What's the proper shape of a black hole?

Question

When we observe a circle we see a different shape than someone moving with high speed relative to us. They see an ellipse. Still we say the proper shape of the object is a circle.

Now a black hole doesn't have a shape, because it literally is a 1D hole in space, but the metric around it can be considered as some kind of object too. If you fall in the metric (though coordinate independent) looks differently than the metric seen by an observer stationary above the horizon. Falling in, you fall in proper time onto the center. This is a small time usually. If you stay stationary above the horizon you can stay far longer than the time measured by your in-falling partner.

So both metrics vary. The proper time measured by the stationary observer is different from the proper time of the in-falling one.

How does the proper metric of a black hole look like? What is the equivalent of the frame in which a circle is at rest and shows it's proper shape?

Answer 1

A metric in general relativity describes distances between different points in a spacetime manifold. The manifold is the underlying connectedness of points (the topology) while the metric gives a meaning to distance. The "shape" of spacetime is given by the metric and manifold together: this is in a sense the proper shape.

This is somewhat different from shapes we are used to in everyday life. There shapes are embedded in normal flat spacetime and might look different from different angles or to different observers without actually being different. In general relativity there is no outside where you can stand and watch the entire shape of spacetime, so every observer will unavoidably see things "from inside". But the shape of the spacetime is still constant and observer-independent.

There is no frame where where you can see a black hole in an undistorted way. The black shadow is not even the event horizon, but the photon sphere... which has a visual size different from what the coordinates would say in an uncurved spacetime. The proper shape of the spacetime is a 3+1 dimensional manifold with a particular metric. It does not look like anything we can see, since we cannot see four-dimensional objects even normally, and here the object is infinite too.

That metrics are observer independent is a key thing in relativity: the spacetime metric is the underlying reality that can then look very different to different observers. It is often useful to select coordinates that make the metric work better mathematically, but this is a re-labelling of existing points rather than changing the shape of the world.