5.1 Definition and properties of metric tensors

Metric tensors are the backbone of geometry in curved spaces. They define how we measure distances and angles, bridging the gap between flat and curved geometries. This powerful tool allows us to generalize concepts like length and volume to complex spaces.

In this section, we'll unpack the definition and key properties of metric tensors. We'll explore their symmetry, positive definiteness, and role in index manipulation. Understanding these concepts is crucial for grasping how we quantify geometry in various physical theories.

Metric Tensor Definitions

Fundamental Concepts of Metric Tensors

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• Metric tensor defines distance and angles in a given space
• Represents geometric structure of a manifold
• Enables measurement of lengths, areas, and volumes in curved spaces
• Generalizes the concept of dot product to non-Euclidean geometries
• Denoted by $[g_{ij}](https://www.fiveableKeyTerm:g_{ij})$ in tensor notation, where i and j are indices

Covariant and Contravariant Metric Tensors

• Covariant metric tensor transforms like covariant vectors
  • Denoted by lowercase indices (g_{ij})
  • Used to lower indices of contravariant vectors
• Contravariant metric tensor transforms like contravariant vectors
  • Denoted by uppercase indices (g^{ij})
  • Used to raise indices of covariant vectors
• Relationship between covariant and contravariant metric tensors
  • Inverse of each other: $g^{ik}g_{kj} = \delta^i_j$ (Kronecker delta)

Symmetry and Components of Metric Tensors

• Symmetric tensor property applies to metric tensors
  • Components remain unchanged when indices are swapped: $g_{ij} = g_{ji}$
• Number of independent components in an n-dimensional space
  • Calculated as $\frac{n(n+1)}{2}$ due to symmetry
• Components of metric tensor depend on coordinate system
  • Can vary from point to point in curved spaces
• Metric tensor components determine local geometry (flat, curved, Euclidean, non-Euclidean)

Metric Tensor Properties

Positive Definiteness and Metric Signature

• Positive definite property ensures real-valued distances
  • For any non-zero vector v^i, $g_{ij}v^iv^j > 0$
  • Guarantees that lengths and angles are well-defined
• Metric signature characterizes the nature of the space
  • Riemannian metrics have all positive eigenvalues (+, +, +, ...)
  • Pseudo-Riemannian metrics have mixed signs (--, +, +, ...)
  • Minkowski metric in special relativity has signature (-, +, +, +)

Index Manipulation and Tensor Operations

• Raising and lowering indices changes tensor type
  • Raising index: $v^i = g^{ij}v_j$ (contravariant from covariant)
  • Lowering index: $v_i = g_{ij}v^j$ (covariant from contravariant)
• Metric tensor facilitates contraction of tensors
  • Summing over repeated indices (Einstein summation convention)
• Determinant of metric tensor ($\det(g_{ij})$) crucial for integration
  • Provides measure of volume element in curved spaces

Coordinate Transformations and Invariance

• Metric tensor components transform under coordinate changes
  • Ensures invariance of physical quantities (distances, angles)
• Transformation rule for metric tensor components
  • $g'_{ij} = \frac{\partial x^k}{\partial x'^i}\frac{\partial x^l}{\partial x'^j}g_{kl}$
• Invariant line element $ds^2 = g_{ij}dx^idx^j$ remains unchanged
  • Fundamental to describing geometry independent of coordinates

Common Metric Tensors

Riemannian Metric in Differential Geometry

• Defines smooth inner product on tangent space of manifold
• Allows measurement of distances and angles on curved surfaces
• Components may vary from point to point on the manifold
• Enables study of intrinsic geometry (Gaussian curvature)
• Applications in general relativity and differential geometry

Euclidean Metric in Flat Spaces

• Simplest form of metric tensor for flat spaces
• Components form identity matrix in Cartesian coordinates
  • $g_{ij} = \delta_{ij} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ (for 3D space)
• Pythagoras theorem directly applies: $ds^2 = dx^2 + dy^2 + dz^2$
• Invariant under rotations and translations
• Serves as local approximation for curved spaces (tangent space)

Minkowski Metric in Special Relativity

• Describes geometry of spacetime in special relativity
• Signature (-,+,+,+) reflects one time and three space dimensions
• Components in standard coordinates: $\eta_{\mu\nu} = \text{diag}(-1,1,1,1)$
• Invariant interval: $ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$
• Preserves causality and defines light cone structure
• Lorentz transformations leave Minkowski metric invariant