Math 2.0

Relativity and gravitational theory Definition

Killing horizon

Builds on Lorentzian manifoldKilling vector field

Definition

Let (M,g)(M, g) be a Lorentzian manifold with signature (−,+,…,+)(-,+,\dots,+), and let ξ\xi be a Killing vector field on MM. A Killing horizon of ξ\xi is an embedded null hypersurface H⊆MH \subseteq M with two properties. The field ξ\xi is nowhere zero on HH, and ξ\xi is normal to HH at each point of HH.

Since HH is null, its normal is tangent to HH, so ξ\xi is a null vector tangent to the generators of HH. The function g(ξ,ξ)g(\xi, \xi) vanishes on HH, so its gradient is normal to HH. Hence there is a function κ\kappa on HH, the surface gravity, with ∇a(ξbξb)=−2κ ξaon H.\nabla^a \bigl( \xi^b \xi_b \bigr) = -2 \kappa \, \xi^a \quad \text{on } H.

Remarks

Example: in Minkowski space ℝ1,1\R^{1,1} with coordinates (t,x)(t, x) and metric −dt2+dx2-dt^2 + dx^2, take the boost Killing field ξ=x∂t+t∂x\xi = x \partial_t + t \partial_x. On the null line H={t=x>0}H = \set{t = x > 0} one has ξ=x(∂t+∂x)\xi = x(\partial_t + \partial_x), which is nonzero, null and normal to HH. Here ξbξb=t2−x2\xi^b \xi_b = t^2 - x^2, whose gradient is (−2t,−2x)(-2t, -2x) after raising the index; on HH this is −2ξ-2\xi, so κ=1\kappa = 1. Non-example: the same line HH is not a Killing horizon of the time translation ∂t\partial_t, because ∂t\partial_t is not normal to HH. The surface gravity is constant along each generator of HH. It is constant on all of HH when the horizon is bifurcate. It is also constant when the Einstein equations hold with matter that satisfies the dominant energy condition; this is the zeroth law of black hole mechanics. If ξ\xi is rescaled by a constant c>0c > 0, then κ\kappa is multiplied by cc, so one fixes a normalisation of ξ\xi at infinity.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Levi-Civita connection, concept, no evidence levelLevi-Civita connectionconceptLie derivative, concept, no evidence levelLie derivativeconceptPseudo-Riemannian manifold, concept, no evidence levelPseudo-Riemannian man…conceptSignature of a quadratic form, concept, no evidence levelSignature of a quadra…conceptTangent bundle, concept, no evidence levelTangent bundleconceptKilling vector field, definition, no evidence levelKilling vector fielddefinitionLorentzian manifold, definition, no evidence levelLorentzian manifolddefinitionKilling horizon, definition, no evidence levelKilling horizondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Levi-Civita connection concept
  2. Lie derivative concept
  3. Pseudo-Riemannian manifold concept
  4. Signature of a quadratic form concept
  5. Tangent bundle concept
  6. Killing vector field definition
  7. Lorentzian manifold definition
  8. Killing horizon definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Killing_horizon
About this page
TagZGK44HV177
URIm2://ZGK44HV177
Kinddefinition
WritingW0 stub
MSC83C57, 53C50
DefinesKilling horizon
UsesLorentzian manifold, Killing vector field
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.