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The Lorentz form is not an inner product

Statement refuted

Statement refuted: "the Lorentz form η of a Lorentz vector space is an inner product, i.e. a positive-definite symmetric bilinear form" (Real and complex inner product spaces, with the inner product linear in the first argument).

Facts & Assumptions

Given: A Lorentz vector space (V,η) of signature (−,+,+,+), and an η-orthonormal ordered basis of V supplied by Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1).

[L1]

There is an ordered basis (e0,e1,e2,e3) of V with η(ea,eb)=diag⁡(−1,1,1,1)ab for all a,b; a basis with this property is called η-orthonormal (Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)).

[L2]

An inner product ⟨ ⋅ , ⋅ ⟩ on a real or complex vector space satisfies: ⟨v,v⟩ is real and nonnegative, and ⟨v,v⟩=0 if and only if v=0 (Real and complex inner product spaces, with the inner product linear in the first argument).

[L3]

The Euclidean inner product of x,y∈Rn is ⟨x,y⟩=∑k<nxkyk, and it is the positive-definite inner product of the Euclidean space Rn (The Euclidean inner product ⟨x,y⟩=∑k<nxkyk on Rn).

Counterexample

1.1L1choose

By [L1] fix an η-orthonormal ordered basis (e0,e1,e2,e3) of V. Since it is a basis, e0≠0, and by the displayed matrix η(e0,e0)=−1<0.

2.1step 1.1L2L3

An inner product must satisfy ⟨v,v⟩≥0 for every v, with equality only at v=0, by [L2]. The symmetric bilinear form η fails this at the nonzero vector e0, where η(e0,e0)=−1<0; hence η is not positive definite and therefore is not an inner product, refuting the displayed statement. The contrast with [L3] shows that the failure comes from η itself rather than from the definition of an inner product, which the Euclidean inner product satisfies.

3.1step 2.1L1L2L3∎

Consequently η does not induce a norm on V: the Euclidean norm of [L3] is defined as ∥x∥2=⟨x,x⟩, and a norm arising from an inner product in this way requires the nonnegative quantity ⟨v,v⟩ of [L2] for every v, whereas η(e0,e0)=−1 admits no real square root. In particular the positive-definite inner-product theory (Cauchy-Schwarz, norms and orthogonal projection) cannot be applied to η without additional argument.

Remarks

The refutation uses one nonzero vector. It does not claim that η has no useful positivity properties at all: on the orthogonal complement of a timelike vector the restriction of η is positive definite, as Worked orthogonal complement of a timelike displacement is positive definite shows, and that is a statement about a subspace rather than about η itself.

Depends on

Dependency tree · 2 levels

Sources