LAGRANGIAN FORMULATION OF MOND; MOND FIELD IN PERTURBED SPHERICAL SYSTEMS by REIJIRO MATSUO Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Dissertation Adviser: Dr. Glenn Starkman Department of Physics CASE WESTERN RESERVE UNIVERSITY August 2010 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of Reijiro Matsuo candidate for the Doctor of Philosophy degree*. Dr. Glenn Starkman (chair of the committee) Dr. Barry Wessels Dr. Corbin Covault Dr. Harsh Mathur June 29, 2010 Contents 1 Introduction 1 2 Basic theory of MOND and its phenomenological implications 5 2.1 TheOriginalFormulationofMOND . 5 2.2 MONDandrotationcurvefitting . 7 2.3 TheLagrangianFormulationofMOND . 8 2.4 RelativisticFormulationsofMOND . 10 2.5 Confrontations with MOND, X-ray emitting Clusters . ... 14 2.6 Confrontations with MOND, the Bullet Cluster . .. 16 2.6.1 ConditionsonaVectorPerturbation . 18 2.6.2 NumericalCode.......................... 21 2.6.3 LensingandConvergencemap. 22 2.6.4 Model ............................... 26 2.6.5 Result ............................... 31 3 MONDfieldandperturbedsphericalsystems 32 3.1 GeneralSetups .............................. 32 3.2 PerturbationsinsideaMassShell . 35 3.2.1 GeneralRelativity . 35 3.2.2 MOND............................... 37 3.2.3 APerturbativecalculation . 37 3.2.4 Numericalcalculations . 40 3.2.5 TestingtheCode ......................... 41 3.3 PerturbationsonaShellandoutsideaShell . ... 43 3.3.1 ShieldingandAnti-Shieldingeffects . 49 4 Concludingremarksandimplications 54 3 A Numerical Code 57 A.1 GeneralEquations ............................ 57 A.2 Lattice................................... 58 A.3 InitialConditions ............................. 58 A.4 BoundaryConditions........................... 59 A.5 Discretization of the Field and Implementation of the Numerical Code 59 B Initialconditionfortheshellwithopening 64 References 68 i List of Figures 1 GalacticrotationcurveforNGC2903 . 6 2 The comparison of MOND dynamical mass and observed baryonic mass forX-rayclusters ............................. 16 3 The convergence map and X-ray image of the Bullet Cluster . .. 18 4 The geometric configuration for a point mass gravitational lensing . 23 5 The convergence map: p =0.00 and p =0.50 ............. 27 6 The convergence map: p =0.90 and p =0.99.............. 28 7 The convergence map: rh = 200kpc and rh = 100kpc .......... 29 8 The convergence map: rh = 25kpc and rh = 10kpc ........... 30 9 Mass distribution: a point mass and a spherical shell . .... 33 10 Mass distribution: a system with multiple shells . ..... 34 11 Testing the code: Comparison to the perturbative solution ...... 41 12 Testing the code: Dependence on the number of lattice sites ..... 42 13 The acceleration on the point particle as a function of mass...... 43 14 The acceleration of the point particle as a function of the distance . 44 15 The field inside a perturbed shell with perturbation of order l =1.. 48 16 The field inside a perturbed shell with perturbation of order l =2.. 49 17 The field inside a perturbed shell with perturbation of order l =3.. 50 18 The acceleration on the interior shell as a function of its mass for perturbation with l =1.......................... 52 19 The acceleration on the interior shell as a function of its mass for perturbation given by an outer shell with an opening . .. 53 20 The configuration for discretized field in the numerical code ..... 60 1 U 2 21 | | as a function of U .................... 64 2 ν2 ν2 1 ( 2 ) | | − 22 Theinfinitesimalringontheshell . 67 ii List of Tables 1 The mass and position of baryon matter for the Bullet Cluster.... 31 iii Dedication To my friend, M. Shiro. iv Acknowledgements I would like to thank Dr. De-Chang Dai, with whom much of the work in section 2.6 and 3.2 was done. I also want to thank Dr. Yi-zen Chu and Dr. Pascal Vaudrevange for their friendship and counsel. Finally, I wish to thank my advisor, Professor Glenn Starkman, for his guidance, encouragement and limitless support in all phases of my work. v LAGRANGIAN FORMULATION OF MOND; MOND FIELD IN PERTURBED SPHERICAL SYSTEMS by REIJIRO MATSUO Abstract We investigate how perturbation of a spherically symmetric system would affect the gravitational field. In particular, we study systems of a finite-mass test particle inside a spherical shell, perturbed and unperturbed spherical shells. For a system of test particle inside a spherical shell, we find non-vanishing accel- eration for that test particle in both GR and MOND. In GR, the acceleration is highly suppressed, and physically insignificant. In MOND, on the contrary, the acceleration of the point particle can be a significant fraction of the field just outside of the spherical shell. For a multiple-shell we show that perturbation field within the shell is screened by the spherically symmetric component of the mass, and is reduced as the spherically symmetric component is increased. However, for a very light inner shell, the perturbation to the field can be en- hanced. The enhancement is typically larger for smaller inner shells, and the perturbed field can be amplified by almost a factor of 2. The relevance to the effect of external fields on galaxy dynamics is discussed. vi 1 Introduction Most astrophysical computations concern a local system embedded in a much larger background system. For example, the Solar System is embedded in Milky Way, a galaxy in a cluster, a cluster in a supercluster, etc. To compute analytically the external field effect on a local system due to its environment for a real mass distri- bution is often very difficult. One often invokes a simplified geometric structure as a toy model and hopes that the model will capture physical reality. For a system of a galaxy in a cluster, the relevant geometric structure may be a point mass inside a spherical shell. If the system possess perfect spherical symmetry, that is to say a zero-mass test particle in the presence of a spherically symmetric background mass distribution, computation of the external field on the test particle is trivial. One would invoke Birkhoff’s theorem. Birkhoff’s Theorem (BT) is the generalization from Newtonian gravity to General Relativity (GR) of Gauss’ Law for gravity. This theroem states that, in GR any spherically symmetric solution of the Einstein field equations in vacuum must be stationary and asymptotically flat. As a consequence, the metric exterior to a spherically symmetric mass distribution must be a Schwarzschild metric. One important corollary of BT is that the metric inside a spherical shell (or inside the innermost of a concentric sequence of such shells) is the Minkowski metric. Thus, as a consequence of Birkhhoff’s Theorem (or Gauss’s Law), the gravitational field anywhere inside any spherical mass shell vanishes for both GR and the Newtonian theory. However, in realistic situations, the “test particles” probing a gravitational field have a finite mass. Often, this mass is not small at all, as in the case of a large galaxy inside a cluster. Therefore, unless the test particle is at the center of the distribution, its presence perturbs the system from spherical symmetry, spoiling the assumptions underlying BT. In Newtonian gravity, there is a fortunate coincidence that the result for a zero- 1 mass test particle also holds for a finite mass particle. In the context of the inverse- square-law of Newtonian gravity, these results are easily understood. For example, at a point inside a spherical shell, the force on a test particle from any thin ring on the shell is precisely balanced by the force due to a ring subtending the same angle but directly opposite the first ring, even if the test particle is not at the center. This is because the decrease in the gravitational force due to one ring being farther away from the test particle is precisely balanced by the increase in area (and so mass) of that farther-away ring. We also note that, since the superposition principle holds in Newtonian gravity, if the shell itself is perturbed, the external field effect is entirely determined by the perturbation. In GR, the force on a test particle inside a spherical mass shell may not actually vanish. General relativity is a metric theory. Massive bodies change the geometry around them. A massive test particle distorts the space around it and thus a ”spherical shell” ceases to be a spherical shell unless the test particle is at its center. In the subsequent section, we shall show that a spherical shell indeed exerts a force on a massive test particle. However, for a typical astrophysical parameters of a galaxy and a cluster, the external field induced in this way is negligibly small. In general, for both Newtonian gravity and GR, if the shell-particle system pos- sesses approximate spherical symmetry, BT holds approximately, and the external field can practically be ignored. It would be desirable if such an extrapolation could be implemented in the theory of Modified Newtonian Dynamics (MOND). The original formulation of MOND [1, 2, 3], is an alternative to dark-matter in which the missing gravity problem of galaxies is solved by altering the gravitational force law rather than by introducing new unseen forms of matter. In this naive formulation, the MOND field can be thought of as a rescaling of the Newtonian field with a particular algebraic expression which allows the gravitational field to scale 1 2 as as r− instead of r− at large distances from a bounded mass distribution. Much 2 improved fits to galactic rotation curves are obtained with this formulation applied to isolated galaxies[9, 10, 11, 12, 13, 14, 15, 20, 27]. In this naive picture, a spherical shell does not exert any force on a point mass within it, since the field configuration is a rescaling of Newtonian gravity. If the spherical shell itself is perturbed, the external field resulting from the perturbation would be larger than in the Newtonian case. This is because the field of a bounded mass distribution scales as 1 in MOND, and ∼ r the field due to the aspherical part may persist to greater distances.