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The Poincaré group is another name for the inhomogeneous Lorentz group (Weinberg 1972, p. 28) and corresponds to the group of inhomogeneous Lorentz transformations, also ...

The contravariant four-vector arising in special and general relativity, x^mu=[x^0; x^1; x^2; x^3]=[ct; x; y; z], (1) where c is the speed of light and t is time. ...

A loose term for a true statement which may be a postulate, theorem, etc.

A tensor-like object which reverses sign under inversion. Given a transformation matrix A, A_(ij)^'=det|A|a_(ik)a_(jl)A_(kl), where det is the determinant. A pseudotensor is ...

A pullback is a general categorical operation appearing in a number of mathematical contexts, sometimes going under a different name. If T:V->W is a linear transformation ...

Let R be a commutative ring. A category C is called an R-category if the Hom-sets of C are R-modules.

Let V and W be vector spaces over a field F, and let T:V->W be a linear transformation. Assuming the dimension of V is finite, then dim(V)=dim(Ker(T))+dim(Im(T)), where ...

The constant ratio by which all distances are increased (or decreased) in a similarity. A similarity with ratio of magnification equal to 1 is called an isometry.

If two algebraic plane curves with only ordinary singular points and cusps are related such that the coordinates of a point on either are rational functions of a ...

Suppose for every point x in a manifold M, an inner product <·,·>_x is defined on a tangent space T_xM of M at x. Then the collection of all these inner products is called ...