**Theorem 6.3 (Criterion for Lorentz Frames)**

The following are equivalent:

**(a) **A coordinate system ^{i} in Minkowski space M is Lorentz at the point p

**(b) **If x is any frame such that, at p, G = diag[1, 1, 1, -c^{2}], then the columns of the change-of-coordinate matrix

D_{j}^{i} | = | ^{i}x ^{j} |

satisfy

- column i, column j =

where the inner product is defined by the matrix G.

**(c) ** = diag[1, 1, 1, -c^{2}]

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