# Innermost stable circular orbit for rotating black holes + length contraction

I viewed many relevant threads but didn't quite find the answer. I don't really know what I'm talking about so I'm mostly looking for a lead on how I should be thinking about this

From my understanding a Kerr black hole with a high rate of rotation can have a very close innermost stable circular orbit. For my question I think the orbit doesn't actually matter, neither does the proximity, it's just a mechanism to visualize the effects of an extreme gravitational field.

Also compared to an external observer in a non gravitational field, the Lorentz Factor of the extreme gravitational field goes towards infinity for both time dilation and increases the length of space in the radial direction to the body that is the source of the gravitational field.

If an observer places themselves into increasing closer orbits to this black hole, they should be able to measure how long a single orbit takes. Compared to the distant observer, this time difference should be just the Lorentz factor.

The question I have is shouldn't the observer in orbit be able to measure how long 1 orbit takes them in their own reference frame? And from this information shouldn't they be able to calculate the maximum possible diameter of the black hole from their perspective?

If the black hole is absolutely massive at a 10 light-year circumference , and they're in a circular orbit corresponding to a Lorentz factor of about 10, they would be forced to conclude that the black hole is only a little more than 1 light year in circumference.

From the perspective of an observer entering increasingly closer circular orbits, doesn't the black hole appear to become arbitrarily small?

I think the orbit wouldn't matter because that should be a property of the gravitational field and not of the orbit- so shouldn't an observer falling towards a blackhole observe that the event horizon is shrinking to occupy zero volume before they can fall through it?

Ultimately I'm having trouble understanding the mechanism of how an observer is able to cross an event horizon since both observers run into infinities before that point is passed.

AngryGroceries wrote: "If an observer places themselves into increasing closer orbits to this black hole, they should be able to measure how long a single orbit takes. Compared to the distant observer, this time difference should be just the Lorentz factor."

That's compared to a ZAMO at the same radial distance as the circular orbit. Relative to the distant observer the factor increases by the lapse function $$\surd g^{\rm tt}$$, see here in the text below equation (3.5).

AngryGroceries asked: "The question I have is shouldn't the observer in orbit be able to measure how long 1 orbit takes them in their own reference frame?"

This question already has an answer, see orbital period and velocity around a Kerr black hole