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The constant Minkowski metric as a smooth time-oriented Lorentzian manifold

Example

Let $M : = R^{4}$ with its standard smooth structure, for which the identity map $x = (x^{0}, x^{1}, x^{2}, x^{3})$ is a global smooth chart (Euclidean spaces and Euclidean open subsets as smooth manifolds). Define the family $A = (A_{p})_{p \in M}$ by declaring its components in this chart to be the constants

$A_{a b} : = diag (- 1, 1, 1, 1)_{a b}, that is A_{p} (v, w) = - v^{0} w^{0} + v^{1} w^{1} + v^{2} w^{2} + v^{3} w^{3}$

for $v = \sum_{a} v^{a} \partial_{a} ∣_{p}$ , $w = \sum_{a} w^{a} \partial_{a} ∣_{p}$ . Then $(M, A)$ is a smooth Lorentzian four-manifold in the sense of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model; the field $N : = \partial / \partial x^{0}$ is a time orientation, so $(M, A, N)$ is a time-oriented smooth Lorentzian four-manifold; and at every $p$ the future-directed causal vectors are exactly the nonzero $v$ with $v^{0} \geq ∣ v_{v e c} ∣ = ((v^{1})^{2} + (v^{2})^{2} + (v^{3})^{2})^{1 / 2}$ . In particular $v_{(1)} : = (1, 0, 0, 0)$ is future timelike, $v_{(2)} : = (1, 1, 0, 0)$ is future null and $v_{(3)} : = (0, 1, 0, 0)$ is spacelike.

Facts & Assumptions

Given: $M = R^{4}$ with its global identity chart and the field $A$ with components $diag (- 1, 1, 1, 1)$ in that chart.

[L1]

A smooth covariant $2$ -tensor field is a family of bilinear forms whose components are $C^{\infty}$ in every chart of the maximal atlas (Smooth covariant 2-tensor fields on a smooth manifold); components in an atlas determine the field, and smoothness in one atlas implies smoothness in every chart (Chart independence and the transformation law of a smooth covariant 2-tensor field).

[L2]

A Lorentz form of signature $(-, +, +, +)$ on a $4$ -dimensional real vector space is a symmetric nondegenerate bilinear form with inertia $(3, 1, 0)$ (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space), and it admits an orthonormal basis with matrix $diag (- 1, 1, 1, 1)$ (Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)).

[L3]

$(M, A, N)$ is time-oriented when $N$ is a smooth vector field with $A_{p} (N_{p}, N_{p}) < 0$ for all $p$ ; then a nonzero causal $v$ is future-directed exactly when $A_{p} (v, N_{p}) < 0$ and its future cone has the description proved in Classification of future and past causal vectors by a timelike field (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L4]

The constant field $\partial_{0}$ is a smooth vector field: in the global chart it has the constant component functions $(1, 0, 0, 0)$ , so smoothness holds by the component criterion for vector fields (A smooth vector field is a smooth section of the tangent bundle, Smoothness of a vector field is equivalent to smooth coordinate components).

Verification

technique · direct

1.1L1L2given

The component functions of $A$ in the global chart are the constants $A_{a b} = diag (- 1, 1, 1, 1)_{a b}$ , hence $C^{\infty}$ ; by [L1] the family $A$ is therefore a smooth covariant $2$ -tensor field, and the component matrix is symmetric, invertible with $\det = - 1$ , and has three positive and one negative diagonal entries, so each $A_{p}$ is symmetric, nondegenerate and has inertia $(3, 1, 0)$ , i.e., is a Lorentz form of signature $(-, +, +, +)$ by [L2]. Hence $(M, A)$ is a smooth Lorentzian four-manifold.

2.1step 1.1L3L4

The vector field $N = \partial_{0}$ has constant components $(1, 0, 0, 0)$ in the global chart, so it is smooth by [L4], and $A_{p} (N_{p}, N_{p}) = - 1 < 0$ at every $p$ ; by [L3] this makes $(M, A, N)$ a time-oriented smooth Lorentzian four-manifold.

3.1step 2.1L3algebra

For a nonzero $v = \sum_{a} v^{a} \partial_{a} ∣_{p}$ , $A_{p} (v, v) = - (v^{0})^{2} + ∣ v_{v e c} ∣^{2}$ , so $v$ is causal exactly when $∣ v^{0} ∣ \geq ∣ v_{v e c} ∣$ (equality being the null case), that is, when $v^{0} \geq ∣ v_{v e c} ∣$ or $v^{0} \leq - ∣ v_{v e c} ∣$ ; among these, by [L3] and step 2.1, $v$ is future-directed exactly when $A_{p} (v, N_{p}) = - v^{0} < 0$ , that is, when $v^{0} > 0$ . Hence the future-directed causal vectors are exactly the nonzero $v$ with $v^{0} \geq ∣ v_{v e c} ∣$ , which is the flat case of the description proved in [L3] for the field $N$ .

4.1step 3.1algebra∎

Evaluating the description of step 3.1: $v_{(1)} = (1, 0, 0, 0)$ has $v_{(1)}^{0} = 1 > 0 = ∣ v_{(1), v e c} ∣$ , so $v_{(1)}$ is future timelike; $v_{(2)} = (1, 1, 0, 0)$ has $A_{p} (v_{(2)}, v_{(2)}) = - 1 + 1 = 0$ with $v_{(2)} \neq 0$ and $v_{(2)}^{0} = 1 = ∣ v_{(2), v e c} ∣$ , so $v_{(2)}$ is future null; $v_{(3)} = (0, 1, 0, 0)$ has $A_{p} (v_{(3)}, v_{(3)}) = 1 > 0$ , so $v_{(3)}$ is spacelike.

Remarks

Choice convention. The smooth vector fields of this example are the tangent-bundle sections of A smooth vector field is a smooth section of the tangent bundle, whose canonical smooth structure on $T M$ assumes the axiom ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ); this example inherits that assumption and uses no further choice principle.

The constant Minkowski metric as a smooth time-oriented Lorentzian manifold

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources