Metric tensor

The metric tensor gμνg_{\mu\nu} is the tensor associated with the Scalar product in spacetime (specifically Minkowski space). It is used to describe the concept of distance in spacetime, hence the name "metric". It is defined as

gμν=(−1000010000100001)g_{\mu\nu}=\begin{pmatrix}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix}

and is applied to two four-vectors as

a⋅b=−a0b0+a⋅b=gμνaμbνa\cdot b=-a_{0}b_{0}+\mathbf{a}\cdot \mathbf{b}=g^{\mu\nu}a_{\mu}b_{\nu}

The signature of this metric tensor is (−++ +)(-++\ +). The signature (+−− −)(+--\ -) is also a valid choice and leads to the same results. In the latter case, the scalar product would be

a⋅b=a0b0−a⋅ba\cdot b=a_{0}b_{0}-\mathbf{a}\cdot \mathbf{b}

The choice must be kept consistent to avoid mismatched formulas.

Properties

  • The metric tensor converts between covariant and contravariant four-vectors: aμ=gμνaνa_{\mu}=g_{\mu\nu}a^{\nu}.
  • The induced product a⋅ba\cdot b is a relativistic invariant. By extension, so is the induced norm ∣a∣2=a⋅a=gμνaμaν|a|^{2}=a\cdot a=g^{\mu\nu} a_{\mu}a_{\nu}.