The Kerr Spacetime

Posted on Posted in General Relativity, Navigating the Mainstream, Physics

Abstract. A brief and very basic introduction is given to Kerr spacetime. The metric is presented, and the various horizon surfaces and their physical meanings are discussed. An overview of the geometric and topological structure of maximally extended Kerr spacetime is given, using the waterfall analogy. Frame dragging, closed time-like curves, naked singularities, and the cosmic censorship hypothesis are introduced.

In today’s article, I will go beyond the simplistic Schwarzschild metric and introduce you to another solution to the Einstein equations, the Kerr solution. While the Schwarzschild solution describes a region of spacetime characterised by a spherical, stationary mass devoid of any other features, the Kerr metric allows us to add another property, being angular momentum; it thus describes a mass that rotates around a stationary axis, and is axially symmetric with respect to that axis. The Kerr solution is still a vacuum solution, i.e. it describes spacetime in the exterior vacuum of such a central body, as well as black holes that – in addition to mass – can now also carry angular momentum. As we shall see, the geometry and topology of this spacetime has a rich structure with some quite nonintuitive features.

One problem which will become immediately obvious once I write down for you the Kerr metric is that it is mathematically quite a bit more complicated than the very simple Schwarzschild solution. We will find that it contains off-diagonal terms, and that the metric coefficients themselves are also more complex; this unfortunately results in any actual calculations performed with this metric to be at best lengthy and tedious, at worst possible only numerically. I will thus refrain from focussing too much on the maths here, and rather focus on presenting the geometric features in plain language.

The Kerr metric has the form

(1)   \begin{equation*} \displaystyle{ds^2=\left ( 1-\frac{r_s r}{\rho^2} \right )dt^2-\frac{\rho^2}{\Delta }dr^2-\rho^2d\theta ^2-\left ( r^2+\alpha ^2+\frac{r_s r \alpha ^2}{\rho^2} sin^2\theta \right )sin^2\theta d\phi ^2+\frac{2r_s r \alpha sin^2\theta }{\rho^2}dtd\phi } \end{equation*}


(2)   \begin{equation*} \displaystyle{\alpha =\frac{J}{M}} \end{equation*}

(3)   \begin{equation*} \displaystyle{\rho^2=r^2+\alpha ^2 cos^2 \theta } \end{equation*}

(4)   \begin{equation*} \displaystyle{\Delta =r^2-r_sr+\alpha ^2} \end{equation*}

As before, M signifies the total mass of the spacetime, and the new parameter J is a measure of the angular momentum. As you can imagine, doing anything but the simplest of operations with this metric will lead to very tedious computations, which is why this solution is mostly examined using computer algebra systems such as Maple or Mathematica.

To start with, let us examine the horizon structure of Kerr spacetime. Unlike the Schwarzschild solution, which has only one significant surface ( the event horizon ), the Kerr metric has a number of physically important horizons and surfaces. Consider this schematic diagram depicting the structure of a Kerr black hole, which is not to scale [1] :

Fig 1. Horizons and Surfaces of Kerr Black Holes

Firstly, we note that the singularity in the center of this black hole is not point-like, but rather a ring singularity. This is because angular momentum is a conserved quantity – when a rotating star collapses into a Kerr black hole, the angular momentum it carries cannot just vanish into thin air; a single point cannot physically carry angular momentum ( we are only considering the classical case here, and disregard quantum mechanical spin ), instead, the simplest topological structure that allows this is a 1-dimensional ring. Hence, the Kerr singularity is a ring of finite mass, finite radius, but zero thickness – this being the purely classical extrapolation, with the same caveats as explained in my Schwarzschild articles.

Ring Singularity. Due to the conservation of angular momentum, the singularity in a Kerr black hole is not a point, but ring shaped.

In Schwarzschild spacetime, we have defined the event horizon as the surface past which events are no longer causally connected to the rest of the universe, and hence as the surface past which light can no longer escape to infinity. For Kerr black holes, these conditions are true for the outer horizon r_{+}, which is the surface shown in red in figure 1. This type of horizon ( a causal boundary, past which light cannot escape ) is called an absolute horizon. In the above diagram it appears as if the event horizon r_{+} is spherical, but in reality this is not true – it topologically spherical, but not geometrically. Its surface area is

(5)   \begin{equation*} \displaystyle{A=4\pi \left ( r_{+}^{2}+a^2 \right )} \end{equation*}

which is not the same as that of a sphere. If a very far away outside observer was able to see the horizon as a solid surface, it would look more like an oblate ellipsoid than a sphere to him.

Now consider a rocket approaching the Kerr black hole from infinity. In the Schwarzschild case, in order to stand still at any given point outside the horizon, the rocket would have to fire its thrusters only downwards, i.e. radially towards the black hole. In the Kerr case, the rocket will not only be dragged radially downwards, but also tangentially in the direction of rotation of the black hole; in order to stand still, the rocket would thus have to fire its thrusters not only downwards, but also in the direction of rotation. This phenomenon is called frame dragging. To illustrate it, you can perform a very simple experiment : take a linen cloth and spread it out on a flat table. Now take a fork, place it onto the cloth, and start to twist it – you will see that the cloth is being dragged along with the rotation of the fork. A similar principle holds for spacetime itself.

Frame Dragging. The tendency of a test particle to be “dragged along” the direction of rotation of a central body.

Fig 2. Frame Dragging

As the rocket continues to approach the Kerr black hole, it will become increasingly harder to counteract the frame dragging effect, until a point is reached where the rocket would need to expand an infinite amount of thrust to stop itself from being dragged along. This point is called the static limit, or the outer ergosurface, and is shown in yellow in figure 1. Note that this is not an absolute horizon, because it is still possible to escape from the static limit away to infinity; it is just no longer possible to counteract the frame dragging effect. Anything crossing the static limit will inevitably be dragged along in the direction of rotation, no matter how much thrust is expanded.

Static Limit. This is the limit past which it is no longer possible to counteract the frame dragging effect, regardless of how much thrust one expands. The static limit is not an absolute horizon.

Going further past the static limit, there is a region of spacetime from which you can still escape to infinity, but not resist the frame dragging – this region is called the ergosphere, and it is the region in blue in diagram fig 1. It is the precisely the region that lies between the static limit and the ( outer ) event horizon. Its size depends on the amount of angular momentum the Kerr black hole has – if the angular momentum is zero, then there is no frame dragging and no static limit, and we recover the usual Schwarzschild geometry.

Going still further, our rocket would encounter yet another surface called the inner event horizon r_{-}, or Cauchy horizon. A Cauchy horizon is a boundary  which separates a region in which causality holds ( i.e. future events are uniquely determined by past events – as is the case in our “normal” universe ) from a region where this is no longer necessarily the case. What does a region where causality doesn’t hold look like ? Basically, we are looking at a situation where an observer starts at some event ( i.e. at some point in space at an instant in time ), travels for a finite amount of time as measured on his own clock, yet ends up again at the same event he started with, i.e. at the same point at the same time as when he started off ! In essence, such an observer would have gone through a kind of closed “loop” which connects an event to itself – meaning we are in a region of spacetime where events cause themselves, in a manner of speaking, and are not necessarily caused by other events in the past. Such “loops” are called closed time-like curves ( CTCs ). This can be visualised thus [3] :

Fig 3. Closed Time-Like Curve

Closed Time-Like Curves. CTCs are topological constructs which connect events to themselves, while requiring a finite amount of proper time for a traveller to do so. This violates causality, so CTCs are hidden behind a Cauchy horizon.

In the Kerr geometry, CTCs would loop around the immediate vicinity of the ring singularity, and can be entered only by falling into the black hole in specific ways ( I won’t go into this here ). If you fall through the inner horizon in any other way, then, analoguous to the situation in Schwarzschild spacetime, you will enter a region that is a time-reverse version of the black hole – a Kerr white hole. The gravitational effect of a white hole is repulsive, so you would be flung outward and never again be able to re-enter either the outer nor the inner horizon. Just like in Schwarzschild spacetime, this region can be either in a completely separate universe ( singly connected spacetime ), or a distant region in our own universe ( multiply connected spacetime ). To better illustrate this, we use what is called the waterfall analogy – imagine spacetime like a river of water, in which case a Kerr black hole would look like this [4] :

Fig 4. Kerr Black Hole – Waterfall Analogy

Of course, in reality spacetime doesn’t “flow”, so this is just an analogy to make it easier to visualise things. Seen from another angle [5] :

Fig 5. Kerr Black Hole

Just as is the case for Schwarzschild black holes, the Kerr geometry is inherently unstable, and extremely sensitive to even tiny fluctuations; as such, it is highly doubtful that any of the structures we have discussed actually exist below the outer horizon. Of course, the solution is mathematically exact and unambiguous, but that does not mean that such black holes would form in the real world, under real conditions. Like the Schwarzschild metric, the Kerr solution is hence best understood to be a highly idealised model, rather than an exact description of the real world. It enables us to study what types of solutions the Einstein equations permit under certain assumptions.

There is another issue with the Kerr metric that is worth mentioning – if you increase the amount of angular momentum the black hole has, the outer and inner horizons will draw closer together, until they eventually coincide, and then shrink towards the central ring singularity. Given sufficient angular momentum, it is thus possible to make the horizon surfaces coincide with the ring itself, producing a naked singularity, i.e. a singularity that is not hidden behind an event horizon, and hence exposed to the rest of the universe. This poses a lot of problems ( mainly to do with concepts of causality and determinism ), which is why Roger Penrose proposed a conjecture called the cosmic censorship hypothesis – effectively, this would rule out the existence of naked singularities by basically attempting to demonstrate that initial conditions which lead to naked singularities are unphysical, and hence they do not form in the real world. The exact status of this hypothesis is currently uncertain.

And this concludes our – admittedly very brief – overview of the Kerr solution. I have introduced you to the basic features of this metric, and would urge you to go and do some more detailed research on your own, if you are interested.

In my next blog entry, we will take a closer look at an example of a metric that is not a vacuum solution, being the Reissner-Nordström metric. This describes a body with mass and electric charge, but no angular momentum, yet we will find that it shares some of the features discussed here for the Kerr metric.

Stay tuned !


[1], figure 1





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