Question

What is the definition of a tensor (in physics)?

Answer 1

Short answer
- In physics, a tensor is a geometric object that is independent of coordinates and whose components transform under a change of basis (or coordinates) according to a fixed multilinear rule. Because of that transformation law, equations written with tensors are coordinate‑independent (physically meaningful).

More precise/technical
- On a vector space V (or the tangent space of a manifold) a tensor of type (k,l) (also called rank k+l) is an element of V^{⊗k} ⊗ (V*)^{⊗l}. Equivalently, it is a multilinear map
(V*)^k × V^l → R.
- In coordinates x^i, the components T^{i1...ik}_{j1...jl} of a (k,l)-tensor transform under a change x → x' by
T'^{i1...ik}_{j1...jl}
= (∂x'^{i1}/∂x^{a1}) ... (∂x'^{ik}/∂x^{ak}) (∂x^{b1}/∂x'^{j1}) ... (∂x^{bl}/∂x'^{jl}) T^{a1...ak}_{b1...bl}.
This is the defining transformation law.

Common examples and remarks
- Scalars are (0,0)-tensors. Vectors are (1,0)-tensors; covectors (one-forms) are (0,1)-tensors. The metric is typically a symmetric (0,2)-tensor. The stress–energy tensor T^{μν} in relativity is a (2,0)-tensor (often written T_{μν} as (0,2) depending on index placement).
- Operations: tensor product (builds higher-rank tensors), contraction (sums an upper and a lower index to lower rank), index raising/lowering with the metric, symmetrization/antisymmetrization.
- Pseudotensors and tensor densities are related objects that do not transform exactly by the tensor law (they pick up sign or determinant factors under orientation-reversing or non‑unit Jacobian changes).

Intuition
- A tensor encodes multilinear relations between vectors and covectors in a coordinate-free way. The key property is the specific transformation law: if components transform that way, the object is a tensor and physical laws expressed in tensor form do not depend on the observer’s coordinate choice.