170 Inverse Metric
A matrix function that is the inverse of the metric matrix that is used to raise indices on tensors.
definition [d] (Inverse Metric = Contravariant Metric Tensor) The contravariant metric tensor \(g^{ik}\) that is the matrix inverse of the covariant metric \(g_{kj}\):
- \(g^{ik}\, g_{kj} = \delta^{i}_{\ j}\) .
where
- \(g^{ik}\) are the components of the inverse metric.
- \(g_{kj}\) are the components of the covariant metric.
- \(\delta^{i}_{\ j}\) is the Kronecker delta.
- \(i, j, k\) are coordinate indices.
Note:
- \(g_{kj}\) must be nondegenerate so the inverse exists.
- \(g^{ik} = g^{ki}\).
definition [d] (Inverse Metric = Contravariant Metric Tensor) The type-\((2,0)\) tensor used to raise indices,
- \(V^{i} = g^{ij}\, V_{j}\) ,
with \(g^{ik}\, g_{kj} = \delta^{i}_{\ j}\).
where
- \(g^{ij}\) are the components of the inverse metric.
- \(V^{i}\) are the raised, contravariant components of a vector.
- \(V_{j}\) are the lowered, covariant components of a vector.
- \(\delta^{i}_{\ j}\) is the Kronecker delta.
Note:
- lowering uses the covariant metric: \(V_{i} = g_{ij}\, V^{j}\).
- together, \(g_{ij}\) and \(g^{ij}\) identify vectors with covectors.
170.1 Elementary Example
170.1.1 Simple
Let \(g\) be a diagonal metric matrix. The inverse metric \(g^{-1}\) is the matrix inverse of \(g\).
\[ g = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix},\quad g^{-1} = \begin{pmatrix} \tfrac{1}{2} & 0 \\ 0 & 1 \end{pmatrix} \]
\[ g^{ik} g_{kj} = \delta^{i}_{\ j} \]
where
- \(g_{kj}\) are components of the metric.
- \(g^{ik}\) are components of the inverse metric.
- \(\delta^{i}_{\ j}\) is the Kronecker symbol, equal to \(1\) if \(i = j\) and \(0\) otherwise.
170.1.2 General
In three dimensions the same rule inverts a diagonal metric with three entries.
\[ g = \operatorname{diag}(2,1,4),\quad g^{-1} = \operatorname{diag}(\tfrac{1}{2},1,\tfrac{1}{4}) \]
\[ g^{ik} g_{kj} = \delta^{i}_{\ j} \]
where
- \(\operatorname{diag}(a,b,c)\) is the diagonal matrix with those entries.
- raising an index uses \(v^{i} = g^{ij} v_{j}\).
170.2 References
- Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — contravariant metric \(g^{ik}\); \(g^{ik}g_{kj}=\delta^{i}_{j}\).
- Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — inverse metric and index raising.
- Hassani, S. Mathematical Physics, 2nd ed. Springer. — inverse of a nondegenerate metric.
- Alternating Function
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