einstein s 1915 relativistic theory of gravity general relativity will soon be a century old .it is the classical theory of one of the four fundamental forces .it underlies our contemporary understanding of the big bang , black holes , pulsars , quasars , x - ray sources , the final destiny of stars , gravitational waves , and the evolution of the universe itself .it is the intellectual origin of many of the ideas at play in the quest for a unified theory of the fundamental forces that includes gravity .the heart of general relativity is one of the most beautiful and revolutionary ideas in modern science the idea that gravity _ is _ the geometry of curved four - dimensional spacetime .general relativity and quantum mechanics are usually regarded as the two greatest developments of twentieth - century physics .  yet , paradoxically , general relativity so well established , so important for several branches of physics , and so simple in its basic conception is often not represented anywhere in the undergraduate physics curriculum .an informal survey by william hiscock of the course offerings of 32 mid - western research universities found only a handful that offered an intermediate ( junior / senior ) course in general relativity as part of the undergraduate physics curriculum .this has the consequence that many students see gravity first in the context of planetary orbits in basic mechanics and next , if at all , in an advanced graduate course designed in part for prospective specialists in the subject .there might have been an argument for such an organization half a century ago .but there is none today in an era when gravitational physics is increasingly important , increasingly topical , increasingly integrated with other areas of physics , and increasingly connected with experiment and observation . in the author s opinion, every undergraduate physics major should have an opportunity to be introduced to general relativity .its importance in contemporary physics is not the only reason for introducing undergraduates to general relativity .there are others : first , the subject excites interest in students .warped spacetime , black holes , and the big bang are the focus both of contemporary research and of popular scientific fascination .students specializing in physics naturally want to know more .a further argument for undergraduate general relativity is accessibility .as i hope to show in this paper , a number of important phenomena of gravitational physics can be efficiently introduced with just a basic background in mechanics and a minimum of mathematics beyond the usual advanced calculus tool kit .other subjects of great contemporary importance such as high temperature superconductivity or gauge theories of the strong interactions require much more prerequisite information .general relativity can be made accessible to both students and faculty alike at the undergraduate level .it is probably fruitless to speculate on why a subject as basic , accessible , and important as general relativity is not taught more widely as part of the undergraduate physics curriculum .limited time , limited resources , inertia , tradition , and misconceptions may all play a role .certainly it is not a lack of textbooks are a partial list of texts known to the author that treat general relativity in some way at an introductory level .available time is one of the obstacles to introducing general relativity at the undergraduate level .the deductive approach to teaching this subject ( as for most others ) is to assemble the necessary mathematical tools , motivate the field equations , solve the equations in interesting circumstances , and compare the predictions with observation and experiment .this ` math first ' order takes time to develop for general relativity which may not be available to either students or faculty .this article describes a different , ` physics first ' approach to introducing general relativity at the junior / senior level .briefly , the simplest physically relevant solutions to the einstein equation are introduced _ first _ , without derivation , as curved spacetimes whose properties and observable consequences can be explored by a study of the motion of test particles and light rays .this brings the student to interesting physical phenomena as quickly as possible .it is the part of the subject most directly connected to classical mechanics and the part that requires a minimum of ` new ' mathematical ideas . laterthe einstein equation can be motivated and solved to show where the solutions come from . when time is limitedthis is a surer and more direct route to getting at the applications of general relativity that are important in contemporary science .section ii expands very briefly on the importance of general relativity in contemporary physics .section iii outlines the basic structure of the subject .sections iv and v describe the ` math first ' and ` physics first ' approaches to introducing general relativity to undergraduate physics majors .this is not an even - handed comparison .the ` math first ' approach is described only to contrast it with the ` physics first ' approach which is advocated in this paper .section vi illustrates how ideas from classical mechanics can be used to calculate important effects in general relativity .section vii reports the personal experiences of the author in using the ` physics first ' method .gravity is the weakest of the four fundamental forces at accessible energy scales .the ratio of the gravitational force to the electric force between two protons separated by a distance is ( in gaussian electromagnetic units ) gravity might thus seem to be negligible . butthree other facts explain why it is important and where it is important .first , gravity is a universal force coupling to all forms of mass and energy .second , gravity is a long - range force in contrast to the weak and strong forces which are characterized by nucleus - size ranges and below .third , and most importantly , gravity is unscreened .there is no negative `` gravitational charge '' ; mass is always positive .these three facts explain why gravity is the dominant force governing the structure of the universe on the largest scales of space and time the scales of astrophysics and cosmology .the strong and weak forces are short range .the relatively much greater strength of electromagnetic forces ensures that charges will be screened in an electrically neutral universe like ours .only gravity is left to operate on very large scales .relativistic gravity general relativity is important for an object of mass and size when neutron stars and black holes are relativistic objects by this rough criterion .so is our universe if we take to be the present hubble distance and to be the mass within it .figure 1 displays some phenomena for which relativity is important and ones for which it is not .= 3.3 in  general relativity can sometimes be important even when is small provided compensating observational precision can be achieved . for the sun , yet the solar systemis the domain of the precision tests that confirm general relativity to as much as 1 part in  . for the earth ,yet general relativistic effects are important for the operation of the global positioning system ( gps ) .relativistic gravity is also important on the smallest scales considered in contemporary physics those of quantum gravity .these are characterized by the planck length  this is much , much smaller than even the scale of the strong interactions cm . yet, this is the scale which many contemporary explorers believe will naturally characterize the final theory unifying the four fundamental forces including gravity .this is the characteristic scale of string theory .this is the scale that will characterize the union of the two great developments of twentieth century physics general relativity and quantum mechanics .the important point for this discussion is that the last few decades have seen dramatic growth in observational data on the frontier of the very large , and an equally dramatic growth in theoretical confidence in exploring the frontier of the very small .black holes , for example , are no longer a theorist s dream .they have been identified at the center of galaxies ( including our own ) and in x - ray binaries .they are central to the explanations of the most energetic phenomena in the universe such as active galactic nuclei . on even larger scales , it is now a commonplace observation that cosmology has become a data driven science .cosmological parameters once uncertain by orders of magnitude have been determined to accuracies of 10% .adventures into planck scale physics may be mostly in the minds of theorists , but the quest for a unified theory of the fundamental forces including gravity is being pursued with impressive vigor and confidence by a large community .indeed , at the big bang where large and small are one , we should eventually see direct evidence of planck scale physics .  for these reasonsgeneral relativity is increasingly central to today s physics .it is increasingly topical , increasingly connected with experiment and observation , and increasingly integrated with other branches of physicsthis section sketches a few key ideas in general relativity for those who may not be familiar with the theory .the intent is not to offer an exposition of these ideas . that , after all , is properly the task of texts on the subject .rather , the purpose is merely to mention ideas that will occur in the subsequent discussion and to illustrate the simplicity of the conceptual structure of the subject .it s potentially misleading to summarize any subject in physics in terms of slogans . however , the following three roughly stated ideas are central to general relativity .  *_ gravity is geometry ._ phenomena familiarly seen as arising from gravitational forces in a newtonian context are more generally due to the curvature of geometry of four - dimensional spacetime .* _ mass - energy is the source of spacetime curvature ._ mass is the source of spacetime curvature and , since general relativity incorporates special relativity , any form of energy is also a source of spacetime curvature . *_ free mass moves on straight paths in curved spacetime ._ in general relativity , the earth moves around the sun in the orbit it does , not because of a gravitational force exerted by the sun , but because it is following a straight path in the curved spacetime produced by the sun .  making these ideas more precise and more explicitis an objective of any course in general relativity .we mention a few steps toward this objective here .points in four - dimensional spacetime can be located by four coordinates .coordinates are arbitrary provided they label points uniquely . generally , several different coordinate patches are required to label all the points in spacetime .  the geometry of a spacetime is specified by giving the _ metric _ , , where the indicates that the metric is generally a function of all four coordinates .the metric determines the squared distance in four - dimensional spacetime between points separated by infinitesimal coordinate intervals .specifically , where a double sum over and from 0 to 3 is implied .integration of the specfied by this expression gives the distance along curves .metrics satisfy the _einstein equation _ relating a measure of curvature on the left hand side to the energy - momentum tensor of matter on the right .this einstein equation comprises 10 non - linear , partial differential equations for the metric .an important example of a solution to the einstein equation is the schwarzschild geometry giving the metric in the empty space outside a spherically symmetric black hole or star . in standardschwarzschild spherical coordinates this is where is the mass of the black hole or star .this , to an excellent approximation , describes the curved spacetime outside our sun .test particles with masses too small to affect the ambient geometry move on straight paths in it .more precisely , they move between any two points , and , in spacetime on a world line ( curve ) of stationary proper time . the proper time along a world line is the distance along it measured in time units .thus , .( the negative sign is so is positive along the world lines of particles which always move with less than the speed of light ) .curves of stationary proper time are called _geodesics_. the world line of a particle through spacetime from to can be described by giving its coordinates as a function of any parameter that takes fixed values on the end points .for instance using the schwarzschild metric , the principle of stationary proper time takes the form : ^{\frac{1}{2}}=0 \label{threefive}\end{aligned}\ ] ] where a dot denotes a derivative with respect to and means the first variation as in classical mechanics .the variational principle for stationary proper time has the same form as the variational principle for stationary action in classical mechanics .the lagrangian is the integrand of .lagrange s equations are the geodesic equations of motion .their form can be made especially simple by choosing proper time for the parameter . from themone can deduce the conservation of energy , the conservation of angular momentum , and an effective equation for radial motion in the schwarzschild geometry as we will describe in section vi [ cf . ] . with that , one can calculate a spectrum of phenomena ranging from the precession of planetary orbits to the collapse to a black hole .when extended to light rays and implemented in appropriate metrics , the geodesic equations are enough to explore most of the important applications of relativistic gravity displayed in table 1 .+ global positioning system ( gps ) & & & & & lense - thirring precession of a gyroscope + gravitational redshift & & & & & cosmological redshift + bending of light by the sun & & & & & expansion of the universe + precession of mercury s perihelion & & & & & big - bang + shapiro time delay & & & & & cosmic background radiation + gravitational lensing & & & & & the fate of the universe + accretion disks around compact objects & & & & & propagation of gravitational waves + determining parameters of binary pulsars & & & & & operation of gravitational wave detectors + spherical gravitational collapse & & & & & x - ray sources + formation of black holes & & & & & active galactic nuclei + hawking radiation from black holes & & & & & neutron stars + frame - dragging by a rotating body & & & & & +the deductive approach to teaching many subjects in physics is to  1 .  introduce the necessary mathematical tools ; 2 .  motivate and explain the basic field equations ; 3 .solve the field equations in interesting circumstances ; 4 .apply the solutions to make predictions and compare with observation and experiment .  for electromagnetism ,( 1)(4 ) are , _e.g. _ ( 1 ) vector calculus , ( 2 ) maxwell s equations , ( 3 ) boundary value problems , the fields of point particles , radiation fields , etc . , ( 4 ) charged particle motion , circuits , wave guides and cavities , antennas , dielectric and magnetic materials , magnetohydrodynamics a list that could very easily be extended . for gravitation ( 1)(4 )e.g. _ ( 1 ) differential geometry , ( 2 ) the einstein equation and the geodesic equation , ( 3 ) the solutions for spherical symmetry , cosmological models , gravitational waves , relativistic stars , etc .table 1 lists some of the applications of general relativity that constitute ( 4 ) .this deductive order of presentation is logical ; it is the order used by the great classic texts ; and it is the order used in standard graduate courses introducing the subject at an advanced level .but the deductive order does have some drawbacks for an elementary introduction to physics majors in a limited time .differential geometry is a deep and beautiful mathematical subject .however , even an elementary introduction to its basic ideas and methods are not a part of the typical advanced calculus tool kit acquired by physics majors in their first few years .this is ` new math ' .in contrast , the vector calculus central to electromagnetism _ is _ part of this tool kit .it is possible at the undergraduate level to give an introduction to the basic mathematical ideas of manifolds , vectors , dual vectors , tensors , metric , covariant derivative , and curvature . indeed , many students feel empowered by learning new mathematics .but it does take time .it also must be practiced .the author s experience is that many students at this level need considerable exercise before they are able to accurately and efficiently manipulate tensorial expressions and feel at home with the four - dimensional mathematical concepts necessary to formulate einstein s equation .when time is limited , pursuing the deductive order may leave little available for the interesting applications of general relativity .further , solving the einstein equation to exhibit physically relevant spacetime geometries is a difficult matter .their non - linear nature means that there is no known general solution outside of linearized gravity .each new situation , _e.g. _ spherical symmetry , homogeneous and isotropic cosmological models , gravitational plane waves , is typically a new problem in applied mathematics . deriving the solutions only adds to the time expended before interesting applications can be discussed .many of the successful introductory texts in general relativity follow this logical order at various levels of compromise . in the authors opinion , an outstanding example is bernard schutz s classic _ a first course in general relativity _ . in the next sectionwe consider a different way of introducing general relativity to undergraduates .electricity and magnetism are not usually presented in introductory ( freshman ) courses in the deductive order described in the previous section .specifically , we do not usually first develop vector calculus , then exhibit maxwell s equations , then solve for the fields of charges , currents , and radiation , and finally apply these to realistic electromagnetic phenomena .rather , the typical course posits the fields of the simplest physically relevant examples , for instance the electric field of a point charge , the magnetic field of a straight wire , and the electromagnetic plane wave .these are used to build understanding of fields and their interaction with charges for immediate application to demonstrable electromagnetic phenomena .maxwell s ten partial differential equations and their associated gauge and lorentz invariances are better appreciated later , usually in a more advanced course.  general relativity can be efficiently introduced at an intermediate ( junior / senior ) level following the same ` physics first ' model used in introducing electromagnetism . specifically :  1 .exhibit the simplest physically important spacetime geometries _ first _, without derivation ; 2 .derive the predictions of these geometries for observation by a study of the orbits of test particles and light rays moving in them ; 3 .  apply these predictions to realistic astrophysical situations and compare with experiment and observation ; 4 .  motivate the einstein equation and solve it to show where the spacetime geometries posited in ( 1 ) come from .these are essentially the same four elements that comprise the deductive approach described in the previous section , but in a different order .that order has considerable advantages for introducing general relativity at an intermediate level as we now describe :   _ less ` new math ' up front : _ to exhibit a spacetime geometry , the only ` new math ' ideas required are the metric and its relation to distances in space and time . to analyze the motion of test particles in these geometries , only the notions of four - vectors and geodesics are needed .these three new mathematical ideas are enough to explain in detail a wide range of physical phenomena , such as most of those in table 1 .further , these three new mathematical ideas are among the simplest parts of a relativist s tool kit to introduce at an intermediate level .four - vectors are often familiar from special relativity .geodesics viewed as curves of extremal proper time are special cases of lagrangian mechanics .the idea of a metric can be motivated from the theory of surfaces in three - dimensional flat space . a general theory of tensors as linear maps from vectors into the real numbersis not required because only one tensor the metric is ever used ..13 in _ the simplest spacetimes are the most physically relevant : _  *the sun is approximately spherical .* spherical black holes exhibit many characteristic properties of the most general black hole . *the universe is approximately homogeneous and isotropic on scales above 100 megaparsecs . *detectable gravitational waves are weak .these four facts mean that the simplest solutions of the einstein equation are the ones most relevant for experiment and observation .the static , spherically symmetric schwarzschild geometry describes the solar system experimental tests , spherically symmetric gravitational collapse , and spherical black holes .the exactly homogeneous and isotropic friedman - robertson - walker ( frw ) models provide an excellent approximation to the structure and evolution of our universe from the big bang to the distant future .the linearized solutions of the einstein equation about a flat space background describe detectable gravitational waves .a ` physics first ' treatment of general relativity that concentrates on the simplest solutions of the einstein equation is thus immediately relevant for physically realistic and important situations ..13 in _ no stopping before some physics : _ students with different levels of experience , preparation , abilities , and preconceptions will take different lengths of time to acquire the basic concepts of general relativity . beginning with the applicationsguarantees that , wherever the course ends , students will have gained some understanding of the basic physical phenomena ( table 1 ) which make general relativity so important and not simply of a mathematical structure which is the prerequisite for a deductive approach . .13 in _ more concrete , less abstract : _ beginning with the applications rather than the abstract structure of the theory is easier for some students because it is more concrete .beginning with the applications is also a surer way of driving home that general relativity is a part of physics whose predictions can be observationally tested and not a branch of mathematics ..13 in _ fewer compromises : _ the analysis of the motion of test particles and light rays in the simplest geometries can be carried out in essentially the same way as it is done in advanced textbooks . no compromises of method or generality are needed ..13 in _ flexibility in emphasis : _ beginning with applications allows enough time to construct courses with different emphases ; for instance on black holes , gravitational waves , cosmology , or experimental tests ..13 in _ closer to the rest of the undergraduate physics curriculum : _ calculations of the orbits of test particles and light rays to explore curved spacetimes are exercises in mechanics .the symmetries of the simplest important solutions imply conservation laws .these can be used to reduce the calculation of orbits to one - dimensional motion in effective potentials .even the content of the einstein equation can be put in this form for simple situations .this allows the intuition and techniques developed in intermediate mechanics to be brought to bear , both for qualitative understanding and quantitative prediction .conversely , this kind of example serves to extend and reinforce an understanding of mechanics .indeed , in the author s experience a few students are surprised to find that mechanics is actually useful for something .the examples discussed in the next section will help illustrate the close connection between calculating geodesics and undergraduate mechanics ..13 in _ fewer prerequisites : _ the close connection with mechanics described above and in the next section means that the only essential prerequisite to a ` physics first ' exposition of general relativity is an intermediate course in mechanics .neither quantum mechanics nor electrodynamics are necessary.some acquaintance with special relativity is useful , but its brief treatment in many first year courses means that it is usually necessary to develop it _ de novo _ at the beginning of an introductory course in general relativity .intermediate mechanics is thus the single essential physics prerequisite .this means that an introductory ` physics first ' course in general relativity can be accessible to a wider range of physics majors at an earlier stage than courses designed to introduce students to other frontier areas ..13 in _ closer to research frontiers : _ beginning with the applications means that students are closer sooner to the contemporary frontiers of astrophysics and particle physics that they can hear about in the seminars , read about in the newspapers , and see on popular television programs ..13 in _ more opportunities for undergraduate participation in research : _ the applications of general relativity provide a broad range of topics for students to pursue independent study or even to make research contributions of contemporary interest .more importantly , it is possible to identify problems from the applications that are conceptually and technically accessible to undergraduate physics majors and can be completed in the limited time frame typically available .problems that involve solving for the behavior of test particles , light rays , and gyroscopes are examples , as are questions involving linear gravitational waves , black holes , and simple cosmological models .the ` physics first ' approach to teaching general relativity enables undergraduate participation in research because it treats such applications first ..13 in _ specialized faculty not needed : _ learning a subject while teaching it , or learning it better , is a part of every physics instructor s experience .the absence of previous instruction means that teaching an introductory general relativity course will often be the first exposure to the subject for many faculty .the process of learning by faculty is made easier by a ` physics first ' approach for the same reasons it is easier for students .a wider variety of faculty will find this approach both familiar and manageable .specialists in gravitational physics are not necessary .no approach to teaching is without its price and the ` physics first ' approach to introducing general relativity is no exception .the obvious disadvantage is that it does not follow the logically appealing deductive order , although it can get to the same point in the end .a ` physics first ' approach to introducing general relativity _ may _ not be indicated in at least two circumstances : first , when there is enough time in the curriculum and enough student commitment to pursue the deductive order ; second , when students already have significant preparation in differential geometry . even then however it is a possible alternative .a ` physics first ' approach is probably not indicated when the mathematics is of central interest , as for students concentrating in mathematics , and for physics students who study einstein s theory mainly as an introduction to the mathematics of string theory .this section illustrates how the effective potential method developed in typical undergraduate mechanics courses can be applied to important problems in general relativity . of the several possible illustrations ,just one is considered here the relativistic effects on test particle orbits outside a spherical star or black hole .applications of this include the precession of perihelia of planets in the solar system and the location of the innermost stable circular orbit ( isco ) in an accretion disk powering an x - ray source .these examples also serve to illustrate how near the calculations of such effects are to starting principles in a ` physics first ' approach to general relativity .nothing more than sketches of the calculations are intended . for more detail and the precise meaning of any quantities involved, the reader should consult any standard text .the schwarzschild geometry specified in describes the curved spacetime outside a static , spherically symmetric body of mass .this is the geometry outside the sun to an excellent approximation , and is the geometry outside a spherical black hole .the motion of test particles is specified by the principle of stationary proper time in . the argument in the square bracket of that equationcan be thought of as a lagrangian .lagrange s equations are the geodesic equations .time translation invariance implies a conserved energy per unit rest mass related to .spherical symmetry implies a conserved angular momentum per unit rest mass and are defined as an energy and angular momentum per unit rest energy rather than per unit rest mass as here . with that definition of both terms on the right hand side ofwould be divided by .that is the minor exception alluded to in the previous footnote . ] for orbits in the equatorial plane which is proportional to .a third integral of the motion arises just from the definition of proper time .this integral can be combined with the other to to find an effective energy integral for the radial motion :  [ sixone ] where   here , is the schwarzschild radial coordinate of the test particle and   is the proper time along its world line .the mass of the test particle is absent from these expressions .it cancels out because of the equality of gravitational and inertial mass .eq . has the same form as the energy integral for a newtonian central force problem .the first two terms in the effective potential have the same form as a newtonian gravitational potential with a newtonian centrifugal barrier .the third term provides a general relativistic correction to the newtonian effective potential .figure 2 shows its effects .circular orbits illustrate the importance of these effects .newtonian gravity permits only one stable circular orbit for each .but in general relativity there are _ two _ circular orbits for values of such at that used in figure 2 .there is a stable circular orbit at the _ minimum _ of the effective potential such as one approximating the orbit of the earth in its progress around the sun .in addition there is an unstable circular orbit at the radius of the _ maximum _ of the effective potential .the radii of the stable circular orbits are easily found from : ^{\frac{1}{2 } } \bigg\ } .\label{circ}\ ] ] for sufficiently small there are _ no _ stable circular orbits .that is because the effective potential is everywhere attractive for low .in contrast to newtonian physics , general relativity therefore implies that there is an _innermost ( smallest ) stable circular orbit _ ( isco ) whose radius is which is times the characteristic radius of the black hole .the isco is important for the astrophysics of black holes .the spectra of x - ray sources exhibit lines whose observed shapes can in principle be used to infer the properties of the black hole engine .the shape of the line is determined by several factors but importantly affected by the gravitational redshift .that is maximum for radiation from parts of the accretion disk closest to the black hole , ie from the isco .this defines the red end of the observed line .another important prediction of general relativity derivable from the effective potential is the shape of bound orbits such as those of the planets .the shape of an orbit in the equatorial plane ( ) may be specified by giving the azimuthal angle as a function of .the orbits close if the total angle swept out in the passage away from the inner turning point and back again is .this can be calculated from and the angular momentum integral .writing and using these two relations gives an expression for as a function of , , and which can be integrated .the result is ^{-1/2 }  \label{prec}\ ] ] where the integral is from the radius of the inner turning point to the outer one .when the relativistic term in is absent , for all and .that is the closing of the keplerian ellipses of newtonian mechanics .the relativistic correction to makes the orbit precess by small amount on each pass . to lowest order in this is in the solar system the precession is largest for mercury but still only per century .the confirmation of that prediction is an important test of general relativity .the purpose of this section was not to explain or even emphasize the two effects of general relativity on particle orbits that were described here .rather it was to show three things .first , that a standard technique developed in undergraduate mechanics can be used to calculate important predictions of general relativity .the second purpose was to show how close these important applications can come to starting principles in a ` physics first ' approach to teaching general relativity .introduce the schwarzschild geometry , use the principle of stationary proper time to find the geodesic equations or their integrals , use the effective potential method to qualitatively and quantitatively understand important properties of the orbits e.g. and .that is just three steps from the basic ideas of metric and geodesics to important applications .third , both of the applications treated here can be immediately related to contemporary observation and experiment .particle orbits in the schwarzschild geometry are not the only important problems in which the effective potential method is useful .motion in the geometry of a rotating black hole , the motion of test light rays in the schwarzschild geometry , and the evolution of the friedman - robertson - walker cosmological models are further examples where it can be usefully applied .a one quarter ( 28 lectures ) ` physics first ' course in general relativity has been a standard junior / senior elective for physics majors at the university of california , santa barbara for approximately thirty years . in the limited span of a quarterthe author is usually able to review special relativity , motivate gravity as geometry , derive the orbits in the schwarzschild geometry in detail , describe the experimental tests , introduce black holes , and develop the friedman - robertson - walker cosmological models .a semester provides more opportunities .  at santa barbara this course is routinely taught by faculty from many different areas of physics general relativity of course , but also elementary particle physics and astrophysics .it has been taught by both theorists and experimentalists . for some of these colleagues teachingthis course was their first serious experience with general relativity .they usually report that they were successful and enjoyed it .  in the author s experience studentsare excited by general relativity and motivated to pursue it .often it is their first experience with a subject directly relevant to current research .it is one of the few contemporary subjects that can be taught without quantum mechanics or electromagnetism .the author has written a text based on the ` physics first ' approach which comes with a solutions manual available to instructors for the approximately 400 problems of graded levels of difficulty .spergel _ et al ._ , _ first year wilkinson microwave anisotropy probe ( wmap ) observations : determination of cosmological parameters _ , _ ap. j. suppl . _* 148 * , 175194 ( 2003 ) ; j.l .tonry , , _ cosmological results from high - z supernovae _ , _ ap . j. _ * 594 * , 124 ( 2003 ) .see , e.g. c.s .reynolds and m.a .nowak , _ fluorescent iron lines as a probe of astrophysical black hole systems _ , _ physics reports _ * 377 * , 389 ( 2003 ) ; a.c .fabian , k. iwasawa , c.s .reynolds , a.j .young , _ broad iron lines in active galactic nuclei _ , _ publications of the astronomical society of the pacific _ * 112 * , 1145 ( 2000 ) .shapiro , _ solar system tests of general relativity _, in _ general relativity and gravitation : 12th international conference on general relativity and gravitation _ ed by n. ashby , d.f .bartlett , and w. wyss , ( cambridge university press , cambridge , 1990 ) , 313 - 330 .

Abstract: einstein s general relativity is increasingly important in contemporary physics on the frontiers of both the very largest distance scales ( astrophysics and cosmology ) and the very smallest(elementary particle physics ) . this paper makes the case for a ` physics first ' approach to introducing general relativity to undergraduate physics majors .


this paper is devoted to the numerical study of a macroscopic version of the vicsek model which describes swarming behavior .this macroscopic model has been derived in from the microscopic vicsek model .the goal of this work is to provide a numerical validation of the macroscopic model by comparing it with simulations of the microscopic model .  the vicsek model is widely used to describe swarming behavior such as flock of birds , schools of fish ( in this case the model is combined with an attractive - repulsive force ) or recently the motion of locusts .in this model , individuals have a constant velocity and they tend to align with their neighbors . despite the simplicity of the model , a lot of questions remain open about it .a first field of research concerns phase transitions within the model depending on the level of noise .another question arises from the long time dynamics of the model : is there convergence to a stationary state of the system ? from another perspective ,since collective displacements in natural environment can concern up to several million individuals , it is natural to look for a macroscopic version of the vicsek model . on the one hand ,macroscopic models constitute powerful analytical tools to study the dynamics at large scales . on the other hand ,the related numerical schemes are computationally much more efficient compared with particle simulations of a large number of interacting agents . in , a macroscopic vicsek model ( mv ) has been derived from a large scale limit of the microscopic vicsek model .the macroscopic model is obtained from a rigorous perturbation theory of the original vicsek model .another macroscopic model is obtained in based on more phenomenological closure assumptions .the mv model presents several specificities which make the model interesting .first , it is a non - conservative hyperbolic system and secondly it involves a geometric constraint .these are the consequences at the macroscopic level of two specificities of the microscopic model : the total momentum is not conserved by the particle dynamics and the speed of the particles is constant .the first property is an intrinsic property of self - propelled particles and the second property is a usual assumption in the models of collective displacements .up to our knowledge the theory of such systems is almost empty .non - conservative systems have been studied in the literature but none of them involve geometric constraints .  in this work , since a theoretical framework for such systems is not available , we adopt several approaches .first , we introduce a conservative formulation for the 1d formulation of the mv model which is equivalent to the initial one for smooth solutions only . with this conservative formulation, we can use standard hyperbolic theory to build riemann problem solution and shock capturing schemes . the numerical scheme based on the conservative formulationis called _ conservative method_. but since the equivalence with the original formulation is only valid for smooth solutions , there is no guarantee that the conservative formulation gives the right answer at shocks .for this reason , we introduce another formulation of the mv model where the constraint is treated through the relaxation limit of an unconstrained conservative system .this formulation leads to a natural numerical scheme based on a splitting between the conservative part of the equation and the relaxation .this scheme will be referred to as the _ splitting method_. for comparison purposes , two other numerical schemes are also used , an _ upwind scheme _ and a _ semi - conservative _ one ( where only the mass conservation equation is treated in a conservative way ) .the numerical simulations of the mv model reveal that the numerical schemes all agree on rarefaction waves but disagree on shock waves . to determine the correct solution, we use the microscopic model in a regime where its solution is close to that of the macroscopic model . in practice, this corresponds to regimes where the number of particles per domain of interaction is high .the splitting method turns out to be in good agreement with particle simulations of the microscopic model , by contrast with the other schemes .in particular , for an initial condition with a contact discontinuity , the solution given by the conservative form is simply a convection of the initial condition whereas the splitting method and the particle simulations agree on a different and more complex solution .these results show first that the mv model well describes the microscopic model in the dense regime .secondly , that the correct formulation of the mv model is given by the limit of a conservative equation with a stiff relaxation term .the theoretical and numerical studies of the mv model highlight the specificity of non - conservative hyperbolic models with geometric constraints .more theoretical work is necessary in order to understand why the splitting method matches the microscopic model whereas the other methods do not . in particular , an extension of the theory developed in to non - conservative relaxed modelswould be highly desirable .the outline of the paper is as follows : first , we present the vicsek and mv models in section [ sec : presentation_mv ] .then , we analyze the mv model and give two different formulations of the model in section [ sec : macro_equa ] . we develop different numerical schemes based on these formulations and we use them to numerically solve different riemann problems in section [ sec : num_simu_cva_macro ] .finally , we compare simulations of the microscopic model with those of the macroscopic system in the same situations in section [ sec : micro_cva ] . finally , we draw a conclusion .at the particle level , the vicsek model describes the motion of particles which tend to align with their neighbors .we denote by the position vector of the particle and by its velocity with a constant speed ( ) . to simplify , we suppose that the particles move in a plane .therefore and . the vicsek model at the microscopic levelis given by the following equations ( in dimensionless variables ) : where is the identity matrix and the symbol denotes the tensor product of vectors . is the intensity of noise , is the brownian motion and is the direction of mean velocity around the particle defined by : where defines the radius of the interaction region . equation ( [ cont_orient_bis ] ) expresses the tendency of particles to move in the same direction as their neighbors . the operator is the orthogonal projector onto the plane perpendicular to .it ensures that the speed of particles remains constant .this model is already a modification of the original vicsek model , which is a time - discrete algorithm .the macroscopic vicsek model ( mv ) describes the evolution of two macroscopic quantities : the density of particles and the direction of the flow .the evolution of and is governed by the following equations : where , and are some constants depending on the noise parameter .the expressions of , and are given in appendix [ sec : coeff ] .by contrast with the standard euler system , the two convection coefficients and are different .the other specificity of this model is the constraint .the operator ensures that this constraint is propagated provided that it is true at the initial time . the passage from ( [ cont_pos_bis])-([cont_orient_bis ] ) to ( [ eq : macro_rho])-([eq : macro_omega])-([eq : macro_constraint ] )is detailed in .we note that vortex configurations are special stationary solutions of this model in two dimensions ( see appendix [ sec : moulin ] ) . up to our knowledge , this is the first swarming model which have such analytical solutions .to study model ( [ eq : macro_rho])-([eq : macro_omega])-([eq : macro_constraint ] ) , we first use the rescaling . then equations ( [ eq : macro_rho])-([eq : macro_omega])-([eq : macro_constraint ] ) are written : with and . in the sequel , we drop the primes for clarity .we refer to the appendix [ sec : coeff ] for the computation of and and we just mention that we have ( see figure [ fig : div_c1 ] ) :   in two dimensions , we can use a parameterization of in polar coordinates : .therefore , equations ( [ eq : macro2_rho])-([eq : macro2_omega ] ) can be rewritten as : in this section , we suppose that and are independent of meaning that we are looking at waves which propagate in the x - direction . under this assumption ,the system reads : with .\ ] ] the characteristic velocities of this system are given by \ ] ] with .therefore , the system is strictly _a possible choice of right eigenvectors is the two fields are genuinely nonlinear except at , and at the extrema values of which satisfy : .\ ] ]   and depending on ( in this graph ) . for each curve, there exists a unique extremum ( and ) which corresponds to a degeneracy of the system . ]  the riemann invariants of the system ( [ eq:2d_x_nc ] ) are given by : the integral curve and starting from are given by : these are the rarefaction curves .to select the physically admissible rarefaction curve , we remark that must grow from the left to right states .the proofs of these elementary facts are omitted .the quantities as functions of are depicted in figure [ fig : vp ] .   for non - conservative systems , shock waves are not uniquely defined . however , in the present case , a conservative formulation of the system can be found in dimension 1 .indeed , it is an easy matter to see that , if , system ( [ eq:2d_x_nc ] ) can be rewritten in conservative form : with : however , the functions and are singular when which means that the conservative form is only valid as long as stays away from .the conservative form ( [ eq:1dcons_x ] ) leads to the following rankine - hugoniot conditions for shock waves : two states and are connected by a shock wave traveling at a constant speed if they satisfy :   we can combine the two equations of the system ( [ eq : rh ] ) to eliminate the constant and this leads to the following expression of the shock curve : this equation must be numerically solved .the entropic part of the shock curve is determined by the requirement that must satisfy the lax entropy condition . in figure[ fig : u_lmr ] , we give an example of a solution of a riemann problem obtained by computing the intersection of the shock and rarefaction curves .   and ( solid line for shock waves and dotted line for rarefaction waves ) .in this example , the solution is given by two shock waves . ]we are going to prove that the mv model ( [ eq : macro2_rho])-([eq : macro2_omega])-([eq : macro2_constraint ] ) can be seen as the relaxation limit of a conservative hyperbolic model with a relaxation term .this link will be used later to build a new numerical scheme .more precisely , we introduce the relaxation model : in this model , the constraint is replaced by a relaxation operator .formally , in the limit , we recover the constraint .[ ppo : umv_mv ] the relaxation model ( [ eq : umv_rho])-([eq : umv_rho ] ) converges to the mv model ( [ eq : macro2_rho])-([eq : macro2_omega])-([eq : macro2_constraint ] ) as goes to zero .we define .suppose that as goes to zero : then , which generically implies that ( except where which one assumes to be a negligible set ) .therefore , we have : then since , we have : and consequently when : for a real number to be determined . taking the scalar product of ( [ eq : om_0_alpha ] ) with and using ( [ eq : om_0 ] ) , we find : using the conservation of mass ( ) , we finally have : therefore , the relaxation term satisfies : \omega^0 + o({\varepsilon}).\ ] ] inserting in ( [ eq : umv_rho])-([eq : umv_omega ] ) and taking the limit , we recover the mv model ( [ eq : macro2_rho])-([eq : macro2_omega ] ) at the first order in .  as for the mv model, we can also analyze the hyperbolicity of the left hand side of ( [ eq : umv_rho])-([eq : umv_omega ] ) .the eigenvalues are given by : where denotes the -coordinate of and .the system is hyperbolic if and only if .as we can see in figure [ fig : hyp_umv2 ] , for , is positive for any values of the noise parameter . in particular, this implies that the relaxation model is hyperbolic for every .   depending on . the relaxation model ( [ eq : umv_rho])-([eq : umv_rho ] )is hyperbolic when the speed is below this curve . at the limit , andtherefore the relaxation model is hyperbolic for any in this limit . ]we propose four different numerical schemes to solve the mv model .the first two schemes originate from the discussions of the previous section , the two other one are based on the non - conservative form of the mv model .+ we use the following notations : we fix a uniform stencil ( with ) and a time step .we denote by the value of the mass and flux direction at the position and at time .   herewe use the conservative form of the mv model ( [ eq:1dcons_x ] ) : with and .we use a roe method to discretize this equation : where the intermediate flux is given by : and is the jacobian of the flux : \ ] ] calculated at the mean value   as mentioned earlier , the conservative form is only valid when does not cross a singularity or ( i.e. ) .nevertheless , numerically we can still use the formulation ( [ eq : cons ] ) when changes sign .moreover , since is an even function , this only gives . to determine the sign of , we use an auxiliary value which we update with the upwind scheme ( [ eq : upwind ] ) .the sign of is then determined using the sign of .the next scheme uses the relaxation model ( [ eq : umv_rho])-([eq : umv_omega ] ) .the idea is to split the relaxation model in two parts , first the conservative part : and then the relaxation part : this system reduces to : since this equation only changes the vector field in norm ( i.e. ) , we can once again reduce this equation to : equation ( [ eq : omega2 ] ) can be explicitly solved : with .we indeed take the limit of this expression and replace the relation term by a mere normalization : .the conservative part is solved by a roe method with a roe matrix computed following page 156 .we present two other numerical schemes based on the non - conservative formulation of the mv model .  *( i ) upwind scheme *  the method consists to update the value of with the formula : where and are ( respectively ) the positive and negative part of , defined such that and and are computed using an explicit diagonalization of .  *( ii ) semi - conservative scheme *  one of the problem with the upwind scheme is that it does not conserve the total mass ( ) . in order to keep this quantity constant in time, we use the equation of conservation of mass ( [ eq : macro2_rho ] ) in a conservative form : with .therefore , a conservative numerical scheme associated with this equation would be : where is the numerical estimation of the flux at the interface between and . to estimate numerically this flux , we use the following formula with : where the intermediate value is given by and the first line of the absolute value of .  for the estimation of the angle , we keep the same scheme as for the upwind scheme .this numerical scheme uses one conservative equation ( for the mass ) and a non - conservative equation ( for the angle ) .it is thus referred to as the _ semi - conservative _ scheme .   to compare the various numerical schemes, we use a riemann problem as initial condition .we choose solutions which consist of a rarefaction wave ( figure [ fig : rare11 ] ) or a single shock wave ( figures [ fig : choc1]-[fig : choc111 ] ) .we take the following parameters : , the length of the domain is units and the discontinuity for the riemann problem is at ( the middle of the domain ) .the simulations are run during two time units with a time step and a space step .for these values , the courant number ( ) is .we use homogeneous neumann conditions as boundary conditions .  for the rarefaction wave ,we take : the numerical schemes capture well the theoretical solution ( see figure [ fig : rare11 ] ) .  for the shock wave ,we choose : for these values , the shock speed computed with ( [ eq : hugo ] ) is .the results of the numerical simulations using the four schemes are given in figure [ fig : choc1 ] .the numerical solutions are in accordance with the theoretical solution given by the conservative formulation for all the numerical schemes .nevertheless , the conservative scheme is in better accordance with this solution . for the other schemes , the shock speed differs slightly . + a second example of a shock waveis computed using the following initial condition : the solutions given by the numerical schemes are very different .only the conservative method is in agreement with the solution given by the conservative formulation .but the conservative formulation is not necessary the right one .indeed , in the next section , particle simulations show that the right solution is not given by the conservative formulation but rather by the splitting method .  ) given by a rarefaction curve ( solid line ) and the numerical solutions ( points ) , ( blue ) and ( green ) as functions of space .the simulations are run during 2 time units , with a time step and a space step ( cfl= ) . ]  ) ]  ) ]in this part , we would like to validate the macroscopic vicsek model by the simulation of the microscopic vicsek .the macroscopic model relies on the fact that the particle distribution function is at local equilibrium given by a von mises distribution ( see ) : where is set by the normalization condition where is the modified bessel function of order 0 ] .the goal of this section is to show numerically that the particle distribution of the microscopic vicsek model is close in certain regimes to this von mises distribution .  to this aim , in appendix[ sec : num_scheme_ps ] we propose a numerical scheme to solve system ( [ cont_orient_bis ] ) .the setting for our particle simulations is as follows : we consider a square box with periodic boundary conditions . as initial condition for the position , we choose a uniform random distribution in space .the velocity is initially distributed according to a uniform distribution on the unit circle .  during the simulation ,we compute the empirical distribution of the velocity direction and of the mean velocity of particles .we then compare this empirical distribution with its theoretical distribution given by ( [ eq : m_von_mises ] ) . in figure[ fig : distri_speed ] , we give an example of a comparison between the distribution of the velocity direction and the theoretical distribution predicted by the theory .( with ) compared with its theoretical distribution after time units of simulation .right figure : the corresponding particle simulation. parameters of the simulation : , ( domain size ) , number of particles , , , , . ]( with ) compared with its theoretical distribution after time units of simulation .right figure : the corresponding particle simulation .parameters of the simulation : , ( domain size ) , number of particles , , , , . ]( [ eq : total_flux ] ) for different values of . parameters of the simulation : , ( domain size ) , number of particles , radius of interaction , unit time , the simulations are run during unit time . ]  since the distribution of velocity converges , we have a theoretical value for the mean velocity .we denote by the mean velocity of particles and the theoretical value given by the stationary distribution : at least locally in , we have that . in figure[ fig : mean_speed ] , we compare the two distributions for different values of the noise and we can see that the two distributions are in good agreement .we also observe a smooth transition from order ( ) to disorder ( ) as it has been measured in the original vicsek model .the situation is different when we look at a larger system .we still have convergence of the velocity distribution of particles to a local equilibrium , but the mean direction now depends on .therefore the mean velocity of the particles in all the domain differs from the expected theoretical value ( [ eq : m_von_mises])-([eq : total_flux ] ) .we illustrate this phenomena in figure [ fig : mean_speed_diff ] : we fix the density of particles and we increase the size of the box .as we can observe , the mean velocity ( [ eq : total_flux ] ) has a smaller value when the size of the box increases .this phenomena has been previously observed in .the mean velocity can also differ from the expected theoretical value ( [ eq : m_von_mises])-([eq : total_flux ] ) when the density of particles is low . in figure[ fig : mean_speed_diff2 ] , we fix the size of the box ( ) and we increase the density of particles ( the density is given by the number of particles inside the circle of interaction ) . at low density ,the mean velocity is much more smaller than the theoretical prediction .but as the density of particles increases , the mean velocity grows ( see also ) and moreover converges to . because of that , a dense regime of particles has to be used in the following in order to numerically compare the microscopic model with the mv model .( [ eq : total_flux ] ) for different values of .we use different domain sizes and we keep the same density of particles . as the domain size increases , the total flux decreases which means that particles are less aligned globally .parameters of the simulations : ( domain size ) , number of particles , radius of interaction , time units , the simulations are run during time units . ]( [ eq : total_flux ] ) for different values of .we change the density of particles ( given by the mean number of neighbors in unit of radius of interaction ) .when we increase the mean number of neighbors , particles are more aligned .parameters of the simulations : ( domain size ) , number of particles and , radius of interaction , time units , the simulations are run during time units . ]   we now compare the evolution of the two macroscopic quantities and for the two models . we have seen that the different schemes applied to the macroscopic equation can give different solutions ( see figure [ fig : choc111 ] ) .therefore , we expect that particle simulations will indicate what is the physically relevant solution of the macroscopic equation .we first briefly explain how we proceed to run the particle simulations of a riemann problem ( see also appendix [ sec : num_scheme_ps ] ) .first , we have to choose a left state , a right state and the noise parameter .then we distribute a proportion of particles uniformly in the interval  ] .then , we generate velocity distribution for the particles according to the distribution ( [ eq : m_von_mises ] ) with on the left side and on the right side .we use the numerical scheme given in appendix [ sec : num_scheme_ps ] to generate particle trajectories . to make the computation simpler , we choose periodic boundary conditions . therefore the number of particles is conserved . as a consequence ,there are two riemann problems corresponding to discontinuities at and at or ( which is the same by periodicity ) .we use a particle - in - cell method to estimate the two macroscopic quantities : the density and the direction of the flux ( which gives ) . in order to reduce the noise due to the finite number of particles , we take a mean over several simulations to estimate the density and ( simulations in our examples ) .  in figure[ fig : exple_2d ] , we show a numerical solution for the following riemann problem : using particle simulations and the macroscopic equation .we represent the solutions in a 2d representation .since the initial condition is such that the density and the direction are independent of the -direction , we only represent and along the -axis in the following figures .  in figure [fig : choc_d02_particle_split ] , we represent the two solutions ( the particle and the macroscopic one ) with only a dependence in the -direction .three quantities are represented : the density ( blue ) , the flux direction ( green ) and the variance of the angle distribution ( red ) .the macroscopic model supposes that the variance of should be constant everywhere .nevertheless , we can see that the variance is larger in regions where the density is lower . for and , we see clearly the propagation of a shock in the middle of the domain and a rarefaction at the boundary .the cpu time for one numerical solution at the particle level is about seconds with the parameters given in figure [ fig : choc_d02_particle_split ] . for the macroscopic equation ,the cpu time is about second which represents a cost reduction of three orders of magnitude compared with the particle simulations .since we have to take a mean over many particle simulations , the cost reduction is even larger .  in figure[ fig : choc111_particle ] , we use the same riemann problem to set the initial position as in figure [ fig : choc111 ] both with ( [ eq : riemann_shock2 ] ) .we use a larger domain in ( space units ) in order to avoid the effect of the periodic boundary condition .the upwind scheme and the semi - conservative method are clearly not in accordance with the particle simulations . moreover , the splitting method is in better agreement with the particle simulation since the shock speed is closer to the values given by the particle simulations than that predicted by the conservative scheme .finally , our last simulations concern a contact discontinuity .we simply initialize with : i.e. we reflect the angle with respect to the x - axis across the middle point . a natural solution for this problemis the contact discontinuity propagating at speed : with when and when .this is the solution provided by the conservative scheme ( figure [ fig : theta_pm ] ) .but surprisingly , the splitting method and the particle simulation agree on a different solution .indeed , the solutions given by the particles and the splitting method are in fairly good agreement with each other , which seems to indicate that the `` physical solution '' to the contact problem ( [ eq : theta_pm ] ) is not given by the conservative formulation ( [ eq : theta_pm_sol ] ) but by a much more complex profile .the constraint of unit speed drastically changes the profile of the solution compared with what would be found for a standard system of conservative laws .computed with particle simulations ( left ) and the macroscopic equations ( right ) .we initialize with a riemann problem ( [ eq : ic_shock ] ) .numerical parameters for the particle simulations : particles , , , , , we take a mean over computations .numerical parameters for the macroscopic model : ( cfl=0.416 ) , we use the splitting method .the simulations are run during 2 time units . ]   computed with particle simulations ( left ) and the macroscopic equations ( right ) .we initialize with a riemann problem ( [ eq : ic_shock ] ) .numerical parameters for the particle simulations : particles , , , , , we take a mean over computations .numerical parameters for the macroscopic model : ( cfl=0.416 ) , we use the splitting method .the simulations are run during 2 time units . ]+  ) with computed using the macroscopic model and particle simulations of the microscopic model ( see figure [ fig : choc_d02_particle_split ] ) .numerical parameters for the macroscopic model : ( cfl=0.778 ) .numerical parameters for the particle simulation : particles , , , , and .we take a mean over simulations .the simulations are run during 6 time units . since , fluctuations are higher ( see figure [ fig : mean_speed_diff ] ) , we have to increase the density of particles to reduce this effect . ]+ ) computed with the conservative method ( top ) , the splitting method ( down ) and with particle simulations ( dots ) . numerical parameters for the macroscopic model : ( cfl=0.416 ) .numerical parameters for the particle simulations : particles , , , , , .we take a mean over simulations .the simulations are run during 2 time units.,title="fig : " ]   + ) computed with the conservative method ( top ) , the splitting method ( down ) and with particle simulations ( dots ) .numerical parameters for the macroscopic model : ( cfl=0.416 ) .numerical parameters for the particle simulations : particles , , , , , .we take a mean over simulations .the simulations are run during 2 time units.,title="fig : " ]in this work , we have numerically studied both the microscopic vicsek model and its macroscopic version . due to the geometric constraint that the velocity should be of norm one , the standard theory of hyperbolic systems is not applicable .therefore , we have proposed several numerical schemes to solve it . by comparing the numerical simulations of the microscopic and macroscopic equations, it appears that the scheme based on a relaxation formulation of the macroscopic model , used in conjunction with a splitting method is in good agreement with particle simulations .the other schemes do not show a similar good agreement .in particular , with an initial condition given by a contact discontinuity , the splitting method and the microscopic model provide a similar solution which turn to be much more complex than what we could be expected .these results confirm the relevance of the macroscopic vicsek model .since the cpu time is much lower with the macroscopic equation , the macroscopic vicsek model is an effective tool to simulate the vicsek dynamics in a dense regime of particles .many problems are still open concerning the macroscopic vicsek model .we have seen that the splitting method gives results which are in accordance with particle simulations .but , we have to understand why this particular scheme captures well the particle dynamics better than the other schemes .since the macroscopic equation has original characteristics , this question is challenging .another point concerns the particle simulations .we have seen that the particles density has a strong effect on the variance of the velocity distribution .when the density is low , the variance is larger .the macroscopic equation does not capture this effect since the variance of the distribution is constant .works in progress aims at taking into account this density effect .[ sec : coeff ]  the analytical expression of the coefficient involved the distribution of the local equilibrium ( [ eq : m_von_mises ] ) .the two other coefficients ( and ) involve also the solution of the following elliptic equation : + { \mathrm{e}}^{x / d } g =  -(1-x^2)^{3/2 } { \mathrm{e}}^{x / d},\ ] ] on the interval .if we define the function and , these macroscopic coefficients can be written as : in the above expressions , we can see that .now we are going to explore two asymptotics of when the parameter is small or large .[ lem : g ] let be the solution of equation ( [ eq : el_fort ] ) .we have the asymptotics : +   o(d),\\   \label{eq : g_d_large }   g & \stackrel{d\rightarrow \infty}{\sim } & -\frac{1}{2}\,\sqrt{1-x^2 } +   \frac{1}{12\,d } x \sqrt{1-x^2 } + o\left(\frac{1}{d}\right ) .\end{aligned}\ ] ]  introducing the hilbert space : we have already seen in that there exists a unique solution of ( [ eq : el_fort ] ) .moreover this solution is negative .  to derive the asymptotic behavior of depending on , we first develop ( [ eq : el_fort ] ) : + ( 1-x^2)\,\frac{1}{d } \partial_x g -   \frac{g}{1-x^2 } = ( 1-x^2)^{1/2}.\ ] ] when , we have : on the interval $ ] for all . since belongs to , we also have the boundary condition , so converges to when .+ to derive the next order of convergence in the limit , we normalize with , which gives : + ( 1-x^2 ) \partial_x \widetilde{g } -   d\,\frac{\widetilde{g}}{1-x^2 } = ( 1-x^2)^{1/2}.   \end{aligned}\ ] ] in the limit , we deduce that : which has an explicit solution : .since , we have .numerically , we find that but the proof is still open .this formally proves ( [ eq : g_d_low ] ) .when , ( [ eq : el_fort_dev ] ) gives : - \frac{g_0}{1-x^2 } = ( 1-x^2)^{1/2}.   \end{aligned}\ ] ] a simple calculation shows that is a solution of ( [ eq : d_inf ] ) . + to derive the next order of convergence , we look at the difference , which satisfies ( see ( [ eq : el_fort_dev ] ) and ( [ eq : d_inf ] ) ) : + ( 1-x^2)\,\frac{1}{d } \partial_x v -   \frac{v}{1-x^2 } = -(1-x^2)\,\partial_x g_0.\ ] ] in the limit , satisfies :  -   \frac{v}{1-x^2 } = -\frac{1}{2 } x\sqrt{1-x^2}.  \end{aligned}\ ] ] a simple calculation shows that is solution of ( [ eq : g_1_inf ] ) .therefore we formally have the expression ( [ eq : g_d_large ] ) in the proposition .  in figure[ fig : g ] , we compute numerically the function ( [ eq : el_fort ] ) .we use a finite element method with a space step .the two asymptotics of when and are computed in figure [ fig : g1 ] .[ ppo : c_12 ] the two coefficients and defined ( resp . ) by the equations ( [ eq : c_1 ] ) and ( [ eq : c_2 ] ) satisfy the following asymptotics :   we have an explicit expression for the coefficient using the change of unknowns : where .the expressions of ( [ eq : c_1_asymp_0 ] ) and ( [ eq : c_1_asymp_inf ] ) are simply deduced by a taylor expansion of the last expression .  for the coefficient , we insert the development of ( [ eq : g_d_large ] ) in expression ( [ eq : c_2 ] ) .  the behavior of when is more difficult to analyze .the density probability used in formula ( [ eq : c_2 ] ) becomes singular in this limit .nevertheless , due to the expression of , the density converges to a dirac delta at which explains why . to capture the next order of convergence , we need to find the second order correction of in the limit which is not availablehowever , numerically we find that :   in figure [ fig : div_c1 ] , we numerically compute the coefficients , and their asymptotics .   of ( [ eq : el_fort ] ) for different values of the parameter .we have the following asymptotics ( see lemma [ lem : g ] ) : and . ]   of when .the red curve is the theoretical asymptotic limit : ( see lemma [ lem : g ] ) .right figure : the first correction of when .the red curve is the theoretical asymptotic : ( see lemma [ lem : g]).,title="fig : " ]  of when .the red curve is the theoretical asymptotic limit : ( see lemma [ lem : g ] ) .right figure : the first correction of when .the red curve is the theoretical asymptotic : ( see lemma [ lem : g]).,title="fig : " ]   and ( solid lines ) and their two asymptotics ( dashed lines ) ( see proposition [ ppo : c_12 ] ) computed with .,title="fig : " ]  and ( solid lines ) and their two asymptotics ( dashed lines ) ( see proposition [ ppo : c_12 ] ) computed with .,title="fig : " ]in this appendix , a vortex configuration is exhibited as a stationary solution of the mv model ( [ eq : macro_rho])-([eq : macro_omega])-([eq : macro_constraint ] ) in dimension 2 .a stationary state of the mv model has to satisfy : introducing the polar coordinates , , , where and , we are able to formulate the proposition :  [ ppo : vortex ] the following initial condition is a stationary state of the mv model ( [ eq : stationnary ] ) : where is a constant .  with the expression of and given by ( [ eq : init_vortex ] ) , the divergence of the mass is zero and the gradient of is orthogonal to , therefore the system ( [ eq : stationnary ] ) reduces to : or in polar coordinates : since , we can easily check that the solution of this equation is given by .in the limit , an explicit euler method for the differential system ( [ cont_pos_bis])-([cont_orient_bis ] ) imposes a restriction time step condition of .therefore , we develop an implicit scheme for this system . the idea is to go back to the original vicsek model ( see ) .we use the formulation : where and is the average velocity ( [ cv_moy_3_bis ] ) .when , we recover exactly the original vicsek model .( [ eq : mid_point ] ) can in fact be solved explicitly .first , we have to remember that belongs to the unit circle ( i.e. ). then we use that is the orthogonal projection of on the orthogonal plan of .therefore and are on the circle with center and radius ( see figure [ fig : tech_circ ] ) .this fully defines since and are the two intersection points of the unit circle and the circle . denoting the angle of the unit vector , we easily check that we have in terms of angles : to take into account the effect of the noise , we simply add a random variable : where is a standard normal distribution independent of .    1 .choose a riemann problem and .2 .  initiate particles according to the distributions and .3 .  let evolve the particles in time using the time - discretization ( [ eq : discrete_micro_angle ] ) of equation ( [ cont_orient_bis ] ) .4 .  compute the mass and the direction of the flux using particle - in - cell method in order to compare the simulation with the one of the mv model .

Abstract: the vicsek model is a very popular individual based model which describes collective behavior among animal societies . a macroscopic version of the vicsek model has been derived from a large scale limit of this individual based model . in this work , we want to numerically validate this macroscopic vicsek model ( mv ) . to this aim , we compare the simulations of the macroscopic and microscopic models one with each other . the mv model is a non - conservative hyperbolic equation with a geometric constraint . due to the lack of theory for this kind of equations , we derive several equivalents for this system leading to specific numerical schemes . the numerical simulations reveal that the microscopic and macroscopic models are in good agreement provided that we choose one of the proposed formulations based on a relaxation of the geometric constraint . this confirms the relevance of the macroscopic equation but it also calls for a better theoretical understanding of this type of equations .   * key words : * individual based model , hyperbolic systems , non - conservative equation , geometric constraint , relaxation , splitting scheme  * ams subject classifications : * 35q80 , 35l60 , 35l65 , 35l67 , 65m60 , 82c22 , 82c70 , 82c80 , 92d50  * acknowledgments : * the authors wish to thank pierre degond for his support and his fruitful suggestions . they also would like to thank guy theraulaz , jacques gautrais and richard bon for helpful discussions . this work was supported by the french agence nationale pour la recherche ( anr) in the frame of the contract anr-07-blan-0208 - 03 entitled panurge.


as astronomical and cosmological observations always demand higher sensitivities , importance of large focal planes with detectors at cryogenic temperatures ever increases .the observation of the cosmic microwave background ( cmb ) polarization , which is one of the best probes for studying the early universe  , is an outstanding example where such cryogenic large detector arrays are needed .recently , a few cmb polarization experiments have reported the detection of the faint sub - degree - scale b - mode signal , the odd - parity mode of the cmb polarization pattern  .the use of a large array with an order of 10,000 polarization detectors , such as the focal plane of polarbear-2 ( pb-2 )  , spt-3 g  , and bicep-3  , is essential for characterizing the b - mode power spectrum .the need of a large focal plane leads to a challenge in the thermal design , as large thermal load is expected from a large optical window . in order to keep the detector sensivitity high enough , efficient infra - red ( ir ) filters should be considered .one promising solution to this problem is to introduce an alumina plate as an ir filter with high thermal conductivity  .the alumina filter absorbs the incident ir emission efficiently .the absorbed power is conducted from the central region of the filter to the edge , which is thermally connected to the cryogenic stage .the thermal conductance of alumina in the temperature range between 50 and 100 k is very high ; three orders of magnitude as large as those of conventional plastic filters , such as ptfe , nylon , and black polyethylene .the temperature gradient and optical loading of the alumina plate are thus very small even with a large diameter , which is ideal as the ir filter material .a remaining technological challenge on the use of alumina ir filter is to establish ar coatings for a large frequency coverage .alumina is known to be highly reflective in the millimeter wavelength .we thus need an ar coating on each surface of the alumina disc .ar coating on such a large surface is easily peeled off owing to the thermal shock at cold temperatures . the aforementioned next - generation cmb projects demand multi - layer ar coatings on large ir filters with a typical diameter of 500 mm .in the previous studies , the polarbear group attempted to develop a two - layer ar coating method with epoxy on an alumina surface , with a diameter of 50 mm  . however , the epoxy and alumina cracked during thermal cycling . to avoid the cracking , stress - relief grooves were adopted to reduce the stress between alumina and the epoxy layer . this technology was also used in the bicep-3  group , which succeeded in making an single - layer ar coating with epoxy for a diameter of about .however , this method is expensive . also , it is not easy to maintain the thickness uniformity of the ar layers on the thin alumina plate .alternatively , the actpol  group succeeded in making two- and three - layer ar coatings on silicon lenses , whose diameters were approximately 300 mm .they tried and succeeded using the technology of subwavelength grating ( swg ) .it is theoretically possible to apply this technology to an alumina surface ; however , it is difficult because the dicing blade is subject to wear so that groove pitch and depth are changed .it is also difficult to make a large ingot of silicon .machining the material is also very expensive .yet another method studied for the spider project is a polyimide sheet as an ar coating on the sapphire with a diameter of about 250 mm .the spider method is very easy to perform and less expensive than the method with epoxy or swg . finally , the first application of thermal sprayed mullite as ar coating was made on a 50 mm diameter alumina disc  .it is also possible to tune the dielectric constant of a thermal sprayed layer by spraying microspheres and alumina powder with different ratios . a three - layer thermal sprayed broadband ar coating that achieves over a 100 % fractional bandwidth has been demonstrated on a 50 mm diameter alumina disc  ..  in this study , we newly employ the polyimide foam coating on the mullite coating described above to establish a two - layer ar coating method .we apply this technique to a thin and large alumina disc with a 460 mm diameter as an ir filter , and demonstrate the performance of the ar coating .we fabricated a 2 mm thick and 460 mm diameter alumina filter , and demonstrated the performance of the new ar coating .our development focuses on the use in future cmb measurements but the method developed can also be used for other applications . in section [ sec : filterdesign ] we describe the design of a large - diameter alumina ir filter with our new ar coating method .section [ sec : materialproperties ] explains our measurements of material properties .the fabrication process is detailed in section [ sec : fabrication ] .section [ sec : characterization ] shows our characterization of the ar coatings on a large - diameter alumina ir filter .we discuss our results in section [ disc ] and give conclusions in section [ sec : conclusion ] .to be concrete , our ar - coating deveopment targets next - generation cmb experiments , where typical requirements on the ir filter and the ar coating are as follows :   * the filter diameter must be extendable to 450 mm or larger ; * the ar coating is robust against thermal shock , so that it is usable in cryogenic temperatures ; * the uniformity of the additional optical path length due to the ar coating should be less than , where is the wavelength of the incoming electromagnetic wave ; * transmittance should be above 95 % at the detection bands , which we assume 95 ghz and 150 ghz with a fractional bandwidth for each ; * the 3 db cutoff frequency should be below 1 thz .the filter diameter requirement satisfies the requirement for a large focal plane with o(10000 ) superconducting millimeter - wave sensors .the requirement on robustness against thermal shock is important for use at low temperatures .the thickness of the ar coating is designed to be so that the reflected beam is canceled .the uniformity must be better than , which corresponds to a transmittance uncertainty of 1 % .the requirement on transmittance in detection bands comes from the design sensitivity of experiment .the requirement on the cutoff frequency is derived to reduce the ir emission whose main frequency range is 10 - 100 thz .figure [ fig : af_config ] shows a schematic view of two - layer ar coatings on a large - diameter alumina disc .the design satisfies the typical requirements mentioned above .the ar coatings consist of the thermally - sprayed mullite , which has the index of refraction ( ior ) of 2.5 as the first layer and polyimide foam ( called skybond foam  hereafter ) as the second layer .a thin layer of low - density polyethylene ( ldpe ) is placed in between two ar layers as a glue layer .we employ the mullite as the first layer because the coefficient of thermal expansion ( cte ) is close to that of alumina .the idea of using the skybond foam for the second layer is a natural extension of the polyimide sheet adopted by spider , which was explained in the previous section .an important advantage of the polyimide foam is that the ior can be controlled by changing the filling factor of the polyimide , which gives us an additional tuning knob to optimize the ar coating properties for given observation frequencies .our example design for this study is optimized to achieve the highest integrated efficiency for simultaneous measurements centered at 95 ghz and 150 ghz with a fractional bandwidth for each . herewe fix the thickness of the alumina and the ior of mullite , while the ior of the skybond foam and the thicknesses of mullite and the skybond foam layers are free parameters . for the design in fig .[ fig : af_config ] , the optimized ior for the skybond foam is 1.43 .measurements of the properties of each ar layer as well as those of alumina we have used in this study are described in the next section ., width=249 ]we have measured basic properties of alumina , mullite and the skybond foam .the measurements include thicknesses , iors , and the loss tangent values .table [ material ] summarizes our results .the ior and the loss tangent are obtained from transmission measurements .a detailed description of the measurement system is available elsewhere  . the beam at millimeter wavelength ( 72 - 108 ghz and 108 - 148 ghz ) is produced by a signal generator with a sixfold and eightfold frequency multiplier .the frequency range of the signal generator is between 12 and 18 ghz .the beam is collimated by an ultra - high - molecular - weight polyethylene lens and the measured sample is placed immediately after the aperture .the transmitted radiation is detected by a diode detector .the signal is chopped at 13 hz for modulation with a lock - in amplifier , and the modulated signal is obtained using a daq .the detector is moved along the optical axis for more than a half wavelength to subtract the effect of a standing wave in the measurement setup .we measure the power with and without the sample and take the ratio between them to obtain the transmittance .the measured transmittance is fitted with the fresnel formula  . in the following sections , we describe more details on each material .  .basic properties of alumina and ar materials .the thickness , the ior , and the loss tangent are listed . our measurements for and are carried out both at room temperature and at the temperature of liquid nitrogen , whereas the thickness values are only measured at room temperature .the errors include both statistical and systematic uncertainties .the systematic errors mainly arise from the uncertainty of the ar thicknesses and temperature variations during the measurements .[ material ] [ cols="^,^,^,^,^ " , ]   we measure the transmittance of the alumina filter with the same setup described in section [ sec : materialproperties ] at room temperature and at 81 k cooled with liquid nitrogen . since the uniformity measurements at room temperature gurantee that the fabrication process is independent of the size of the filter , to perform measurements with liquid nitrogen , we fabricated a sample with 50 mm in diameter to avoid mechanical conflict with the measurement system .figure [ fig : ar ] shows the result .the black lines show fit results with the emt .the estimated average transmittances are 95.9 % and 95.8 % in the 95 ghz and 150 ghz bands , respectively .  , width=321 ]we estimate the emissivity , , from the following equation : where , and are the absorption , the power reflectance , and the transmittance of the filter , respectively .the reflectances calculated from the parameters given in table [ material ] are 2.1 % and 1.1 % at 95 ghz and 150 ghz bands .the estimated emissivities are 2.7 % and 4.2 % in the 95 ghz and 150 ghz bands , respectively .the transmittance of the alumina filter at sub - millimeter wavelengths is measured using a fourier transform spectrometer ( fts ) with an insb bolometer .a mercury lamp is used as a broadband thermal source .the detailed description of the fts is found elsewhere  .the transmittance is measured from 200 ghz to 1,500 ghz at two different temperatures of 19 k and 298 k. we prepare the alumina filters with a diameter of 20 mm for the fts measurements .the measured spectrum at 19 k has a higher transmittance than the spectrum at 298 k as shown in fig .[ fig : ir ] .the 3 db cutoff frequencies of the filter are estimated as 400 and 650 ghz at 295 and 19 k , respectively .these 3 db cut off frequencies are sufficiently lower than our design goal ., width=359 ]we have demonstrated that the ar coating method with mullite and skybond foam satisfies all the requirements listed in section [ sec : filterdesign ] , which are typical in next - generation cmb experiments . although we have focused on the measurements at 95 ghz and 150 ghz , a future cmb satellite mission litebird  for example requires wider frequency ranges from 40 ghz to 400 ghz using two telescopes .skybond foam is tunable in ior and can thus be considered as a candidate technology for such a space mission .we have newly applied the emt in millimeter wave measurements to have the better surface model , which were not adopted in previous works .the fit results are in better agreement with data .the performances of the new ar coating method described in this study are as good as those with epoxy with grooves .the fabrication time of the method with mullite and skybond foam is much shorter than the epoxy coating , which is a clear advantage .finally , this method can in principle be applied to curved surfaces , so that applications for lenses should also be considered .we have newly introduced polyimide foam ( skybond foam ) as the ar coating material .we show that the ior of skybond foam is controlled between 1.1 and 1.7 by changing the filling factor .the tunable ior allows us to design a multi - layer ar coating much easier .we have made a two - layer ar coating with thermally - sprayed mullite as the first layer and skybond foam as the second layer on a 2 mm - thick alumina plate with 460 mm in diameter .we have confirmed that sample uniformity in the ior and thickness is satisfactory for 150 ghz measurements .we have also confirmed that the ar coating is robust against thermal cycles .the measured transmittances of this filter at 81 k are 95.8 % for 95 ghz and 95.9 % for 150 ghz .the measured 3 db cut - off frequency is 650 ghz at 19 k. the results satisfy requirements on ir filters for next - generation cmb polarization measurements where a large focal - plane and a large window are needed .we thank hiroshi matsuo and tom nitta for their help with thz measurements .we are also grateful to satoru igarashi , iwao murakami , and keishi toyoda for their help in setting up the measurement .we would like to express our gratitude to adrian lee , osamu tajima , suguru takada , akito kusaka , hiroshi yamaguchi , junichi suzuki , yuichi watanabe and yuuko segawa .we would like to thank the kek cryogenics science center for the support .yuki inoue was supported by advanced research cource in sokendai ( the graduate university for advanced studies ) , and by academia sinica under grants no .cda-105-m06 in taiwan .kek authors were supported by mext kakenhi grant numbers 21111002 , 15h05891 , and jsps kakenhi grant numbers 13j03626 , 24740182 , 24684017 , 15h03670 , 24111715 and 26220709 . finally , this work was supported by the jsps core - to - core program , a. advanced research networks .99 m.kamionkowski , a.kosowsky and a.stebbins , `` a probe of primordial gravity waves and vorticity , '' phys . , 2058 - 2061(1997 ) .u. seljak and m. zaldarriaga,``signature of gravity waves in the polarization of the microwave background , '' springer us . , 2054 ( 1997 ) .the part number and output frequency of synthesized frequency generator are agillent 83711b and 12 - 18 ghz , respectively .the frequency multiplier is made by amtechs corporation , http://www.amtechs.co.jp .e. hecht , optics fourth edition ( addison wesley , 2002 ) .     o. levy . and d.stroud . maxwell garnett theory for mixtures of anisotropic inclusions : application to conducting polymers ." physical review b , , 8035 ( 1997 ) . h. matsuo , a. sakamoto , and s. matsushita , fts measurements of submillimeter - wave atmospheric opacity at pampa la bola " , pasj , 359 - 366 ( 1998 ) .t. matsumura _ et al ._ , j. low . temp .phys .  *176 * , 733 ( 2014 ) doi:10.1007/s10909 - 013 - 0996 - 1 [ arxiv:1311.2847 [ astro-ph.im ] ] .t. matsumura _ et al ._ , journal of low temperature physics ( 2016 ) , 10.1007/s10909 - 016 - 1542 - 8 .

Abstract: we have developed a novel two - layer anti - reflection ( ar ) coating method for large - diameter infrared ( ir ) filters made of alumina , for the use at cryogenic temperatures in millimeter wave measurements . thermally - sprayed mullite and polyimide foam ( skybond foam ) are used as the ar material . an advantage of the skybond foam is that the index of refraction is chosen between 1.1 and 1.7 by changing the filling factor . combination with mullite is suitable for wide - band millimeter wave measurements with sufficient ir cutoff capability . we present the material properties , fabrication of a large - diameter ir filter made of alumina with this ar coating method , and characterizations at cryogenic temperatures . this technology can be applied to a low - temperature receiver system with a large - diameter focal plane for next - generation cosmic microwave background ( cmb ) polarization measurements , such as polarbear-2 ( pb-2 ) .