The geometry of a pseudo-Euclidean space is consistent despite some properties of Euclidean space not applying, most notably that it is not a
metric space as explained below. The
affine structure is unchanged, and thus also the concepts
line,
plane and, generally, of an
affine subspace (
flat), as well as
line segments.
Positive, zero, and negative scalar squares of A
null vector is a vector for which the quadratic form is zero. Unlike in a Euclidean space, such a vector can be non-zero, in which case it is self-
orthogonal. If the quadratic form is indefinite, a pseudo-Euclidean space has a
linear cone of null vectors given by . When the pseudo-Euclidean space provides a model for
spacetime (see
below), the null cone is called the
light cone of the origin. The null cone separates two
open sets, respectively for which and . If , then the set of vectors for which is
connected. If , then it consists of two disjoint parts, one with and another with . Similarly, if , then the set of vectors for which is connected. If , then it consists of two disjoint parts, one with and another with .
Interval The quadratic form corresponds to the square of a vector in the Euclidean case. To define the
vector norm (and distance) in an
invariant manner, one has to get
square roots of scalar squares, which leads to possibly
imaginary distances; see
Square root of negative numbers. But even for a
triangle with positive scalar squares of all three sides (whose square roots are real and positive), the
triangle inequality does not hold in general. Hence terms
norm and
distance are avoided in pseudo-Euclidean geometry, which may be replaced with
scalar square and
spacetime interval respectively. Though, for a
curve whose
tangent vectors all have scalar squares of the same sign, the
arc length is defined. It has important applications: see
Proper time, for example.
Rotations and spheres The
rotations group of such space is the
indefinite orthogonal group , also denoted as without a reference to particular quadratic form. Such "rotations" preserve the form and, hence, the scalar square of each vector including whether it is positive, zero, or negative. Whereas Euclidean space has a
unit sphere, pseudo-Euclidean space has the
hypersurfaces and . Such a hypersurface, called a
quasi-sphere, is preserved by the appropriate indefinite orthogonal group.
Symmetric bilinear form The quadratic form gives rise to a
symmetric bilinear form defined as follows: \langle x, y\rangle = \tfrac12[q(x + y) - q(x) - q(y)] = \left(x_1 y_1 + \ldots + x_k y_k\right) - \left(x_{k+1}y_{k+1} + \ldots + x_n y_n\right). The quadratic form can be expressed in terms of the bilinear form: . When , and are
orthogonal vectors of the pseudo-Euclidean space. This bilinear form is often referred to as the
scalar product, and sometimes as "inner product" or "dot product", but it does not define an
inner product space and it does not have the properties of the
dot product of Euclidean vectors. If and are orthogonal and , then is
hyperbolic-orthogonal to . The
standard basis of the real -space is
orthogonal. There are no ortho
normal bases in a pseudo-Euclidean space for which the bilinear form is indefinite, because it cannot be used to define a
vector norm.
Subspaces and orthogonality For a (positive-dimensional) subspace of a pseudo-Euclidean space, when the quadratic form is
restricted to , following three cases are possible: • is either
positive or negative definite. Then, is essentially
Euclidean (up to the sign of ). • is indefinite, but non-degenerate. Then, is itself pseudo-Euclidean. It is possible only if ; if , which means than is a
plane, then it is called a
hyperbolic plane. • is degenerate. One of the most jarring properties (for a Euclidean intuition) of pseudo-Euclidean vectors and flats is their
orthogonality. When two non-zero
Euclidean vectors are orthogonal, they are not
collinear. The intersections of any Euclidean
linear subspace with its
orthogonal complement is the
{{math|{0}}} subspace. But the definition from the previous subsection immediately implies that any vector of zero scalar square is orthogonal to itself. Hence, the
isotropic line generated by a
null vector is a subset of its orthogonal complement . The formal definition of the orthogonal complement of a vector subspace in a pseudo-Euclidean space gives a perfectly well-defined result, which satisfies the equality due to the quadratic form's non-degeneracy. It is just the condition : or, equivalently, all space, which can be broken if the subspace contains a null direction. While subspaces
form a lattice, as in any vector space, this operation is not an
orthocomplementation, in contrast to
inner product spaces. For a subspace composed
entirely of null vectors (which means that the scalar square , restricted to , equals to ), always holds: : or, equivalently, . Such a subspace can have up to
dimensions. For a (positive) Euclidean -subspace its orthogonal complement is a -dimensional negative "Euclidean" subspace, and vice versa. Generally, for a -dimensional subspace consisting of positive and negative dimensions (see ''
Sylvester's law of inertia for clarification), its orthogonal "complement" has positive and negative dimensions,⊥) = d
0, which itself is a direct consequence of the definition of U''⊥ and Sylvester's law, but it would be better to find a source. --> while the rest ones are degenerate and form the intersection.
Parallelogram law and Pythagorean theorem The
parallelogram law takes the form : q(x) + q(y) = \tfrac12(q(x + y) + q(x - y)). Using the
square of the sum identity, for an arbitrary triangle one can express the scalar square of the third side from scalar squares of two sides and their bilinear form product: : q(x + y) = q(x) + q(y) + 2\langle x, y \rangle. This demonstrates that, for orthogonal vectors, a pseudo-Euclidean analog of the
Pythagorean theorem holds: : \langle x, y \rangle = 0 \Rightarrow q(x) + q(y) = q(x + y). == Algebra and tensor calculus ==