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We often read in scientific publications about the diameter of massive objects like black holes. These objects are known to distort space and time, in a way that I believe we're not quite certain to fully understand.

So my question is:

What sense does it make to speak about the diameter of an object that distorts geometry?

I mean we may have an idea of their apparent circumference, and the time it would take to go around them. But it's unclear what would happen if an object was to go straight through it. As I understand it, it's not even possible to do so as you can't escape back from a black hole. So are people referring to the diameter the object would have, given its circumference, if it was geometrically flat?

Qmechanic
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Arnaud Mortier
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3 Answers3

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In the case of a black hole, trying to draw a line from the singularity to the edge of the event horizon is pointless in regular spherical coordinates (though, theoretically, you could try in Kruskal-Szekeres coordinates). We define the event horizon radius not as a quantity on its own but the circumference of the event horizon sphere divided by $2\pi$.

For less-extreme objects you can still speak of radius. A star for example does curve space around it that would distort measurements, but the measurements are made locally and by an observer embedded in the spacetime. You, as a human in spacetime, might measure a distorted diameter for the star, but from the perspective of someone off the manifold, you can still pick two antipodal coordinate points on the star’s surface and take their Euclidean distance to be the diameter of the star, even though that distance might not exactly correspond to how a human would measure it.

I think that, physically, it is more meaningful to describe something as it can be perceived at infinity (i.e. at rest relative to it, very far away so we are “outside” the influence of its gravitational effects and what have ye). Close to it, one might experience the geometry differently (see time flow being different closer to/farther from a black hole). In most cases, though, we are far away from the things we’re talking about, if we’re talking about stars or quasars or black holes or alien warp drive spaceships. This perception from infinity might be distorted, but it’s the best thing you can get from a measuring-the-thing-from-a-particular-point perspective.

controlgroup
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Arnaud Mortier asked: "So are people referring to the diameter the object would have, given its circumference, if it was geometrically flat?"

As controlgroup already said, in most coordinates we have proper circumference where the spatial distortion is projected into the radial coordinate. To calculate the proper radius see here, for example:

Wikipedia wrote: the r-coordinate at the body's surface is less than its proper (measurable interior) radius, although for the Earth the difference is only about 1.4 millimetres.

Yukterez
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If you define diameter as circumference divided by $\pi$, then you face the problem that the number $\pi$ is itself defined as circumference divided by diameter. So you have to find a definition of diameter that is independent of $\pi$, based only on intrinsic properties of the geometric object. Let's consider a two-dimensional universe in the form of a round sphere. Note that a sphere by itself has no interior or sides, and cannot be contracted to a point. But as a two-dimensional surveyor, you can triangulate your world to get its total area $A$, and measure a great circle to get its length $L$. With these two you can then define the diameter as the ratio $D\equiv A/L$ and the radius as $R\equiv D/2$.

JanG
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