Minkowski Metric

The Minkowski metric, also called the Minkowski tensor or pseudo-Riemannian metric, is a tensor eta_(alphabeta) whose elements are defined by the matrix

(eta)_(alphabeta)=[-1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1],

(1)

where the convention c=1 is used, and the indices alpha,beta run over 0, 1, 2, and 3, with x^0=t the time coordinate and (x^1,x^2,x^3) the space coordinates.

The Euclidean metric

(g)_(alphabeta)=[1 0 0; 0 1 0; 0 0 1],

(2)

gives the line element

ds^2 = g_(alphabeta)dx^alphadx^beta

(3)

= (dx^1)^2+(dx^2)^2+(dx^3)^2,

(4)

while the Minkowski metric gives its relativistic generalization, the proper time

dtau^2 = eta_(alphabeta)dx^alphadx^beta

(5)

= -(dx^0)^2+(dx^1)^2+(dx^2)^2+(dx^3)^2.

(6)

The Minkowski metric is fundamental in relativity theory, and arises in the definition of the Lorentz transformation as

Lambda^alpha_gammaLambda^beta_deltaeta_(alphabeta)=eta_(gammadelta),

(7)

where Lambda^alpha_beta is a Lorentz tensor. It also satisfies

eta^(betadelta)Lambda^gamma_delta=Lambda^(betagamma)

(8)

eta_(alphagamma)Lambda^(betagamma)=Lambda_alpha^beta

(9)

Lambda_alpha^beta=eta_(alphagamma)Lambda^(betagamma)=eta_(alphagamma)eta^(betadelta)Lambda^gamma_delta.

(10)

The metric of Minkowski space is diagonal with

eta_(alphaalpha)=1/(eta_(alphaalpha)),

(11)

and so satisfies

eta^(betadelta)=eta_(betadelta).

(12)

The necessary and sufficient conditions for a metric g_(munu) to be equivalent to the Minkowski metric eta_(alphabeta) are that the Riemann tensor vanishes everywhere (R^lambda_(munukappa)=0) and that at some point g^(munu) has three positive and one negative eigenvalues.

http://mathworld.wolfram.com/MinkowskiMetric.html