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Minkowski round spheres are not future trapped

Physical scope

An exact flat Lorentzian geometry calculation; it is not an empirical statement and does not claim that Minkowski space contains no other trapped surfaces.

Example

Use the Minkowski model with metric $g = - (d x^{0})^{2} + (d x^{1})^{2} + (d x^{2})^{2} + (d x^{3})^{2}$ and future direction $N = \partial_{x^{0}}$ . For $R > 0$ , let $S = {x^{0} = 0, (x^{1})^{2} + (x^{2})^{2} + (x^{3})^{2} = R^{2}}$ . The future null normals $ℓ = \partial_{x^{0}} + \partial_{r}$ and $k = \partial_{x^{0}} - \partial_{r}$ obey $g (ℓ, k) = - 2$ and have expansions $θ_{ℓ} = 2 / R$ and $θ_{k} = - 2 / R$ . Thus these round spheres fail the future-trapping hypothesis because one expansion is positive.

Example references

Trapping and the asymptotic black-hole region

Null normal pairs and screen geometry

The flat model spacetime: the Minkowski metric on R^4 and its inertial chart

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

Compactness of the future horismos under null completeness

Penrose null geodesic incompleteness

Facts & Assumptions

Given: $R > 0$ , the Minkowski metric and time orientation above, and the null second fundamental form convention.

[F1]

Trapping and the asymptotic black-hole region: $θ_{ℓ} = {tr}_{q} χ_{ℓ}$ with $χ_{ℓ} (X, Y) = g (\nabla_{X} ℓ, Y)$ , and a future trapped surface requires both future null expansions to be negative.

[F2]

The flat model spacetime: the Minkowski metric on R^4 and its inertial chart: in the identity chart, $g = diag (- 1, 1, 1, 1)$ and $N = \partial_{x^{0}}$ is future timelike.

[F3]

Null normal pairs and screen geometry: the null second fundamental form is traced with the positive induced metric $q$ on the spacelike two-surface.

[F4]

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields: in a chart, the metric-selected connection has Christoffel symbols $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ .

[F5]

Compactness of the future horismos under null completeness: its compact-horismos conclusion assumes a compact future trapped surface, so the expansion signs computed here decide whether this particular sphere meets that prerequisite.

[F6]

Penrose null geodesic incompleteness: its incompleteness conclusion likewise assumes a compact future trapped surface; it cannot be applied with this sphere when one future expansion is positive.

Verification

Given: [F1]–[F4] and $R > 0$ .

1.1F2givenalgebra

On the spherical-coordinate region $r > 0$ , the metric is $g = - (d x^{0})^{2} + d r^{2} + r^{2} γ_{S^{2}}$ . Hence $g (ℓ, ℓ) = g (k, k) = 0$ , $g (ℓ, k) = - 2$ , and $g (ℓ, N) = g (k, N) = - 1$ , so both normals are future-directed.

1.2F2F3F4givenalgebra

For angular coordinate fields $E_{A}$ , the Christoffel formula [F4] gives $Γ^{B}_{r A} = \frac{1}{2} (r^{- 2} γ^{B C}) (2 r γ_{C A}) = r^{- 1} δ_{A}^{B}$ and $Γ^{μ}_{0 A} = 0$ . Therefore on $r = R$ , $\nabla_{X} ℓ = R^{- 1} X$ and $\nabla_{X} k = - R^{- 1} X$ for every tangent vector $X$ to $S$ .

2.1F1F3step 1.2givenalgebra

By [F1], $χ_{ℓ} = R^{- 1} q$ and $χ_{k} = - R^{- 1} q$ . Since $S$ is two-dimensional, tracing with $q$ gives $θ_{ℓ} = 2 / R > 0$ and $θ_{k} = - 2 / R < 0$ . Thus the pair of strict negative inequalities required for trapping fails.

3.1F5F6step 2.1∎

The positive expansion means this sphere does not supply the trapped-surface premise in [F5] or [F6]. The calculation makes no assertion about other surfaces in Minkowski space.

Remarks

The calculation rules out trapping only for these standard round spheres. It does not claim that every spacelike two-surface in Minkowski space has an expansion of this sign pattern.

Minkowski round spheres are not future trapped

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources