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A null cone where

In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0.

In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.

A quadratic space (X, q) which has a null vector is called a pseudo-Euclidean space. The term isotropic vector v when q(v) = 0 has been used in quadratic spaces,[1] and anisotropic space for a quadratic space without null vectors.

A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces A and B, X = A + B, where q is positive-definite on A and negative-definite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres: The null cone is also the union of the isotropic lines through the origin.

Split algebras

[edit]

A composition algebra with a null vector is a split algebra.[2]

In a composition algebra (A, +, ×, *), the quadratic form is q(x) = x x*. When x is a null vector then there is no multiplicative inverse for x, and since x ≠ 0, A is not a division algebra.

In the Cayley–Dickson construction, the split algebras arise in the series bicomplex numbers, biquaternions, and bioctonions, which uses the complex number field as the foundation of this doubling construction due to L. E. Dickson (1919). In particular, these algebras have two imaginary units, which commute so their product, when squared, yields +1:

Then
so 1 + hi is a null vector.

The real subalgebras, split complex numbers, split quaternions, and split-octonions, with their null cones representing the light tracking into and out of 0 ∈ A, suggest spacetime topology.

Examples

[edit]

The light-like vectors of Minkowski space are null vectors.

The four linearly independent biquaternions l = 1 + hi, n = 1 + hj, m = 1 + hk, and m∗ = 1 – hk are null vectors and { l, n, m, m∗ } can serve as a basis for the subspace used to represent spacetime. Null vectors are also used in the Newman–Penrose formalism approach to spacetime manifolds.[3]

In the Verma module of a Lie algebra there are null vectors.

References

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Null vector

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In mathematics, a null vector (also known as an isotropic vector) is a vector in a vector space equipped with a quadratic form whose value under that form is zero.[1] Typically, this refers to a non-zero vector ''v'' such that ''q''(''v'') = 0, where ''q'' is the quadratic form; the zero vector is a trivial null vector in any such space. In the special case of positive-definite quadratic forms, such as the Euclidean norm, the only null vector is the zero vector. The term "null vector" is also used in some linear algebra contexts to specifically denote the zero vector, the additive identity of the vector space.[2] Null vectors arise in various settings, including isotropic vectors in quadratic forms, split algebras and Clifford algebras, and special relativity in Minkowski space, where non-zero null vectors represent light-like directions. Further details on these contexts and applications are covered in subsequent sections.

Zero vector in linear algebra

Definition and notation

In linear algebra, the null vector, also known as the zero vector, is the unique element in a vector space VV that acts as the additive identity, satisfying v+0=v\mathbf{v} + \mathbf{0} = \mathbf{v} for every v∈V\mathbf{v} \in V.[3][2] This vector is distinguished from non-zero vectors, as it is the only one that, when represented in coordinates with respect to any basis of VV, has all components equal to zero.[3] The null vector is commonly denoted by 0\mathbf{0}, 0⃗\vec{0}, or simply 00. For instance, in the space Rn\mathbb{R}^n, it takes the form (0,0,…,0)(0, 0, \dots, 0), the tuple consisting of nn zeros.[2][3] In normed vector spaces, the null vector has magnitude zero, satisfying ∥0∥=0\|\mathbf{0}\| = 0.[4] The uniqueness of the null vector arises directly from the vector space axioms, which guarantee exactly one such additive identity in any vector space.[3] Note that while "null vector" most commonly refers to this zero vector in linear algebra, the term can also denote an isotropic vector in contexts involving quadratic forms, where a non-zero vector has zero quadratic norm.[2]

Properties and role in vector spaces

In any vector space VV over a field FF, the zero vector 0\mathbf{0} serves as the additive identity, satisfying v+0=0+v=v\mathbf{v} + \mathbf{0} = \mathbf{0} + \mathbf{v} = \mathbf{v} for every v∈V\mathbf{v} \in V.[5] This property is one of the fundamental axioms defining a vector space.[6] Additionally, the zero vector is unique; if another element a∈V\mathbf{a} \in V satisfies a+v=v\mathbf{a} + \mathbf{v} = \mathbf{v} for all v\mathbf{v}, then a=0\mathbf{a} = \mathbf{0}.[7] The zero vector also exhibits distinctive behavior under scalar multiplication: for any scalar c∈Fc \in F, c⋅0=0c \cdot \mathbf{0} = \mathbf{0}.[8] This follows from the vector space axioms and ensures that scaling the zero vector yields itself regardless of the scalar.[5] In the context of linear combinations, the singleton set {0}\{ \mathbf{0} \} is linearly dependent, as 1⋅0=01 \cdot \mathbf{0} = \mathbf{0} provides a nontrivial relation equaling the zero vector.[9] Moreover, the span of {0}\{ \mathbf{0} \} is the trivial subspace {0}\{ \mathbf{0} \}, the smallest subspace of VV.[10] Geometrically, the zero vector lacks a specific direction, distinguishing it from nonzero vectors that point along lines through the origin.[11] In inner product spaces, this manifests in its orthogonality to every vector, since v⋅0=0\mathbf{v} \cdot \mathbf{0} = 0 for all v\mathbf{v}.[12] It is sometimes regarded as parallel to all vectors in a formal sense, though its undefined direction avoids contradictions in geometric interpretations.[13] Every vector space contains precisely one zero vector, which acts as the origin in coordinate representations, where vectors are identified with tuples of field elements relative to a basis.[14] This role underscores its foundational position in the algebraic structure of vector spaces, enabling the definition of subspaces and linear maps.[15]

Isotropic vectors in quadratic forms

Definition and quadratic norm

In a vector space XX over the real numbers equipped with a quadratic form q:X→Rq: X \to \mathbb{R}, a null vector is defined as a non-zero element x∈Xx \in X such that q(x)=0q(x) = 0.[16] This concept arises in the study of quadratic forms that are not positive definite or negative definite, where the zero set of qq extends beyond the origin. Unlike the zero vector 0\mathbf{0}, which trivially satisfies q(0)=0q(\mathbf{0}) = 0 in any quadratic form due to homogeneity, null vectors are explicitly non-zero and represent directions where the form vanishes.[16] The quadratic form qq is associated with a symmetric bilinear form B:X×X→RB: X \times X \to \mathbb{R} via the relation q(x)=B(x,x)q(x) = B(x, x), where BB is linear in each argument and satisfies B(x,y)=B(y,x)B(x, y) = B(y, x) for all x,y∈Xx, y \in X.[16] In this framework, a null vector xx satisfies B(x,x)=0B(x, x) = 0, and the polarization identity links BB back to qq through B(x,y)=12[q(x+y)−q(x)−q(y)]B(x, y) = \frac{1}{2} [q(x + y) - q(x) - q(y)].[16] This bilinear structure allows null vectors to be characterized algebraically without direct reference to the quadratic norm alone. The collection of all vectors x∈Xx \in X such that q(x)=0q(x) = 0, including the zero vector, forms the isotropic cone of the quadratic form, often denoted C(q)C(q).[17] The non-zero elements of this cone are precisely the null vectors, which span isotropic subspaces when linearly independent sets of such vectors exist. Examples of quadratic forms admitting non-trivial null vectors include indefinite forms, such as those of signature (p,q)(p, q) with p>0p > 0 and q>0q > 0, like the Minkowski metric in four dimensions with signature (1,3)(1, 3).[16] For instance, consider R2\mathbb{R}^2 equipped with q(x,y)=x2−y2q(x,y) = x^2 - y^2; the vector (1,1)(1,1) satisfies q(1,1)=1−1=0q(1,1) = 1 - 1 = 0 and is thus a null vector.[16] In these cases, the quadratic norm q(x)q(x) can take positive, negative, or zero values, enabling the existence of such vectors.

Characterization in inner product spaces

In inner product spaces equipped with an indefinite inner product of signature (p,q)(p, q) where p>0p > 0 and q>0q > 0, a null vector x≠0x \neq 0 is characterized by satisfying ⟨x,x⟩=0\langle x, x \rangle = 0.[18] Such spaces extend the standard positive definite case by allowing the inner product to take both positive and negative values, leading to the existence of non-trivial null vectors that lie on the isotropic cone.[19] A key property of null vectors in these spaces is their self-orthogonality: since ⟨x,x⟩=0\langle x, x \rangle = 0, the vector is orthogonal to itself. This self-orthogonality implies that any subspace spanned by a single null vector is degenerate, as the inner product restricted to that subspace vanishes identically, violating the non-degeneracy condition for the form on the subspace.[18] More generally, subspaces containing null vectors can exhibit degeneracy when the restricted inner product has a non-trivial kernel. The radical of the inner product, defined as the subspace rad(V)={y∈V∣⟨y,z⟩=0 ∀z∈V}\mathrm{rad}(V) = \{ y \in V \mid \langle y, z \rangle = 0 \ \forall z \in V \}, consists entirely of null vectors, since any y∈rad(V)y \in \mathrm{rad}(V) satisfies ⟨y,y⟩=0\langle y, y \rangle = 0. In non-degenerate spaces, where rad(V)={0}\mathrm{rad}(V) = \{0\}, null vectors still exist but do not form the radical; however, in degenerate cases, the radical is a non-trivial isotropic subspace fully comprising null vectors.[19] Vectors in indefinite inner product spaces are classified into positive (⟨x,x⟩>0\langle x, x \rangle > 0), negative (⟨x,x⟩<0\langle x, x \rangle < 0), and null (⟨x,x⟩=0\langle x, x \rangle = 0) directions, with the signature (p,q)(p, q) indicating the numbers of positive and negative eigenvalues in a diagonalization of the inner product. In finite-dimensional spaces over R\mathbb{R}, non-trivial null vectors exist if and only if the signature is indefinite (i.e., p≥1p \geq 1, q≥1q \geq 1) and the dimension n=p+q≥2n = p + q \geq 2, as definite signatures yield no such vectors while indefinite ones guarantee isotropy.[18][19]

Contexts and applications

In split algebras and Clifford algebras

In the theory of quadratic forms over fields, a split algebra arises when the associated quadratic space admits a decomposition into hyperbolic planes, each of which is a two-dimensional space equipped with a bilinear form that permits a pair of orthogonal isotropic vectors generating the plane.[20] Such hyperbolic planes allow for maximal isotropic subspaces—spans of vectors with zero quadratic norm—achieving dimension equal to half the total dimension of the space, thereby maximizing the presence of null vectors within the structure.[20] Within Clifford algebras, null vectors play a central role as generators satisfying the defining relation of the algebra. The Clifford algebra $ Cl(p, q) $ is constructed from a quadratic space of dimension $ n = p + q $ with signature $ (p, q) $, where the algebra is generated by vectors $ v $ obeying $ v^2 = Q(v) \cdot 1 $, and $ Q $ denotes the quadratic form.[21] Consequently, a null vector $ v $, characterized by $ Q(v) = 0 $, satisfies $ v^2 = 0 $, rendering it a non-invertible element that contributes to the ideal structure and representations of the algebra.[21] The Witt decomposition theorem provides a canonical framework for understanding the distribution of null vectors in these settings. Every non-degenerate quadratic space $ (V, q) $ decomposes uniquely as an orthogonal direct sum $ V = V_t \oplus V_h \oplus V_a $, where $ V_t $ is the radical (totally isotropic kernel), $ V_h $ is hyperbolic (a direct sum of hyperbolic planes populated by isotropic vectors), and $ V_a $ is anisotropic (containing no non-zero null vectors).[22] This decomposition highlights how null vectors concentrate in the hyperbolic component $ V_h $, which supports maximal isotropic subspaces and influences the isomorphism class of the associated Clifford algebra $ Cl(p, q) $.[22] A concrete example occurs in the real split algebra $ Cl(1,1) $, the Clifford algebra over the two-dimensional quadratic space $ \mathbb{R}^{1,1} $ with signature $ (1,1) $. This algebra is isomorphic to the complex numbers $ \mathbb{C} $, generated by basis vectors $ e_0 $ and $ e_1 $ satisfying $ e_0^2 = 1 $, $ e_1^2 = -1 $, and $ {e_0, e_1} = 0 $.[23] The null vectors $ e_0 \pm e_1 $ square to zero and span the light lines of the underlying space, forming the isotropic directions that define the algebra's split nature.[23] Historically, the study of null vectors in Clifford algebras gained depth through Élie Cartan's foundational work on spinors and representations. In 1908, Cartan classified the general Clifford algebras $ Cl(p, q) $ as matrix algebras over $ \mathbb{R} $, $ \mathbb{C} $, or the quaternions $ \mathbb{H} $, uncovering the periodicity of order 8 in their structure, which facilitates the analysis of isotropic elements and spinor spaces.[24] His 1913 developments on irreducible representations of simple Lie groups further connected these algebras to spinors, where null vectors underpin the geometric interpretations of higher-dimensional rotations.[24]

In special relativity and Minkowski space

In Minkowski space, the flat four-dimensional spacetime underlying special relativity, the geometry is defined by the quadratic form $ q(x) = -t^2 + x^2 + y^2 + z^2 $ (with the speed of light $ c = 1 $), which has Lorentzian signature (1,3).[25] A vector $ x = (t, x, y, z) $ is null if it satisfies $ q(x) = 0 $, meaning the invariant spacetime interval between events connected by such a vector is zero.[26] This condition arises from the Minkowski inner product $ \langle x, y \rangle = -t t' + x x' + y y' + z z' $, where null vectors have zero norm.[25] Physically, null vectors represent the worldlines of massless particles, such as photons, propagating at the speed of light.[27] Along these paths, the four-velocity $ u^\mu = dx^\mu / d\lambda $ (parameterized by an affine parameter $ \lambda $ rather than proper time, since proper time vanishes for massless particles) satisfies $ \langle u, u \rangle = 0 $, distinguishing them from the timelike four-velocities of massive particles where $ \langle u, u \rangle = -1 $.[28] For example, a photon traveling in the x-direction has displacement vector $ (t, t, 0, 0) $, yielding $ q(x) = 0 $.[25] Null vectors delineate the structure of light cones in spacetime, which emanate from any event and separate causally connected regions.[25] The future light cone consists of all future-directed null vectors from the event, forming its boundary; interior points correspond to timelike vectors ($ q(x) < 0 ),accessibletomassiveparticlesviasubluminalpaths,whileexteriorspacelikevectors(), accessible to massive particles via subluminal paths, while exterior spacelike vectors ( q(x) > 0 $) lie outside causal reach.[26] Past light cones are defined analogously for incoming light rays. This conical geometry enforces causality: influences propagate at or below light speed, with null directions marking the boundary.[25] Lorentz transformations, the symmetry group of Minkowski space, preserve the quadratic form and thus map null vectors to null vectors.[25] For instance, a boost along the x-axis, given by
(t′x′)=(γ−γv−γvγ)(tx), \begin{pmatrix} t' \\ x' \end{pmatrix} = \begin{pmatrix} \gamma & -\gamma v \\ -\gamma v & \gamma \end{pmatrix} \begin{pmatrix} t \\ x \end{pmatrix},
with $ \gamma = 1/\sqrt{1 - v^2} $, leaves the lightlike relation $ x = \pm t $ invariant, ensuring the speed of light is constant in all inertial frames.[25] This preservation underscores the physical equivalence of null directions across observers, central to relativistic invariance.[26]

Additional examples

In the context of indefinite quadratic forms on R2\mathbb{R}^2, a simple example is the form q(x,y)=xyq(x, y) = xy, which arises from the symmetric bilinear form B((x1,y1),(x2,y2))=12(x1y2+x2y1)B((x_1, y_1), (x_2, y_2)) = \frac{1}{2}(x_1 y_2 + x_2 y_1).[20] The null vectors satisfy q(x,y)=0q(x, y) = 0 with (x,y)≠(0,0)(x, y) \neq (0, 0), so xy=0xy = 0, corresponding to non-zero vectors along the coordinate axes, such as (1,0)(1, 0) or (0,1)(0, 1).[20] This form is equivalent to the standard hyperbolic quadratic form x2−y2x^2 - y^2 via a change of basis, illustrating the isotropic nature of the space.[29] Viewing the complex numbers C\mathbb{C} as R2\mathbb{R}^2 via z=x+iy↦(x,y)z = x + i y \mapsto (x, y), one can equip it with an indefinite quadratic form such as q(x,y)=x2−y2q(x, y) = x^2 - y^2.[30] Null vectors then satisfy x2=y2x^2 = y^2 with (x,y)≠(0,0)(x, y) \neq (0, 0), yielding lines of isotropy; for instance, (1,1)(1, 1) corresponds to the complex number 1+i1 + i, and (1,−1)(1, -1) to 1−i1 - i.[30] This perspective highlights the hyperbolic geometry underlying such representations, distinct from the usual positive definite Euclidean norm on C\mathbb{C}.[29] In projective geometry over a field equipped with a quadratic form, null vectors generate isotropic lines, which projectively represent points at infinity.[31] An isotropic line consists of points whose connecting displacement vectors have zero quadratic norm, forming the null cone's generators in the projective space. These structures unify finite and infinite points, as parallel lines in the affine plane intersect at isotropic points on the line at infinity.[31] In lattice theory, null vectors appear prominently in indefinite even unimodular lattices associated with root systems of Lie algebras.[32] For example, the Lorentzian lattice II1,25II_{1,25} of rank 26, related to constructions involving root lattices like those of E8E_8, contains primitive null vectors ρ\rho such that ρ∈ρ⊥\rho \in \rho^\perp.[32] Quotienting by ρZ\rho \mathbb{Z} yields positive definite even unimodular lattices, such as the Leech lattice of rank 24, underscoring the role of null vectors in bridging indefinite and definite structures in Lie theory.[32] Historically, split quaternions, introduced by James Cockle in 1848 as a variant of Hamilton's 1843 quaternions, form a four-dimensional real algebra with basis {1,i,j,k}\{1, i, j, k\} where j2=1j^2 = 1 and k=ij=−jik = i j = -j i.[33] The associated quadratic form is indefinite, q(w+xi+yj+zk)=w2+x2−y2−z2q(w + x i + y j + z k) = w^2 + x^2 - y^2 - z^2, admitting non-zero null elements where q=0q = 0, such as (1,0,1,0)(1, 0, 1, 0).[33] These null elements, including zero divisors and nilpotents, distinguish split quaternions from Hamilton's division algebra and arise in applications like hyperbolic geometry.[34]

References

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  36. Generalized Split Quaternions and Their Applications on Non ...
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