1.1 Einstein’s equation The goal is to ﬁnd a solution of Einstein’s equation for our metric (1), Rµν − 1 2 gµν = 8πG c4 Tµν (3) FIrst some terminology: Rµν Ricci tensor, R Ricci scalar, and Tµν stress-energy tensor (the last term will vanish for the Schwarzschild solution). Looking forward An Introduction to the Riemann Curvature Tensor and Diﬀerential Geometry Corey Dunn 2010 CSUSB REU Lecture # 1 June 28, 2010 Dr. Corey Dunn Curvature and Diﬀerential Geometry Pythagoras, the metric tensor and relativity1 Pythagoras2 is regarded to be the ﬁrst pure mathematician. Normal Vector, Tangent Plane, and Surface Metric 407 Section 56. Section 55. Coordinate Invariance and Tensors 16 X. Transformations of the Metric and the Unit Vector Basis 20 XI. Divergences, Laplacians and More 28 XIII. the single elements % as a function of the metric tensor. Instead, the metric is an inner product on each vector space T p(M). ijvi: It is said that “the metric tensor ascends (or descends) the indices”. This means that any quantity A = Aae a in another frame, Abe b = ∂xb ), at least from the formal point of view. 1 Tensor Analysis and Curvilinear Coordinates Phil Lucht Rimrock Digital Technology, Salt Lake City, Utah 84103 last update: May 19, 2016 Maple code is available upon request. 1 Introduction In this work, a preliminary analysis of the relation between monotone metric tensors on the manifold of faithful quantum states and group actions of suitable extensions of the unitary group is presented. 1.16.32) – although its components gij are not constant. The symmetrization of ω⊗ηis the tensor ωη= 1 2 (ω⊗η+η⊗ω) Note that ωη= ηωand that ω2 = ωω= ω⊗ω. METRIC TENSOR 3 ds02 = ds2 (9) g0 ijdx 0idx0j = g0 ij @x0i @xk dxk @x0j @xl dxl (10) = g0 ij @x0i @xk @x0j @xl dxkdxl (11) = g kldxkdxl (12) The ﬁrst line results from the transformation of the dxiand the last line results from the invariance of ds2.Comparing the last two lines, we have covariant or contravariant, as the metric tensor facilitates the transformation between the di erent forms; hence making the description objective. Dual Vectors 11 VIII. The resulting tensors may, however, prescribe abrupt size variations that Similarly, the components of the permutation tensor, are covariantly constant | |m 0 ijk eijk m e. In fact, specialising the identity tensor I and the permutation tensor E to Cartesian coordinates, one has ij ij User specica-tions can also be formulated as metric tensors and combined with solution-based and geometric metrics. Surface Geodesics and the Exponential Map 425 Section 58. The above tensor T is a 1-covariant, 1-contravariant object, or a rank 2 tensor of type (1, 1) on 2 . METRIC TENSOR: INVERSE AND RAISING & LOWERING INDICES 2 On line 2 we used @x0j @xb @xl @x0j = l b and on line 4 we used g alg lm= a m. Thus gijis a rank-2 contravariant tensor, and is the inverse of g ijwhich is a rank-2 covariant tensor. The Metric Generalizes the Dot Product 9 VII. Since G=M T M, new metric related the quantum geometry, ds̃2 = g AB dx A dxB, (8) where gAB = g ⊗ g . The Robertson-Walker metric with ﬂat spatial sections, ds 2= −dt +a(t)2(dx2 + dy2 + dz2), satisﬁes this condition and its Ricci tensor is consequently diagonal. Metric tensor Taking determinants, we nd detg0 = (detA) 2 (detg ) : (16.14) Thus q jdetg0 = A 1 q ; (16.15) and so dV0= dV: This is called the metric volume form and written as dV = p jgjdx1 ^^ dxn (16.16) in a chart. Deﬁnition:Ametric g is a (0,2) tensor ﬁeld that is: • Symmetric: g(X,Y)=g(Y,X). As we shall see, the metric tensor plays the major role in characterizing the geometry of the curved spacetime required to describe general relativity. Since the metric tensor is symmetric, it is traditional to write it in a basis of symmetric tensors. 1.2 Manifolds Manifolds are a necessary topic of General Relativity as they mathemat- It does, indeed, provide this service but it is not its initial purpose. Therefore we have: r' • r' = r • r from which foUows, applying relation (4): r"ar' = r'Gr and from (9): r'A 'GAr = r'Gr Example 2: a tensor of rank 2 of type (1-covariant, 1-contravariant) acting on 3 Tensors of rank 2 acting on a 3-dimensional space would be represented by a 3 x 3 matrix with 9 = 3 2 Now consider G-1 X. The Formulas of Weingarten and Gauss 433 Section 59. This is the second volume of a two-volume work on vectors and tensors. When no so-lution is yet available, metrics based on the computational domain geometry can be used instead . 1 Pythagoras’ Theorem The Riemann-Christoffel Tensor and the Ricci Identities 443 Section 60. allows the presence of a metric in each manifold and defines all the associated tensors (Riemann, Ricci, Einstein, Weyl, etc.) I feel the way I'm editing videos is really inefficient. Orthogonal coordinate systems have diagonal metric tensors and this is all that we need to be concerned with|the metric tensor contains all the information about the intrinsic geometry of spacetime. Here is a list with some rules helping to recognize tensor equations: • A tensor expression must have the same free indices, at the top and at the bottom, of the two sides of an equality. While we have seen that the computational molecules from Chapter 1 can be written as tensor products, not all computational molecules can be written as tensor products: we need of course that … A quantity having magnitude only is called Scalar and a quantity with 12|Tensors 2 the tensor is the function I.I didn’t refer to \the function (!~)" as you commonly see.The reason is that I(!~), which equals L~, is a vector, not a tensor.It is the output of the function Iafter the independent variable!~has been fed into it.For an analogy, retreat to the case of a real valued function The infimum in (213) is taken over all vector fields u on R N such that the linear transport equation ∂f/∂s + ∇ υ ⋅ (fu) = 0 is satisfied.By polarization, formula (213) defines a metric tensor, and then one is allowed to all the apparatus of Riemannian geometry (gradients, Hessians, geodesics, etc. Since the matrix inverse is unique (basic fact from in the same flat 2-dimensional tangent plane. I have 3 more videos planned for the non-calculus videos. Observe that g= g ijdxidxj = 2 Xn i=1 g ii(dxi)2 +2 X 1≤i