Normal Modes. the True Story

Total Page:16

File Type:pdf, Size:1020Kb

Normal Modes. the True Story Normal modes. The true story. Emil Zak∗ Department of Physics and Astronomy, University College London, London, WC1E 6BT, UK (Dated: January 20, 2016) I. INTRODUCTION KEO in arbitrary curvlinear coordinates was introduced by Yachmenev and Yurchenko [3] and successfully ap- The aim of this article is a comprehensive description plied to four-atomic systems by Chubb, Yachmenev and of normal modes of molecular vibrations. The starting Yurchenko [4]. Parallel analytic progress in the field has point is chosen to be a general molecular system with been made by Szalay [5], [6]. Several authors: Tennyson separated center of mass and an arbitrary embedding of et al. [7], [8], Jensen [9], Csaszar [10] and others also body-fixed axes. This allows to focus on internal de- succeeded to evaluate practical representations for many- grees of freedom only, leaving the problem of rotational body Hamiltonians in an arbitrary embedding of the motion behind. Nevertheless, for the sake of complete- body-fixed frame. In each of these cases one is theoreti- ness we first introduce a general quantum-mechanical cally able to linearize the internal coordinates and subse- Hamiltonian in Eckart-Watson representation, to make quently transfer into normal modes level. Nonetheless we a quick leap into a simplified harmonic description for shall follow the historically first attempt due to its strong internal motion and rigid rotor for rotational degrees of association with normal coordinates. After separating freedom. This regime constitutes a basis for more sophis- the nuclear center of mass, and moving into the Eckart ticated calculations. The first section introduces normal frame which guarantees minimal rotation-vibration cou- modes with emphasis on mathematical precision. Along- pling, some analytic manipulations yield the well known side we stress the points where approximations are made, Watson Hamiltonian: for the reader to be aware of limitations of the current model. Second section exemplifies presented methodol- ogy on a simple system of two masses connected with harmonic springs in 1D. Section V has supplementary 1 X 1 X character and underlines the formal route needed be H^ = M^ µ M^ + P^2 + U^ + V^ (1) 2 i ij j 2 i taken to derive the Watson Hamiltonian, which in turn i;j i is immersed in the framework of normal vibrations. The last section (VI) is a detailed discussion of rigid rotor model. Current review is largely based on classic books where U^ is mass dependent contribution to the potential by Bunker& Jensen [1] and Wilson,Decius&Cross [2] sup- V^ : plied with original papers, and a few author's own com- ments, derivations and theorems. 2 X 2 U^ = −~ µ ≡ −~ T rµ (2) 8 ii 8 II. GENERAL THEORY i Before proceeding to the main topic, a few words of explanation are needed. Section V presents a detailed and µij is the inverse of the dynamical moment of inertia insight into the formal procedure of derivation of the tensor[11]. M^ i = J^i − p^i − L^i stands for the i-th compo- arXiv:1601.04735v1 [quant-ph] 18 Jan 2016 Eckart-Watson form of quantum Hamiltonian for an ar- nent of the total angular momentum of the system and is bitrary many-body system. This is a popular but rather defined by the rovibronic angular momentum, vibrational specific approach, as is constrained to normal coordinates angular momentum and electronic angular momentum, only. Any set of coordinates which is not linearly related respectively. The potential operator V contains interac- to cartesian displacements brings significant complica- tions between all particles in the system, while P^ is the tions. Many efforts were made to evaluate an exact rep- linear momentum operator in normal coordinates repre- resentation of the kinetic energy operator (KEO) in curv- sentation. By imposing Born-Oppenheimer separation of linear coordinates and Eckart embedding. Most recently nuclear and electronic degrees of freedom the molecular an effective numerical procedure for expansion of nuclear Hamiltonian becomes a sum of an electronic and a rovi- brational part: ∗ [email protected]; http://www.homepages.ucl.ac.uk/~ucapejz/ H^ = H^el + H^rv (3) 2 The rovibrational Hamiltonian can be readily expanded later transfer to the rigid rotor approximation we are and arranged in the following fashion: left with internal nuclear motion problem. For a non- linear molecule, or non-linear system of atoms this leaves 1 X 1 X H^ = µe J^2 + P^2+ 3N − 6 internal degrees of freedom. Lets start from a rv 2 αα α 2 i α i fully classical picture. After formulating a suitable rep- 1 X resentation of the Hamilton function, quantization will + (µ − µ ) J^ − p^ J^ − J^ 2 αβ αβ α α β β be performed. From eq. 5 one can infer that the total αβ energy of the non-rotating system is a sum of kinetic and X e ^ X e 2 ^ ^ potential energy: − µααJαp^α + + µααp^α + U + Vnn α α E = T (x_ ) + V (x) (8) (4) Mixing terms appearing in above expression are re- where x = x1; y1; z1; :::; x3N−6; y3N−6; z3N−6 and x_ = sponsible for centrifugal distorsion and Coriolis cou- x_ 1; y_1; z_1; :::; x_ 3N−6; y_3N−6; z_3N−6 are position and veloc- plings. For sufficiently rigid molecules they can be ne- ity vectors of all 3N − 6 atoms, respectively. No net glected. What remains is a purely rotational part added external force is exerted on the system and all internal to internal motion energy operators decoupled from ro- forces are conservative, hence the total energy is equal to tations: the Hamilton function. We should be able to write ad hoc both components in cartesian coordinates: 3N−6 1 X e 2 1 X 2 1 H^rv = µ J^ + P^ + V^nn (5) X X 2 2 αα α 2 i Tcart = miα_i (9) α i 2 i=1 α=x;y;z The task now is to find normal coordinates for a general where α stands for cartesian coordinate in the molecule- many-body system, so that Watson approach may be re- i fixed (body-fixed) frame. It is very instructive to refor- alized. Hence it becomes clear that the choice of this par- mulate the above expression in a matrix form: ticular type of Hamiltonian representation was strongly dictated by its association to normal coordinates. How- 0 1 0 1 3N−6 m 0 0 x_ ever from practical point of view, one usually prefer start 1 X i i T = (_x ; y_ ; z_ ) 0 m 0 y_ (10) with a set of geometrically defined internal coordinates or cart 2 i i i @ i A @ i A i=1 0 0 m z_ cartesian coordinates. Thus, below we present a detailed i i instructions to obtain normal coordinates from arbitrar- and further: ily chosen internal coordinates. Assuming the knowledge of normal coordinates of the system we may expand inter- 1 T = x_ T Tx_ (11) nuclear interaction potential energy as: cart 2 where x is a 3N − 6 dimensional position vector, while T 3N−6 3N−6 1 X 1 X stands for a (3N − 6) × (3N − 6) matrix representation V = λ Q2 + Φ Q Q Q + ::: (6) nn 2 i i 6 ijk i j k of a diagonal metric tensor build from N − 2 diagonal i=1 i;j;k=1 blocks each containing three equal masses mi · 13. It and by truncating this expansion on quadratic terms we is clearly visible that the kinetic energy is a quadratic stay in harmonic approximation, in which normal coor- form in cartesian velocities. This observation will allow dinates were introduced. Now pick up the first two sums us to utilize some tools from quadratic forms theory in in eq. 5 and combine the second one with the truncated section IV. The internuclear potential energy is a func- potential energy. By neglecting all other terms we get: tion of 3N cartesian coordinates of all nuclei. However the functional form of potential energy is usually avail- able in a geometrically defined internal coordinates, i.e. 1 X 1 X H^ = µe J^2 + P^2 + λ Q2 (7) Vnn = Vnn(R1;R2; :::; R3N−6), where Ri is the i-th inter- rv 2 αα α 2 i i i nal coordinate. The reason for that is because the poten- α i tial energy function is in general a complicated function The first term in above Hamiltonian describes the energy of cartesian coordinates. Internal coordinates chosen in- of a rigid rotor - this model is solved and discussed in tuitively by geometrical considerations provide a more detail in section VI. The second term is the target of transparent perspective of types of internal motions and the present article. It is already shown in the final form, associated potential energy. Therefore internal coordi- however no recipe for finding λi and Qi was yet given. nate representation is more suitable at this stage. Nev- Until then this expression should be considered a black- ertheless, the potential energy function is typically also a box. fairly complicated function (but less complicated than in Having separated both center of mass motion and rota- cartesian representation) of internal coordinates. Thus, a tions via embedding in the Eckart body-fixed frame with for non-high-accuracy calculation purposes lowest terms 3 in the Taylor expansion are sufficient: where M is the total mass of the system, and 1 3N−6 1 3N−6 N X X eq X eq eq Vnn = fijRiRj + fijkRiRjRk + ::: (12) R = µ m (y ∆z − z ∆y ) (17) 2 6 x xx i i i i i i;j=1 i;j;k=1 i=1 Note that the minimum of V is set to 0 and it corre- while y and z rotational coordinates are obtained by sponds to equilibrium values of all internal coordinates: cyclic permutation of cartesian coordinates in above def- ~ ~ −1 R = (0; 0; :::; 0) = Req.
Recommended publications
  • Lectures on Differential Geometry Math 240C
    Lectures on Differential Geometry Math 240C John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: [email protected] June 6, 2011 Preface This is a set of lecture notes for the course Math 240C given during the Spring of 2011. The notes will evolve as the course progresses. The starred sections are less central to the course, and may be omitted by some readers. i Contents 1 Riemannian geometry 1 1.1 Review of tangent and cotangent spaces . 1 1.2 Riemannian metrics . 4 1.3 Geodesics . 8 1.3.1 Smooth paths . 8 1.3.2 Piecewise smooth paths . 12 1.4 Hamilton's principle* . 13 1.5 The Levi-Civita connection . 19 1.6 First variation of J: intrinsic version . 25 1.7 Lorentz manifolds . 28 1.8 The Riemann-Christoffel curvature tensor . 31 1.9 Curvature symmetries; sectional curvature . 39 1.10 Gaussian curvature of surfaces . 42 1.11 Matrix Lie groups . 48 1.12 Lie groups with biinvariant metrics . 52 1.13 Projective spaces; Grassmann manifolds . 57 2 Normal coordinates 64 2.1 Definition of normal coordinates . 64 2.2 The Gauss Lemma . 68 2.3 Curvature in normal coordinates . 70 2.4 Tensor analysis . 75 2.5 Riemannian manifolds as metric spaces . 84 2.6 Completeness . 86 2.7 Smooth closed geodesics . 88 3 Curvature and topology 94 3.1 Overview . 94 3.2 Parallel transport along curves . 96 3.3 Geodesics and curvature . 97 3.4 The Hadamard-Cartan Theorem . 101 3.5 The fundamental group* .
    [Show full text]
  • 3. Introducing Riemannian Geometry
    3. Introducing Riemannian Geometry We have yet to meet the star of the show. There is one object that we can place on a manifold whose importance dwarfs all others, at least when it comes to understanding gravity. This is the metric. The existence of a metric brings a whole host of new concepts to the table which, collectively, are called Riemannian geometry.Infact,strictlyspeakingwewillneeda slightly di↵erent kind of metric for our study of gravity, one which, like the Minkowski metric, has some strange minus signs. This is referred to as Lorentzian Geometry and a slightly better name for this section would be “Introducing Riemannian and Lorentzian Geometry”. However, for our immediate purposes the di↵erences are minor. The novelties of Lorentzian geometry will become more pronounced later in the course when we explore some of the physical consequences such as horizons. 3.1 The Metric In Section 1, we informally introduced the metric as a way to measure distances between points. It does, indeed, provide this service but it is not its initial purpose. Instead, the metric is an inner product on each vector space Tp(M). Definition:Ametric g is a (0, 2) tensor field that is: Symmetric: g(X, Y )=g(Y,X). • Non-Degenerate: If, for any p M, g(X, Y ) =0forallY T (M)thenX =0. • 2 p 2 p p With a choice of coordinates, we can write the metric as g = g (x) dxµ dx⌫ µ⌫ ⌦ The object g is often written as a line element ds2 and this expression is abbreviated as 2 µ ⌫ ds = gµ⌫(x) dx dx This is the form that we saw previously in (1.4).
    [Show full text]
  • Hamilton's Ricci Flow
    The University of Melbourne, Department of Mathematics and Statistics Hamilton's Ricci Flow Nick Sheridan Supervisor: Associate Professor Craig Hodgson Second Reader: Professor Hyam Rubinstein Honours Thesis, November 2006. Abstract The aim of this project is to introduce the basics of Hamilton's Ricci Flow. The Ricci flow is a pde for evolving the metric tensor in a Riemannian manifold to make it \rounder", in the hope that one may draw topological conclusions from the existence of such \round" metrics. Indeed, the Ricci flow has recently been used to prove two very deep theorems in topology, namely the Geometrization and Poincar´eConjectures. We begin with a brief survey of the differential geometry that is needed in the Ricci flow, then proceed to introduce its basic properties and the basic techniques used to understand it, for example, proving existence and uniqueness and bounds on derivatives of curvature under the Ricci flow using the maximum principle. We use these results to prove the \original" Ricci flow theorem { the 1982 theorem of Richard Hamilton that closed 3-manifolds which admit metrics of strictly positive Ricci curvature are diffeomorphic to quotients of the round 3-sphere by finite groups of isometries acting freely. We conclude with a qualitative discussion of the ideas behind the proof of the Geometrization Conjecture using the Ricci flow. Most of the project is based on the book by Chow and Knopf [6], the notes by Peter Topping [28] (which have recently been made into a book, see [29]), the papers of Richard Hamilton (in particular [9]) and the lecture course on Geometric Evolution Equations presented by Ben Andrews at the 2006 ICE-EM Graduate School held at the University of Queensland.
    [Show full text]
  • GEOMETRIC INTERPRETATIONS of CURVATURE Contents 1. Notation and Summation Conventions 1 2. Affine Connections 1 3. Parallel Tran
    GEOMETRIC INTERPRETATIONS OF CURVATURE ZHENGQU WAN Abstract. This is an expository paper on geometric meaning of various kinds of curvature on a Riemann manifold. Contents 1. Notation and Summation Conventions 1 2. Affine Connections 1 3. Parallel Transport 3 4. Geodesics and the Exponential Map 4 5. Riemannian Curvature Tensor 5 6. Taylor Expansion of the Metric in Normal Coordinates and the Geometric Interpretation of Ricci and Scalar Curvature 9 Acknowledgments 13 References 13 1. Notation and Summation Conventions We assume knowledge of the basic theory of smooth manifolds, vector fields and tensors. We will assume all manifolds are smooth, i.e. C1, second countable and Hausdorff. All functions, curves and vector fields will also be smooth unless otherwise stated. Einstein summation convention will be adopted in this paper. In some cases, the index types on either side of an equation will not match and @ so a summation will be needed. The tangent vector field @xi induced by local i coordinates (x ) will be denoted as @i. 2. Affine Connections Riemann curvature is a measure of the noncommutativity of parallel transporta- tion of tangent vectors. To define parallel transport, we need the notion of affine connections. Definition 2.1. Let M be an n-dimensional manifold. An affine connection, or connection, is a map r : X(M) × X(M) ! X(M), where X(M) denotes the space of smooth vector fields, such that for vector fields V1;V2; V; W1;W2 2 X(M) and function f : M! R, (1) r(fV1 + V2;W ) = fr(V1;W ) + r(V2;W ), (2) r(V; aW1 + W2) = ar(V; W1) + r(V; W2), for all a 2 R.
    [Show full text]
  • Differential Geometry Lecture 17: Geodesics and the Exponential
    Differential geometry Lecture 17: Geodesics and the exponential map David Lindemann University of Hamburg Department of Mathematics Analysis and Differential Geometry & RTG 1670 3. July 2020 David Lindemann DG lecture 17 3. July 2020 1 / 44 1 Geodesics 2 The exponential map 3 Geodesics as critical points of the energy functional 4 Riemannian normal coordinates 5 Some global Riemannian geometry David Lindemann DG lecture 17 3. July 2020 2 / 44 Recap of lecture 16: constructed covariant derivatives along curves defined parallel transport studied the relation between a given connection in the tangent bundle and its parallel transport maps introduced torsion tensor and metric connections, studied geometric interpretation defined the Levi-Civita connection of a pseudo-Riemannian manifold David Lindemann DG lecture 17 3. July 2020 3 / 44 Geodesics Recall the definition of the acceleration of a smooth curve n 00 n γ : I ! R , that is γ 2 Γγ (T R ). Question: Is there a coordinate-free analogue of this construc- tion involving connections? Answer: Yes, uses covariant derivative along curves. Definition Let M be a smooth manifold, r a connection in TM ! M, 0 and γ : I ! M a smooth curve. Then rγ0 γ 2 Γγ (TM) is called the acceleration of γ (with respect to r). Of particular interest is the case if the acceleration of a curve vanishes, that is if its velocity vector field is parallel: Definition A smooth curve γ : I ! M is called geodesic with respect to a 0 given connection r in TM ! M if rγ0 γ = 0. David Lindemann DG lecture 17 3.
    [Show full text]
  • Differential Geometry Lecture 18: Curvature
    Differential geometry Lecture 18: Curvature David Lindemann University of Hamburg Department of Mathematics Analysis and Differential Geometry & RTG 1670 14. July 2020 David Lindemann DG lecture 18 14. July 2020 1 / 31 1 Riemann curvature tensor 2 Sectional curvature 3 Ricci curvature 4 Scalar curvature David Lindemann DG lecture 18 14. July 2020 2 / 31 Recap of lecture 17: defined geodesics in pseudo-Riemannian manifolds as curves with parallel velocity viewed geodesics as projections of integral curves of a vector field G 2 X(TM) with local flow called geodesic flow obtained uniqueness and existence properties of geodesics constructed the exponential map exp : V ! M, V neighbourhood of the zero-section in TM ! M showed that geodesics with compact domain are precisely the critical points of the energy functional used the exponential map to construct Riemannian normal coordinates, studied local forms of the metric and the Christoffel symbols in such coordinates discussed the Hopf-Rinow Theorem erratum: codomain of (x; v) as local integral curve of G is d'(TU), not TM David Lindemann DG lecture 18 14. July 2020 3 / 31 Riemann curvature tensor Intuitively, a meaningful definition of the term \curvature" for 3 a smooth surface in R , written locally as a graph of a smooth 2 function f : U ⊂ R ! R, should involve the second partial derivatives of f at each point. How can we find a coordinate- 3 free definition of curvature not just for surfaces in R , which are automatically Riemannian manifolds by restricting h·; ·i, but for all pseudo-Riemannian manifolds? Definition Let (M; g) be a pseudo-Riemannian manifold with Levi-Civita connection r.
    [Show full text]
  • Chapter 7 Geodesics on Riemannian Manifolds
    Chapter 7 Geodesics on Riemannian Manifolds 7.1 Geodesics, Local Existence and Uniqueness If (M,g)isaRiemannianmanifold,thentheconceptof length makes sense for any piecewise smooth (in fact, C1) curve on M. Then, it possible to define the structure of a metric space on M,whered(p, q)isthegreatestlowerboundofthe length of all curves joining p and q. Curves on M which locally yield the shortest distance between two points are of great interest. These curves called geodesics play an important role and the goal of this chapter is to study some of their properties. 489 490 CHAPTER 7. GEODESICS ON RIEMANNIAN MANIFOLDS Given any p M,foreveryv TpM,the(Riemannian) norm of v,denoted∈ v ,isdefinedby∈ " " v = g (v,v). " " p ! The Riemannian inner product, gp(u, v), of two tangent vectors, u, v TpM,willalsobedenotedby u, v p,or simply u, v .∈ # $ # $ Definition 7.1.1 Given any Riemannian manifold, M, a smooth parametric curve (for short, curve)onM is amap,γ: I M,whereI is some open interval of R. For a closed→ interval, [a, b] R,amapγ:[a, b] M is a smooth curve from p =⊆γ(a) to q = γ(b) iff→γ can be extended to a smooth curve γ:(a ", b + ") M, for some ">0. Given any two points,− p, q →M,a ∈ continuous map, γ:[a, b] M,isa" piecewise smooth curve from p to q iff → (1) There is a sequence a = t0 <t1 < <tk 1 <t = b of numbers, t R,sothateachmap,··· − k i ∈ γi = γ ! [ti,ti+1], called a curve segment is a smooth curve, for i =0,...,k 1.
    [Show full text]
  • Estimates on the Metric Tensor
    Estimates on the Metric Tensor David Glickenstein February 14, 2000 1. Intro duction. This is a summary of some easy estimates on the metric tensor of a manifold if the sectional curvatures are b ounded. It generally follows [1, section 1.8]. 2. De nitions. We will estimate derivatives of the metric in geo desic p olar co ordinates. Supp ose wehave a Riemannian manifold M; g. We consider geo desic p olar co ordinates, 2 2 i j and in these co ordinates the metric can b e written as g = dr + r g i j d d . We also are going to need to take a geo desic ton M which is parametrized by arclength, 0 and we will say that P = 0, Q = r , and T t= t. 0 In the sequel, we shall talk ab out vector elds V along the curve . When we write V t, it is understo o d to mean r V t, where r is the Riemannian connection on M . T t We shall also restrict ourselves to r such that r is in the ball around P where exp is P a di eomorphism inside the injectivity radius. 3. Relevant Theorems. We will use Jacobi elds and the Index Inequality. We rst should de ne the Index of a vector eld. De nition 3.1. The index at r of a vector eld V t along a curve is Z r 0 0 I V; V = [gV t;V t + g RT t;V tT t;V t] dt r 0 d where T t= is the tangential vector eld to .
    [Show full text]
  • Arxiv:1406.5413V3
    The tangent bundle exponential map and locally autoparallel coordinates for general connections on the tangent bundle with application to Finsler geometry Christian Pfeifer∗ Institute for theoretical physics,Leibniz Universit¨at Hannover, Appelstrasse 2, 30167 Hannover, Germany We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalise normal coordinates known from metric geometry to Finsler geometric manifolds and it turns out that they are identical to the Douglas-Thomas normal coordinates introduced earlier. We expand the Finsler Lagrangian of a Finsler space- time in these new coordinates and find that it is constant to quadratic order. The quadratic order term comes with the non-linear curvature of the manifold. From physics these coor- dinates may be interpreted as the realisation of an Einstein elevator in Finslerian spacetime geometries. I. INTRODUCTION Locally autoparallel coordinates respectively locally geodesic coordinates constructed with help of the exponential map are an important tool in metric and affine connection geometry for various proofs, calculations and derivations. Their most valuable properties are that the connection coeffi- cients vanish at the origin of the coordinate system and that autoparallels of the affine connection respectively geodesics become straight lines. In gravitational physics the existence of coordinates arXiv:1406.5413v3 [math-ph] 27 Nov 2015 which are optimally adapted to the Lorentzian metric geometry of spacetime, Riemann normal coordinates, is interpreted as the realization of the Einstein equivalence principle.
    [Show full text]
  • Introduction to RIEMANNIAN GEOMETRY
    Introduction to RIEMANNIAN GEOMETRY Gert Heckman Radboud University Nijmegen [email protected] May 23, 2019 1 Contents Preface 2 1 Classical Differential Geometry 5 1.1 SmoothCurvesinEuclideanSpace . 5 1.2 Submanifolds of Euclidean Space . 7 1.3 Geodesics on Submanifolds of Euclidean Space . 9 1.4 HypersurfacesinEuclideanSpace . 10 2 Vector Bundles and Connections 13 2.1 TensorProducts.......................... 13 2.2 VectorBundles .......................... 16 2.3 Connections ............................ 18 2.4 CurvatureofaConnection . 22 2.5 ParallelTransportandHolonomy . 26 2.6 FundamentalGroup ....................... 28 2.7 FlatConnectionsandMonodromy. 34 3 Riemannian Geometry 38 3.1 Riemannian Manifolds . 38 3.2 Levi-Civita Connection and Riemann Curvature . 41 3.3 SectionalCurvature. 47 3.4 RicciCurvature .......................... 50 3.5 Geodesics ............................. 53 3.6 TheSecondFundamentalForm . 60 3.7 TheLaplace–Beltramioperator . 65 3.8 Historicalremarks......................... 68 2 Preface These are lecture notes for an introductory course on Riemannian geometry. The readers are supposed to be familiar with the basic notions of the theory of smooth manifolds, such as vector fields and their flows, differential forms, integration of volume forms and the theorem of Stokes. This material has been covered in the first semester of the third year by Ioan M˘arcut [15]. In addition, it will be helpful if the readers are familiar with the classical differential geometry of curves (the formulas of Frenet) and surfaces (various curvatures and the Theorema Egregium on the intrinsic nature of the Gauss curvature) in R3. Such a course is offered at our university in the second semester of the second year, based on a book by Andrew Pressley [19] and lecture notes by the author [11].
    [Show full text]
  • Arxiv:Gr-Qc/9712092V2 20 Dec 1999
    DESY 97-254 ANL-HEP-PR-97-99 MZ-TH/97-38 A Closed Formula for the Riemann Normal Coordinate Expansion Uwe M¨uller Institut f¨ur Physik, Johannes-Gutenberg-Universit¨at Mainz, Staudingerweg 7, D-55099 Mainz, Germany e-mail: [email protected] Christian Schubert High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439-4815, USA and Institut f¨ur Physik, Humboldt Universit¨at zu Berlin, Invalidenstr. 110, D-10115 Berlin, Germany e-mail: [email protected] Anton E. M. van de Ven II Institut f¨ur Theoretische Physik, Universit¨at Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany e-mail: [email protected] arXiv:gr-qc/9712092v2 20 Dec 1999 Abstract We derive an integral representation which encodes all coeffi- cients of the Riemann normal coordinate expansion and also a closed formula for those coefficients. In gauge theory, one often uses Fock-Schwinger gauge [1, 2] to achieve manifest covari- ance in the calculation of effective actions, anomaly densities, or other quantities. Fock- Schwinger gauge “centered at 0” can be defined by the condition µ y Aµ(y) = 0. (1) Locally in some neighbourhood of 0, this condition can be solved in terms of the following integral representation connecting the gauge potential and the field strength tensor [3, 4], 1 ρ Aµ(y)= y dvvFρµ(vy) (2) Z0 (see our eqs. (9-13)). More explicitly, in this gauge one can express the coefficients of the Taylor expansion of A at x = 0 in terms of the covariant derivatives of F , ∞ n ∞ n (y · ∂x) ρ (y · Dx) ρ Aµ(y)= y Fρµ(0) = y Fρµ(0).
    [Show full text]
  • Fermi Normal Coordinates
    General Relativity Fall 2019 Lecture 12: Curvature; Fermi normal coordinates Yacine Ali-Ha¨ımoud October 10th, 2019 RIEMANN = 0 , SPACETIME IS FLAT 2 µ ν If spacetime is flat, there exists a globally inertial coordinate system, such that ds = ηµν dx dx . In thees coordinates, the second derivatives of the metric vanish, thus the Riemann tensor vanishes identically. This proves the , implication. Let us now prove the ). We saw that the change of a vector upon parallel transport around a small closed loop is proportional to the Riemann tensor times the area of the loop. Equivalently, we could have shown that the difference in a vector parallel transported along two different paths is proportional to the Riemann tensor times the area of the loop enclosed by the two paths. Therefore, if the Riemann tensor vanishes everywhere, parallel transport is independent of the path (assuming the manifold is simply connected). This holds even for a finite path. To prove this, subdivide a finite loop of area A into N loops of area (A=N). If Riemann vanishes, then the difference in parallel transport per loop (for small enough loop) scales as (A=N)α, where the index α > 1. Hence the total error on a finite path scales as N(A=N)α / Aα=N α−1. This goes to zero as N ! 1. Suppose the Riemann vanishes everywhere. At a given point p 2 M, it is always possible to find an orthonormal basis of the tangent space, fe(µ)g. The fact that the basis is orthonormal means that g(e(µ); e(ν)) = ηµν at p.
    [Show full text]