Intro to differential geometry and general relativity s waner

... and Proper Time 94 12. The Stress Tensor and the Relativistic Stress-Energy Tensor 100 13. Two Basic Premises of General Relativity 109 14. The Einstein Field Equations and Derivation of Newton&apos ;s ... coordinates of sums or scalar multiples of tangent vectors, simply take the corresponding sums and scalar multiples of the coordinates. In other words: (v+w) i = v i + w i a...

Ngày tải lên: 17/03/2014, 14:28

... CONTENTS 1. Preliminaries: Distance, Open Sets, Parametric Surfaces and Smooth Functions 2. Smooth Manifolds and Scalar Fields 3. Tangent Vectors and the Tangent Space 4. Contravariant and Covariant ... Tensor 11. A Little More Relativity: Comoving Frames and Proper Time 12. The Stress Tensor and the Relativistic Stress-Energy Tensor 13. Two Basic Premises of General Relativit...

Ngày tải lên: 27/03/2014, 11:52

... and Proper Time 94 12. The Stress Tensor and the Relativistic Stress-Energy Tensor 100 13. Two Basic Premises of General Relativity 109 14. The Einstein Field Equations and Derivation of Newton&apos ;s ... coordinates of sums or scalar multiples of tangent vectors, simply take the corresponding sums and scalar multiples of the coordinates. In other words: (v+w) i = v i + w i a...

Ngày tải lên: 24/04/2014, 17:08

... same subspace, and hence the normal vector is a constant. The Gauss map is thus a constant, mapping all of Y onto a single point in S n−1 . 2. Suppose that Y is the sphere of radius R (say centered ... physical theories as Yang-Mills fields. This chapter seems to present the most difficulty conceptually for the student. Chapter V discusses the general theory of connections on fibe r bundl...

Ngày tải lên: 17/03/2014, 14:28

... E 3 . Recast as such, this result likewise becomes a special case of the so-called rigidity theorem; cf. Section 1.7. Recast as such, these two theorems together constitute a special case (that where ... the necessary conditions R qijk =0inΩ are also sufficient for the existence of such an immersion Θ. Besides, if Ω is connected, this immersion is unique up to isometries of E 3 . This means t...

Ngày tải lên: 17/03/2014, 14:24

... orthonormal bases and commutators (§2), parallel transport and geodesics (§3), and the Riemann curvature tensor (§4). 2 Orthonormal Bases, Tetrads, and Commutators A vector basis is said to be orthonormal ... tetrad. Thus, an observer carries along an orthonormal bases that we call the observer s tetrad. This basis is the natural one for splitting vectors, one-forms, and ten...

Ngày tải lên: 17/03/2014, 14:28

... respect to some set of basis vectors. A basis is any set of vectors which both spans the vector space (any vector is a linear combination of basis vectors) and is linearly independent (no vector ... indices will usually label components of vectors and tensors. This is why there are parentheses around the indices on the basis vectors, to remind us that this is a collection of vectors,...

Ngày tải lên: 17/03/2014, 14:28

... appendices. Chapters 2 and 3 together, without the appendices, usually take me about three weeks as well. Chapter 4 deals with transformations of random variables and vectors, and also lists the most ... sequence of sets in must contain sets that are the same. Atö first sight this seems to conflict with the implicit assumption that there always exist countably infinite sequences of d...

Ngày tải lên: 31/03/2014, 15:23

... Two sets S and T are equal if each element of S is an element of T and vice versa. This is denoted, S = T. Inequality is S = T , of course. S is a subset of T , S ⊆ T , if every element of S is ... the sets S j . The intersection of two sets, S ∩ T , is the set whose elements are in both of the two sets. In other words, the intersection of two sets in the set of element...

Ngày tải lên: 31/03/2014, 16:19