THE STRANGE UNIVERSE OF ROTATING BLACK HOLES

"The macroscopic geometry of relativity has many special features. Features that suggest a hidden complex-manifold origin and certain deep, underlying physical connections between ordinary space-time relations and the complex linear superposition of quantum mechanics."

Roger Penrose, The Complex Geometry of the Natural World

Einstein's theory of general relativity was published between 1915 and 1916. This discovery represented such a radical change in the way we understand the Universe that more than a century later we still struggle to assimilate some of its consequences. A little over a month after the last publication of the theory, German physicist Karl Schwarzschild found an exact solution to Einstein's equations for the case of a static, spherically symmetric black hole. However, real black holes formed by stellar collapse are not static but possess angular momentum. Finding the solution for the case of a rotating spherical black hole took more than 20 years and involved dozens of pages of highly complex calculations. This solution, known as the Kerr metric, has important differences from the static Schwarzschild metric and, as we shall see, contains astonishing phenomena.

In 1965, physicists Newman and Janis made a surprising discovery: if we start from the Schwarzschild metric and allow the metric to take complex values, we can transform the stationary Schwarzschild black hole into a rotating black hole in just 4 easy steps! 20 years of searching and dozens of pages of complex calculations replaced by a few pages of elementary ones! In this article, we will study the strange properties of rotating black holes, their connection with quantum gravity theories, and the possible consequences of the Newman-Janis discovery. Is this just a coincidence or a mathematical trick, or is there really a complex physical structure yet to be discovered?

The strange properties of rotating black holes

The metric of a rotating black hole called the Kerr metric can be described as follows:

where the denominators are given by:

The four coordinates of space-time are expressed in polar coordinates: t= time r= radial distance theta= polar angle fi= azimuthal angle:

The constant "a" is a value that determines the rotation speed of the black hole.

Although this mathematical expression is quite intimidating, we are only interested in two values that can be easily calculated:

1. The values of r for which the metric diverges (tends to infinity)

2. The value of r where the time coefficient dt2 changes sign

Point 1 determines the position of the event horizon and the singularity, while point 2 determines the behavior of the spatial and temporal components. Recall that the Lorentzian plane metric is given by:

The positive components are spatial, and the negative components are time-like. A body can move freely through the spatial components, but the time-like component always "flows" in the same direction and "pulls" all bodies in that direction. The above points will determine the three main zones of a Kerr black hole: the event horizon, the ergosphere, and the singularity .

The event horizon

The points where the metric diverges are the values of r for which the denominator (delta or rho) approaches zero. It's easy to see that for delta these values are:

We then have two values of r for which the metric tends to infinity: r+ and r-. Here we encounter the first strange feature: rotating black holes have two event horizons: one outer (r+) and one inner (r-).

The ergosphere

To analyze the value of r where the coefficient dt2 changes sign, we simply analyze said coefficient:

The above expression becomes positive when:

The roots of the above equation are: GM+-PI (G2M2-a2cos2fi). Now we come to the second strange feature: if we compare this value with that of the position of the event horizon, we find that the coefficient of the time dimension becomes positive before reaching the r+ horizon! This zone between this exterior point where the time part of the metric changes sign and the exterior r+ horizon is called the ergosphere, and as we shall see, it has some astonishing characteristics.

So, from this point outside the black hole, time behaves as if it were a spatial dimension. But then, if all the components of the metric are positive, what coordinate does the time function serve? If we look at the last coefficient of the metric, we can see that by expanding the square, we obtain a cross component dtdfi of negative value. This component begins to behave like the time variable and can be interpreted as an interconnection between both components. It's as if time and the angle of rotation have swapped roles! But how should we interpret this?

To analyze this, we will study the trajectory of a photon traveling radially toward the center of the Kerr black hole. Since the metric is zero for a photon, we set ds2=0, and taking the equatorial plane, we set theta=PI/2 and rho=0:

Solving this expression we obtain the equation:

At the edge where the ergosphere begins, the coefficient dt is zero, so dtt = 0. The solutions to the previous equation are:

These two expressions lead us to the third strange feature of these rotating monsters: the first expression represents the speed of a photon moving in the opposite direction to the black hole's spin and shows us something incredible: the instantaneous speed is zero. This means that, seen from the outside, the instantaneous speed of the photon is zero, which means that, in the opposite direction to the black hole's spin , the edge of the ergosphere is rotating at the speed of light!

Can anyone imagine something like this? As we'll see below, this actually means that nothing can move in the opposite direction to the black hole's spin.

The second expression refers to the speed of a photon moving in the same direction as the black hole's spin. The photon can only move at a certain speed, and this speed coincides with the black hole's spin.

From this it follows that no body can be at rest in the ergosphere, all bodies must move in the direction of rotation of the black hole which means that

The black hole is dragging space-time along as it spins! In the ergosphere, the coordinate fi flows relentlessly forward, and nothing can move in the opposite direction. In other words, the spatial coordinate fi begins to behave like a temporal coordinate! The black hole's spin behaves like a coordinate that marks the passage of time.

The central singularity

When we calculated the event horizon positions earlier, we forgot to analyze the other denominator, rho. For rho, the metric diverges when r2+a2sin2theta=0.

Now we come to the fourth strange feature: the singularity is not located at a point at r=0! The solution to the previous expression includes all points that are located at r=0 and also satisfy theta=PI/2 . This last condition is fulfilled by all points in the equatorial plane of radius a, that is,

The central singularity is not a point but a ring!

The singularity ring is a "time-like" surface, meaning it's a physically realizable trajectory in time. If we were to stand right next to the ring and follow a closed path around it, we would travel along the temporal dimension, so we would return to the starting point, back in time! Now that would truly be time travel to the past!

As if this were not exotic enough, a cosmic traveler could theoretically pass through the interior of the ring, which could lead him to a different space-time.

For obvious reasons, the possible existence of closed timelike curves is a serious problem for any real physical system, and physicists are convinced that they cannot exist in our Universe. Although these circular singularities are solutions to the equations of general relativity, there is strong evidence that these types of singularities are unstable, so it is highly unlikely that they can occur in real black holes.

Different areas of a rotating black hole

Extreme black holes: A gateway to quantum gravity?

Kerr black holes hold another surprise. If we increase the black hole's rotational speed (the value of the parameter "a"), we reach a limit where general relativity seems to collapse: the two horizons merge into one, the entropy value disappears, and the temperature of the hole tends to zero. How is this possible? Physicists believe that these phenomena indicate a discontinuity and are a sign that the classical theory of general relativity cannot predict what happens when "a" reaches the highest allowed value. These black holes with the highest possible angular momentum are called extreme black holes.

These black holes have been the subject of intense theoretical work, which has revealed something very interesting: despite GR predicting a vanishing entropy in these black holes, our most important candidate theories of quantum gravity predict that the entropy exactly matches that predicted by the Bekenstein-Hawking formula. Specifically, the equations of string theory have a generic low-energy solution that predicts this entropy. Furthermore, string theory explains the origin of the microstates that generate this entropy through certain conserved charges (see this article ). This leads us to wonder: Are extreme black holes a gateway to quantum gravity phenomena?

The Newman-Janis algorithm

Finally, we will briefly describe another fascinating discovery about these strange objects. Newman and Janis discovered that if we start from the static Schwarzschild metric and allow the metric to take complex values , then we can transform the Schwarzschild metric into the Kerr metric in just four simple steps. This surprising fact may be just a mathematical curiosity or hide something very profound about the nature of spacetime.

Without going into technical details, the four steps would be the following (starting from the Schwarzschild metric written in advanced Eddington-Finkelstein coordinates):

1. Writing the metric in the form of zero-metric tetrahedra

2. Extend the coordinates r, u so that they can take complex values and perform the following transformation:

3. Calculate the resulting metric

4. Take only the real values of the resulting metric.

The key points are steps 2 and 4: while in step 2 we allow the metric to enter the "complex world," in step 4 we return to the "real world" by taking only the real values of the resulting metric. The most striking thing about this result, apart from incredibly simplifying the calculations, is that it suggests a relationship between general relativity and a complex structure of space-time. Future theoretical and experimental advances will tell us whether this is just a mathematical coincidence or a fascinating new physical structure.

Sources:

The Kerr spacetime: A brief introduction , Complex Spacetimes and the Newman-Janis Trick