tensors tensor transformation law matrix physical quantities tensor
More generally, most physical quantities are best described as (components of) tensors. So to transform from one frame to another, we use the well-known tensor transformation law





T


θ
′


ι
′

⋯

κ
′




α
′


β
′

⋯

ζ
′



=

Λ


α
′







μ



Λ


β
′







ν


⋯

Λ


ζ
′







ρ



Λ


θ
′







σ



Λ


ι
′







υ


⋯

Λ


κ
′







ϕ



T

σ
υ
⋯
ϕ


μ
ν
⋯
ρ




{\displaystyle T_{\theta '\iota '\cdots \kappa '}^{\alpha '\beta '\cdots \zeta '}=\Lambda ^{\alpha '}{}_{\mu }\Lambda ^{\beta '}{}_{\nu }\cdots \Lambda ^{\zeta '}{}_{\rho }\Lambda _{\theta '}{}^{\sigma }\Lambda _{\iota '}{}^{\upsilon }\cdots \Lambda _{\kappa '}{}^{\phi }T_{\sigma \upsilon \cdots \phi }^{\mu \nu \cdots \rho }}


where 




Λ


χ
′







ψ




{\displaystyle \Lambda _{\chi '}{}^{\psi }}

 is the reciprocal matrix of 




Λ


χ
′







ψ




{\displaystyle \Lambda ^{\chi '}{}_{\psi }}

. All tensors transform by this rule.
