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String Action Gauge Metric Pdf Download ((NEW))
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Measure string action (the height of the string above the fret) at any point by placing the gauge behind the string. The string height markings are at increments of .010\" (ten thousandths of an inch). When the bottom of a mark aligns with the bottom of the string, that measurement is the string height at that point.
The global conformal gauge is playing the crucial role in string theory providing the basis for quantization. Its existence for two-dimensional Lorentzian metric is known locally for a long time. We prove that if a Lorentzian metric is given on a plain then the conformal gauge exists globally on the whole \\({{\\mathbb {R}}}^2\\). Moreover, we prove the existence of the conformal gauge globally on the whole worldsheets represented by infinite strips with straight boundaries for open and closed bosonic strings. The global existence of the conformal gauge on the whole plane is also proved for the positive definite Riemannian metric.
The (super)string theory attracts much interest in physics and mathematics for the last fifty years (see, e.g., [1,2,3]). It is usually considered as the basis for construction of the unified quantum theory of all fundamental interactions including gravity. The crucial role in the theory is played by the conformal gauge for a metric of Lorentzian signature in which it is conformally flat. In fact, almost all results in string theory are obtained using the assumption that the conformal gauge exists on the whole string worldsheets represented by infinite strips with straight boundaries. For example, the covariant and light cone quantizations use Fourier series which exist only if the conformal gauge is applied on the whole string worldsheet.
The local existence of the conformal gauge is well known for a long time (see, for \\({\\mathcal{{C}}}^2\\)-metric, e.g., [4, Ch. I, 6.1], or [5, Ch. I, 3.1]). This gauge and boundary value problems were also considered in [6]. The local existence of the gauge is proved by writing down equations for transformation functions and considering their integrability conditions which guarantee the existence of solution in some neighbourhood of an arbitrary point. However it is not enough. In string theory, it is assumed that the conformal gauge exists on an infinite strip with straight boundaries. There are two subtle questions: does the conformal gauge exist on the whole strip and can the boundaries be made straight In the present paper, we answer these questions affirmatively. The transition from local to global considerations is based on the global existence theorem for the solution of the Cauchy problem for two-dimensional hyperbolic differential equations with varying coefficients (see, e.g., [7], book IV, ch. I). This theorem is highly nontrivial, but allows one to make global statements.
The local existence of the conformal gauge (isothermal coordinates) for positive definite Riemannian metric is also known in mathematics for a long time. The proof for analytic metric is given e.g. in [4, Ch. I, 6.4] and [5, Ch. I, 3.4] and for \\({\\mathcal{{C}}}^3\\)-metric e.g. in [9, Theorem 2.5.14]. In the present paper, we extend the proof to the whole Euclidean plane.
In the next section, we introduce notation and write down equations of motion with boundary conditions. Afterwards we consider infinite, open, and closed strings in subsequent sections, respectively. Finally, we analyze the Riemannian two-dimensional metric.
where brackets denote the usual scalar product in \\({{\\mathbb {R}}}^{1,D-1}\\). Now the embedding (1) defines the Lorentzian metric on the string worldsheet interior \\({{\\mathbb {U}}}\\) with signature \\((+-)\\).
with identified boundaries. There are many ways to identify smoothly the boundaries (6). In string theory, we, first, impose the conformal gauge on the metric on the same strip (6) and, second, impose the smooth periodicity conditions
where \\(\\phi (x)\\) is some sufficiently smooth function, on the whole string worldsheet. The aim of the present paper is to prove that this conformal gauge can be imposed on the same strips (5) and (6) both for open and closed strings with the same straight boundaries.
Here and in what follows, \\({{\\mathbb {U}}}\\) denotes either the whole Euclidean plane (infinite string) or an open set (strip) on the plane \\((\\tau ,\\sigma )\\in {{\\mathbb {U}}}\\subset {{\\mathbb {R}}}^2\\) (open or closed string), where the induced metric is nondegenerate. The boundaries \\(\\partial {{\\mathbb {U}}}\\) of open string, on which the metric is degenerate, are considered separately.
In contrast to the local theorem (see, e.g., [4, 5]) stating the existence of the conformal gauge only in some neighborhood of every point, the above theorem is global in a sense that it provides the existing of the conformal gauge for a Lorentzian metric given on the whole plane \\(x\\in {{\\mathbb {R}}}^2\\).
Now we consider an open string whose worldsheet \\({\\overline{{{\\mathbb {U}}}}}\\) is an infinite strip on the plane \\((\\tau ,\\sigma )\\in {{\\mathbb {R}}}^2\\) with two, probably, curved boundaries: the left \\(\\gamma _{\\textsc {l}}\\) and right \\(\\gamma _{\\textsc {r}}\\) boundaries. The induced metric on the boundaries is degenerate, and results of the previous section must be revised. First, we assume that metric is not degenerate and return to this problem later.
So, if the metric is not degenerate on the boundaries of an open string worldsheet, then there exists such global \\({\\mathcal{{C}}}^1\\) coordinate transformation that the transformed metric is conformally flat (32) on the whole vertical strip with straight boundaries \\({\\tilde{\\sigma }}=0\\) and \\({\\tilde{\\sigma }}=\\pi \\). This statement follows from Theorems 4.1 and 5.1 because the conformal transformation is a diffeomorphism, and diffeomorphisms form a group.
We showed in the previous section that there is the global diffeomorphism \\((\\tau ,\\sigma )\\mapsto ({\\hat{\\tau }},{\\hat{\\sigma }})\\) which maps an arbitrary infinite strip with timelike boundaries on the vertical strip with straight boundaries where metric becomes conformally flat. The same procedure can be performed for the fundamental domain of a closed string. Without loss of generality, we assume that boundaries go through points \\({\\hat{\\sigma }}=\\pm \\pi \\), as is usually supposed in string theory. Then the boundary identification is written as the periodicity condition for every value of the timelike coordinate \\({\\hat{\\tau }}\\):
The existence of vector fields t and s provides sufficient conditions for the existence of the conformal gauge on the whole Euclidean plane \\({{\\mathbb {R}}}^2\\) for metrics separated from zero and infinity (30, and analog of Theorem 4.1 holds.
The Euclidean version of string theory is used in the path integral formulation of quantum string theory, which assumes summation over Riemannian surfaces of different genera. The Riemannian surfaces cannot be covered by a single coordinate chart, and therefore we cannot talk about the conformal gauge on the whole Riemannian surface. The results of the present section guarantee the existence of the conformal gauge on the whole coordinate chart which is diffeomorphic to \\({{\\mathbb {R}}}^2\\). Previous theorems provide sufficient conditions for the existence of the conformal gauge only in some sufficiently small neighbourhood of each point of the manifold.
It was assumed for many years that there exists the global conformal gauge in string theory though this statement was proved only locally. In fact, almost all results were obtained under validity of this assumption which turns out to be true quite unexpectedly at least to the author. We proved the global existence of the conformal gauge for infinite, open, and closed strings. The transition from local to global statement is based on the global existence of the solution of the Cauchy problem for a two-dimensional hyperbolic equation with varying coefficients [7] and is far from being obvious.
As a byproduct, we proved global existence of the conformal gauge for a general two-dimensional Lorentzian metric defined on the whole plane \\({{\\mathbb {R}}}^2\\) which is not necessarily induced by an embedding and is well known locally for a long time (see, e.g. [4, 5]).
The existence theorem is also proved for a Riemannian positive definite metric defined on the whole Euclidean plane. It generalizes previous results providing the existence of the conformal gauge in some sufficiently small neighbourhood of each point.
2. Introducing Differential Geometry: PDF Manifolds: Topological spaces, differentiable manifolds and maps between manifolds. Tangent Spaces: tangent vectors, vector fields, integral curves and the Lie derivative. Tensors, covectors and one-forms. Differential Forms: the exterior derivative, de Rahm cohomology, integration and Stokes' theorem. 3. Introducing Riemannian Geometry: PDF The metric; Riemannian and Lorentzian manifolds, the volume form and the Hodge dual. The Maxwell action. Hodge theory. Connections and the covariant derivative, curvature and torsion, the Levi-Civita connection. The divergence theorem. Parallel transport, normal coordinates and the exponential map, holonomy, geodesic deviation. The Ricci tensor and Einstein tensor. Connection 1-forms and curvature 2-forms. 4. The Einstein Equations: PDF The Einstein-Hilbert action, the cosmological constant; diffeomorphisms and the Bianchi identity; Minkowski, de Sitter and anti-de Sitter spacetimes; Symmetries and isometries, Killing vectors, conserved quantities; Asymptotics of spacetime, conformal transformations and Penrose diagrams; Coupling matter, the energy-momentum tensor, perfect fluids, spinors, energy conditions; Cosmology. 5. When Gravity is Weak: PDF The Linearised theory, gauge symmetry, the Newtonian limit; Gravitational waves, de Donder gauge, transverse traceless gauge, LIGO; Gravitational wave production, binary systems, the quadrupole formula, gravitational wave sources. 6. Black Holes: PDF The Schwarzschild solution, Birkhoff's theorem, Eddington-Finkelstein Coordinates, Kruskal diagrams and Penrose diagrams, weak cosmic censorship; The Reissner-Nordstrom solution, Cauchy horizons and strong cosmic censorship, Extremal black holes; The Kerr solution, global structure, the ergoregion, the Penrose process and superradiance, no hair theorems. Problem SheetsJoĆ£o Melo has put together a preparatory worksheet, based on Chapter 1 of the lectures notes, to help refresh your understanding of geodesics before the course begins. It can be downloaded here. 153554b96e
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