# Flat Spacetime at Event Horizon of Black Hole?

• btouellette
In summary: Near the horizon, the metric is equivalent to the flat-space metric. Curvature is determined by taking second derivatives of the space-time metric.
btouellette
I've been working through Leonard Susskind's "The Theoretical Minimum" lecture series (which are a fantastic introduction to the topics covered by the way) and a couple of his comments confused me when he was covering the Kruskal-Szekeres metric/coordinates in General Relativity.

The end of the relevant section is at 28:48 here:

He uses the Schwarzschild metric (ignoring the scale factor of Rs)
$$d\tau^2 = (1-1/r) dt^2 - dr^2 / (1-1/r) - r^2 d\Omega^2$$
and redefines it in terms of the proper distance from the event horizon (ρ) and a new time coordinate (ω = t/2) to arrive at the metric
$$d\tau^2 = F(\rho) \rho^2 d\omega^2 - d\rho^2 - r(\rho)^2 d\Omega^2$$
and as we approach the event horizon where ρ→0 the metric approaches
$$d\tau^2 = \rho^2 d\omega^2 - d\rho^2 - (1+\rho^2/4)^2 d\Omega^2$$

It makes sense to me that as ρ→∞ the metric approximates the metric of flat space and I can see that at the event horizon the metric is basically the same as the flat space metric using hyperbolic polar coordinates but I'm not sure how to interpret that information and don't have any intuition. There are both tidal forces and curvature at the event horizon so the spacetime is not flat. Is this just related to the facts that hyperbolic polar coordinates can be used to define hyperbolas of constant r on a spacetime diagram which are equivalent to a constant relativistic acceleration and that an observer in a static position outside a black hole is undergoing a constant acceleration to cancel out the acceleration due to gravity?

Spacetime is not flat at the event horizon. Near the event horizon, the metric may look similar to the flat-space metric, but given any point in any spacetime, we can *always* choose coordinates such that the metric has that form near that point. To determine whether spacetime is flat, you need to compute the curvature, which involves taking second derivatives of the metric.

Oh! That makes perfect sense and actually really helps me tie in the last few lectures with the first half of the series where he covers that topic explicitly. Thank you!

The space-time metric can be approximated by the Rindler metric near the horizon.

## 1. What is flat spacetime at the event horizon of a black hole?

Flat spacetime at the event horizon of a black hole refers to the region just outside the event horizon where the curvature of spacetime is relatively small and can be described by the principles of special relativity. This means that the effects of gravity are not as strong as inside the event horizon, where spacetime is significantly curved.

## 2. How does the concept of flat spacetime apply to black holes?

In general relativity, the curvature of spacetime is directly related to the distribution of matter and energy. At the event horizon of a black hole, the curvature is relatively small due to the extreme curvature caused by the mass of the black hole. This results in a region of flat spacetime just outside the event horizon.

## 3. Why is flat spacetime important in understanding black holes?

Understanding flat spacetime at the event horizon of a black hole is important because it helps us to understand the behavior of matter and energy near the black hole. It also helps us to better understand the effects of gravity on spacetime and how it can be distorted by massive celestial objects like black holes.

## 4. Can anything escape from the flat spacetime region at the event horizon of a black hole?

In general relativity, nothing can escape from within the event horizon of a black hole. However, in the region of flat spacetime just outside the event horizon, it is possible for matter and energy to escape if they have enough energy to overcome the gravitational pull of the black hole.

## 5. How does the concept of flat spacetime at the event horizon relate to the information paradox?

The information paradox is a theoretical puzzle in which information about matter that falls into a black hole seems to disappear. The concept of flat spacetime at the event horizon plays a role in understanding this paradox, as it suggests that the information may be stored on the event horizon itself in the form of Hawking radiation. However, this is still a topic of debate among scientists.

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