# What's the significance of a Killing horizon?

A Killing horizon is defined as a null hypersurface generated by a Killing vector, which is then null at that surface. Some often cited examples come from the Kerr spacetime, where the Killing vector $$\partial_t$$ becomes null at the ergosphere; one can also take a linear combination of that with $$\partial_\phi$$, which is null at the event horizon.

From this it would seem that Killing horizons are related to "special" surfaces of the spacetime, although not always: Minkowski spacetime has Killing horizons at any point, due to its high degree of symmetry. Can we make some general statements about Killing horizons besides what the definition says? Are they related to event horizons in any predictable way? If I have a metric and I find a Killing horizon, what can I say about it?

• I am not sure that Minkowski has a Killing horizon at any point, only at the lightcone May 20 '20 at 21:08
• @spiridon_the_sun_rotator but you can translate that to any point, can't you? May 20 '20 at 21:18
• You're right, observer can be located at any point of the spacetime May 20 '20 at 21:22
• It is worth pointing out that the ergosphere is not a Killing horizon. Although $\partial_t$ is null there, the ergosphere is not a null hypersurface. Only the event horizon is a Killing horizon in this case (ignoring the interior) with Killing vector $\partial_t + \Omega_H \partial_\phi$ May 21 '20 at 16:50

The "special" surfaces of spacetime defined by Killing horizons are null hypersurfaces. A null hypersurface which is a hypersurface whose normal vector at every point is a null vector (with respect to the local metric tensor).

The "boring" and trivial example is a light cone, as already mentioned. EDIT: from the comments, it is true that this statement is not true for Minkowski spacetime. Then I must say I am not sure when this applies.

In terms of other applications, I can think of two, even though they are quite interconnected: black holes event horizons, and surface gravity $$\kappa$$. A nice set of slides with useful discussions about this can be found here:

### Black holes

Taken verbatim from here:

[...] by Hawking’s rigidity theorem the event horizon of a stationary, asymptotically flat black hole spacetime (supplemented by certain additional assumptions, see [19] for a review), is a Killing horizon. In fact, one often uses the notion of a Killing horizon to provide a quasi-local definition of an equilibrium black hole.

I just wanted to show the above as a "quantitative" connection to the event horizon, as you asked. In this case, then, you can see that a physical meaning (albeit asympotically) of a Killing horizon is that it corresponds to the event horizon.

But black holes are also entering the picture through surface gravity. See below.

### Surface gravity

Surface gravity has a meaning in Newtonian/classical gravity, which is not the same in GR. Maybe the same name was used, historically, because one wanted to define the same object. But the two things have differences nowadays. Especially in black holes.

The physical meaning of the GR surface gravity $$\kappa$$ of a static Killing horizon is the acceleration, as exerted at infinity, needed to keep an object at the horizon. Mathematically, if $$k^{a}$$ is a suitably normalised Killing vector, then the surface gravity is defined by

$$k^a ∇_a k^b = \kappa k^b ,$$

where the equation is evaluated at the horizon. Specific solutions for black hole metrics are listed here.

Surface gravity is "physically" interesting because it is related to the temperature of Hawking radiation $$T_{\mathrm{H}}$$:

$$T_{\mathrm{H}} = \frac{\hbar c\kappa}{2 \pi k_{\mathrm{B}}}.$$

• OK, this sounds like asking too much but is there a bumper sticker for Killing horizons the way there is one for event horizons? Can I say something short like I can say the following about event horizons: "things cannot return to their previous causal patch of Penrose diagram after crossing this surface"? Sure, the definition of Killing horizon itself is pretty short but something in terms of how things behave around it rather than how mathematical objects behave around it if that makes sense. May 21 '20 at 7:29
• Don’t know about that sorry May 21 '20 at 16:45
• For Minkowski space, a light cone is not a Killing horizon because there is no Killing vector tangent to it. Instead, people usually talk about Rindler space which is a wedge of Minkowski space, and its horizon is the point where a boost Killing vector becomes null. May 21 '20 at 16:52
• First small comment: the light cone is not a Killing horizon but the surface $x^2-t^2=0$ is, right? Anyway, thanks for the answer! I was mostly interested in the connection to event horizons, and the Hawking theorem seems like a good place to start. May 21 '20 at 21:32
• Yes. It turns out the derivation for the Minkowski case is on the Wikipedia page for Killing horizons May 21 '20 at 21:35