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What makes us assign such meaning to these quantities is matching in the limit when there are only small amounts of dilute matter (which can be treated by linearized gravity). In that limit these asymptotic quantities exactly match the meaning of total linear and angular momentum of the dilute matter as defined in your classical-mechanics course. So a similar asymptotic field as for the Kerr metric would be generated by a cloud of dilute matter with a total mass $M$ and rotating with a total angular momentum $Ma$. So that is what makes you believe that there is something rotating in Kerr space-time.

So what is, really, rotating in Kerr space-time? The curvature singularity in the center of the space-time is a ring, and it would be very easy to say it is a rotating ring carrying the entire angular momentum $Ma$. But consider a very compact object very close to a black hole such as a neutron star. There you cannot assign the angular momentum only to the matter, an increasingly larger portion is stored nonlocally in the gravitational field, the space-time is rotating as well. On the other hand, the gravitational field (the space-time) would never rotate on its own, it will only rotate if the neutron star is rotating as well. This is true for any situation with matter - the field and its source rotate in tandem, each having a significant contribution to the angular momentum.

So what about black holes? There is really no rigorous argument to extend the observations of the last paragraph to the black hole, though. But consider this: in a rather precise sense, the nonsingular part of the black hole field "causes" the existence of the curvature singularity inside field. You can slice space-time in a particular way, put one of the slices not including the black hole singularity in the computer and let it evolve the slice into the future according to GR. To your surprise, it will spontaneously evolve to the singularity (in the case of the Kerr space-time it would actually evolve a crumpled singularity called the Cauchy horizon). So in this sense it is the field that causes the singularity to exist rather than the other way around. So I am inclined to say that black holes can be seen as the limit where the fraction of mass and angular momentum in the gravitational field has actually converged to the entire mass and angular content of the space-time.

What makes us assign such meaning to these quantities is matching in the limit when there are only small amounts of dilute matter (which can be treated by linearized gravity). In that limit these asymptotic exactly match the meaning of total linear and angular momentum of the dilute matter as defined in your classical-mechanics course. So a similar asymptotic field as for the Kerr metric would be generated by a cloud of dilute matter with a total mass $M$ and rotating with a total angular momentum $Ma$. So that is what makes you believe that there is something rotating in Kerr space-time.

So what is, really rotating in Kerr space-time? The curvature singularity in the center of the space-time is a ring, and it would be very easy to say it is a rotating ring carrying the entire angular momentum. But consider a very compact object very close to a black hole such as a neutron star. There you cannot assign the angular momentum only to the matter, an increasingly larger portion is stored nonlocally in the gravitational field, the space-time is rotating as well. On the other hand, the gravitational field (the space-time) would never rotate on its own, it will only rotate if the neutron star is rotating as well. This is true for any situation with matter - the field and its source rotate in tandem, each having a significant contribution to the angular momentum.

So what about black holes? There is really no rigorous argument to extend the observations of the last paragraph to the black hole, though. But consider this: in a rather precise sense, the nonsingular part of the black hole field "causes" the existence of the inside field. You can slice space-time in a particular way, put one of the slices not including the black hole singularity in the computer and let it evolve the slice into the future according to GR. To your surprise, it will spontaneously evolve to the singularity (in the case of the Kerr space-time it would actually evolve a crumpled singularity called the Cauchy horizon). So in this sense it is the field that causes the singularity to exist rather than the other way around. So I am inclined to say that black holes can be seen as the limit where the fraction of mass and angular momentum in the gravitational field has actually converged to the entire mass and angular content of the space-time.

What makes us assign such meaning to these quantities is matching in the limit when there are only small amounts of dilute matter (which can be treated by linearized gravity). In that limit these asymptotic quantities exactly match the meaning of total linear and angular momentum of the dilute matter as defined in your classical-mechanics course. So a similar asymptotic field as for the Kerr metric would be generated by a cloud of dilute matter with a total mass $M$ and rotating with a total angular momentum $Ma$. So that is what makes you believe that there is something rotating in Kerr space-time.

So what is, really, rotating in Kerr space-time? The curvature singularity in the center of the space-time is a ring, and it would be very easy to say it is a rotating ring carrying the entire angular momentum $Ma$. But consider a very compact object very close to a black hole such as a neutron star. There you cannot assign the angular momentum only to the matter, an increasingly larger portion is stored nonlocally in the gravitational field, the space-time is rotating as well. On the other hand, the gravitational field (the space-time) would never rotate on its own, it will only rotate if the neutron star is rotating as well. This is true for any situation with matter - the field and its source rotate in tandem, each having a significant contribution to the angular momentum.

So what about black holes? There is really no rigorous argument to extend the observations of the last paragraph to the black hole, though. But consider this: in a rather precise sense, the nonsingular part of the black hole field "causes" the existence of the curvature singularity inside. You can slice space-time in a particular way, put one of the slices not including the black hole singularity in the computer and let it evolve the slice into the future according to GR. To your surprise, it will spontaneously evolve to the singularity (in the case of the Kerr space-time it would actually evolve a crumpled singularity called the Cauchy horizon). So in this sense it is the field that causes the singularity to exist rather than the other way around. So I am inclined to say that black holes can be seen as the limit where the fraction of mass and angular momentum in the gravitational field has actually converged to the entire mass and angular content of the space-time.

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But whatWhat makes us assign such meaning to these quantities is by matching toin the limit when there are only small amounts of dilute matter, which (which can be treated by linearized gravity). In that limit these asymptotic exactly match the meaning of total linear and angular momentum of the dilute matter as defined in your classical-mechanics course. So a similar asymptotic field as for the Kerr metric would be generated by a cloud of dilute matter with a total mass $M$ and rotating with a total angular momentum $Ma$. So that is what makes you believe that there is something rotating in Kerr space-time.

But what makes us assign such meaning to these quantities is by matching to the limit when there are only small amounts of dilute matter, which can be treated by linearized gravity. In that limit these asymptotic exactly match the meaning of total linear and angular momentum of the dilute matter as defined in your classical-mechanics course. So a similar asymptotic field as for the Kerr metric would be generated by a cloud of dilute matter with a total mass $M$ and rotating with a total angular momentum $Ma$. So that is what makes you believe that there is something rotating in Kerr space-time.

What makes us assign such meaning to these quantities is matching in the limit when there are only small amounts of dilute matter (which can be treated by linearized gravity). In that limit these asymptotic exactly match the meaning of total linear and angular momentum of the dilute matter as defined in your classical-mechanics course. So a similar asymptotic field as for the Kerr metric would be generated by a cloud of dilute matter with a total mass $M$ and rotating with a total angular momentum $Ma$. So that is what makes you believe that there is something rotating in Kerr space-time.

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So let us first ask ourselves: What makes us believe that anything is rotating in the Kerr space-time?

The answer is that we go very far away from the black hole and look at the asymptotics of the gravitational field as it becomes weaker and weaker. From there we can read off in a well defined way a set of mass and current multipoles "contained in the space-time". The lowest-order multipoles can be defined almost unambiguously and they would be interpreted as "total linear momentum in the space-time" and "total angular momentum in the space-time". Then what you read of for the Kerr metric is an angular momentum of the magnitude $J =Ma$.

But what makes us assign such meaning to these quantities is by matching to the limit when there are only small amounts of dilute matter, which can be treated by linearized gravity. In that limit these asymptotic exactly match the meaning of total linear and angular momentum of the dilute matter as defined in your classical-mechanics course. So a similar asymptotic field as for the Kerr metric would be generated by a cloud of dilute matter with a total mass $M$ and rotating with a total angular momentum $Ma$. So that is what makes you believe that there is something rotating in Kerr space-time.

But as larger amounts of dense matter are added, you cannot just sum all the momenta of the matter particles in a space-time to get the total linear and/or angular momentum. The problem is that the space-time is curved and you stop being able to easily do things such as sum two vectors at different points of the space-time. But another issue is that no matter how hard you try to sum the contributions of momentum in the matter, you keep missing something, it just does not add up to the asymptotic momenta. It turns out that somehow the gravitational field itself carries momentum. But pinning down how and where this energy and momentum of the gravitational field is notoriously difficult. There simply is not a universally valid formula for that, the energy and angular momentum of the gravitational field is stored non-locally.

So what is, really rotating in Kerr space-time? The curvature singularity in the center of the space-time is a ring, and it would be very easy to say it is a rotating ring carrying the entire angular momentum. But consider a very compact object very close to a black hole such as a neutron star. There you cannot assign the angular momentum only to the matter, an increasingly larger portion is stored nonlocally in the gravitational field, the space-time is rotating as well. On the other hand, the gravitational field (the space-time) would never rotate on its own, it will only rotate if the neutron star is rotating as well. This is true for any situation with matter - the field and its source rotate in tandem, each having a significant contribution to the angular momentum.

So what about black holes? There is really no rigorous argument to extend the observations of the last paragraph to the black hole, though. But consider this: in a rather precise sense, the nonsingular part of the black hole field "causes" the existence of the inside field. You can slice space-time in a particular way, put one of the slices not including the black hole singularity in the computer and let it evolve the slice into the future according to GR. To your surprise, it will spontaneously evolve to the singularity (in the case of the Kerr space-time it would actually evolve a crumpled singularity called the Cauchy horizon). So in this sense it is the field that causes the singularity to exist rather than the other way around. So I am inclined to say that black holes can be seen as the limit where the fraction of mass and angular momentum in the gravitational field has actually converged to the entire mass and angular content of the space-time.