inner product tensors Lorentz transformations Lorentz transformation tensor
One can extend this idea to tensors of higher order, for a second order tensor we can form the invariants:





T

α






α


,

T

α






β



T

β






α


,

T

α






β



T

β






γ



T

γ






α


=

invariant scalars

,


{\displaystyle T^{\alpha }{}_{\alpha },T^{\alpha }{}_{\beta }T^{\beta }{}_{\alpha },T^{\alpha }{}_{\beta }T^{\beta }{}_{\gamma }T^{\gamma }{}_{\alpha }={\text{invariant scalars}},}


similarly for higher order tensors. Invariant expressions, particularly inner products of 4-vectors with themselves, provide equations that are useful for calculations, because one does not need to perform Lorentz transformations to determine the invariants.
