A tensor is a mathematical object that generalizes scalars, vectors, and matrices to higher dimensions.Imagine you want to describe the tension inside a piece of rubber: you pull it along, across, or diagonally – the force responds differently. A tensor is like a 'summary table' of all such interactions at once. It stores information about how a material responds to stress in any direction. Simply put, if a number is a point, a vector is an arrow, then a tensor is a whole system of interconnected arrows.
The history of tensors begins with works on differential geometry and mechanics. Gauss and Riemann laid the foundations, but the systematic calculus was developed by Ricci-Curbastro and Levi-Civita in 1884–1900. In 1915, Albert Einstein used their apparatus to formulate general relativity, where gravity manifests as the curvature of spacetime, described by the Ricci tensor and the Einstein tensor. James Clerk Maxwell had already in 1865 essentially written his equations in tensor form, though he didn't use the term. Hendrik Lorentz in 1904 derived transformations that became the basis for understanding the tensor nature of physical laws in special relativity.
The transformation law for tensor components when transitioning to new coordinates x'^μ = x'^μ(x^ν) is given by the product of Jacobian matrices. For example, for a tensor of rank (1,1): T'^μ_ν = (∂x'^μ/∂x^α) (∂x^β/∂x'^ν) T^α_β. Maxwell's equations in tensor form: ∂_ν F^{μν} = μ_0 J^μ, where F^{μν} is the electromagnetic field tensor, J^μ is the 4-current. Einstein field equations: G_{μν} + Λ g_{μν} = (8πG/c^4) T_{μν}, where G_{μν} is the Einstein tensor expressing spacetime curvature through the metric tensor and its derivatives. Tensors also describe stress and strain in mechanics: σ_{ij} = C_{ijkl} ε^{kl}, where σ is the stress tensor, ε is the strain tensor, and C is the fourth-rank elasticity tensor.
Without tensors, we wouldn’t be able to accurately predict how a bridge deforms under the weight of a train. And it was tensors that helped Einstein describe how massive bodies warp spacetime – like a heavy ball on a stretched sheet.
| r | variable | tensor-to-scalar ratio — dimensionless ratio |
| P_t | variable | power spectrum of tensor perturbations — power spectrum |
| P_s | variable | power spectrum of scalar perturbations — power spectrum |
| w_1 | variable | weight for a — weight |
| a | variable | first value |
| w_2 | variable | weight for b — weight |
| b | variable | second value |
| H_{\text{HFI}} | variable | magnetic hyperfine interaction Hamiltonian — hamiltonian joule |
| \boldsymbol{\mu}_I | variable | nuclear magnetic moment — magnetic moment ampere square metre |
| \mathbf{A} | variable | hyperfine coupling tensor — tensor joule per ampere square metre |
| \mathbf{S} | variable | electron spin operator — spin operator |
| \cdot | operator | scalar product of vectors — dot product |
