Table of Contents
- Introduction
- Motivation and Applications of Tensor Calculus
- Scalars, Vectors, and Tensors: A Unified Language
- Index Notation and Einstein Summation Convention
- Covariant and Contravariant Components
- Transformation Laws for Tensors
- The Metric Tensor and Index Manipulation
- Tensor Operations: Addition, Outer Product, and Contraction
- Covariant Derivative and Connection Coefficients
- Christoffel Symbols and Geodesics
- Riemann Curvature Tensor
- Ricci Tensor and Ricci Scalar
- Tensor Densities and Volume Elements
- Applications in Physics
- Conclusion
1. Introduction
Tensor calculus is the language of modern physics and differential geometry. It allows us to express laws of nature — such as general relativity — in a coordinate-independent way. Tensors generalize scalars and vectors, and tensor operations provide a consistent framework for dealing with curved spaces, multiple dimensions, and complex physical interactions.
2. Motivation and Applications of Tensor Calculus
Why do we need tensor calculus?
- Describes physical laws in any coordinate system
- Used in general relativity, continuum mechanics, fluid dynamics, and electromagnetism
- Necessary for studying curved manifolds like spacetime
- Enables rigorous and compact mathematical formulation
3. Scalars, Vectors, and Tensors: A Unified Language
- Scalars: Single-valued, coordinate-independent quantities (e.g., temperature \( T \))
- Vectors: Ordered set of numbers transforming linearly under coordinate changes
- Tensors: Generalizations of vectors; a rank-\((m, n)\) tensor has \(m\) contravariant and \(n\) covariant indices
A second-rank tensor \( T^\mu_{\ \nu} \) transforms as:
\[
T’^\mu_{\ \nu} = \frac{\partial x’^\mu}{\partial x^\alpha} \frac{\partial x^\beta}{\partial x’^\nu} T^\alpha_{\ \beta}
\]
4. Index Notation and Einstein Summation Convention
Einstein notation simplifies tensor expressions:
- Repeated indices imply summation
- \( A^i B_i = \sum_i A^i B_i \)
Free indices (those not repeated) must match on both sides of an equation.
5. Covariant and Contravariant Components
- Contravariant: \( A^i \) — transforms opposite to coordinate basis
- Covariant: \( A_i \) — transforms like basis covectors (dual basis)
This distinction is crucial in general geometry.
6. Transformation Laws for Tensors
For a vector:
\[
A’^i = \frac{\partial x’^i}{\partial x^j} A^j
\]
For a covector:
\[
A’_i = \frac{\partial x^j}{\partial x’^i} A_j
\]
This ensures tensorial quantities behave consistently across frames.
7. The Metric Tensor and Index Manipulation
The metric tensor \( g_{\mu\nu} \) defines the inner product of vectors and distances in spacetime.
- It allows lowering and raising indices:
\[
A_\mu = g_{\mu\nu} A^\nu, \quad A^\mu = g^{\mu\nu} A_\nu
\]
- It encodes the geometry (flat, curved, etc.)
8. Tensor Operations: Addition, Outer Product, and Contraction
- Addition: Only tensors of same rank and type
- Outer product: Combines tensors into higher-rank ones
- Contraction: Sum over a pair of one upper and one lower index:
\[
T^\mu_{\ \mu} = \text{Trace}
\]
This operation reduces the rank by 2.
9. Covariant Derivative and Connection Coefficients
The covariant derivative generalizes partial derivatives to curved spaces.
For a vector \( V^\mu \):
\[
\nabla_\nu V^\mu = \partial_\nu V^\mu + \Gamma^\mu_{\nu\sigma} V^\sigma
\]
Where \( \Gamma^\mu_{\nu\sigma} \) are Christoffel symbols — not tensors.
10. Christoffel Symbols and Geodesics
Christoffel symbols are defined as:
\[
\Gamma^\rho_{\mu\nu} = \frac{1}{2} g^{\rho\sigma} \left( \partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} – \partial_\sigma g_{\mu\nu} \right)
\]
They appear in the geodesic equation:
\[
\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0
\]
This describes free-fall in curved spacetime.
11. Riemann Curvature Tensor
The Riemann tensor measures how much vectors are rotated by parallel transport:
\[
R^\rho_{\ \sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} – \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} – \Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma}
\]
If \( R^\rho_{\ \sigma\mu\nu} = 0 \), the space is flat.
12. Ricci Tensor and Ricci Scalar
Contracting the Riemann tensor:
- Ricci tensor: \( R_{\mu\nu} = R^\alpha_{\ \mu\alpha\nu} \)
- Ricci scalar: \( R = g^{\mu\nu} R_{\mu\nu} \)
These appear in Einstein’s equations and determine how curvature sources gravitational effects.
13. Tensor Densities and Volume Elements
The volume element in curved space involves the determinant of the metric:
\[
dV = \sqrt{|g|} \, d^4x
\]
Tensor densities adjust for coordinate volume scaling and are used in action integrals and field theory.
14. Applications in Physics
- General relativity: Field equations use Ricci tensor and scalar curvature
- Electromagnetism: Expressed using the antisymmetric electromagnetic field tensor
- Fluid dynamics: Stress-energy tensors encode momentum and pressure
- Quantum field theory: Tensor fields represent interactions in spacetime
15. Conclusion
Tensor calculus is the backbone of modern theoretical physics. From describing gravitational waves to the dynamics of the early universe, it offers a consistent and powerful framework for expressing physical laws in any coordinate system.
Understanding tensors, their transformations, and operations like covariant differentiation and curvature is essential for anyone venturing into relativity, cosmology, and high-energy physics.
