# What does it mean that the event horizon is a global concept?

In Chapter 6 of Spacetime and Geometry by S. Carroll, he says

‘‘Because the event horizon is a global concept, it might be difficult to actually locate one when you are handed a metric in an arbitrary set of coordinates.’’.

I understand what local and global variables are, and I have a rough understanding of the previous sentence. However, I would like to understand it even more. What does it mean that the event horizon is a ‘‘global concept’’ and why does this make it harder to locate it, given a particular coordinate system?

Formally defined, an event horizon bounds a region which is causally disconnected from future null infinity. To truly know if a boundary meets this definition, requires knowing a good amount of the complete future history of the spacetime: local knowledge of the event horizon and its surroundings for part of the history won't tell you if it is indeed an event horizon. This is what is meant by the event horizon being a global concept.

• I suspect we don't know enough about blackholes and Hawking radiation to judge if they are true event horizons then and don't leak informational content while dissipating/evaporating? Jun 21, 2020 at 3:46
• @WorldSEnder math tells us where the event horizon is for an eternal black hole, and real black holes are very close to eternal. That doesn't mean that we won't get all of the information back after 10^100 years, of course, and quantum mechanics say we should. Jun 21, 2020 at 12:46

I will essentially say the same thing as in John's answer, but in a different way, and maybe this will be easier to understand (judge for yourself).

"Global," here, does not mean global variables, it means that you need to know more than just the geometry local to the event horizon to know that you are at an event horizon. Specifically, you need to know that there is a singularity somewhere else (e.g., at $$r=0$$ in a Schwarzschild spacetime).

If you are at some point $$x^\mu$$ in spacetime, and you need to know properties of the spacetime non-local to $$x^\mu$$ in order to infer property $$X$$, then $$X$$ is a global feature of the spacetime. Thus, event horizons are a global feature of your geometry.

Another way to understand this is to recognize that the event horizon is not a geometrically significant point in your spacetime. It makes no sense to talk about the event horizon unless you add the concept of causality to your physics. Contrast this to the singularity, which by Einstein's theory is explicitly intrinsically geometric in nature. Thus, if you have a metric and change your system of coordinates, then no change will disguise the breakdown of your metric at the singularity, since it is a geometric feature, but the event horizon has no such privilege: spacetime is smooth at the event horizon, and will look like any other patch of spacetime (assuming you can see neither the effects of the black hole singularity "nearby" nor the AMPS firewall, which may or may not exist...). Hence, given a black hole spacetime in some arbitrary system of coordinates, it may not be easy to spot that some particular point $$r_X$$ defined in the metric somewhere is an event horizon, not at first glance anyway.

Finally, to understand how my answer is, really, identical to John's answer it simply needs to be made manifest that time, obviously, is a spacetime coordinate. Thus talking of "globally geometric properties of your spacetime" is really just saying "future histories" in a different way.