on scale . that the binary hierarchy model was a sirsn , and suggested two other possible models for sirsn , one of which is the poisson line process model . has then proved important properties of the poisson line process with appropriate speeds : it does yield a random metric space and this space is a geodesic space . moreover , in dimension two , the geodesics are almost everywhere unique , the geodesics are locally of finite mean - length , and the subnetwork obtained from the routes connecting points of an independent poisson point process has finite length in a compact set .the latter properties establish a `` pre - sirsn '' result , but fall short of the full definition .  as a candidate fora sirsn , poisson line process model then fall in a large category : networks derived from geodesic spaces .indeed , from any geodesic space , we may build a spatial network by associating to any two points the geodesic(s ) connecting them .it is not obvious how one might determine when such a network is a sirsn .  in section [ notations ] , we give a precise definition of a sirsn and of poisson line processes .we also present other notation and definitions , and recall some known results , in particular that the poisson line process with speed limits yields a random metric on .we then give a few basic properties of -paths , that is paths in this metric space . in section [ time_diam ] ,we give sharp bounds on the random diameter for this metric of a euclidean ball , with a few generalizations .these estimates will be an important tool in several subsequent proofs . in section[ unique ] , we prove that geodesics between a given pair of points are almost surely unique , in any dimension. lemma [ tour ] will play a central role : we introduce the notion of `` many directions '' , and the lemma states that geodesics have many directions at relevant points .a consequence is that any path using the same lines as a geodesic will contain these points .we will then conclude by noticing that geodesics between the same pair of points almost surely use the same lines ( lemma [ sameseg ] ) .in section [ expect_geo ] , we prove that geodesics have finite mean euclidean length .alternatively , we may see the result as supplying a stochastic control of the euclidean diameter of balls for the metric generated by the poisson line process .section [ ldn ] contains the last and most important component of the proof that the poisson line process generates a sirsn .intuitively , we establish a sharp control of the total length of the intersection of all infinitely many geodesics minus a ball around each endpoint , with a given ball .this corresponds to the fact that all these geodesics coalesce before hitting the ball and split after leaving it .bounds are given using the pigeon - hole principle and the fact that relevant geodesics must use the few fast lines that hit the ball .finally , section [ conclusion ] suggests a few potential directions of future inquiry .we follow s ( ) notation whenever possible .  we write for the ball with center and radius .we first give the precise definition of a sirsn .suppose that is an atom - free measure on a measurable space .recall that a poisson point process of intensity measure is a random set of points such that for any , the number of points in is a poisson variable with intensity , and the number of points in disjoint are independent . then a sirsn is defined as follows .[ defsirsn ] a sirsn is a process that associates to any two points and in random routes such that :  between two specified points and , there is almost surely a unique route .it is a finite - length path connecting and .  for a finite number of points in , consider the subnetwork formed by the random routes connecting all and .then is statistically invariant under translation , rotation and change of scale : if is a euclidean similarity of , then has the same distribution as .let be the length of a route between two points at unit distance .then < \inftyl k ] .consider the following parameterization of a line : it is given by a direction and a localization .the direction is an element of the projective space .it then defines a hyperplane normal to this direction , through a special point the origin of .the localization is a point on this hyperplane .the line is then the line through this point normal to this hyperplane .hence , writing for an hyperplane of , we may parameterize the set of lines by .notice that if we want to keep track of the topology of the set of lines , the product should be twisted , but we only need measure - theoretical properties , so we stick to the simpler direct product .recall that the projective space may be seen as the sphere with opposite points identified .the projective space then inherits the natural metric on the sphere , namely the distance between two pairs of antipodal points is the angle between the pair of segments joining them , in radians .up to a null - measure set , the projective space may be more simply viewed as a hemisphere .  with this parameterization , and writing for the unit ball in ,we may write as a product measure on :   to make a clearer reference to it , we write for the probability measure on . for a set of lines , we write for the measure of the set of directions of lines in that go through the origin . in particular , for a subset of and a point of , the value ) ] .we denote the _ silhouette _ of , that is the random set in made of all the lines of , that is .we also write for the random closed set in made of all the lines in such that .  we may then define -paths .[ pipath ] a finite - time -path is a locally lipschitz path in respecting the speed limits imposed by .more precisely , it is a continuous valued function & \to & \mathbb{r } ^d,\end{aligned}\ ] ] with finite , such that for almost all  ] , we define the _ intersection length _ of and as ) ) ] . * in particular ,almost surely , for all geodesics , we have the following equality and decompositions : * if the support of is included in , that is , we say that is -path .we abuse notation by writing . * similarly , we write if and .  finally , a few more generic notation .we call _ internal -net _ of a subset of any maximal subset of such that for all . for a set , we write for its -widening , that is the minkowski sum .we denote the maximal speed of a set of lines by .that is , .notice that if those lines all hit a compact set , this supremum is a maximum .we abuse notation by writing ) ] of is less than .* may be covered by balls of radius .then there is a depending only on , , and such that , for any , with probability , the diameter of on \setminus[f]}]-diameter of is bounded from above by on the event that this construction is possible .now to control the probability of this event , we need :  * a bound on the number of pairs of balls in the same -net ; * a bound on the probability that the fastest line hitting two such balls but not is slower than .we obtain the first bound by using the formula .indeed , each -net of a -ball is then the same as a -net of a radius ball .since the balls centered on the points of such a -net are disjoint , and all included in a ball of radius , a volume argument shows that there at most balls in each -net . so that there are at most balls in , and at most pairs of balls in the same -net .we now consider these lines that hit both of two -balls and in an internal -net of a -ball , but that do not hit .we have seen that they were in  ] are in a cone of angle such that . so that by formula ( [ amettreintro ] ) :\bigr ) & \leq & ( 4 \alpha ) ^{1 - d}.\end{aligned}\ ] ]  since , we have .moreover , with the -volume of a unit -ball , we have \cap\bigl [ b(y , r_{n+1 } ) \bigr ] \cap [ f ] ^c \bigr ) & \leq & \nu_{d-1 } \bigl ( ( 4 \alpha ) ^{1 - d } - \omega\bigr ) \biggl ( \frac{r_{n+1}}{2 } \biggr ) ^{d-1 } \\ & \,\hat{= } &  c_{\alpha , \omega } r_{n+1}^{d-1},\end{aligned}\ ] ] using the notation to summarize and emphasize that this bound depends on and , but not on , , or any other feature of and .we thus get \cap\bigl [ b(y , r_{n+1 } ) \bigr ] \cap [ f ] ^c \bigr ) \leq v_n \bigr ] & \leq&\exp\biggl ( - c_{\alpha , \omega } \frac { r_{n+1}^{d-1}}{v_n^{\gamma-1 } } \biggr).\end{aligned}\ ] ]  multiplying by the number of relevant pairs of ball , we then obtain that the construction is possible except on an event of probability at most   taking and replacing in bound ( [ prob_cons_bound ] ) , we see that this becomes less than for , so that does not depend on or except through and .  replacing and with their value in the bound ( [ gen_bound_a ] ), we get where the first parentheses correspond to and is finite since . we have thus proved formula ( [ td ] ) of the theorem .the moment ( [ morexp ] ) is simple integration : & \leq & c + \frac{1}{\varepsilon_{\max } } \int_{0}^{\varepsilon_{\max } } \exp\biggl ( \delta\biggl(t_{r , { \accentset{\circ}{n } } } \biggl ( \ln\frac { 1}{\varepsilon } \biggr ) ^{\afrac{1}{\gamma- 1 } } \biggr)^{\gamma- 1 } \biggr ) \,\mathrm { d}\varepsilon \\ & \leq & c + \frac{1}{\varepsilon_{\max } } \int_{0}^{\varepsilon_{\max } } \varepsilon^{- \delta t_{r , { \accentset{\circ}{n}}}^{\afrac{1}{\gamma- 1}}}\ , \mathrm{d}\varepsilon \\ & < & \infty.\end{aligned}\ ] ] here , we use and corresponds to the integral between and  , bounded by the value of for .[ ... ] if no lines are forbidden , that is if , then can be taken as big as we wish , so that the lines used all hit as small a widening of as we want .we may slightly generalize the theorem : instead of using a forbidden area , we could use different conditions for which lines to accept .the important property is that we must have enough relevant lines hitting pairs of balls in a -net .there are a few optimisations that could be used to gain slightly in the constants .for example , we have written the proof with one subnet inside each ball .if we had used a single -net of a correctly widened , we would have only about pairs of ball , allowing a bigger .the result stays essentially the same , however , as proven by the following proposition .[ optimal_diameter ] for any two points and , their -distance does not have a moment with higher exponent on the time than in formula ( [ morexp ] ) .that is , for any , for any , we have & = & \infty.\end{aligned}\ ] ]  say that .a path from to has to go from to the border of .hence , it must use lines hitting for a length at least . so that is controlled by the fastest line hitting .now , & = & \exp\bigl ( - c(r ) v_{\varepsilon}^{1 - \gamma } \bigr )  = \varepsilon\end{aligned}\ ] ] with where depends only on .so that on an event of probability at least , we have the bound .hence , for some positive constant depending on , and  : & \geq&\int_{0}^1 \exp\biggl ( - c_3 \biggl ( \ln\frac{1}{\varepsilon } \biggr)^{\eta/ ( \gamma- 1 ) } \biggr)\,\mathrm{d } \varepsilon \\ & = & \infty,\end{aligned}\ ] ] since .in this section , we prove that the network satisfies the property 1 of a sirsn , that is that between two specified points and in , there is almost surely only a unique route . since routes are the -geodesics , this is equivalent to almost sure uniqueness of the geodesic .the case in dimension has already been established in s ( ) paper . the following proof , on the other hand works in all dimensions more than , as stated in theorem [ unique_geodesics ] .note that it does not work in dimension .the strategy is the following :  * we introduce a concept of _ many directions _ , with the following property .if a -path has many directions near a point , any finite -path supported by the same lines contains .* we show that almost surely all geodesics have many directions near all the ends of their component segments , except for the two extremal points .this is the step where is needed .* we show that for specified and in , almost surely all geodesics from to are supported exactly on the same lines . *so that almost surely all such geodesics contain the same segment ends , and this will prove they are the same .the author thinks the proof is very technical for something that looks clear enough , but could not find an easier way .maybe the need to work with tree - like paths in dimension at least three is the reason why there is no obvious argument .hopefully , the concept of many directions can be useful elsewhere .we first state two technical lemmas we need for the proofs in section [ tech ] .we introduce the notion of `` many directions '' and give some cases where paths have many directions in section [ manydir ] , culminating in lemma [ tour ] .we prove that geodesics must use the same lines in section [ samelines ] , and end the proof of uniqueness in section [ asunique ] .the first lemma yields a control on the proportion of balls in a nested set that are hit by lines faster than a threshold appropriately scaling with their size .[ nested_balls ] let .let be a scale factor .we write .let for be a set of nested balls with for some .for some , define define now as the number of balls smaller than that are hit by lines faster than , and as the number of balls that are hit by lines faster than , that is , \neq\varnothing\bigr\ } , \\i_n & = & \ # \bigl\ { i \leq n : \exists k > 0 : [ b_i ] \cap \mathcal{v } _ { i - k } \neq\varnothing\bigr\ } \leq\sum _ { i=0}^{n-1 } h_i.\end{aligned}\ ] ] then \ } ] = ( 1 - p ) \omega_{d-1 } r_0^{d-1 } / ( 2 v_0^{\gamma- 1}) \beta n \delta n\beta n \delta n ] is less than .there are at least balls that are hit by at most lines of speed between and , for some fixed .moreover , all those lines have independent , uniformly random directions .the will yield the we need to continue the proof , that is ,   we deal with these properties in reverse order , and first consider property 3 .  since the sets of lines are disjoint , the events that more than lines in hit are mutually independent ( in ) , as are their directions .moreover , since the are balls , the direction of the lines of the isotropic poisson line process that hit it are uniformly random .so that the number of such events is a binomial random variable where is the probability of a single event .the chernoff bound for a binomial is , for , & \leq&\biggl ( \biggl ( \frac{q}{\delta } \biggr ) ^{\delta } \biggl(\frac{1-q}{1 - \delta } \biggr)^{1 - \delta } \biggr ) ^n.\end{aligned}\ ] ] here , it suffices to take . for big enough , if , for example , we take , the above bound is negligible with respect to .  nowthe probability of a single event is the probability that a poisson random variable with parameter not depending on is bigger than .we then just choose so that .now for property 2 .  since the sets of lines \setminus[b_{i+1}] ] is less than are independent .arguing as for property 3 , we merely have to choose such that the probability of a single event is less than .now , by scaling , the sets may all be covered by the same number of balls with radius , and centres in .moreover , is connected .now the -widening of is included in , and since , we know that is at distance at least of this widening .so that the maximum solid angle of viewed from any point of the widening is a decreasing function of , hitting zero when goes to infinity .thus , for big enough , we may apply lemma [ tform ] with . replacing in bound ( [ td ] ) with to ensure , we find that property 2 is ensured if we choose   this choice of ensures that .  as for property 1 , we apply lemma [ nested_balls ] .we substitute in equation ( [ res_tower ] ) , and use  .we obtain \\ & & \qquad \leq\bigl(c \ln(\alpha ) \alpha^{-\sfrac{6}5}\bigr)^n \\ & & \qquad  = o\bigl(\alpha^{-n}\bigr).\end{aligned}\ ] ]  let us now consider a geodesic from a point outside to a point inside .since the balls are nested , they are all crossed by the geodesic .even the sets introduced in the proof of property 2 are all crossed , that is we must pass from the spherical boundary to the smaller boundary .  if the three properties are satisfied , then there are at least indices for which the conditions are simultaneously satisfied .for such an , by property 2 the geodesic will go from to in time at most .since the two boundaries are apart , the geodesic must use a line with speed more than within .now , and , by property 1 , there is no line faster than in .so that the geodesic must use a line in  ] , so that constraint ( [ constraint ] ) must be satisfied for each . in particular , on the event ( [ n_time ] ) , the following constraint is satisfied for all  ] , with the conventions and : \setminus\bigl[b(x , r_{l_{i-1}})\bigr ] \bigr ) \geq v_{l_{i+1 } - 1 } \bigr\}.\end{aligned}\ ] ]  notice that the events do not depend only on , but on the whole sequence of .notation is easier this way . in particular, the former paragraph reads & \leq&\sum_{k=0}^m \sum_{0=l_0 < \cdots <l_k \leq m } \mathbb{p } [ b_0 \cap b_1 \cap\cdots\cap b_{k-1 } \cap a_k ] .\end{aligned}\ ] ]  let us consider the filtration generated by the lines intersecting for increasing , that is } ) ] .we may then compute & = & 1 - \exp\bigl ( - c r_{l_i}^{d-1 } v_{l_{i+1 } - 1 } ^{-(\gamma - 1 ) } \bigr)\nonumber \\[-2pt ] & \leq & c r_{l_i}^{d-1 } v_{l_{i+1 } - 1 } ^{-(\gamma - 1)}\nonumber \\[-2pt ] & = & c r_{l_i}^{d-1 } 2^{-(\gamma- 1)(l_{i+1 } - l_i -1 ) } v_{l_i } ^{-(\gamma - 1)}\nonumber \\[-2pt ] & = & 2^{-(\gamma- 1)(l_{i+1 } - l_i -1 ) } c r_{l_i}^{d- \gamma } t_n^{\gamma- 1 } \\[-2pt ] & \leq&2^{-(\gamma-1)(l_{i+1 } - l_i -1 ) } c r_{0}^{d- \gamma } t_n^{\gamma- 1}\qquad\mbox{since } r_0 \leq r_{l_i}\nonumber \\[-2pt ] & = & 2^{-(\gamma-1)(l_{i+1 } - l_i ) } p(r_0 ) \qquad\mbox{with}\nonumber \\[-2pt ] p(r_0 )  & = & 2^{\gamma-1 } c r_{0}^{d- \gamma } t_n^{\gamma- 1}.\nonumber\end{aligned}\ ] ]  recalling the convention , we obtain \\[-2pt ] & & \qquad  \leq\mathbb{p } [ b_0 ] \mathbb{p } [ b_1 | b_0 ] \cdots\mathbb{p } [ a_k | b_0 , b_1,\dots , b_{k-1 } ] \\[-2pt ] & & \qquad  \leq\prod_{i=0}^{k } \mathbb{p } [ a_i ] \qquad\mbox{by { ( \ref{pai } ) } } \\[-2pt ] & & \qquad  \leq\prod_{i=0}^{k } 2^{-(\gamma-1)(l_{i+1 } - l_i ) } p(r_0)\qquad\mbox{by { ( \ref{unp } ) } } \\[-2pt ] & & \qquad  = 2^{-(\gamma- 1)(m+1 ) } p(r_0)^{k+1}.\end{aligned}\ ] ]  substituting into bound ( [ bornelxy ] ) , we obtain & \leq&\sum_{k=0}^m \sum_{0=l_0 < \cdots < l_k \leq m } 2^{-(\gamma- 1)(m+1 ) } p(r_0)^{k+1}\nonumber \\ \label{sum } & = & \sum_{k=0}^m \pmatrix{m \cr k } 2^{- ( m+1)(\gamma-1 ) } p(r_0)^{k+1 } \\ & = & 2^{- ( m+1)(\gamma-1 ) } p(r_0 ) \bigl(1 + p(r_0 ) \bigr)^m\nonumber \\ & \leq&\bigl ( 2^{-(\gamma-1 ) } \bigl(1 + p(r_0)\bigr ) \bigr ) ^{m+1}. \label{reborne}\end{aligned}\ ] ]  let .we now choose our free parameters : so that   substituting into bound ( [ reborne ] ) , we get & \leq&2^{-(m+1 ) \kappa } \\ & \leq&2^{-(n+1)}.\end{aligned}\ ] ]  since \leq2^{-(n+1)} ] , but <\infty ] . notice that scale equivariance yields similar results if the long - distance network is defined by removing balls of any fixed radius instead of radius .the main argument relies on the pigeon - hole principle : a geodesic getting close to a prescribed point must use one of a very few fast lines close to that point , and must use them again to draw away . and by uniqueness of geodesics , two geodesics with two common points must agree between those points .[ finite_intensity ] let . with the above notation ,let be the length of the long - distance network in . for , with probability at least ,this length is less than , for constants , and depending only on and .consequently , there is a finite moment of exponential form : for any , & < & \infty.\end{aligned}\ ] ] in particular , has finite mean .since is countable , by theorem [ unique_geodesics ] , almost surely all geodesics between its points are unique . in the proof ,we only consider such geodesics and sub - geodesics , so that we assume uniqueness .consider the balls and .we call their set difference the _ border_.  now if a point , then .hence , geodesics with an endpoint in make no contribution to .we have   to hits for the first time at and the last time at .it hits for the first time at and the last time at . ]hence , geodesics making contributions to are structured in the following way , illustrated in figure [ local_geo ] .* they hit for the first time at point on the corresponding sphere .* then they hit for the first time at point on the corresponding sphere .* then they hit for the last time at point on the corresponding sphere .* then they hit for the last time at point on the corresponding sphere .in particular , the contribution to is included in the sub - geodesic .now , uniformly on and , the time between and is bounded by the time diameter of .so that , by theorem [ tform ] , with probability , we have \\[-9pt]\nonumber & \,\hat{= } &  t_{2/3 , 1 } \biggl ( \ln\frac{2 } { \varepsilon\cdot \varepsilon_{\max } } \biggr)^{\afrac{1}{\gamma- 1}}.\end{aligned}\ ] ]  we call _ fast lines _ the lines faster than .conversely , lines slower than are _ slow lines_. we write for the intersection of all those fast lines with the ball .  the number of fast lines hitting is a poisson variable with parameter for explicit constants and , with .we recall that the moment generating function of such a poisson variable is = \exp ( \lambda_{\varepsilon } ( e^t - 1))$ ] , and use chernoff bound to get & \leq & e^{\lambda_{\varepsilon } ( e^t - 1 ) } e^{- t c_3 \lambda_{\varepsilon } } \\[-2pt ] & = & \exp\bigl ( \lambda_{\varepsilon } ( c_3 - 1 - c_3 \ln c_3 ) \bigr ) \qquad\mbox{with } e^t = c_3 \\[-2pt ] & = & \biggl(\frac{c_1 } { \varepsilon } \biggr)^{c_2 ( c_3 - 1 - c_3 \ln c_3 ) } \\[-2pt ] & \leq&\frac{\varepsilon}{2},\end{aligned}\ ] ] with chosen big enough to have .  .the solid curve connecting and is the geodesic .the geodesic has a common point with the black dotted geodesic on a fast ( black ) line when first crossing the border , and another when last crossing the border .hence , they agree between and . any othergeodesic contributing to would meet one of the geodesics in the family in the same way .]  hence , with probability at least , there are at most fast lines hitting .with probability at least both this event and are true .we assume both from now on .  since , the intersection of with slow lines has length at most .so that , since and ( resp . , and ) are at least apart, the geodesic must have fast segments for length at least between and ( resp . , and ) , that is , m_1 ( \mathcal{v } _{ \varepsilon } \cap g_{vz } ) & \geq & \tfrac{1}{6}.\end{aligned}\ ] ]  now & \leq&\frac{4 c_3 \lambda_{\varepsilon}}{3 } \\[-2pt ] & = & c_4 \ln\biggl ( \frac{c_1}{\varepsilon } \biggr),\end{aligned}\ ] ] so that   hence , by the pigeon - hole property as illustrated in figure [ pigeonhole ] , we may find a maximal family of at most geodesics such that :  * is a geodesic between and in , passing through , , and defined as for and above . * any geodesic contributing to crosses one of the when first and last crossing the border , that is , there are points and such that and .  by uniqueness of geodesics , and coincide on .in particular , they coincide on .hence , the intersection is included in the finite number of geodesics .we may then conclude by separating contributions from fast and slow lines :   [ issirsn ] the network made of the time geodesics is a sirsn .property 1 of a sirsn is a consequence of almost sure uniqueness of geodesics between two points , that is theorem [ unique ] in dimension at least , and theorem 4.4 . in s ( ) article in dimension .property 2 of a sirsn is because the underlying poisson line process is invariant by translation and rotation . as for change of scale ,the underlying poisson line process is invariant by a transformation where scale is multiplied by and speed by .hence , all paths have their time length multiplied by the same , so that the geodesics are the same and is invariant .property 3 of a sirsn is theorem [ teo_len ] .property 4 of a sirsn is theorem [ finite_intensity ] .we have established that the improper poisson line process with adequate speed limits yield a sirsn .  along the way , a few questions have been raised .is there an easier , more natural way to prove uniqueness of geodesics ?what are the tightest moments of the euclidean length of a geodesic ? when can we generalize this construction using geodesics from a random geodesic metric space ?  on a more general note , we may wonder which property of our network translate to general sirsns , or to sirsns made of geodesics of a metric space .for example , it should be easy to show that not all sirsns have the equivalent of lemma [ tour ] or corollary [ strengthening ] : it is certainly not true for the binary hierarchy model by aldous .we might also raise a few typical questions in stochastic geometry .is there only one geodesic connecting a prescribed point to infinity , like in dimension two ?in many models , infinite geodesics have an asymptotic direction . for sirsns ,this property looks unlikely , and characterizing the random walk of the angle as a function of the -distance to looks worthwhile .what is the law of a typical cell in the tessellation generated by the network connecting the points of , an intensity poisson point process ?  finally ,another somewhat tangential direction of research would be to study more closely the properties of the random metric space .for example , being a sirsn entails coalescence of geodesics , a very hyperbolic - like property . we may also draw comparisons with a well - known random metric space such as the brownian map [ see , e.g. , ] .the brownian map is a random metric space homeomorphic to the sphere .it has hausdorff dimension .all its geodesics minus their endpoints is of hausdorff dimension .the cut - locus of its geodesics starting from a given point has hausdorff dimension , and the topology of an open continuous tree .on the other hand , we have shown that our metric space is homeomorphic to  .it should be easy , by scaling arguments , to show that its hausdorff dimension is , which is bigger than .in particular , with and , we have the same dimensions as the brownian map .if any geodesic can be appropriately approximated by geodesics between points of poisson point processes , it should also be easy to show that the geodesics minus their endpoints is of dimension .however , the cut - locus might have a very different behaviour .i would like to thank wilfrid kendall for introducing us to the problem during a talk , and helpful discussions since then .remarks from a referee have greatly improved the presentation of the paper .

Abstract: aldous has introduced a notion of scale - invariant random spatial network ( sirsn ) as a mathematical formalization of road networks . intuitively , those are random processes that assign a route between each pair of points in euclidean space , while being invariant under rotation , translation , and change of scale , and such that the routes are not too long and mainly lie on `` main roads '' .   the only known example was somewhat artificial since invariance had to be added using randomization at the end of the construction . we prove that the network of geodesics in the random metric space generated by a poisson line process marked by speeds according to a power law is a sirsn , in any dimension .  along the way , we establish bounds comparing euclidean balls and balls for the random metric space . we also prove that in dimension more than two , the geodesics have `` many directions '' near each point where they are not straight .   ./style / arxiv - general.cfg