Positive, zero, and negative scalar squares

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 { x | q(x) = 0 }. 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,^[4] respectively for which q(x) > 0 and q(x) < 0. If k ≥ 2, then the set of vectors for which q(x) > 0 is connected. If k = 1, then it consists of two disjoint parts, one with x₁ > 0 and another with x₁ < 0. Similarly, if n − k ≥ 2, then the set of vectors for which q(x) < 0 is connected. If n − k = 1, then it consists of two disjoint parts, one with x_n > 0 and another with x_n < 0.

Interval

The quadratic form q 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 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 O(q), also denoted as O(k, n − k) without a reference to particular quadratic form.^[5] Such "rotations" preserve the form q 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 { x | q(x) = 1 } and { x | q(x) = −1 }. Such a hypersurface, called a quasi-sphere, is preserved by the appropriate indefinite orthogonal group.

Symmetric bilinear form

The quadratic form q gives rise to a symmetric bilinear form defined as follows:

$${\displaystyle \langle x,y\rangle ={\tfrac {1}{2}}[q(x+y)-q(x)-q(y)]=\left(x_{1}y_{1}+\ldots +x_{k}y_{k}\right)-\left(x_{k+1}y_{k+1}+\ldots +x_{n}y_{n}\right).}$$

The quadratic form can be expressed in terms of the bilinear form: q(x) = ⟨x, x⟩.

When ⟨x, y⟩ = 0, then x and y 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 x and y are orthogonal and q(x)q(y) < 0, then x is hyperbolic-orthogonal to y.

The standard basis of the real n-space is orthogonal. There are no orthonormal 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^[6] U of a pseudo-Euclidean space, when the quadratic form q is restricted to U, following three cases are possible:

1. q|_U is either positive or negative definite. Then, U is essentially Euclidean (up to the sign of q).
2. q|_U is indefinite, but non-degenerate. Then, U is itself pseudo-Euclidean. It is possible only if dim U ≥ 2; if dim U = 2, which means than U is a plane, then it is called a hyperbolic plane.
3. q|_U 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 {0} subspace. But the definition from the previous subsection immediately implies that any vector ν of zero scalar square is orthogonal to itself. Hence, the isotropic line N = ⟨ν⟩ generated by a null vector ν is a subset of its orthogonal complement N^⊥.

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 dim U + dim U^⊥ = n due to the quadratic form's non-degeneracy. It is just the condition

U ∩ U^⊥ = {0} or, equivalently, U + U^⊥ = all space,

which can be broken if the subspace U contains a null direction.^[7] 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 N composed entirely of null vectors (which means that the scalar square q, restricted to N, equals to 0), always holds:

N ⊂ N^⊥ or, equivalently, N ∩ N^⊥ = N.

Such a subspace can have up to min(k, n − k) dimensions.^[8]

For a (positive) Euclidean k-subspace its orthogonal complement is a (n − k)-dimensional negative "Euclidean" subspace, and vice versa. Generally, for a (d₊ + d₋ + d₀)-dimensional subspace U consisting of d₊ positive and d₋ negative dimensions (see Sylvester's law of inertia for clarification), its orthogonal "complement" U^⊥ has (k − d₊ − d₀) positive and (n − k − d₋ − d₀) negative dimensions, while the rest d₀ ones are degenerate and form the U ∩ U^⊥ intersection.

Parallelogram law and Pythagorean theorem

The parallelogram law takes the form

${\displaystyle q(x)+q(y)={\tfrac {1}{2}}(q(x+y)+q(x-y)).}$

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:

${\displaystyle q(x+y)=q(x)+q(y)+2\langle x,y\rangle .}$

This demonstrates that, for orthogonal vectors, a pseudo-Euclidean analog of the Pythagorean theorem holds:

${\displaystyle \langle x,y\rangle =0\Rightarrow q(x)+q(y)=q(x+y).}$

Angle

Generally, absolute value |⟨x, y⟩| of the bilinear form on two vectors may be greater than √|q(x)q(y)|, equal to it, or less. This causes similar problems with definition of angle (see Dot product § Geometric definition) as appeared above for distances.

If k = 1 (only one positive term in q), then for vectors of positive scalar square:

$${\displaystyle |\langle x,y\rangle |\geq {\sqrt {q(x)q(y)}}\,,}$$

which permits definition of the hyperbolic angle, an analog of angle between these vectors through inverse hyperbolic cosine:^[9]

$${\displaystyle \operatorname {arcosh} {\frac {|\langle x,y\rangle |}{\sqrt {q(x)q(y)}}}\,.}$$

It corresponds to the distance on a (n − 1)-dimensional hyperbolic space. This is known as rapidity in the context of theory of relativity discussed below. Unlike Euclidean angle, it takes values from [0, +∞) and equals to 0 for antiparallel vectors.

There is no reasonable definition of the angle between a null vector and another vector (either null or non-null).